Handbook of Combinatorial Optimization [Volume 3, 2 ed.] 9781441979964, 9781441979971

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Panos M. Pardalos Ding-Zhu Du Ronald L. Graham Editors

Handbook of Combinatorial Optimization Second Edition

1 3Reference

Handbook of Combinatorial Optimization

Panos M. Pardalos • Ding-Zhu Du Ronald L. Graham Editors

Handbook of Combinatorial Optimization Second Edition

With 683 Figures and 171 Tables

Editors Panos M. Pardalos Department of Industrial and Systems Engineering University of Florida Gainesville, FL, USA

Ronald L. Graham Department of Computer Science and Engineering University of California, San Diego La Jolla, CA, USA

Ding-Zhu Du Department of Computer Science The University of Texas at Dallas Richardson, TX, USA

ISBN 978-1-4419-7996-4 ISBN 978-1-4419-7997-1 (eBook) ISBN Bundle 978-1-4614-3975-2 (print and electronic bundle) DOI 10.1007/978-1-4419-7997-1 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013942873 1st edition: © Springer Science+Business Media New York 1999 (Volumes 1–3 and Supplement Volume A), 2005 (Supplement Volume B) 2nd edition: © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Many practical applications that include machines, people, or different resources that are indivisible, give rise to optimization models with discrete decision variables. On the other hand, nonconvexities (e.g., complementarity) in optimization models can be formulated using discrete variables. In past years, we witnessed several developments in combinatorial optimization. These developments came along with progress in related fields such as complexity theory, approximation algorithms, as well as with advances in software and hardware technology. The first edition of the Handbook of Combinatorial Optimization received a warm welcome from the scientific community. We are happy to introduce the second edition of the Handbook of Combinatorial Optimization. Some material in the first edition has been revised and several new chapters have been added. We are grateful to all authors and referees for their valuable time. Moreover, we thank the publisher and the production team for helping with editing the handbook. We hope that this book will continue to be a helpful tool for individuals working in the field of combinatorial optimization. Acknowledgements. We would like to acknowledge the help of our students during the years of editing the handbook. We have also been partially supported by Air Force, NSF, and DTRA grants. Panos M. Pardalos Ding-Zhu Du Ronald L. Graham

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Contents

Volume 1 A Unified Approach for Domination Problems on Different Network Topologies . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thang N. Dinh, D. T. Nguyen and My T. Thai

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Advanced Techniques for Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . Wolfgang Bein

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Advances in Group Testing . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yongxi Cheng

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Advances in Scheduling Problems . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Ming Liu and Feifeng Zheng Algebrization and Randomization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Liang Ding and Bin Fu Algorithmic Aspects of Domination in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Gerard Jennhwa Chang Algorithms and Metaheuristics for Combinatorial Matrices . . . . . . . . . . . . . . . 283 Ilias S. Kotsireas Algorithms for the Satisfiability Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 John Franco and Sean Weaver Bin Packing Approximation Algorithms: Survey and Classification . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Edward G. Coffman, J´anos Csirik, G´abor Galambos, Silvano Martello and Daniele Vigo Binary Unconstrained Quadratic Optimization Problem . . . . . . . . . . . . . . . . . . . 533 Gary A. Kochenberger, Fred Glover and Haibo Wang

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Combinatorial Optimization Algorithms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 Elisa Pappalardo, Beyza Ahlatcioglu Ozkok and Panos M. Pardalos Combinatorial Optimization in Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Samira Saedi and O. Erhun Kundakcioglu Combinatorial Optimization Techniques for Network-Based Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Oleg Shirokikh, Vladimir Stozhkov and Vladimir Boginski

Volume 2 Combinatorial Optimization in Transportation and Logistics Networks . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 Chrysafis Vogiatzis and Panos M. Pardalos Complexity Issues on PTAS . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 Jianer Chen and Ge Xia Computing Distances Between Evolutionary Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 747 Bhaskar DasGupta, Xin He, Tao Jiang, Ming Li, John Tromp, Lusheng Wang and Louxin Zhang Connected Dominating Set in Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 Hongjie Du, Ling Ding, Weili Wu, Donghyun Kim, Panos M. Pardalos and James Willson Connections Between Continuous and Discrete Extremum Problems, Generalized Systems, and Variational Inequalities . . . . . . . . . . . . . . 835 Carla Antoni, Franco Giannessi and Fabio Tardella Coverage Problems in Sensor Networks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899 Mihaela Cardei Data Correcting Approach for Routing and Location in Networks . . . . . . . . 929 Boris Goldengorin Dual Integrality in Combinatorial Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 Xujin Chen, Xiaodong Hu and Wenan Zang Dynamical System Approaches to Combinatorial Optimization . . . . . . . . . . . 1065 Jens Starke Efficient Algorithms for Geometric Shortest Path Query Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Danny Z. Chen

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Energy Efficiency in Wireless Networks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 Hongwei Du, Xiuzhen Cheng and Deying Li Equitable Coloring of Graphs . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199 Ko-Wei Lih Faster and Space Efficient Exact Exponential Algorithms: Combinatorial and Algebraic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249 Dongxiao Yu, Yuexuan Wang, Qiang-Sheng Hua and Francis C. M. Lau Fault-Tolerant Facility Allocation .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293 Hong Shen and Shihong Xu Fractional Combinatorial Optimization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311 Tomasz Radzik

Volume 3 Fuzzy Combinatorial Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357 Panos M. Pardalos, Emel Kızılkaya Aydogan, Feyza Gurbuz, Ozgur Demirtas and Birce Boga Bakirli Geometric Optimization in Wireless Networks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415 Weili Wu and Zhao Zhang Gradient-Constrained Minimum Interconnection Networks . . . . . . . . . . . . . . . 1459 Marcus Brazil and Marcus G. Volz Graph Searching and Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1511 Anthony Bonato and Boting Yang Graph Theoretic Clique Relaxations and Applications . . . . . . . . . . . . . . . . . . . . . . 1559 Balabhaskar Balasundaram and Foad Mahdavi Pajouh Greedy Approximation Algorithms . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1599 Peng-Jun Wan Hardness and Approximation of Network Vulnerability . . . . . . . . . . . . . . . . . . . . 1631 My T. Thai, Thang N. Dinh and Yilin Shen Job Shop Scheduling with Petri Nets . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667 Hejiao Huang, Hongwei Du and Farooq Ahmad Key Tree Optimization . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713 Minming Li Linear Programming Analysis of Switching Networks . . . . . . . . . . . . . . . . . . . . . . 1755 Hung Q. Ngo and Thanh-Nhan Nguyen

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Map of Geometric Minimal Cuts with Applications . . . . . . . . . . . . . . . . . . . . . . . . . 1815 Evanthia Papadopoulou, Jinhui Xu and Lei Xu Max-Coloring .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1871 Juli´an Mestre and Rajiv Raman Maximum Flow Problems and an NP-Complete Variant on Edge-Labeled Graphs . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1913 Donatella Granata, Raffaele Cerulli, Maria Grazia Scutell`a and Andrea Raiconi Modern Network Interdiction Problems and Algorithms . . . . . . . . . . . . . . . . . . . 1949 J. Cole Smith, Mike Prince and Joseph Geunes Network Optimization .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1989 Samir Khuller and Balaji Raghavachari

Volume 4 Neural Network Models in Combinatorial Optimization.. . . . . . . . . . . . . . . . . . . 2027 Mujahid N. Syed and Panos M. Pardalos On Coloring Problems . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095 Weifan Wang and Yuehua Bu Online and Semi-online Scheduling . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2191 Zhiyi Tan and An Zhang Online Frequency Allocation and Mechanism Design for Cognitive Radio Wireless Networks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2253 Xiang-Yang Li and Ping Xu Optimal Partitions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2301 Frank K. Hwang and Uriel G. Rothblum Optimization in Multi-Channel Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 2359 Hongwei Du, Weili Wu, Lidong Wu, Kai Xing, Xuefei Zhang and Deying Li Optimization Problems in Data Broadcasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2391 Yan Shi, Weili Wu, Jiaofei Zhong, Zaixin Lu and Xiaofeng Gao Optimization Problems in Online Social Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 2455 Jie Wang, You Li, Jun-Hong Cui, Benyuan Liu and Guanling Chen Optimizing Data Collection Capacity in Wireless Networks . . . . . . . . . . . . . . . . 2503 Shouling Ji, Jing (Selena) He and Yingshu Li

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Packing Circles in Circles and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2549 Zhao Zhang, Weili Wu, Ling Ding, Qinghai Liu and Lidong Wu Partition in High Dimensional Spaces . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2585 Zhao Zhang and Weili Wu Probabilistic Verification and Non-approximability . . . . . . . . . . . . . . . . . . . . . . . . . 2625 Mario Szegedy and Kashyap Kolipaka Protein Docking Problem as Combinatorial Optimization Using Beta-Complex . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2685 Deok-Soo Kim

Volume 5 Quadratic Assignment Problems . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2741 Rainer E. Burkard Reactive Business Intelligence: Combining the Power of Optimization with Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2815 Roberto Battiti and Mauro Brunato Reformulation–Linearization Techniques for Discrete Optimization Problems .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2849 Hanif D. Sherali and Warren P. Adams Resource Allocation Problems . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2897 Naoki Katoh, Akiyoshi Shioura and Toshihide Ibaraki Rollout Algorithms for Discrete Optimization: A Survey . . . . . . . . . . . . . . . . . . . 2989 Dimitri P. Bertsekas Simplicial Methods for Approximating Fixed Point with Applications in Combinatorial Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015 Chuangyin Dang Small World Networks in Computational Neuroscience .. . . . . . . . . . . . . . . . . . . . 3057 Dmytro Korenkevych, Jui-Hong Chien, Jicong Zhang, Deng-Shan Shiau, Chris Sackellares and Panos M. Pardalos Social Structure Detection . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3089 Kai Yang, Weili Wu, Wei Zhang, Tzuwei Hsu and Lidan Fan Steiner Minimal Trees: An Introduction, Parallel Computation, and Future Work .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3133 Frederick C. Harris Jr. and Rakhi Motwani Steiner Minimum Trees in E3 : Theory, Algorithms, and Applications.. . . . 3179 J. MacGregor Smith

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Tabu Search .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3261 Fred Glover and Manuel Laguna Variations of Dominating Set Problem . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3363 Liying Kang Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3395

Contributors

Warren P. Adams Department of Mathematical Sciences, Clemson University, Clemson, SC, USA Beyza Ahlatcioglu Ozkok Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Istanbul, Turkey Farooq Ahmad Information Technology, University of Central Punjab, Lahore, Pakistan Carla Antoni Naval Academy, Livorno, Italy Emel Kızılkaya Aydogan Faculty of Engineering, Department of Industrial Engineering, Erciyes University, Kayseri, Turkey Birce Boga Bakirli Faculty of Engineering, Department of Industrial Engineering, Gazi University, Ankara, Turkey Balabhaskar Balasundaram School of Industrial Engineering and Management, Oklahoma State University, Stillwater, OK, USA Roberto Battiti LION Lab, Universit`a di Trento, and Lionsolver Inc., Trento, Italy Wolfgang Bein Department of Computer Science, University of Nevada, Las Vegas, Las Vegas, NV, USA Dimitri P. Bertsekas Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, USA Vladimir Boginski Industrial and Systems Engineering Department, University of Florida, Shalimar, FL, USA Anthony Bonato Department of Mathematics, Ryerson University, Toronto, ON, Canada Marcus Brazil Department of Electrical and Electronic The University of Melbourne, Parkville, VIC, Australia

Engineering,

Mauro Brunato LION Lab, Universit`a di Trento, and Lionsolver Inc., Trento, Italy Yuehua Bu Department of Mathematics, Zhejiang Normal University, Jinhua, People’s Republic of China xiii

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Contributors

Rainer E. Burkard Institute of Optimization and Discrete Mathematics, Graz University of Technology, Graz, Austria Mihaela Cardei Computer and Electrical Engineering and Computer Science Department, Florida Atlantic University, Boca Raton, FL, USA Raffaele Cerulli Department of Mathematics, University of Salerno, Fisciano (SA), Italy Gerard Jennhwa Chang Department of Mathematics, National Taiwan University, Taipei, Taiwan Danny Z. Chen Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, USA Guanling Chen Department of Computer Science, University of Massachusetts, Lowell, MA, USA Jianer Chen Department of Computer Science and Engineering, Texas A&M University, College Station, TX, USA Xujin Chen Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, People’s Republic of China Xiuzhen Cheng Department of Computer Science, George Washington University, Washington, D.C., USA Yongxi Cheng School of Management, Xi’an Jiaotong University, Xi’an, Shaanxi, People’s Republic of China Jui-Hong Chien Biomedical Engineering Department, University of Florida, Gainesville, FL, USA Edward G. Coffman Department of Computer Science, Columbia University, New York, NY, USA J´anos Csirik Department of Computer Algorithms and Artificial Intelligence, University of Szeged, Szeged, Hungary Jun-Hong Cui Department of Computer Science and Engineering, University of Connecticut, Storrs, CT, USA Chuangyin Dang Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong Bhaskar DasGupta University of Illinois, Chicago, IL, USA Ozgur Demirtas Department of Business Administration, Erciyes University, Kayseri, Turkey Liang Ding Department of Computer Science, The University of Texas-Pan American, Edinburg, TX, USA Ling Ding Institute of Technology, University of Washington Tacoma, Tacoma, WA, USA

Contributors

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Thang N. Dinh Department of Computer Science, Information Science and Engineering, University of Florida, Gainesville, FL, USA Hongjie Du Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Hongwei Du Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, People’s Republic of China Lidan Fan Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA John Franco School of Electronic and Computing Systems, University of Cincinnati, Cincinnati, OH, USA Bin Fu Department of Computer Science, The University of Texas-Pan American, Edinburg, TX, USA G´abor Galambos Faculty of Education, Department of Computer Science, University of Szeged, Szeged, Hungary Xiaofeng Gao Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, People’s Republic of China Joseph Geunes Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA Franco Giannessi University of Pisa, Pisa, Italy Fred Glover OptTek Systems, Inc, Boulder, CO, USA Boris Goldengorin Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA Donatella Granata Department of Statistics, Probability and Applied Statistics, University of Rome “La Sapienza”, Rome, Italy Feyza Gurbuz Faculty of Engineering, Department of Industrial Engineering, Erciyes University, Kayseri, Turkey Frederick C. Harris Jr. Department of Computer Science & Engineering, University of Nevada, Reno, NV, USA Jing (Selena) He Department of Computer Science, Kennesaw State University, Kennesaw, GA, USA Xin He Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA Tzuwei Hsu Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Xiaodong Hu Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, People’s Republic of China

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Contributors

Qiang-Sheng Hua Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, People’s Republic of China Hejiao Huang Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, People’s Republic of China Frank K. Hwang National Chiao Tung University (Retired), Hsinchu, People’s Republic of China Toshihide Ibaraki The Kyoto College of Graduate Studies for Informatics, Kyoto, Japan Shouling Ji Department of Computer Science, Georgia State University, Atlanta, GA, USA Tao Jiang Department of Computer Science and Engineering, University of California, Riverside, CA, USA Liying Kang Department of Mathematics, Shanghai University, Shanghai, People’s Republic of China Naoki Katoh Department of Architecture and Architectural Engineering, Graduate School of Engineering, Kyoto University, Kyoto, Japan Samir Khuller Department of Computer Science, University of Maryland, College Park, MD, USA Deok-Soo Kim Department of Industrial Engineering, Hanyang University, Seongdong-ku, Seoul, South Korea Donghyun Kim Department of Mathematics and Computer Science, North Carolina Central University, Durham, NC, USA Gary A. Kochenberger School of Business, University of Colorado, Denver, CO, USA Kashyap Kolipaka Department of Computer Science, Rutgers, The State University of New Jersey, Piscataway, NJ, USA Dmytro Korenkevych Industrial and System Engineering Department, University of Florida, Gainesville, FL, USA Ilias S. Kotsireas Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada O. Erhun Kundakcioglu Department of Industrial Engineering, University of Houston, Houston, TX, USA Manuel Laguna Leeds School of Business, University of Colorado, Boulder, CO, USA Francis C. M. Lau Department of Computer Science, The University of Hong Kong, Hong Kong, People’s Republic of China

Contributors

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Lei Lei Department of Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA Deying Li Department of Computer Science, Renmin University, Beijing, People’s Republic of China Ming Li Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada Minming Li Department of Computer Science, City University of Hong Kong, Hong Kong, People’s Republic of China Xiang-Yang Li Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA Yingshu Li Department of Computer Science, Georgia State University, Atlanta, GA, USA You Li Department of Computer Science, University of Massachusetts, Lowell, MA, USA Ko-Wei Lih Institute of Mathematics, Academia Sinica, Taipei, Taiwan Benyuan Liu Department of Computer Science, University of Massachusetts, Lowell, MA, USA Ming Liu School of Economics & Management, Tongji University, Shanghai, People’s Republic of China Qinghai Liu College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, People’s Republic of China Zaixin Lu Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Silvano Martello DEI “Guglielmo Marconi”, University of Bologna, Bologna, Italy Juli´an Mestre School of Information Technologies, The University of Sydney, Sydney, NSW, Australia Rakhi Motwani Department of Computer Science & Engineering, University of Nevada, Reno, NV, USA Hung Q. Ngo Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA D. T. Nguyen Department of Computer & Information Science & Engineering, University of Florida, Gainesville, FL, USA Thanh-Nhan Nguyen Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA

xviii

Contributors

Foad Mahdavi Pajouh School of Industrial Engineering and Management, Oklahoma State University, Stillwater, OK, USA Evanthia Papadopoulou Faculty of Informatics, Universit`a della Svizzera italiana, Lugano, Switzerland Elisa Pappalardo Dipartmento di Matematica e Informatica, Universita di Catania, Catania, Italy Panos M. Pardalos Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA Mike Prince Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA Tomasz Radzik Department of Informatics, Kings College London, London, UK Andrea Raiconi Department of Mathematics, University of Salerno, Fisciano (SA), Italy Balaji Raghavachari Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Rajiv Raman Indraprasta Institute of Information Technology, Delhi, India Uriel G. Rothblum Faculty of Industrial Engineering and Management Science, Technion - Israel Institute of Technology, Haifa, Israel Chris Sackellares Research Department, Optima Neuroscience, Gainesville, FL, USA Samira Saedi Department of Industrial Engineering, University of Houston, Houston, TX, USA Maria Grazia Scutell`a Department of Computer Science, University of Pisa, Pisa, Italy Hong Shen School of Computer Science, The University of Adelaide, Adelaide, SA, Australia School of Information Science and Technology, Sun Yat-sen University, Guangzhou, People’s Republic of China Yilin Shen Department of Computer Science, Information Science and Engineering, University of Florida, Gainesville, FL, USA Hanif D. Sherali Grado Department of Industrial and Systems Engineering (0118), Virginia Polytechnic Institute and State University, Blacksburg, VA, USA Yan Shi Department of Computer Science and Software Engineering, University of Wisconsin - Platteville, Platteville, WI, USA Deng-Shan Shiau Research Department, Optima Neuroscience, Alachua, FL, USA

Contributors

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Akiyoshi Shioura Department of System Information Sciences, Graduate School of Information Sciences, Tohoku University, Sendai, Japan Oleg Shirokikh Industrial and Systems Engineering Department, University of Florida, Gainesville, FL, USA J. Cole Smith Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA J. MacGregor Smith Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst, MA, USA Jens Starke Department of Mathematics & Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark Vladimir Stozhkov Industrial and Systems Engineering Department, University of Florida, Gainesville, FL, USA Mujahid N. Syed Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA Mario Szegedy Department of Computer Science, Rutgers, The State University of New Jersey, Piscataway, NJ, USA Zhiyi Tan Department of Mathematics, Zhejiang University, Hangzhou, People’s Republic of China Fabio Tardella Sapienza University of Rome, Rome, Italy My T. Thai Department of Computer Science, Information Science and Engineering, University of Florida, Gainesville, FL, USA John Tromp CWI, Amsterdam, The Netherlands Daniele Vigo DEI “Guglielmo Marconi”, University of Bologna, Bologna, Italy Chrysafis Vogiatzis Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA Marcus G. Volz TSG Consulting, Melbourne, VIC, Australia Peng-Jun Wan Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA Haibo Wang College of Business Administration, Texas A&M International University, Laredo, TX, USA Jie Wang Department of Computer Science, University of Massachusetts, Lowell, MA, USA Lusheng Wang Department of Computer Science, City University of Hong Kong, Hong Kong Weifan Wang Department of Mathematics, Zhejiang Normal University, Jinhua, People’s Republic of China

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Contributors

Yuexuan Wang Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, People’s Republic of China Sean Weaver U.S. Department of Defense, Ft. George G. Meade, MD, USA James Willson Department of Computer Science, The University of New Mexico, Albuquerque, NM, USA Lidong Wu Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Weili Wu Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Ge Xia Department of Computer Science, Lafayette College, Easton, PA, USA Kai Xing Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Jinhui Xu Department of Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA Lei Xu Department of Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA Ping Xu Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA Shihong Xu School of Computer Science, The University of Adelaide, Adelaide, SA, Australia Boting Yang Department of Computer Science, University of Regina, Regina, SK, Canada Kai Yang Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Dongxiao Yu Department of Computer Science, The University of Hong Kong, Hong Kong, People’s Republic of China Wenan Zang Department of Mathematics, The University of Hong Kong, Hong Kong An Zhang Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou, People’s Republic of China Jicong Zhang Industrial and System Engineering Department, University of Florida, Gainesville, FL, USA Louxin Zhang Department of Mathematics, National University of Singapore, Singapore Wei Zhang Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA

Contributors

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Xuefei Zhang Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA Zhao Zhang College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, People’s Republic of China Feifeng Zheng Glorious Sun School of Business and Management, Donghua University, Shanghai, People’s Republic of China Jiaofei Zhong Department of Mathematics and Computer Science, University of Central Missouri, Warrensburg, MO, USA

A Unified Approach for Domination Problems on Different Network Topologies Thang N. Dinh, D. T. Nguyen and My T. Thai

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dominating Set Variants and Generalizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Positive Influence Dominating Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multiple-Hop PIDS Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Generalized Dominating Set Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Approximability of Domination Problems in General Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hardness of Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Approximation Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Domination Problems in Special Graph Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Power-Law Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dense Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Finding Optimal Solutions in Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Scalable Algorithm for Multiple-Hop PIDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 VirAds: Multiple-Hop PIDS Greedy Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Experimental Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 3 5 6 7 7 18 20 21 27 27 29 30 32 37 38

Abstract

This chapter studies approximability and inapproximability for a class of important domination problems in different network topologies. In addition to the well-known connected dominating set, total dominating set, more general forms

T.N. Dinh () • M.T. Thai Department of Computer Science, Information Science and Engineering, University of Florida, Gainesville, FL, USA e-mail: [email protected]; [email protected] D.T. Nguyen Department of Computer & Information Science & Engineering, University of Florida, Gainesville, FL, USA e-mail: [email protected] P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 24, © Springer Science+Business Media New York 2013

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of the problems, in which each node is required to be dominated by more than one of its neighbors, are also considered. An example is the positive influence dominating set (PIDS) problem, originated from the context of influence propagation in social networks. The PIDS problem seeks for a minimal set of nodes P such that all other nodes in the network have at least a fraction  > 0 of their neighbors in P . Furthermore, domination problems can be hybridized to form new problems such as T-PIDS and C-PIDS, the total version and the connected version of PIDS. The goal of the chapter is to narrow the gaps between the approximability and inapproximability of those domination problems. Going beyond the classic O.log n/ results, the chapter presents the explicit constants in the approximation factors to obtain tighter approximation bounds. While the first part of the chapter focuses on the general topology, the second part presents improved approximation results in different network topologies including powerlaw networks, social networks, and treelike networks.

1

Introduction

The dominating set problem in graph asks for a minimum size subset of vertices in which each vertex is either in the dominating set or adjacent to some vertex in the dominating set. It is one of the most widely studied topics in graph theory. In a survey by Hanes et al. [1], over 1,200 papers on the subject were listed in the bibliography. Domination problems have found many practical applications in building routing virtual backbones, power management, topology control, broadcast scheduling in sensor networks, and many others [2–9]. Recently, a variation of dominating set, called positive influence dominating set (PIDS), has been applied to identify influential users in social networks for viral marketing [10–12]. The approximability of the dominating set problem is the same with that of the set cover problem [13]. The work of Slavik [14] states that both problems can be approximated within a factor ln n  ln ln n C .1/, where n is the number of vertices in the dominating set instance (or the number of elements in the set cover instance). The lower bound on the approximability is given by Feige [13] in which set cover cannot be approximated within a factor .1  o.1// ln n, unless NP has slightly superpolynomial time algorithms.1 Hence, the approximation factor for the DS problem is tight up to a small sub-logarithm factor. However, this is not true for many variants of the DS problem, in which the gaps between lower bounds and upper bounds on the approximability are rather large. In many domination problems, only the asymptotic approximation factor O.log n/ is given without stating the explicit constant in the factor. In the case of the PIDS problem, the best hardness result is only APX-hard [11], while the best approximation factor is O.log n/ [16]. This chapter is an effort to narrow down the approximability gaps for variants of the DS problem in some important graph classes, revealing the hidden constant in the approximation factors. In addition, 1 Under the typical assumption that P ¤ NP, the best known inapproximability result is c  ln n [15] for some constant c > 0.

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inapproximability of numerous domination problems can be achieved under the same framework in which parameters in the Feige’s construction for set cover [13] are fine-tuned. For special graph classes, better approximation factors can be obtained by exploiting the graph topologies. A common theme in this chapter is that approximability results are first proved for the PIDS problem, and then necessary modification to achieve the same results for other variants is presented. Organization. The chapter continues with a brief summary on variants of the DS problem and their current approximability states. After that, the approximability of the considered domination problems in general graphs is presented. The rest of the chapter studies the domination problems in special graph classes including powerlaw graphs (social networks), dense graphs, and trees. Finally, a scalable algorithm for multiple-hop PIDS problem is introduced. Experiments on real-world social networks are presented to demonstrate the application of the dominating problem in finding influential users in social networks.

2

Dominating Set Variants and Generalizations

The most famous variant of the DS problem is the connected dominating set problem. The problem seeks for a minimum size DS which induces a connected subgraph. The problem has the same inapproximability .1  o.1// ln n of the DS problem. Guha and Khuller [17] give an H./ C 2 D ln n C O.1/ approximation algorithm for finding minimum CDS, where H.n/ D 1 C 1=2 C : : : C 1=n is the harmonic function. Another well-studied variant of DS is total dominating set. The total dominating set has an addition condition on the subgraph induced by the dominating set that the subgraph has no isolated vertices. The problem has the same hardness of the DS problem and can be approximated within a factor ln n C O.1/ [18]. In addition to the well-known connected dominating set and total dominating set, more generalization forms of the problems, in which each nodes is required to be dominated by more than one of its neighbors, are also considered. The t-tuple DS problem seeks for a subset S  V in graph G D .V; E/ so that each vertex is either in S or adjacent to at least t vertices in S . The problem admits a hardness of approximation factor .1  "/ ln n and an ln. C 1/ C 1 approximation [19–21]. A generalization of the t-tuple DS problem is the k-connected t-tuple dominating set problem [22] in which the subgraph is induced by the t-tuple DS. There is a 6 C ln 5=2.k 1/C5=k approximation algorithm for the problem on UDG. No hardness results other than NP-completeness for the problem were presented in literature.

2.1

Positive Influence Dominating Set

The positive influence dominating set problem emerges in the context of social networks, in which a set of influential users are sought for to propagate the influence to other users. Regularly, individuals tend to be influenced by the opinions/behaviors

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of their relatives, friends, and colleagues. For example, children whose parents smoked are twice as likely to begin smoking between 13 and 21 [23], and peer pressure accounts for 65 % reasons for binge drinking, a major health issue, by children and adolescents [24]. Moreover, the tendency of a user to adopt a behavior increases together with the number of his neighbors that follow that behavior. The motivation behind the PIDS problem is to exploit the relationships and influences among individuals in social networks might offer considerable benefit to both the economy and society. As an example, positive impacts of intervention and education programs on a properly selected set of initial individuals can diffuse widely into society via various social contacts: face to face, phone calls, email, social networks, and so on. Formally, let G D .V; E/ be a graph that represents the social network and an influence factor 0 <  < 1, in the PIDS problem one wishes to find a minimum set of vertices P , so that every other vertices in G has at least a fraction  of their neighbors in P . Denote by N.v/ the set of neighbors of a vertex v 2 V and d.v/ D jN.v/j the degree of v. The problem is defined in graph theory language as follows. POSITIVE INFLUENCE DOMINATING SET (PIDS) Input: An undirected graph G D .V; E/ with influence factor 0 <  < 1. Problem: Find a subset P  V such that 8u 2 V nP W jN.u/ \ P j  d.u/. Vertices in P are said to dominate or influence their neighbors in V nP . The constant  is called the influence factor, since it determines for each vertex the minimum number of neighbors to include in the PIDS. The studied problem in [10–12] is a special case of the above problem with  D 1=2. It is rather difficult to show the inapproximability for the PIDS problem. The best hardness result is only APX-hard [11], i.e., there is a certain constant " > 0 that approximating the problem within a factor 1 C " is NP-hard. This lower bound on the approximability is far from the typical lower bounds O.ln n/ in many domination problems. In contrast to t-tuple DS and any DS problems with fixed domination requirements, adding edges to a PIDS instance during a reduction will change the domination requirements. The “dynamic” threshold d.v/, rather than one fixed t, makes the PIDS problem resist to proof approaches for known domination problems. The literature on identifying influential users begins with Domingos and Richardson [25] who were the first to study the propagation of influence in networks. Kempe et al. [26, 27] formulated the influence maximization problem as an optimization problem. Later, Leskovec et al. [28] study the influence propagation in a different perspective in which they aim to find a set of nodes in networks to detect the spread of virus as soon as possible. Similarly, the connected and total versions of the PIDS problem can be defined as follows. TOTAL POSITIVE INFLUENCE DOMINATING SET (T-PIDS) Input: An undirected graph G D .V; E/ with influence factor 0 <  < 1. Problem: Find a subset PT  V such that 8u 2 V W jN.u/ \ PT j  d.u/.

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CONNECTED POSITIVE INFLUENCE DOMINATING SET (C-PIDS) Input: An undirected connected graph G D .V; E/ with influence factor 0 <  < 1. Problem: Find a subset PC  V such that 8u 2 V nPC W jN.u/ \ PC j  d.u/ and PC induces a connected subgraph of G. Similarly, total connected positive influence dominating set (TC-PIDS) problem asks for a connected and total PIDS PT C of G.

2.2

Multiple-Hop PIDS Problem

The PIDS problem can be extended to allow multiple-hop influence. The multiplehop PIDS problem [29] seeks for a minimal cost seeding, measured as the number of nodes, to massively and quickly spread the influence to the whole network (or a large segment of the network). The influence is limited to the nodes that are within d hops from the seeding for some constant d  1. In other words, the influence is expected to spread to the whole networks within d propagation rounds. Locally Bounded Diffusion Model. Let R0  V be the subset of vertices selected to initiate the influence propagation, called the seeding. Also a vertex v 2 R0 is said to be a seed. The propagation process happens in round, with all vertices in R0 are influenced (thus active in adopting the behavior) at round t D 0. At a particular round t  0, each vertex is either active (adopted the behavior) or inactive, and each vertex’s tendency to become active increases when more of its neighbors become active. If an inactive vertex u has more than d d.u/e active neighbors at round t, then it becomes active at round t C 1. The process goes on for a maximum number of d rounds, and a vertex once becomes active will remain active until the end. An initial set R0 of vertices is a d -seeding if R0 can make all vertices in the networks active within at most d rounds. The influence factor 0 <  < 1 decides how widely and quickly the influence propagates through the network. Influence factor  reflects real-world factors such as how easy to share the content with others or some intrinsic benefit for those who initially adopt the behavior. In case  D 1=2, the model is also known as the majority model that has many application in distributed computing, voting system [30], etc. Problem Definition. The multiple-hop PIDS problem is defined as follows: Given an undirected graph G D .V; E/ modeling a social network and an influence factor 0 <  < 1, find in V a minimum size d -seeding, i.e., a subset of vertices, that can activate all vertices in the network within at most d rounds. Influence propagation with a limited number of hops (multiple-hop PIDS) as well as a special case of T-PIDS, when  D 1=2, were first considered in Wang et al. [10] in which they iteratively find dominating sets until forming a T-PIDS. Feng et al. [16] showed NP-completeness for the PIDS problem, when  D 1=2.

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The APX-hardness and an O.log n/ approximation algorithm for T-PIDS problem were introduced in [11]. Under the condition that the geometric representation of a unit disk graph is given and the maximum degree is bounded by a constant, Zhang et al. [31] devised a polynomial time approximation scheme (PTAS) for the t-latency bounded information propagation.

2.3

Generalized Dominating Set Problems

Another way to generalize the PIDS problem is to replace the domination requirement d.v/ for vertex v with a function rv .x/ of d.v/, called the threshold function. This yields the so-called generalized DS problem. GENERALIZED DOMINATING SET (GDS) Input: An undirected graph G D .V; E/ and a family of functions rv .x/ for v 2 V. Problem: Find a subset P  V such that 8v 2 V nP W jN.v/ \ P j  rv .d.v//. Different family of frv .x/gv2V corresponds to different domination problems, for example, rv .x/ D 1 gives the usual dominating set; rv .x/ D x gives the PIDS problem; rv .x/ D t, where t is a positive constant, gives the t-TUPLE DOMINATING SET problem [20, 21]; and rv .x/ D cv , where cv are positive constants, gives the FIXED THRESHOLD DOMINATING SET (FTDS) problem. Moreover, for scalefree networks that degree sequences p follow power-law distribution, the threshold functions can take any of the form x or log x, etc. Notice that using a function rv .x/ instead of a simple constant cv for vertex v (the case of FTDS) allows the domination requirement of v to be adjusted accordingly when the network evolves or is argumented. It is important to treat GDS as a family of problems but not a single problem. Different families of threshold functions might lead to very different approximation algorithm and inapproximability results. For example, setting rv .x/ D x yields the MINIMUM VERTEX COVER that has simple two-approximation algorithm and 2  " lower bound under the unique game conjecture [32], while most domination problems achieve only O.log n/ approximation algorithms. In the same way that T-PIDS, C-PIDS, and TC-PIDS are formulated, the following problems are defined by considering the generalized threshold function: • TOTAL GENERALIZED DOMINATING SET (T-GDS). • CONNECTED GENERALIZED DOMINATING SET (C-GDS). • k-CONNECTED GENERALIZED DOMINATING SET (kC-GDS), in which the subgraph induced by the dominating set is k-vertex connected, i.e., there are at least k-vertex-disjoint paths between any vertices in the dominating set, where k is an integral constant. C-GDS is a special case of kC-GDS with k D 1. • TOTAL CONNECTED GENERALIZED DOMINATING SET (TC-GDS). • k-CONNECTED TOTAL GENERALIZED DOMINATING SET (kCT-GDS).

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Threshold Function. Not all threshold functions yield meaningful domination problems. For example, if rv .x/ > x, then there is no way to satisfy the requirement jN.v/ \ P j > rv .d.v// > d.v/. Only monotone increasing threshold function rv .x/ that satisfies the following conditions is considered in the rest of the chapter: 1. 0 < rv .x/ < x 8x > 0. 2. 0  rv .x/  rv .x  1/  1 8x > 0. rv .x/ < 1. 3. lim x!1 x The first condition makes the problem “nontrivial” to solve (to be precise, it makes the problem hard to approximate within O.log n/). The second and third conditions guarantee the growing rate of the function to be linear or sublinear.

3

Approximability of Domination Problems in General Graphs

This section presents inapproximability results (lower bound) for domination problems in families GDS, T-GDS, kC-GDS, and TC-GDS followed by approximation algorithms (upper bound). All the results also hold for PIDS, C-PIDS, and T-PIDS, as they are the special cases of the generalized dominating set problems considered in this section.

3.1

Hardness of Approximation

First, the hardness ln   O.ln ln / is proved by studying the problems in B-bounded graphs (Sect. 3.1.2). Then the proofs are extended to obtain the hardness .1=2  "/ ln n in Sect. 3.1.3. Finally, the hardness result for multiple-hop PIDS is derived through a reduction from the one-hop PIDS problem. The proofs require setting parameters in Feige’s reduction for set cover [13]. The hardness results of set cover cannot be applied in a black-box fashion since it leads to weaker inapproximability O.log B/ and O.log n/ for B-bounded graph and general graph. That is, they cannot provide the explicit constants in the inapproximability factors. The challenge lies on bounding the size of added vertices in our reductions with the size of the optimal set cover. Two important quantities that are not mentioned in the Feige’s reduction are the maximum capacity of a set and the maximum frequency of a point. The approximation factor and hardness of approximation of domination problems are summarized in Tables 1 and 2. The new results shown in this chapter are the following: • The proof that domination problems belong to the families GDS, T-GDS, C-GDS, and TC-GDS can be approximated within ln  C O.1/ but cannot be approximated within . 12  o.1// ln n, unless NP  DTIME(nO.log log n/ ). • On B-bounded degree graphs, none of the problems can be approximated within ln B  O .ln ln B/, under the standard assumption P ¤ NP . As a consequence,

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Table 1 Hardness of approximation for domination problems in graphs Problem

Original

T-(*)

kC-(*)

kCT-(*)

t -tuple DS

.1  "/ ln n[20]

.1  "/ ln n

Positive Influence DS

APX-hard,  D 12 [12] .1  "/ ln Œ? . 12  "/ ln nŒ? .1  "/ ln n

–

.1  "/ ln n, k D 1 k D 1 [20], 8kŒ? APX-hard, k D 1 [11] .1  "/ ln Œ? . 12  "/ ln nŒ? .1  "/ ln nŒ?

.1  "/ ln n, k D 1 k D 1 [20], 8kŒ? –

Fixed Threshold DS

.1  "/ ln Œ? . 12  "/ ln nŒ? .1  "/ ln n

.1  "/ ln Œ? . 12  "/ ln nŒ? .1  "/ ln nŒ?

Symbol Œ? means results are derived from this chapter. The hardness .1  "/ ln  is proved under the typical assumption P ¤ NP, while the hardness . 12  "/ ln n is proved with the assumption NP 6 DTIME.nO.log log n/ / Table 2 Hardness of approximation for domination problems in graph with maximum degree  Problem

Original

T-(*)

kC-(*)

kCT-(*)

Dominating Set

ln   O .ln ln / [33]

ln   O .ln ln / [33]

t -tuple DS

ln   O .ln ln /Œ? ln   O .ln ln /Œ? ln   O .ln ln /Œ?

ln   O .ln ln /Œ? ln   O .ln ln /Œ? ln   O .ln ln /Œ?

ln   O .ln ln / k D 1 [33], 8 k Œ? ln   O .ln ln /Œ? ln   O .ln ln /Œ? ln   O .ln ln /Œ?

ln   O .ln ln / k D 1 [33], 8 k Œ? ln   O .ln ln /Œ? ln   O .ln ln /Œ? ln   O .ln ln /Œ?

Positive Influence DS Fixed Threshold DS

Symbol Œ? means results is derived from this chapter. The hardness are proved with the assumption P ¤ NP

the considered problems are not approximated within ln   O .ln ln /, unless P = NP. • If the network is scale-free, i.e., the degree sequence follows a powerlaw distribution or the network is dense, the greedy algorithm obtains a constant factor approximation algorithm for “PIDS-like” domination problems. • In networks with tree topologies, it is possible to find the optimal solution in linear time for all considered domination problems.

3.1.1 Feige’s Reduction for Set Cover This part briefly summarizes the Feige’s proof for set cover (SC) which is essential to understand the incoming proofs. Feige presented a reduction from a k-prover proof system for a MAX 3SAT-5 instance  that is a conjunctive normal form formula consists of n variables and 5n 3 clauses of exactly three literals. The verifier interacts with k provers and asks provers different questions based on a random string r; each question involves l=2 clauses and l=2 variables. If the formula  is

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satisfiable, then the provers have a strategy that cause the verifier accepts for all random strings. If only a .1  "/ fraction of the clauses in  are simultaneously satisfiable, then for all strategies of the provers, the verifier weakly accepts with a probability at most k 2  2cl , where c is a constant that depends only on ". The core of the SC gadget is a partition system B.m; L; k; d /, where B is a ground set of m points. The partition system is a collection of L D 2l partitions P1 ; : : : ; PL of B, each partition Pi has exactly k disjoint subsets pi;1 ; : : : ; pi;k . Any cover of m points in B requires at least d D .1  22 /k ln m subsets. The condition to make constructing such a system possible is that k < 3 lnlnlnmm . Let R D .5n/l denote the number of possible random strings for the verifier. Make R copies of partition system B. Let Br denote the copy of the partition r associated with the random string r and pi;j the copy of set pi;j in Br . The instance of SC in the Feige’s reduction is now presented. The universal set [  ˚ UD Br contains N D jUj D mR points; and the set system is S D Sq;a;i q;a , r2R

where i can be deduced from syntax of .q; a/. Each set Sq;a;i corresponds to [ a question-answer pair .q; a/ of the i th prover and Sq;a;i D par r ;i where .q;i /2r

.q; i / 2 r means on random string r, the i th prover receives question q, and ar is the assignment of variables extracted from a. As long as k 2 2cl < k 3 ln82 m , the hardness result will be .1  k4 / ln m, i.e., if formula  is satisfiable, then mR points in U can be covered by kQ subsets, and if only .1"/ fraction of the clauses are simultaneously satisfiable, the minimum set cover has size at least .1  k4 / ln m kQ. Here, Q is the set of all nl .5=3/l=2 possible questions. The condition can be satisfied with l > c1 .5 log k C 2 log ln m/. The hardness ratio .1 f .k// ln m of the set cover is obtained from the following key lemma. Lemma 1 (Lemma 4.1 [13]) If  is satisfiable, then the above set of N D mR points can be covered by kQ subsets. If only a .1  "/ fraction of the clauses in  are simultaneously satisfiable, the above set requires .1  2f .k//kQ ln m subsets in order to be covered, where f .k/ ! 0 as k ! 1. Note that ln m D .1  "/ ln N by the setting of n; l; and m in the proof. Thus, the final hardness ratio is .1  "/ ln N , where N D jUj. However, choosing different settings of n; l, and m yields different hardness ratios. Upper bounds for quantities that appear in later proofs are presented for convenience. • The number of subsets jSj  jQj22l . Since, for each question q 2 Q, there are at most 22l answers of 2l bit length. • The maximum size of a subset S D max jS j  m3l=2 . Since each i and q 2 Q S 2S

there are at most 3l=2 random strings r such that the verifier makes query q to the i th prover and jpar r ;i j  m.

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• The maximum frequency of a point (element) in U: f  k2l . Because, for a pair .q; i /, each partition par r ;i is included at most 2l times, plus each point in Br appears in exactly k partitions.

3.1.2 Hardness of Approximation: O.ln / The hardness results ln O.ln ln / for domination problems are shown indirectly by proving the hardness ln B  O.ln ln B/ in the graphs with degree bounded by a constant B. In the first part, hardness results for PIDS and its variations are established. In the second part, the hardness of domination problems in the families GDS, T-GDS, kC-GDS, and d BD-GDS is proved with subtle modifications on the reductions in the first part. Positive Influence Dominating Set. The inapproximability of PIDS and its variants is given in the following theorem. Theorem 1 Neither PIDS, T-PIDS, nor kC-PIDS can be approximated within ln B  O.ln ln B/ in B-bounded graphs, unless P=NP. The proof is based on a reduction from an instance of the bounded set cover problem (SCB ) to an instance of PIDS problem whose degrees are also bounded by B 0 D B poly log B. BOUNDED SET COVER

(SCB )

Input: A set system .U; S/, where U D fe1 ; e2 ; : : : ; en g, is a universe and S is a collection of subsets of U. Each subset in S has at most B elements and each point belongs to at most B subsets, for a predefined constant B > 0. Problem: Find a cover that is a subfamily C  S of sets whose union is U with the minimum number of subsets. For a sufficient large constant B0 > 0, set cover problem where each set has at most B > B0 elements is hard to approximate to within a factor of ln B O.ln ln B/, unless P D NP [34]. The proof [34] maps an instance of GAP  SAT1; to an instance F D .U; S/ of set cover with S  B. Parameters l; m in Feige’s construction [13] are fixed B to .ln ln B/ and , respectively. Since the parameters can be found in poly log.B/ constant time using brute-force search, the assumption for the hardness is P ¤ NP instead of NP 6 DTIME(nO.log log n/ ). The produced instance has the following properties that will be used later in our proofs: • jUj D mnl poly logB; jSj D nl poly log B. • S  B; FU  poly log B for sufficient large B. A sufficient large constant B0 gives us f  poly log.B/  B for all B  B0 . t u

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11

Fig. 1 Reduction from SCB to PIDS (left) and T-PIDS (right)

SCB -PIDS Reduction. For each instance F D .U; S/ of SCB , construct a graph H D .V; E/ as follows (Fig. 1): • Construct a bipartite graph with the vertex set U [ S and edges between S and all elements x 2 S , for each S 2 S. • Add a set D consisting of t vertices and a set D 0 with same number of vertices, say D D fx1 ; x2 ; : : : ; xt g and D 0 D fx10 ; x20 ; : : : ; xt0 g. The value of t will be determined later. • Connect xi to xi0 ; 8i D 1 : : : t to force the selection of xi in the optimal PIDS.  • Connect each vertex ej 2 U to d 1 f .ej /e  1 and each vertex Sk 2 S  to d 1 jSk je vertices in D, where f .ej / is the frequency of point ej . The connection minimizes the degree differences of vertices in D. Lemma 2 The size difference between the optimal PIDS of H and the optimal SCB of F is exactly the cardinality of D, i.e., OPTPIDS .H/ D OPTS C .F / C t. Proof Let P be an optimal PIDS of H. Since either xi or xi0 must be selected into P , and xi0 2 P can be always replaced with xi inside P , thus, it is safe to assume that D 0 \ P D ; and D  P . By the construction, each vertex Sk 2 S has enough required neighbors in P , while each vertex ei 2 U needs at least one more neighbors in P or it has to be selected. Since all vertices in U must be adjacent to at least one vertex in S, it is possible to replace each vertex ei 2 P with one of its neighbors in S without increasing the size of P . Thus, w.l.o.g. assumes that the optimal solution will contain vertices in S but not in U. Hence, P n D must induce a cover for F D .U; S/. In other words, OPTPIDS .H/  OPTS C .F / C t. Besides, given a cover C  S for .U; S/, it is easy to check that C [ D gives a PIDS for H. Thus, OPTS C .F / C t  OPTPIDS .H/ that completes the proof. t u

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The requirements to transfer hardness results of set cover to PIDS problem are to keep the degree of vertices in H bounded by B 0 and keep t sufficiently small in comparison with the optimal solution of the SCB instance in order to derive the ratio. Lemma 3 There exists a construction of H with t  O.B poly log B/.

OPTSC ln2 B

and B 0 D .H/ D

Proof First, the total degree of vertices in D, vol.D/, is bounded. For two sets of vertices A and B, define .A; B/ the set of edges crossing between them. vol.D/ D j.D; D 0 /j C j.D; U/j C j.D; S/j X X   jSk je C f .ej /  1e D jDj C d d 1 1 S 2S e 2U k



2 jSjB C jSj C t D 1

The above inequalities hold because of

X Sk 2S

8Sk 2 S. Select t D

jU j . Since each set in B ln2 B SC  t. OPTS C  jUB j ; hence, OPT ln2 B



j

 2 B C 1 jSj C t: 1

jSk j D

X

(1)

f .ej / and jSk j  B;

ej 2U

S can cover at most B elements, it follows

that To have a valid construction of H, it is sufficient that t:B 0  vol.D/. Thus, select 0 B satisfying B0  

1 t 



  2 B C 1 jSj C t 1  B ln2 B nl poly log B 2 B C1  B poly log B: 1 mnl poly log B

Hence, setting B 0 D B poly log B gives us the desired construction of H.

(2) t u

Theorem 2 There exist constants B1 ; c1 such that for every B 0  B1 , it is NP-hard to approximate the PIDS problem in graphs with degrees bounded by B 0 within a factor of ln B 0  c1 ln ln B 0 . Proof Assume that there exists an algorithm that finds a PIDS of size at most ln B 0  c1 ln ln B 0 the optimal size in graph with degrees bounded by B 0 . Then the contradiction is obtained by showing how to approximate the S CB problem with ratio ln B  c0 ln ln B. Selecting sufficient large B1 is not difficult and shall be ignored to make the proof simpler. Let F D .U; S/ be an instance of SCB . Construct an instance H of PIDS problem using the reduction SCB -PIDS. From (2), there exists constant ˇ > 0

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13

so that B 0  B lnˇ B. The approximation for PIDS returns a solution of size at most .ln B 0  c1 ln ln B 0 /OPTPIDS . Then, that can be converted into a solution of SCB by excluding vertices in D (see Lemma 2) and obtain a cover of size at most .ln B 0  c1 ln ln B 0 /.OPTS C C t/  t OPTS C ln2 B    1 OPTS C :  ln B C ˇ ln ln B  c1 ln.ln B C .ln ln B// C O ln B

.ln B 0  c1 ln ln B 0 /OPTS C C .ln B 0  c1 ln ln B 0 /

Select c1 D c0 C ˇ C 1. The solution for SCB problem is then smaller than ln B  c0 ln ln B times OPTSC which implies P = NP. t u Theorem 3 It is NP-hard to approximate T-PIDS and kC-PIDS problems in graphs of bounded degree B 0 > B2 within a factor of ln B 0  c2 ln ln B 0 for some constants B2 ; c2 > 0.

Proof The reduction SCB -PIDS shall be adjusted to achieve the same hardness result. The following adjustments are made to guarantee total and k-connected properties: • On top of the subgraph induced by D, construct .2 C O.1//k-regular Ramanujan graphs so that the subgraph has vertex expansion at least k [35]. At the same time, connect S to D so that each vertex in S has at least k neighbors in D. It is easy to check that the subgraph induced by the solution to the dominating set problem is k-vertex connected.  • Connect a vertex xi 2 D with d 1 d.xi /e other nodes in D, balancing nodes’ degrees in D, so that D can dominate themselves. Thus, the degree of each node  j d 1 B 0 e, there are in D is roughly multiplied by a constant. Since t D B jU ln2 B always enough vertices in D to connect xi to. The rest of the proof is straightforward. t u Generalized Dominating Set Problems. By modifying the construction in the hardness proof for PIDS, the hardness results for GDS and its variants are obtained in the following theorem. Theorem 4 Domination problems in families GDS, T-GDS, and kC-GDS with threshold functions rv .x/ cannot be approximated within ln B  O.ln ln B/ in B-bounded graphs, unless P = NP. Proof Given a B-bounded graph G D .V; E/ and a functions rv .x/ satisfying the rv .x/ . By the third condition, conditions of a threshold function, let  D max lim v2V x!1 x

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 < 1 and  D 12 . C 1/ < 1. Since  >  , there exists an absolute constant x0 > 0 such that rv .x/ <  x 8x > x0 and 8v 2 V . Assume through the proof that B is sufficiently large so that l D .log log B/ > x0 . The main idea in the reduction from dominating set with domination requirement rv .d.v// is to add connections from v to some a.v/ more vertices that are guaranteed to be in the optimal solution so that the remaining domination requirement on v becomes one for vertices in U and zero for vertices in S. It is necessary to find xv 2 N so that  cv .xv / D rv .d.v/ C xv /  xv 2

.0; 1 .1; 0

if v 2 U if v 2 S:

(3)

 d.v/e; x0 g, cv .v /  0. Claim For v D maxfd 1

Proof cv .v / D  .d.v/ C v /  xv D .d.v/ 

1 v / 

< 0.

t u

Since • cv .0/ D rv .d.v// > 0. • cv D cv .x C 1/  cv .x/ 2 .1; 0. • cv .v / < 0. There exists some 0  xv  v such that cv .xv / satisfies Eq. 3. Notice that x0 is substantially smaller than B. Hence, the same construction in the case of PIDS with influence factor  will work for the general cases as well. The only exception is that vertices in v 2 .S [ U/ where xv is connected to vertices in D. t u The following result is a direct corollary of the hardness result for the bounded degree case. Theorem 5 Unless P = NP, domination problems in families GDS, T-GDS, and kC-GDS cannot be approximated within a factor of ln  O.ln ln /, where  is the maximum degree.

3.1.3 Hardness of Approximation: O.ln n/ Ideally, since the maximum degree  in the graph can be as large as the number of vertices n, it is natural to expect that the PIDS problem cannot be approximated within a factor ln n. However, the hardness result for PIDS with the best parameter settings is only .1=2  o.1// ln n. It is conjectured that there is a 1=2 ln n approximation for the PIDS and T-PIDS problem. Generalized Dominating Set Problems Theorem 6 Domination problems in families GDS, T-GDS, and kC-GDS with threshold functions rv .x/ cannot be approximated within 12 .1  o.1// ln n where   n is the number of vertices, unless NP  DTIME nO.log log n/ .

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Fig. 2 Reduction from SCB to GDS, T-GDS, and kC-GDS

Proof The same gadget in Fig. 2 is used to prove for the three general problems. From the gadget in the proof of Theorem 4, it is not necessary to keep degree of vertices in the gadget bounded. Thus, vertices in D are all connected to each other. Let ; x0 ; v be the same parameters as in the proof of Theorem 4. The sufficient conditions to make the construction feasible are: • (GDS): To be able to connect each vertex v 2 .S [ U/ to v vertices in D: 

 jDj D O

max

v2.S[U /v



 S 1

 D O.S / D O.m3l=2 /

• (kC-GDS): To add at least k edges from each vertex in S to vertices in D:   jDj D O.kjSj/ D O.jSj/ D O nl .5=3/l=2 22l D O.nl 2.l/ / • (T-GDS): Vertices in D have to dominate themselves. Since v  maxfd 1 d.v/e; x0 g, this can be satisfied when 0  1 jDj  1 D O @ 1   jDj

X

1 v A :

v2.S[U /

Or equivalently 2

jDj D O

X v2.S[U /

! d.v/ C x0 .jSj C jUj/

D

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T.N. Dinh et al.

DO 2

X

! d.v/ C jSj C jUj D O.m R k 2l /

v2U

To summarize, the sufficient condition so that the dominating set of the graph is k-connected and total at the same time is jDj D O.m2.l/ C nl 2.l/ C .mRk2l /1=2 /:

(4)

Notice that, from the proof of Theorem 2, the hardness ratio is in the form .1 k4 /kQ ln mCjDj . kQCjDj

2l

In the Feige’s reduction, jDj D O.S / D O..5n/ " 2.l/ / that yields the .1  "/ ln n hardness ratio for set cover but makes the hardness ratio of the domination problem gets arbitrary close to 1. Fortunately, it is possible to reduce the maximum degree by setting m D .5n/cl with a small constant c > 0. The consequence is that ln m is no longer ln N C O.1/; thus, the inapproximability ratio is reduced. The optimal setting to get the best inapproximability ratio is to set m D .5n/l.1"/ 1" for some " > 0. Then, N D mR D .5n/l.2"/ , or m D N 2" . Hence, from (4), it is sufficient that 2.l/ jDj D nl l " D o.Q/ n2 for sufficiently large l and n. Hence, the hardness ratio will be   .1  k4 /kQ ln m C o.Q/ 5 ln m: > 1 kQ C o.Q/ k The number of vertices in the graph is nH D 2jDj C jSj C jUj < .m3l=2 / C nl 22l

 l=2 5 C .5n/2l" < 2jUj D 2N: 3

Thus, the hardness ratio is at least      " " 5 5 1 1 nH 1=2 42" ln 1 ln nH .1/ > .1"/ ln nH 1 > 1 k 2 k 2 2" 2 for sufficiently large k and sufficiently small ".

t u

Theorem 7 Domination problems in families GDS, T-GDS, and kC-GDS with threshold functions rv .x/ D O.log x/ cannot be approximated within   .1  o.1// ln n where n is the number of vertices, unless NP  DTIME nO.log log n/ . Proof When rv .x/ D O.log x/, the size of jDj is only O.log S /. The original settings in the Feige’s reduction will give the ratio .1  "/ ln nH . t u

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17

Multiple-Hop PIDS Problem. The hardness is shown through a reduction from the one-hop PIDS problem to the multiple-hop PIDS problem with d  2. The hardness result follows immediately by the Theorem 2 in the previous section. Given a graph G D .V; E/ as an instance of the PIDS problem, construct an instance G 0 D .V 0 ; E 0 / of the PIDS problem as follows (and as illustrated in Fig. 5). Replace each edge .u; v/ 2 E by a gadget called transmitter. The transmitter connecting vertices u and v is a chain of d  1 path, named uv1 to uvd 1 . The vertex u is connected to uv1 , uv1 is connected to uv2 and so on, and vertex uvd 1 is connected to v. Each vertex uvi , i D 1 : : : d  1, is connected to all flash points. An example for transmitter is shown in Fig. 3. The transmitter is designed so that if all flash points and vertex u are selected at the beginning, then vertex uvd 1 will be activated after d  1 rounds. Hence, the number of activated neighbors of v after d  1 rounds will equal the number of selected neighbors of v in the original graph (Fig. 4). Finally, each edge .wp ; zp / is replaced with a transmitter. In order to activate all dummy vertices zp after d rounds, assume, w.l.o.g., that all flash points must be selected in an optimal solution. The following lemma follows directly from the construction. Lemma 4 Every solution of size k for the one-hop (d D 1) PIDS problem in G induces a solution of size k C c./ for the d -hop PIDS problem in G 0 . On another direction, the following lemma holds. Lemma 5 An optimal solution of size k 0 for the d -hop PIDS problem induces a size k 0  c./ solution for the one-hop PIDS problem in G.

Proof For a transmitter connecting u to v, if the solution of the d -hop PIDS problem contains any of the intermediate vertices uv1 ; : : : ; uvd 1 , that intermediate vertex can be replaced with either u or v to obtain a new solution of same size (or less). Hence, assume, w.l.o.g., that none of the intermediate vertices are selected. Therefore, all flash points must be selected in order to activate the dummy vertices. It is easy to see that the solution of d -hop PIDS excluding the flash points will be a solution of one-hop PIDS in G with size k 0  c./. t u Note that the number of vertices in G 0 is upper bounded by d n2 , i.e., ln jV 0 j < 2lnjV j C lnd . Thus, using the same arguments used in the proof of Theorem 6, it can be shown that a . 14  "/ ln n approximation algorithm leads to a . 12  "/ ln n approximation algorithm for the one-hop PIDS problem (contradicts Theorem 6). Theorem 8 The PIDS problem cannot be approximated within . 14  "/ log n for d  1, unless NP  DTIME.nO.log log n/ /.

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Fig. 3 The transmitter gadget

Fig. 4 Gap reduction from one-hop PIDS to d -hop PIDS

3.2

Approximation Algorithm

This part presents briefly how approximation algorithms for studied domination problems can be obtained via a simple reduction to the constrained multiset multicover (CMM) problem. In turn, the MMC problem can be seen as an integer covering problem. Thus, the domination problems benefit directly from advances in integer linear programming and the integer covering problems in particular. Generalized Dominating Sets. It is easy to show that all domination problems in the generalized class can be approximated within a factor ln  C O.1/.

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19

Theorem 9 Given a graph G D .V; E/, there exist O ..jV j C jEj/ log log jV j/ algorithms that approximate GDS within H.2/ and T-GDS within H./. Proof The proof begins with the definition of the constrained multiset multicover problem (CMM). CONSTRAINED MULTISET MULTICOVER (CMM) Input: A set cover instance .U; S/. Each point e has an integer requirement re and occurs in a set S with arbitrary multiplicity, denoted m.S; e/. Moreover, associate a cost, cS , with each set S 2 S. Problem: The minimum cost subcollection which fulfils all elements’ cover requirements provided each multiset is picked at most once. Lemma 6 ([36]) There is a natural greedy algorithm that finds a constrained multiset P multicover within an Hk factor of the optimal solution, where k D maxS e m.S; e/. The GDS problem on the graph G D .V; E/ can be reduced to the following instance of CMM: • U D feu W u 2 V g. • The cover requirement of eu is set to ru D dru .d.u//e. • S D fSv W v 2 V g, where Sv contains feu W u 2 N.v/g plus rv copies of ev . That is, m.Sv ; eu / D 1; 8u 2 N.v/, and m.Sv ; ev / D drv .d.v//e. It follows that the GDS problem can be approximated within   H max d.v/ C rv .d.v//  H.2/ D H./ C O.1/: v2V

In case of T-GDS, the only difference in the reduction is that each multiset Sv contains all the neighbors of v, but not any copies of v. The approximation ratio is, hence, H./. Implementation Issue: Straightforward implementations might incur high time complexity. In each step, the greedy algorithms select the node with the highest coverage which is bounded by O.n/. Hence, using van Emde Boas priority queue [37] to maintain and update nodes’ coverages, the total time complexity can be brought down to O ..jV j C jEj/ log log jV j/. Details are presented in the Algorithm 1. t u Corollary 1 There are polynomial time algorithms that approximate PIDS and T-PIDS problems within ratios ln  C 43 and ln  C 1, respectively. Proof Apply Theorem 9 with rv .x/ D x and use the approximation H.n/  ln  the approximation ratios for PIDS and T-PIDS can be rewritten as  n C 0:58; t u ln  C 43 and .ln  C 1/.

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Connected Generalized Dominating Sets. The connected version of GDS problem requires either submodular theory or a conversion to the Steiner tree problem to obtain the approximation factor as presented in the proof of the following theorems. Theorem 10 There is a polynomial time algorithm that finds a weighted C-GDS of size at most 1:55 ln  C 52 time that of the minimum C-GDS. Proof The same approach for connected dominating problem in [17] can be applied. In the first stage, a GDS is found using the greedy heuristic described in Theorem 9. In the second stage, the vertices in the DS are connected using Steiner tree algorithm. Since a GDS is also a (normal) dominating set, adding the optimal solution of C-GDS to the found GDS in the first stage gives a Steiner tree of the GDS. Hence, using a c-approximation algorithm for Steiner tree to connect vertices in a GDS yields a c .H./ C ln 2 C 1/ approximation algorithm for C-GDS problem. Apply the best known approximation ratio for Steiner tree problem c  1:55 by Robins and Zelikovsky [38]; the two-stage algorithm admits a 1:55 ln  C 2:5 approximation algorithm. t u Theorem 11 There are polynomial time algorithms that approximate unweighted C-GDS, and TC-GDS within ratios H.3/ and H.2/, respectively. Proof The greedy algorithm presented in [11] can be extended to work with arbitrary threshold function rv .x/ instead of rv .x/ D dx=2e. The analysis involves a non-submodular potential function that can be overcome with techniques in [39, 40]. Though, it is unclear if the technique can be extended for the weighted cases. t u

4

Domination Problems in Special Graph Classes

This section studies hardness and approximation algorithms for domination problems in special graph classes. In most cases, better approximation algorithms with improved ratio can be obtained. For example, almost all domination problems in power-law networks admit trivial constant factor approximation algorithms as shown later. In planar graphs and disk graphs, there exist polynomial time approximation scheme (PTAS) that approximate the optimal solution within a factor arbitrary close to one [41–46]. In the context of wireless sensor networks (WSNs), distributed .1 C "/ approximation algorithm on unit disk graphs (UDGs) with O.log jGj/ rounds is known [47]. However, PTAS is not known for the weighted version of the problem on UDGs. The latest result for that problem is a 5 C " approximation algorithm [48].

A Unified Approach for Domination Problems on Different Network Topologies

4.1

21

Power-Law Networks

Many social, biological, and technology networks including OSNs display a nontrivial topological feature: their degree sequences can be well approximated by a power-law distribution [49]. Many optimization problems that are hard on general graphs can be solved much more efficient in power-law graphs [50, 51]. The analysis in this part uses the well-known P .˛; ˇ/ model [52] in which there are y vertices of degree x, where x and y satisfy log y D ˛  ˇ log x. In other words, e˛ jfv W d.v/ D xgj D y D ˇ : x Basically, ˛ is the logarithm of the size of the graph and ˇ is the log-log growth rate of the graph. Graphs with the same ˇ value often express the same behaviors and common characteristics. Hence, it is natural to categorize all graphs sharing a ˇ value into a family of graphs and regard to ˇ as a constant (not a part of the input). Without affecting the conclusions, real numbers will be used instead of rounding down to integers. The error terms can be easily bounded and are sufficiently small in the proofs. ˛ The maximum degree in a P .˛; ˇ/ graph is e ˇ . The number of vertices and edges are 8 ˛ ˆ < .ˇ/e if ˇ > 1 e ˛ if ˇ > 1 ; nD  ˛e˛ ˇ ˆ x ˇ : e xD1 if ˇ < 1 1ˇ ˛

eˇ X

˛

where .ˇ/ D

P1

1 i D1 i ˇ

81 ˛ ˆ < 21 .ˇ˛  1/e if ˇ > 2 e if ˇ D 2 m D 12 x ˇ  4 ˛e2˛ ˆ x ˇ : 1 e xD1 if ˇ < 2 2 2ˇ ˛

eˇ X

˛

is the Riemann zeta function.

Theorem 12 Minimum PIDS is APX-hard, i.e., there is a positive constant " depends only on the log-log growth rate ˇ such that approximating PIDS within 1 C " is NP-hard [53]. Theorem 13 In a power-law graph G 2 P .˛; ˇ/, the size of the optimal PIDS

OPTPIDS

8 if ˇ < 1 < .nˇ / D .n= log n/ if ˇ D 2 :

.n/ if ˇ > 2:

Proof 1. Let k be the size of the optimal PIDS. Note that all nodes of degree more than k= must be selected (otherwise, the number of selected neighbors will exceed k). The number of nodes with degree larger than k will be

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T.N. Dinh et al. ˛ ˇ

˛ ˇ

xDk=

xDk=

e e X X ˛ e˛ e˛ D e ˛ .ln e ˇ  ln k/: k> > ˇ x x

(5)

Solving the above relation gives us k > .e ˛ / D .nˇ /. 2. Similarly, if ˇ D 2, then k D .e ˛ / D .n=log n/. 3. The lower bound is obtained through a dual-setting approach. Consider the following linear program and its dual of the PIDS problem:

LP: min

X

xv

v2V

s. t. rv xv C

DP: max X

X

ru yu 

u2V

xu  rv

s. t. ru yu C

u2N.v/

X

zv

v2V

X

yv  zu  1

(6)

v2N.u/

 xu  1

zv  0

xu  0

yv  0

where ru D du . Note that for the integral versions of LP (16) and DP (16), both settings ru D du and ru D ddu e yield the same optimal solutions; however, setting ru D du simplifies the approximation ratio analysis. Set yu D  8u 2 V . The goal is to look for value of  that achieves the tightest lower bound on the size of the optimal PIDS. To satisfy constraints in the dual, set zu D maxf. C 1/du   1; 0g. Then the objective value becomes DP D 

X

du 

u2V

D 

X

X

maxf. C 1/du   1; 0g

(7)

u2V

du 

u2V

X

.. C 1/du   1/

(8)

du > . /

where ./ denotes . C 1/1  1 . 1 Substituting  D .C1/ into (8), this leads to  e X  1 e˛ X e˛ e˛  x x ˇ DP D . C 1/ xD1 x ˇ  xˇ x x>    .ˇ  1/ ˛ X 1 e ˛ e˛ D  e  : . C 1/  x ˇ1 x ˇ x> ˛ ˇ

˛

˛

(9)

(10)

Except for at most be ˇ c points  D 1; 2; : : : ; be ˇ c, the derivative of the objective function, ddDP  , is defined. Moreover, at those integral points, both one-sided limits,

A Unified Approach for Domination Problems on Different Network Topologies

23

lim DP and lim DP , agree, i.e., DP is a continuous function everywhere with

 !i 

respect to .

 !i C

Lemma 7 For every  2 .i; i C 1/; i 2 NC , the derivative satisfies 1 d DP D 2 d 

 .ˇ  1/ ˛ X e ˛ e  C1 x ˇ1 x>

d DP d

is defined and

! :

(11)

By (11), there exists a fixed dividing point x0 2 NC that depends only on ˇ, d DP satisfying ddDP  ./  0; 8 < x0 and d  ./ < 0; 8 > x0 . Since DP is continuous everywhere, it obtains the global maximum value at  D x0 . It is needed to show that the value of DP at  D x0 is .n/, and since the objective of the primal is lower bounded by DP, it follows that the size of the minimum PIDS will be at least .n/. ! X e˛  .ˇ  1/ ˛ X e ˛ C e  . C 1/ x ˇ1 xˇ x>x0 x>x0 ! ! P 1 X e˛ X 1 xx0 x ˇ ˛   .ˇ/  e  1 n D .n/: u t xˇ xˇ .ˇ/ x>x xx

1 DP .x0 / D x0

0

0

Using the same approach in Theorem 13 gives the similar bounds for T-PIDS in the following theorem. Theorem 14 In a power-law graph G 2 P .˛; ˇ/, the size of the optimal T-PIDS.  OPTTPIDS D

.n/ if ˇ < 1 or ˇ > 2

.n=log n/ if ˇ D 1:

Theorem 15 In a power-law graph G 2 P .˛; ˇ/, the size of the optimal C-PIDS  OPTCPIDS D

.n/ if ˇ > 2

.n=log n/ if ˇ D 1:

If networks have optimal PIDS/T-PIDS/C-PIDS of .n/ size, clearly, any algorithms that produce valid PIDS/T-PIDS/C-PIDS will be constant factor approximation algorithms. Theorem 16 Given a power-law P .˛; ˇ/ graph G D .V; E/: 1. ˇ < 1: PIDS is not in APX (cannot be approximated within a constant factor), while T-PIDS admits a constant factor approximation algorithm. 2. ˇ > 2: There exist constant factor approximation algorithms for PIDS, T-PIDS, and C-PIDS problems.

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Theorem 17 Given a power-law network G 2 P .˛; ˇ/, with ˇ > 2 and constant 0 <  < 1, any d -seeding is of size at least .n/. Proof The proof consists of two parts. It is shown in the first part that the volume, i.e., the total degree of vertices, of any d -seeding must be .m/. The second part shows that any subset of vertices S  V with volume vol.S / D .m/ in a powerlaw network with power-law exponent ˇ > 2 will imply that jS j D .n/. Thus, the theorem follows. In the first part, there are two following cases. Case  > 12 : Let S D R0 be the optimal solution for the PIDS problem on G D .V; E/, and S D R0 ; R1 ; R2 ; : : : ; Rd are vertices that become active at round 0; 1; 2; : : : ; d , respectively (see Fig. 5). Thus, fRi gdiD0 form a partition of V . Moreover, for each 1  t  d the following inequality holds: 1 0 d [  @ j.Rt ; Ri /j  Rj /j C 2j.Rt ; Rt /jA j.Rt ; 1   i D1 j Dt C1 t 1 [

(12)

where .A; B/ denotes the set of edges connecting one vertex in A to one vertex in  B. This can be seen as at least a fraction 1 among edges incident with the vertices activated in round t must be incident with active vertices in the previous rounds. Sum up all inequalities in (12) for t D 1 : : : d . This yields 1 0 d d [  X@ j.Rt ; Ri /j  Rj /j C 2j.Rt ; Rt /jA : j.Rt ; 1   t D1 t D1 i D1 j Dt C1

d X

t 1 [

After eliminating the common factors in both sides, d 1 X i D0



d [

j.Ri ;

Rt /j

t Di C1

d 1 d d 1 [ X  X j.Ri ; Rt /j C 2 j.Rt ; Rt /j: 1   j D1 t Dj C1 t D1

And after some algebra,

vol.R0 /  j.R0 ;

d [ t D1

Rt /j

A Unified Approach for Domination Problems on Different Network Topologies

25

Fig. 5 The influence propagation in the network

 ,

d 1 d d [ X 2  1 X j.Ri ; Rt /j C 2 j.Rt ; Rt /j 1   j D1 t Dj C1 t D1

 j.R0 ; V /j  j.R0 ; R0 /j 1 3  4 X 2  1 jEj C j.Rt ; Rt /j: 1 1   t D1 d



(13)

Hence, when  > 1=2, vol.R0 /  21 1 jEj D .m/ for any d -seeding R0 . 1 Case   2 : An edge is said to be active if it is incident to at least one active vertex. At round t D 0, there are at most vol.R0 / active edges, those who are incident to R0 . Equation 12 implies that the number of active edges in each round increases at most 1 times. After d rounds, the number of active edges will be bounded by vol.R0 / d . Since all edges are active at the end, the following inequality holds: vol.R0 /  d jEj: The second part of the proof shows that if a subset S  V has vol.S / D .m/, then jS j D .n/ whenever the power-law exponent ˇ > 2. Assume that vol.S /  cm, for some positive constant c. The size of S is minimum when S contains only the

26

T.N. Dinh et al.

highest degree vertices of V . Let k0 be the minimum degree of vertices in S in that extreme case, it follows that ˛ ˇ

˛ ˇ

kD1

kDk0

e e c X e˛ 1 X e˛ cm D k ˇ  vol.S /  k ˇ: 2 k 2 k

Simplifying two sides, it follows that kX 0 1 kD1

1 k ˇ1

˛ ˇ

 .1  c/

e X kD1

1 D .1  c/ .ˇ  1/: k ˇ1

Since, the zeta function .ˇ  1/ converges for ˇ > 2, there exists a constant k;ˇ that depends only on  and ˇ that satisfies k;ˇ X kD1

1 k ˇ1

> .1  c/ .ˇ  1/:

Obviously, k0  k;ˇ . Thus, the number of vertices that are in S is at least 0 1 ˛ k;ˇ eˇ X X e˛ 1 A n D .n/: D @1  kˇ kˇ

kDk;ˇ

The last step holds because the sum a constant.

kD1

Pk;ˇ

1 kD1 k ˇ

is bounded by a constant since k;ˇ is t u

In both cases  > 1=2 and   1=2, the size of a d -seeding set is at least

.n/. However, there is a clear difference in the propagation speed with respect to d between two cases. When  < 1=2, the number of active edges can increase exponentially (but is still bounded if d is a constant), and it is likely that the number of active vertices also exponentially increases. In contrast, when  > 1=2, exploding in the number of active edges (and hence active vertices) is impossible as the volume of the d -seeding is tied to the number of edges m by a fixed constant 21 , regardless 1 of the value of d . Theorem 18 In power-law networks with power exponent ˇ > 1, there are O.1/ approximation algorithms for the PIDS problem for bounded value of d  1.

A Unified Approach for Domination Problems on Different Network Topologies

4.2

27

Dense Graphs

p Lemma 8 If PT is a T-PIDS of G D .V; E/, then jPT j  . jV j C jEj/. Proof Let k D jPT j be the size of a T-PIDS. All Sv 2 V nPT must be adjacent to at least one vertex in PT . Thus, jV j  jPT j C v2PT jN.v/nPT j. Moreover, for each vertex v 2 PT ; jN.v/ \ PT j  jN.v/j ) jN.v/nPT j  1  jN.v/ \ PT j  1 .k 

 1/. Therefore, nkCk

1 1   2 2  1 .k  1/ D k C k:   

(14)

Divide edges in E into three categories: (1) edges whose both ends are in PT , (2) edges whose exactone  end is in PT , and (3) edges whose both ends are not in PT . There are at most k2 edges of type 1. For a vertex v 2 PT , the number of type 2 edges incident to v is at most 1  .k  1/ since v is adjacent to at most k  1 vertices .k  1/ D in PT . Hence, the number of type 2 edges is upper bounded by 2k 1  2.1/ k  . For each vertex u … PT , the number of type 3 edges incident to u is at  2

most

1 

times the number of type 2 edges incident to u. Therefore, the number of 2 k  type 3 edges is at most .1/ 2 . 2 Adding all three types of edges together yields the following: !   ! 2.1  / .1  /2 k 1 k jEj  1 C C D 2 :  2  2 2 p It follows from (14) and (15) that jPT j D k D . jV j C jEj/.

(15)

t u

The bound is tight, i.e., there is a graph in which the minimum T-PIDS has size p .k 1/c

. jV j C jEj/. For example, consider a “hairy” clique of n D k Ck  b 1  k  1 vertices and m D 2 C k  b  .k  1/c edges by connecting each vertex in a clique

.k 1/c leaf nodes (Fig. 6). The minimum T-PIDS will be the clique of size k to b 1  p itself that is of size k D . n C m/. Theorem 19 For a dense graph G D .V; E/ with jEj D .jV j2 /, there exist constant factor approximation algorithms for PIDS, T-PIDS, and C-PIDS problems.

4.3

Finding Optimal Solutions in Trees

In trees, it is possible to find optimal GDS and T-GDS in polynomial time. However, designing such algorithms in a linear-time fashion is not too obvious. This part

28

T.N. Dinh et al.

Fig.p6 A PIDS (left) may consist of only one node, while a T-PIDS (right) must contain at least O. jV j C jEj/

Fig. 7 GDS-TREE(G)

Fig. 8 T-GDS-TREE(G)

presents two depth-first search-based (DFS) algorithms in Figures 7 and 8 for GDS and T-GDS, respectively. Notice that the solution for kC-GDS and TC-GDS problems on trees is trivial; the optimal solution is simply the set of all nonleaf nodes.

A Unified Approach for Domination Problems on Different Network Topologies

29

Theorem 20 Optimal GDS and T-GDS in trees can be found in linear time. Proof At a given step, P =T denotes the current GDS/T-GDS. For each v 2 V , define the functions rp .v; P / D drv .d.v//e.1  1P .v//  jN.v/ \ P j and rt .v; T / D ( 1 if x 2 A; drv .d.v//e  jN.v/ \ T j, where 1A .x/ D 0 if x … A: Functions rp .v; P /; rt .v; T / determine the minimum numbers of v’s neighbors to include into the optimal solutions. A node u with rp .u; P / > 0 or rt .u; T / > 0 is called uncovered; otherwise, u is called covered. Assume that the tree is rooted at some vertex u. For an edge .u; v/, if u is visited before v, then u is the parent of v and v is a child of u. Correctness. Proof by induction that each selection step is optimal. GDS: Assume that all the selections made so far are optimal. Assume node u, the parent of v is selected in step 3, Fig. 7. From rp .v; P / > 0, it follows that 1. v … P or else rp .v; P / < 0. 2. rp .v; P / D 1 if not v has already been selected by the end of GDS-VISIT(v). To cover v, it is necessary to either select v, u or some children of v. However, since all nodes in the subtree rooted at v have been covered, there will be no extra benefit in selecting v or its children. Formally, if an optimal solution selects v or its children, it is always possible to replace the selected vertex with u and obtain a new optimal PIDS. In case rp .u; P / > 1, the only choice is to select u. T-GDS: After T-GDS-VISIT(v; pv) finishes, v always becomes covered. Assume the selection of vertices into T is optimal so far. During the visit of a node u, if rt .u; T / > 0, pu , the parent of u is selected whenever pu … T . Since pu might cover other vertices, while selecting, children of u will not affect any uncovered vertices other than u. Finally, children of u might be selected to fully covered u (but only after pu is selected). Time Complexity. Values of rp .v; P / and rt .v; T / can be maintained in O.jV j/. After a new vertex is added, it is necessary to update rp .:/ and/or rt .:/ values of that node and all its neighbors. Each node is added at most once; hence, the total cost has the same order with the total degree of all vertices, i.e., 2jV j. Adding the time O.jV j/ taken by the DFS traversal, the overall time complexities are still O.jV j/. t u

5

Scalable Algorithm for Multiple-Hop PIDS

In order to understand the influence propagation when the number of propagation hops is bounded, this section introduces VirAds, an efficient algorithm for the PIDS problem. After that, VirAds’s performance is examined through experiments in realworld large-scale social networks.

30

5.1

T.N. Dinh et al.

VirAds: Multiple-Hop PIDS Greedy Algorithm

With the huge magnitude of OSN users and data available on OSNs, scalability becomes the major problem in designing algorithm for PIDS. VirAds is scalable to network of hundred of millions links and provides high-quality solutions in our experiments. Before presenting VirAds, let us consider a natural greedy for the PIDS problem in which the vertex that can activate the most number of inactive vertices within d hops is selected in each step. This greedy is unlikely to perform well on practice for the following two reasons. First, at early steps, when not many vertices are selected, every vertex is likely to activate only itself after being chosen as a seed. Thus, the algorithm cannot distinguish between good and bad seeds. Second, the algorithm suffers serious scalability problems. To select a vertex, the algorithm has to evaluate for each vertex v how many vertices will be activated after adding v to the seeding, e.g., by invoking an O.m C n/ breadth-first search procedure rooted at v. In the worst-case when O.n/ vertices are needed to evaluate, this alone can take O.n.m C n//. Moreover, as shown in the previous section, the seeding size can be easily .n/; thus, the worst-case running time of the naive greedy algorithm is O.n2 .m C n//, which is prohibitive for large-scale networks. As shown in Algorithm1, our VirAds algorithm overcomes the mentioned problems in the naive greedy by favoring the vertex which can activate the most number of edges (indeed, it also considers the number of active neighbor around each vertex). This avoids the first problem of the naive greedy algorithm. At early steps, the algorithm behaves similar to the degree-based heuristics that favors vertices with high degree. However, when a certain number of vertices are selected, VirAds will make the selection based on the information within d -hop neighbor around the considered vertices rather than only one-hop neighbor as in the degree-based heuristic. The scalability problem is tackled in VirAds by efficiently keeping track of the following measures for each vertex v: • rv : The round in which v is activated .e/ • Nv : The number of new active edges after adding v into the seeding .a/ • Nv : The number of extra active neighbors v needs in order to activate v .i/ • rv W The number of activated neighbors of v up to round i where i D 1 : : : d Given those measures, VirAds selects in each step the vertex u with the highest .e/ .a/ effectiveness which is defined as Nu C Nu . After that, the algorithm needs to update the measures for all the remaining vertices. .e/ Except for Nv , all other measures can be effectively kept track of in only O..m C n/d / during the whole algorithm. When a vertex u is selected, it causes a chain reaction and activates a sequence of vertices or lower the rounds in which vertices are activated. New activated vertices are successively pushed into the queue Q for further updating much like what happens in the Bellman-Ford shortest-paths algorithm. Note that for each node u 2 V , changing of ru can cause at most d update .:/ for rw where w is a neighbor of u. For all neighbors of u, the total number of update .:/ is, hence, O.d  d.u//. Thus, the total time for updating rw 8w 2 V in VirAds will be at most O..m C n/  d /.

A Unified Approach for Domination Problems on Different Network Topologies

31

Algorithm 1: VirAds – Viral Advertising in OSNs Input: Graph G D .V; E/, 0 <  < 1, d 2 NC Output: A small d -seeding .e/ .a/ Nv d.v/; Nv   d.v/; rv d C 1; v 2 V ; .i/

;; rv D 0; i D 0::d; P while there exist inactive vertices do repeat u argmaxv…P fNv.e/ C Nv.a/ g; .e/

Recompute Nv as the number of new active edges after adding u. until u D argmaxv…P fNv.e/ C Nv.a/ g; P P [ fug; Initialize a queue: Q f.u; rv /g; 0; ru foreach neighbor x of u do .a/ .a/ Nx maxfNx  1; 0g; while Q ¤ empty do .t; oldRoundt / Q:pop./; foreach neighbor w of t do foreach i D rt to minfoldRoundt  1; rw  2g do .i/ .i/ rw D rw C 1; .i/

if .rw    dw / ^ .rw  d / ^ .i C 1 < d / then foreach neighbor x of w do .a/ .a/ Nx max Nx  1; 0g; rw D i C 1; if w … Q then Q:push..w; rw //; Output P ;

.e/

.e/

To maintain Nv , the easiest approach is to recompute all Nv . This approach, called Exhaustive Update, is extremely time-consuming as discussed in the naive .e/ greedy. Instead, Nv is updated only if “necessary.” In details, vertices are stored in a max priority queue in which the priority is their effectiveness. In each step, the .e/ vertex u with the highest effectiveness is extracted, and Nu is recomputed. If after updating, u still has the highest effectiveness, u is then selected. Otherwise, u is pushed back to the priority queue, the new vertex with the highest effectiveness is considered, and so on. The PIDS problem can be easily shown to be NP-hard by a reduction from the set cover problem. Thus, there are only two possible choices: either designing heuristics which have no worst-case performance guarantees or designing approximation algorithms which can guarantee the produced solutions are within a certain factor from the optimal. Formally, a ˇ-approximation algorithm for a minimization

32

T.N. Dinh et al.

(maximization) problem always returns solutions that are at most ˇ times larger (smaller) than an optimal solution. Unfortunately, there is unlikely an approximation algorithm with factor less than O.log n/ as shown in next section. However, under the assumption that the network is power-law, our VirAds is an approximation algorithm for PIDS with a constant factor. Theorem 21 In power-law networks, VirAds is an O.1/ approximation algorithm for the PIDS problem for bounded value of d . The theorem follows directly from the result in previous section that the optimal solution has size at least .n/ in power-law networks. Thus, the ratio between the VirAds’s solution and the optimal solution is bounded by a constant.

5.2

Experimental Study

This section presents experiments on OSNs to show the efficiency of VirAds algorithm in comparison with simple degree centrality heuristic. In addition, the experiments give insight into the trade-off between the number of times the information is allowed to propagate in the network and the seeding size.

5.2.1 Comparing to Optimal Seeding One advantage of the discrete diffusion model over probabilistic ones [26, 27] is that the exact solution can be found using mathematical programming. This enables us to study the exact behavior of the seeding size when the number of propagation hop varies. The PIDS problem can be formulated as a 01 integer linear programming (ILP) problem below. minimize

X

xv0

(16)

xvd  jV j

(17)

v2V

subject to

X v2V

X

xwi 1 C d  d.v/exvi 1  d  d.v/e xvi

w2N.v/

8v 2 V; i D 1 : : : d

(18)

xvi  xvi 1

8v 2 V; i D 1 : : : d

(19)

xvi 2 f0; 1g

8v 2 V; i D 0 : : : d

(20)

8 7

IP (Optimal) Max Degree VirAds

6 5 4

b

8

Seeding size (percent)

a Seeding size (percent)

A Unified Approach for Domination Problems on Different Network Topologies

7.6

33

IP (Optimal) Max Degree VirAds

7.2 6.8 6.4

3

6 1

1.5

2

2.5

3

3.5

4

1.5

1

2

c

7.9

Seeding size (percent)

Number of Rounds (d)

7.8

2.5

3

3.5

4

Number of Rounds (d)

IP (Optimal) Max Degree VirAds

7.7 7.6 7.5 7.4 1

1.5

2

2.5

3

3.5

4

Number of Rounds (d)

Fig. 9 Seeding size (in percent) on Erdos’s collaboration network. VirAds produces close to the optimal seeding in only fractions of a second (in comparison to 2-day running time of the IP(optimal)). (a)  D 0:4. (b)  D 0:6. (c)  D 0:8

( where

xvi

D

0 if v is inactive at round i

. 1 otherwise The objective of the ILP is to select a minimum number of seeds at the beginning. The constraint (2) guarantees all nodes are activated at the end, while (3) deals with propagation condition; the constraint (4) is simply to keep vertices active once they are activated. The ILP problem is solved on Erdos collaboration networks, the social network of the famous mathematician [54]. The network consists of 6,100 vertices and 15,030 edges. The ILP is solved with the optimization package GUROBI 4.5 on Intel Xeon 2.93 Ghz PC and setting the time limit for the solver to be 2 days. The running time of the IP solver increases significantly when d increases. For d D 1; 2; and 3, the solver return the optimal solutions. However, for d D 4, the solver cannot find the optimal solutions within the time limit and returns suboptimal solutions with relative errors at most 15 %. The optimal (or suboptimal) seeding sizes are shown in Fig. 9a–c, for  D 0:4; 0:6 and 0:8, respectively. VirAds provides close-to-optimal solutions and performs much better than Max Degree. Especially, when  D 0:8, the VirAds’s

34 Table 3 Sizes of the investigated networks

T.N. Dinh et al.

Vertices Edges Avg. degree

Physics

Facebook

Orkut

37,154 231,584 12.5

90,269 3,646,662 80.8

3,072,441 223,534,301 145.5

seeding is only different with the optimal solutions by one or two nodes. In addition, VirAds only takes fractions of a second to generate the solutions. As shown in Theorem 17, the seeding takes a constant fraction of nodes in the network. For Erdos collaboration network, the seeding consists of 3.8–7 % the number of nodes in the networks. Further, the seeding can consist as high as 20–40 % nodes in the network for larger social networks in next section. Although the mathematical approach can provide accurate measurement on the optimal seeding size, it cannot be applied for larger networks. The rest of our experiments measures the quality and scalability of our proposed algorithm VirAds on a collection of large networks.

5.2.2 Large Social Networks All the algorithms are tested on networks of various sizes including coauthors network in Physics sections of the e-print arXiv [26], Facebook [55], and Orkut [56], a social networking run by Google. Links in all three networks are undirected and unweighted. The sizes of the networks are presented in Table 3. Physics: The physics coauthors network shall be referred as Physics network or simply Physics. Each node in the network represents an author, and there is an edge between two authors if they coauthor one or more papers. Facebook dataset consists 52 % of the users in the New Orleans [55]. Orkut dataset is collected by performing crawling in last 2006 [56]. It contains about 11.3 % of Orkut’s users. 5.2.3 Solution Quality in Large Social Networks The VirAds algorithm is compared with the following heuristic Random method in which vertices are picked up randomly until forming a d -seeding and Max Degree method in which vertices with highest degree are selected until forming a d -hop seeding. Finally, VirAds is compared with its naive implementation, called Exhaustive Update, in which after selecting a vertex into the seeding, the effectiveness of all the remaining vertices are recalculated. With more accurate estimation on vertex effectiveness, Exhaustive Search is expected to produce higherquality solutions than those of VirAds. The seeding size with different number of propagation hop d when  D 0:3 is shown in Fig. 10. To our surprise, VirAds even performs equal or better than Exhaustive Update despite that it uses significantly less effort to update vertex effectiveness. VirAds has smaller seeding in Physics than Exhaustive Update; both of them give similar results for Facebook, while Exhaustive Update cannot finish

45 40

b Random Max Degree Exhaustive Update VirAds

35

Seeding size (percent)

a Seeding size (percent)

A Unified Approach for Domination Problems on Different Network Topologies

30 25 20 15 1

2

3 4 Number of Rounds(d)

Seeding size (percent)

c

50 45 40 35 30 25 20 15 10 5 0

5

6

50 45 40 35 30 25 20 15 10 5 0

35

Random Max Degree Exhaustive Update VirAds

1

2

3 4 Number of Rounds(d)

5

6

Random Max Degree VirAds

1

2

3

4

5

6

Number of Rounds(d)

Fig. 10 Seeding size when the number of propagation hop d varies ( D 0:3). VirAds consistently has the best performance (a) Physics. (b) Facebook. (c) Orkut

on Orkut after 48 h and was forced to terminate. Sparingly update, the vertices’ effectiveness turns out to be efficient enough since the influence propagation is locally bounded. In addition, the seeding produced by VirAds is almost two times smaller than those of Random. The gap between VirAds and Max Degree is narrowed when the number of maximum hops increases. Hence, selecting nodes with high degrees as seeding is a good long-term strategy, but might not be efficient for fast propagation when the number of hops is limited. In Facebook and Orkut, when d D 1, Max Degree has 60–70 % more vertices in the seeding than VirAds. In Physics, the gap between VirAds and the Max Degree is less impressive. Nevertheless, VirAds consistently produces the best solutions in all networks.

5.2.4 Scalability The running time of all methods at different propagation hop d is presented in Fig. 11. The time is measured in second and presented in the log scale. The running times increase slightly together with the number of propagation rounds d and are proportional to the size of the network. The Exhaustive Update has the worst running time, taking up to 15 min for Physics and 20 min for Facebook. For Orkut, the algorithm cannot finish within 2 days, as mentioned. The three remaining

Running time (second)

a

T.N. Dinh et al.

1000 Random Max Degree Exhaustive Update VirAds

100 10 1 0.1

b

10000

Running time (second)

36

1000

Random Max Degree Exhaustive Update VirAds

100 10 1 0.1

0.01 2

3 4 Number of Rounds(d)

c

1000

Running time (second)

1

100

5

6

1

2

3 4 Number of Rounds(d)

5

6

Random Max Degree VirAds

10 1

2

3 4 Number of Rounds(d)

5

6

Fig. 11 Running time when the number of propagation hop d varies ( D 0:3). Even for the largest network of 110 million edges, VirAds takes less than 12 min (a) Physics. (b) Facebook. (c) Orkut

algorithms VirAds, Max Degree, and Random take less than 1 s for Physics and less than 10 s for Facebook. Even on the largest network Orkut with more than 220 million edges, VirAds requires less than 12 min to complete.

5.2.5 Influence Factor The performance of VirAds and the other method at different influence factor  is presented. The number of propagation rounds d is fixed to 4. The size of d -seeding sets are shown in Fig. 12. VirAds is clearly still the best performer. The seeding sizes of VirAds are up to five times smaller than those of Max Degree for small  (although it’s hard to see this on the charts due to small seeding sizes). Since all tested networks are social networks with small diameter, the seeding sizes go to zero when  is close to zero. The exception is the Physics, in which the seeding sizes do not go below 10 % the number of vertices in the networks even when  D 0:05. A closer look into the Physics network reveals that the network contains many isolated cliques of small sizes (2, 3, 4, and so on) which correspond to authors that appear in only one paper. In each clique, regardless of the threshold , at least one vertex must be selected; thus the, seeding size cannot

A Unified Approach for Domination Problems on Different Network Topologies

b 100 Random Max Degree Exhaustive Update VirAds

80 60 40 20 0 0

0.2

0.4

0.6

0.8

1

Relative size of d-Spreader [percent]

Relative size of d-Spreader [percent]

a

37

90 Random Max Degree Exhaustive Update VirAds

80 70 60 50 40 30 20 10 0 0

0.2

Influence factor ρ

0.4

0.6

0.8

1

Influence factor ρ

Relative size of d-Spreader [percent]

c 100 90 80 70 60 50 40 30 20 10 0

Random Max Degree VirAds

0

0.2

0.4

0.6

0.8

1

Influence factor ρ

Fig. 12 Seeding size at different influence factors  (the maximum number of propagation hops is d D 4) (a) Physics. (b) Facebook. (c) Orkut

get below the number of isolated cliques in the networks. To eliminate the effect of isolated cliques, a possible approach is to restrict the problem to the largest component in the network.

6

Conclusion

This chapter studies approximability and inapproximability for important variants of the dominating set problems in different network topologies. In general networks, the considered domination problems are hard to approximate within a factor .1=2  o.1// ln n, and there exist approximation algorithms with factor ln n C O.1/ for the problems. Thus, there is still a 1=2 ln n gap between the lower bound and upper bound for the approximability of those problems. Note that Srinivasan [57] jnj n gives an ln OPT C O.ln ln OPT / approximation factor for DS which provides a better approximation factor when the optimal solution is relatively plarge to the input size. In addition, the sizes of PIDS and T-PIDS are at least jV j C jEj; thus, it is conjectured that PIDS and T-PIDS problems can be approximated within a factor 1=2 ln n C O.1/.

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22. W. Shang, P. Wan, F. Yao, X. Hu, Algorithms for minimum m-connected k-tuple dominating set problem. Theor. Comput. Sci. 381(1–3), 241–247 (2007) 23. K.G. Hill, J.D. Hawkins, R.F. Catalano, R.D. Abbott, J. Guo, Family influences on the risk of daily smoking initiation. J. Adolesc. Health 37(3), 202–210 (2005) 24. J.B. Standridge, R.G. Zylstra, S.M. Adams, Alcohol consumption: an overview of benefits and risks. South. Med. J. 97(7), 664–672 (2004) 25. P. Domingos, M. Richardson, Mining the network value of customers, in KDD ’01: Proceedings of The 7th ACM SIGKDD International Conference on Knowledge Discovery and Datamining (ACM, New York, 2001), pp. 57–66 ´ Tardos, Maximizing the spread of influence through a social 26. D. Kempe, J. Kleinberg, E. network, in KDD’03: Proceedings of the 9th ACM SIGKDD international conference on Knowledge discovery and data mining (ACM, New York, 2003), pp. 137–146 27. D. Kempe, J. Kleinberg, E. Tardos, Influential nodes in a diffusion model for social networks, in International Colloquium on Automata, Languages and Programming ’05 (Springer, Berlin/Heidelberg, 2005), pp. 1127–1138 28. J. Leskovec, A. Krause, C. Guestrin, C. Faloutsos, J. VanBriesen, N. Glance, Cost-effective outbreak detection in networks, in ACM SIGKDD Conference on Knowledge Discovery and Data Mining ’07 (ACM, New York, 2007), pp. 420–429 29. T.N. Dinh, N.T. Dung, M.T. Thai, Cheap, easy, and massively effective viral marketing in social networks: Truth or fiction? in Proceedings of the 23rd ACM conference on Hypertext and Social Media, HT ’12 (ACM, Milwaukee, 2012) 30. D. Peleg, Local majority voting, small coalitions and controlling monopolies in graphs: A review, in SIROCCO’96: Colloquium on Structural Information and Communication Complexity, Weizmann Science Press of Israe, Jerusalem, Israel, (1996), pp. 152–169 31. W. Zhang, Z. Zhang, W. Wang, F. Zou, W. Lee, Polynomial time approximation scheme for t-latency bounded information propagation problem inwirelessnetworks. J. Comb. Optim., 1–11 (2010). doi:10.1007/s10878-010-9359-x. 32. S. Khot, O. Regev, Vertex cover might be hard to approximate to within 2-[epsilon]. J. Comput. Syst. Sci., 74(3), 335–349 (2008) 33. M. Chleb´ık, J. Chleb´ıkov´a, Approximation hardness of dominating set problems in bounded degree graphs. Inf. Comput. 206(11), 1264–1275 (2008) 34. L. Trevisan, Non-approximability results for optimization problems on bounded degree instances, in ACM Symposium on Theory of Computing ’01 (ACM, New York, 2001), pp. 453–461 35. N. Kahale, Eigenvalues and expansion of regular graphs. J. ACM, 42, 1091–1106 (1995) 36. S. Rajagopalan, V.V. Vazirani, Primal-dual rnc approximation algorithms for (multi)-set (multi)-cover and covering integer programs, in Annual IEEE Symposium on Foundations of Computer Science ’93 (IEEE Computer Society, Washington, DC, 1993), pp. 322–331 37. E.B. van Peter, R. Kaas, E. Zijlstra, Design and implementation of an efficient priority queue. Math. Syst. Theory 10, 99–127 (1977) 38. G. Robins, A. Zelikovsky, Tighter bounds for graph steiner tree approximation. SIAM J. Discret. Math. 19, 122–134 (2005) 39. L. Ruan, H. Du, X. Jia, W. Wu, Y. Li, K. Ko, A greedy approximation for minimum connected dominating sets. Theor. Comput. Sci. 329, 325–330 (2004) 40. D.-Z. Du, R.L. Graham, P.M. Pardalos, P.-J. Wan, W. Wu, W. Zhao, Analysis of greedy approximations with nonsubmodular potential functions, in Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms, SODA ’08 (Society for Industrial and Applied Mathematics, Philadelphia, 2008), pp. 167–175 41. B.S. Baker, Approximation algorithms for np-complete problems on planar graphs. J. ACM 41, 153–180 (1994) 42. S. Khanna, R. Motwani, Towards a syntactic characterization of ptas, in Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, STOC ’96 (ACM, New York, 1996), pp. 329–337

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43. D. Eppstein, Diameter and treewidth in minor-closed graph families. Algorithmica 27(3), 275–291 (2000) 44. M. Grohe, Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23, 613–632 (2003) 45. E.D. Demaine, M. Hajiaghayi, Bidimensionality: new connections between fpt algorithms and ptass, in Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, SODA ’05 (Society for Industrial and Applied Mathematics, Philadelphia, 2005), pp. 590–601 46. M. Gibson, I. Pirwani, Algorithms for dominating set in disk graphs: Breaking the log n barrier, in European Symposia on Algorithms 2010, ed. by M. de Berg, U. Meyer. Volume 6346 of Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2010), pp. 243–254 47. A. Czygrinow, M. Hanckowiak, W. Wawrzyniak, Fast distributed approximations in planar graphs, in Distributed Computing, ed. by G. Taubenfeld. Volume 5218 of Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2008), pp. 78–92 48. D. Dai, C. Yu, A 5+[epsilon]-approximation algorithm for minimum weighted dominating set in unit disk graph. Theor. Comput. Sci. 410(8–10), 756–765 (2009) 49. M. Girvan, M.E. Newman, Community structure in social and biological networks. PNAS 99(12), 7821–7826 (2002) 50. C. Gkantsidis, M. Mihail, A. Saberi, Conductance and congestion in power law graphs, in SIGMETRICS ’03: Proceedings of the International Conference on Measurements and Modeling of Computer Systems (ACM, New York, 2003), pp. 148–159 51. A. Ferrante, G. Pandurangan, K. Park, On the hardness of optimization in power-law graphs. Theor. Comput. Sci. 393(1–3), 220–230 (2008) 52. W. Aiello, F. Chung, L. Lu, A random graph model for power law graphs. Exp. Math. 10, 53–66 (2000) 53. Y. Shen, D.T. Nguyen, Y. Xuan, M.T. Thai, New techniques for approximating optimal substructure problems in power-law graphs. Theor. Comput. Sci. (2011) 54. A. Barabasi, H. Jeong, Z. Neda, E. Ravasz, A. Schubert, T. Vicsek, Evolution of the social network of scientific collaborations. Phys. A 311(3–4), 590–614 (2002) 55. B. Viswanath, A. Mislove, M. Cha, K.P. Gummadi, On the evolution of user interaction in facebook, in WOSN’09 (ACM, New York, 2009) 56. A. Mislove, M. Marcon, K.P. Gummadi, P. Druschel, B. Bhattacharjee, Measurement and analysis of online social networks, in IMC’07, San Diego, CA (ACM, New York, 2007) 57. A. Srinivasan, Improved approximations of packing and covering problems, in Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, STOC ’95 (ACM, New York, 1995), pp. 268–276

Advanced Techniques for Dynamic Programming Wolfgang Bein

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A Few Standard Introductory Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 A Brief Introduction to Dynamic Programming: Fibonacci, Pascal, and Making Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Chained Matrix Multiplication Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Shortest Paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Knapsack Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Binary Search Tress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Pyramidal Tours for the Traveling Salesman Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Open-ended Dynamic Programming: Work Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Intricate Dynamic Programming: Block Deletion in Quadratic Time. . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Dynamic Program for Complete Block Deletion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Computing Block Deletion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Total Monotonicity and Batch Scheduling. ................................................... P 5.1 The Problem 1js  batchj wi Ci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 List Batching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Monge Property and Total Monotonicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The SMAWK and LARSCH Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Matrix Searching Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Online Matrix Searching Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Algorithm LARSCH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Standard Type Process: Pt for t Even (INTERPOLATE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Standard Type Process: Pt for t Odd (REDUCE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Quadrangle Inequality and Binary Search Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Decomposition Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Online Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 44 44 46 47 50 52 54 54 58 58 59 62 63 63 64 66 68 68 72 73 75 75 76 76 78 80

W. Bein Department of Computer Science, University of Nevada, Las Vegas, Las Vegas, NV, USA e-mail: [email protected] P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 28, © Springer Science+Business Media New York 2013

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8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 88 88

Abstract

This is an overview over dynamic programming with an emphasis on advanced methods. Problems discussed include path problems, construction of search trees, scheduling problems, applications of dynamic programming for sorting problems, server problems, as well as others. This chapter contains an extensive discussion of dynamic programming speedup. There exist several general techniques in the literature for speeding up naive implementations of dynamic programming. Two of the best known are the Knuth-Yao quadrangle inequality speedup and the SMAWK/LARSCH algorithm for finding the row minima of totally monotone matrices. The chapter includes “ready to implement” descriptions of the SMAWK and LARSCH algorithms. Another focus is on dynamic programming, online algorithms, and work functions.

1

Introduction

Dynamic programming, formally introduced by Richard Bellman (August 26, 1920–March 19, 1984) at the Rand Corporation in Santa Monica in the 1940s, is a versatile method to construct efficient algorithms for a broad range of problems. As with many tools which evolved over time, the original paper sounds antiquated. In his monograph “Dynamic Programming,” Bellman [25] describes the principle of optimality, which is central to dynamic programming: “An optimal policy has the property that whatever the initial state and initial decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.” Not exactly what is found in textbooks today. But then in those days the focus was more on stochastic processes and not so much on combinatorial optimization. The term “programming” is outdated as well, as it does not refer to programming in a modern programming language. Instead, programming means planning by filling in tables. The term “linear programming” derives in the same way. Over the decades, dynamic programming has evolved and is now one of the “killer” techniques in algorithmic design. And, of course, the laptop computer on which this chapter was written is more powerful than anyone in Bellman’s days could have imagined. Today, the emphasis is on how to organize dynamic programming in a way that makes it possible to solve massive problems (which – again – would have never been considered in Bellman’s days) in reasonable time. The focus is on dynamic programming speedup. Such speedup comes from carefully observing what values are essential and need to be computed and which might be unnecessary. Keeping

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proper look-up tables on the side can accomplish this sometimes. But there are many situations where there are inherent monotonicity properties which can be exploited to only calculate a fraction of what is necessary in a simple-minded approach. The goal of this chapter is to briefly introduce dynamic programming, show some of the diversity of problems that can be tackled efficiently with dynamic programming, and – centrally – focus on the issue of dynamic programming speedup. Clearly, the chapter does not cover all of dynamic programming, but there is a section on recommended reading, Sect. 8, which covers some ground. The chapter is organized as follows: Sect. 2 starts with a simple introduction to dynamic programming using a few standard examples, described more tersely than in a textbook and augmented with a general few themes. A reader altogether unfamiliar with dynamic programming might utilize some of the resources given in Sect. 8. Section 3 contains “open-ended dynamic programming,” where a process is updated continuously. This is closely related to the theory of online optimization. In online computation, an algorithm must make decisions without knowledge of future inputs. Online problems occur naturally across the entire field of discrete optimization, with applications in areas such as robotics, resource allocation in operating systems, and network routing. Dynamic programming plays an important role in this kind of optimization. Notably, work functions replace tables or matrices used in offline optimization. Section 4 contains an example from sorting. The intent of this section is twofold: First, lesser-known dynamic programming techniques for sorting are highlighted. Second, here is an example where a straightforward simple-minded implementation might give an algorithm of cubic complexity or worse, and a more intricate setup solves the problem in quadratic time, thus achieving substantial dynamic programming speedup. Section 5 gives an example from scheduling (a batching problem) to illustrate important properties for dynamic programming speedup: the Monge property and total monotonicity. Techniques exploiting these techniques are now standard, and the reader might consult Sect. 8 to see how prolific such techniques are. Speedup is based on two important and intricate algorithms: SMAWK and LARSCH. Use of these is essential for many fast dynamic programming algorithms. However, these algorithms are currently only accessible through the original publications. Section 6 contains “ready to implement” descriptions of the SMAWK and LARSCH algorithms. Another type of speedup is based in the Knuth-Yao quadrangle inequality; this dynamic programming speedup works for a large class of problems. Even though both the SMAWK algorithm and the Knuth-Yao (KY) speedup use an implicit quadrangle inequality in their associated matrices, on second glance, they seem quite different from each other. Section 7 discusses the relation between these kinds of speedup in greater detail. Finally, Sect. 8 is a cross-reference list with other chapters, and Sect. 8 gives concluding remarks.

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2

A Few Standard Introductory Examples

As mentioned earlier, this section has a few introductory examples.

2.1

A Brief Introduction to Dynamic Programming: Fibonacci, Pascal, and Making Change

Many problems can be solved recursively by dividing an instance into subinstances. But a direct implementation of a recursion is often inefficient because subproblems overlap and are recomputed numerous times. The calculation of the Fibonacci numbers provides a simple example: 8 < fn1 C fn2 if n > 1 fn D 1 if n D 1 : 0 n D 0:

(1)

Recursively, the Fibonacci numbers can be calculated in the following way: function f i b.n/ if n D 0 or n D 1 return n else return f i b.n  1/ C f i b.n  2/. This is extremely inefficient; many values will be recalculated; see Fig.1. Instead, one could use an array and would simply fill in values from “left to right,” starting with F0 and F1 and the using Eq. 1 to continue. This way every value is only calculated once. The Pascal triangle for calculating binomial coefficients provides a good example for the use of tables in dynamic programming. Recall the definition of the binomial coefficient:

!

! ! 8 ˆ n  1 n  1 ˆ ˆ C ˆ ˆ k1 k ˆ ˆ ˆ ˆ
0, and there is a knapsack of weight capacity W > 0. One is to fill the knapsack in such a way that the profit of the items chosen is maximized while obeying that the total weight be less than W . Let P Œi; j  D maximum profit, which can be obtained if the weight limit is j and only items from f1; : : : i g may be included,

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Table 3 Example of the progression of the matrix algorithm for the all shortest path problem given in Fig. 4. The operations in the “matrix multiplication” are not the usual “C” and “” but rather “min” and “C” 0 1 0 1 0 1 0 111 0 2 4 3 0 2 4 3 B1 0 1 1C B 3 0 1 3C B 3 0 1 3C .1/ C B C B C I  DDB @1 1 0 1A@ 5 1 0 3A D @ 5 1 0 3A D D 111 0 1 1 4 0 1 1 4 0 0 1 0 1 0 1 0 2 4 3 0 2 4 3 0243 B 3 0 1 3C B 3 0 1 3C B3 0 7 3C .2/ C B C B C D .1/  D D B @ 5 1 0 3A@ 5 1 0 3A D @5 4 0 3A D D 1 1 4 0 1 1 4 0 1 0 1 0 0243 0 2 4 3 0 B3 0 7 3C B 3 0 1 3C B3 .2/ B C B C B D D D@  D 5 4 0 3A @ 5 1 0 3A @5 4140 1 1 4 0 4 0

4140 1 243 0 7 3C C D D .3/ 4 0 3A 140

where 1  i  n and 0  j  W . The following recursion is valid: P Œi; j  D maxfP Œi  1; j ; P Œi  1; j  wi C pi g;

(10)

where P Œ0; j  D 0 and P Œi; j  D 1 when j < 0. Equation 10 says that for a new element i one does not include it (left of max) or one does include it (right of max). If one includes the element, then by the principle of optimality the problem with capacity reduced by the weight of element i on the earlier elements f1; : : : ; i  1g must be optimal. The table P Œi; j  can be filled in either row by row or column by column fashion. The run time of this algorithm is ‚.nW /. Note that this is not a polynomial algorithm for the knapsack problem. The input size is O.n log maxi D1;:::;n wi C log W /, which means that ‚.nW / is exponential in the input size. This is, of course, to be expected as the knapsack problem is N P-hard. The algorithm’s complexity is called pseudo-polynomial. Many dynamic programs give pseudo-polynomial algorithms. Pseudo-polynomial, Strongly Polynomial. Let s be the input to some decision problem …. Let jsjlog be the length of the input – that is, the length of the binary encoding – and let jsjmax be the magnitude of the largest number in s. A problem … is pseudo-polynomially solvable if there is an algorithm for … with run time bounded by a polynomial function in jsjmax and jsjlog . A decision problem is a number problem if there exists no polynomial p such that jsjmax is bounded by p.jsjlog / for all input s. (For example, the decision version of the knapsack problem is a number problem.) Note that by these definitions it immediately follows that an N P-complete non-number problem (such as HAMILTONIAN CIRCUIT, for

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Table 4 An example of the pseudo-polynomial dynamic programming algorithm for the knapsack problem on nine items. The weights are wŒ1 D 1; wŒ2 D 2; wŒ3 D 3; wŒ4 D 4; wŒ5 D 5; wŒ6 D 6; wŒ7 D 7; wŒ8 D 8; wŒ9 D 9 and the profits are pŒ1 D 1; pŒ2 D 2; pŒ3 D 5; pŒ4 D 10; pŒ5 D 15; pŒ6 D 16; pŒ7 D 21; pŒ8 D 22; pŒ9 D 35. The size of the knapsack is 15. The table shows in position .i; j / the maximum profit, which can be obtained if the weight limit is j and only items from f1; : : : i g may be included. Bold entries indicate that in Eq. 10 the second choice is the maximizer, i.e., the new item is considered for inclusion. From the regularbold information the actual solution can be reconstructed: The last entry in the table is 50. Because of the bold type face item i D 9 is included. Thus one looks w9 D 9 many cells to the left in the previous row. The regular type face of 16 in cell .7; 6/ means that item 8 is not included, looking at the cells above neither are items 7 and 6. Item 5 is included, next look-up cell .4; 1/ to see that no more items are included. The solution is f5; 9g with a profit of 50 W

0

1

2

3

4

5

6

7

8

9

11

12

13

14

15

i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9

0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1

1 2 2 2 2 2 2 2 2

1 3 5 5 5 5 5 5 5

1 3 6 10 10 10 10 10 10

1 3 7 11 15 15 15 15 15

1 3 8 12 16 16 16 16 16

1 3 8 15 17 17 21 21 21

1 3 8 16 20 20 22 22 22

1 3 8 17 25 25 25 25 35

1 3 8 18 26 26 26 26 36

1 3 8 18 27 31 31 31 37

1 3 8 18 30 32 36 36 40

1 3 8 18 31 33 37 37 45

1 3 8 18 32 36 38 38 50

example) cannot be solved by a pseudo-polynomial algorithm (unless P D N P). For polynomial p let …p denote the subproblem which is obtained by restricting … to instances with jsjmax  p.jsjlog /. Problem … is called strongly N P-complete if P is in N Pand there exists a polynomial p for which …p is N P-complete. It is easy to see (Table 4): Theorem 1 A strongly N P-complete problem cannot have a pseudo-polynomial algorithm unless P D N P.

2.5

Binary Search Tress

The construction of optimal binary search trees is a classic optimization problem. One is interested in constructing a search tree, in which elements can be looked up as quickly as possible. The first dynamic program for this problem was given by Gilbert and Moore [59] in the 1950s. More formally, given are n search keys with known order Key1 < Key2 <    < Keyn . The input consists of 2n C 1 probabilities p1 ; : : : ; pn and q0 ; q1 ; : : : ; qn . The value of pl is the probability that a search is for the value of Keyl ; such a search is called successful. The value of ql is the probability that a search is for a value between Keyl and KeylC1 (set Key0 D 1

Advanced Techniques for Dynamic Programming

53

4 7

2 (−∞, 2)

(2,4)

9

(4,7) (7,9)

(9,+∞)

Fig. 5 A binary search tree with successful (round) and unsuccessful (rectangular) nodes

and KeynC1 D 1); such a search is called unsuccessful. Note that in the literature the problem is sometimes presented with weights instead of probabilities, in that case the pl and ql are not required to add up to 1. The binary search tree constructed will have n internal nodes corresponding to the successful searches, and nC1 leaves corresponding to the unsuccessful searches. The depth of a node is the number of edges from the node to the root. Denote d.pl / the depth of the internal node corresponding to pl and d.ql / the depth of the leaf corresponding to ql . A successful search requires 1 C d.pl / comparisons, and an unsuccessful search requires d.ql / comparisons. See Fig.5. So, the expected number of comparisons is X

pl .1 C d.pl // C

1ln

X

ql d.ql /:

(11)

0ln

The goal is to construct an optimal binary search tree that minimizes the expected number of comparisons carried out, which is (11). Let Bi;j be the expected number of comparisons carried out in a optimal subtree containing the keys Keyi C1 < Key2 <    < Keyj . Observing that in a search Pj the probability to search in the region between Keyi C1 and Keyj is lDi C1 pl C Pj lDi ql it is clear that the following recurrence holds:

Bi;j

8 if i D j I ˆ < 0; j j X X   D ˆ pl C ql C min Bi;t 1 C Bt;j ; if i < j; : i j2 > j3 > : : : > jnr2 . A pyramidal tour can be found by dynamic programming in ‚.n2 /. To this end, let H Œi; j  be the length of a shortest Hamiltonian path from i to j subject to the condition that the path goes from i to 1 in descending order followed by the rest in ascending order from 1 to j . The reader is encouraged to verify that by the principle of optimality the following recursion holds: 8 H Œi; j  1 C d Œj  1; j  for i < j  1; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < mink t in turn, moving At to the end of the deletion sequence.  Next is given a recurrence to compute the minimum length of a complete block deletion sequence. This recurrence will be used in Algorithm 1 . Theorem 6 Given a permutation x, let ti;j denote the minimum length of any complete block deletion sequence for the sublist xi :::j . The values of ti;j can be computed inductively by the following: Set ti;i D 1, ti;j D 0 for i > j , and for i < j set ( ti;j D

minf1 C ti C1;j ; ti C1;`1 C t`;j g; if 9 ` 2 fi C 1; : : : ; j g such that x` D xi C 1; otherwise. 1 C ti C1;j ; (15)

Proof The proof is by induction on j  i to show that ti;j is computed correctly. The base case is trivial, namely ti;i D 1, because it takes one block deletion to delete a sublist of length 1. For the inductive step, assume that ti;j is the length of a minimum block deletion sequence for xi :::j when j  i < k. For j  i D k, let m D ti;j and let A1 ; : : : ; Am be a corresponding minimum length complete block deletion sequence of xi :::j . By Lemma 3, one can insist that xi D a is an item in Am . Consider now two cases based on whether there is an ` with i < `  j such that x` D a C 1 (The reader might also consult Fig. 9). If the element a C 1 does not occur in the interval fi C 1; : : : ; j g, then the element a is not part of a block in this interval and must be deleted by itself. So, A1 ; : : : ; Am1 is a complete block deletion sequence of xi C1:::j . Note that it must be of optimum length, for if B1 ; : : : ; Br were a shorter complete block deletion sequence of xi C1:::j , then B1 ; : : : ; Br ; fag would be shorter than A1 ; : : : ; Am . Thus, as j  .i C 1/ D k  1, by the induction hypothesis ti C1;j D m  1. So ti;j D 1 C ti C1;j , which is optimum. Now for the case that the element x` D a C 1 does occur in the interval fi C 1; : : : ; j g. This means that a C 1 and possibly other larger values can be included in Am when element a is deleted. If element a C 1 is not included in Am then the same argument used in the previous paragraph shows that m D 1 C ti C1;j . If on the other hand a C 1 is included in Am , then by Lemma 1, for any t; .1  t  m/, At is either completely before or completely after the element a C 1, since Am is deleted after At . By Lemma 2, one can permute the indices so that, for some u < m, 1. if t  u, then At is a subsequence of xi C1:::`1 , and 2. if u < t  m, then At is a subsequence of x`C1:::j .

Advanced Techniques for Dynamic Programming

61

Fig. 9 The recurrence for the ti;j values

Consequently, A1 ; : : : ; Au is a block deletion sequence for xi C1:::`1 and AuC1 ; : : : ; Am  fag is a block deletion sequence for x`C1:::j . Both of these block deletion sequences must be optimum for their respective intervals. That is, for example, if B1 ; : : : ; Br were a shorter complete block deletion sequence for xi C1:::`1 , then B1 ; : : : Br ; AuC1 ; : : : Am would be a complete block deletion sequence for xi :::j , contradicting the minimality of m. By the induction hypothesis, as `  1  .i C 1/ < j  i and j  ` < j  i , it follows that ti C1;`1 D u and t`;j D m  u, so ti;j D ti C1;`1 C t`;j D u C .m  u/ D m.  The resulting dynamic programming algorithm BLOCKDELETION, which is derived from the recurrence of Theorem 6, is shown below. Next the analysis of the run time of algorithm BLOCKDELETION is considered. Let z D .z1 ; : : : ; zn / be the inverse permutation of x, i.e., xi D k if and only if zk D i . Note that z can be computed in O.n/ preprocessing time. Theorem 7 Algorithm BLOCKDELETION has run time O.n2 /.

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Algorithm 1 BLOCKDELETION(X) Let n be the number of elements in x for i 1 to n do t Œi; i  1 for k 2 to n do for i 1 to n  k C 1 do xi C 1; j i Ck 1 Let x` if i < `  j then t Œi; j  minf.1 C t Œi C 1; j /; .t Œi C 1; `  1 C t Œ`; j /g else t Œi; j  1 C t Œi C 1; j  return

Proof To prove the theorem one shows that if tu;v are already known for all i < u  v  j , then ti;j can be computed in O.1/ time. Let m D ti;j for i < j and let At be the subsequence of xi :::j that is deleted at step t of the complete block deletion sequence of xi :::j of length m. By Lemma 3, assume that xi 2 Am . If jAm j > 1, the index ` D zxi C1 , i.e., x` D 1 C xi , can be found in O.1/ time, since one has already spent O.n/ preprocessing time to compute the array z. The recurrence thus takes O.1/ time to execute for each i; j .  Note that to obtain the actual sequence of steps in the optimum complete block deletion sequence one can store appropriate pointers as the ti;j are computed.

4.3

Computing Block Deletion

The reader is reminded that for the block deletion problem one needs to find the minimum length sequence of block deletions to transform a permutation into a monotone increasing list. Next it is shown how one obtains a solution to the block deletion problem for x in O.n2 /-time given that all ti;j are known for 1  i  j  n. Define a weighted acyclic directed graph G with one node for each i 2 f0; : : : ; n C 1g. There is an edge from i to j if and only if i < j and xi < xj , and the weight of that edge is ti C1;j 1 . Simply observe that there is a path h0; i1 ; i2 ; : : : ; ik ; n C 1i in G exactly when xi1 ; : : : xik is a monotone increasing list. Furthermore the weight of the edge hi` ; i`C1 i gives the minimum number of block deletions necessary to delete elements between position i` and i`C1 in x. Thus if there is a block deletion sequence of x of length m, there must be a path from 0 to n C 1 in G of weight m, and vice-versa. Using standard dynamic programming, a minimum weight path from 0 to n C 1 can be found in O.n2 / time.P Let 0 D i0 < i1 <    < i` D nC1 be such a minimum ` weight path, and let w D uD1 tiu1 C1;iu 1 be the weight of that minimum path. Since every deletion is a block deletion, the entire list can be deleted to a monotone list in w block deletions. Thus it follows:

Advanced Techniques for Dynamic Programming

J3

J2 5

J1

63

J1

J5

7

J5

66

J4 14

J3

J4 10

5

72

J2 14

Fig. 10 A batching example. Shown are two feasible schedules for a 5-job problem where processing times are p1 D 3; p2 D 1; p3 D 4; p4 D 2; p5 D 1 and the weights are w1 D w4 D w5 D 1 and w2 D w3 D 2. The encircled values give the sum of weighted completion times of the depicted schedules

Theorem 8 The block deletion problem can be solved in time O.n2 /.

5

Total Monotonicity and Batch Scheduling

Although dynamic programming has been around for decades there have been more recent techniques for making dynamic programming more efficient. This section describes an example from scheduling, where such a technique is applied. Before the technique is shown, the simple traditional dynamic programming for the scheduling problem is given and then the dynamic programming speedup technique is described.

5.1

The Problem 1js  batchj

P

wi Ci

Consider the batching problem where a set of jobs J D fJi g with processing times pi > 0 and weights wi  0, i D 1; : : : ; n, must be scheduled on a single machine, and where J must be partitioned into batches B1 ; : : : ; Br . All jobs in the same batch are run jointly and each job’s completion time is defined to be the completion time of its batch. One assumes that when a batch is scheduled it requires a setup time Ps D 1. The goal is to find a schedule that minimizes the sum of completion times wi Ci , where Ci denotes the completion time of Ji in a given schedule. Given a sequence of jobs, a batching algorithm must assign every job Ji to a batch. More formally, a feasible solution is an assignment of each job Ji to the mti h batch, i 2 f1; : : : ; ng (Fig. 10). The problem considered has the jobs executed sequentially, thus the problem is more precisely referred to as the s-batch problem. There is a different version of the problem not studied here, where the jobs of a batch are executed in parallel, known as the p-batch problem. In that case, the length of a batch is the maximum of the processing P times of its jobs. The s-batch is also denoted in ˛jˇj notation as the 1js-batchj wi Ci problem. Brucker and Albers [6] showed that the

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J1

J2

J3

J4

J5

60 12

J1 4

J1

J3

J2

J4

6

J2

11

14

J4

J3 5

J5

10

J5

51 16

48 14

Fig. 11 List batching. Shown are three schedules for a 5-job problem where all weights are 1 and the processing requirements are p1 = 3, p2 = 1, p3 = 4, p4 = 2, p1 = 1. The encircled values give the sum of weighted completion times of the depicted schedules

P 1js-batchj wi Ci problem is N P-hard in the strong sense by giving a reduction from 3-PARTITION. There is a large body of work on dynamic programming and batching; see the work of Baptiste [12], Baptiste and Jouglet [13], Brucker, Gladky, Hoogeveen, Kovalyov, Potts Tautenhahn, and van de Velde [36], Brucker, Kovalyov, Shafransky, and Werner [37], and Hoogeveen and Vestjens [64], as well as the scheduling text book by Peter Brucker [33]. Batching has wide application in manufacturing (see e.g., [34, 85, 98]), decision management (see, e.g., [74]), and scheduling in information technology (see e.g., [46]). More recent work on online batching is related to the TCP (Transmission Control Protocol) acknowledgment problem (see [20, 49, 67]).

5.2

List Batching

A much easier version of the problem is the list version of the problem where the order of the jobs is given, i.e., mi  mj if i < j . An example is given in Fig. 11. Assume that the jobs are 1; : : : ; n and are given in this order. One can then reduce the list batching problem to a shortest path problem in the following manner: Construct a weighted directed acyclic graph G with nodes i D 1; : : : ; n (i.e., one node for each job) and add a dummy node 0. There is an edge .i; j / if and only if i < j . (See Fig. 12 for a schematic.) Let edge costs ci;j for i < j be defined as

ci;j D

n X `Di C1

! w`

sC

j X `D1

! p` :

(16)

Advanced Techniques for Dynamic Programming

Job 1

Job 1

65

Job 3

Job 2

Job 2

Job 3

Job 4

Job 4

Job 5

Job 5

cij

Fig. 12 Reduction of the list batching problem to a path problem

It is easilyPseen (see [6] for details) that the cost of path < 0; i1 ; i2 ; : : : ; ik ; n > gives the Ci wi value of the schedule which batches at each job i1 ; i2 ; : : : ; ik . Conversely, any batching with cost A corresponds to a path in G with with path length A. A shortest path can be computed in time O.n2 / using the following dynamic program: Let EŒ` D cost of the shortest path from 0 to `; then EŒ` D min fEŒk C ck;l g with EŒ0 D 0; 0k

i

` , E`;k D 1. Furthermore, `  `C1 . 2. If k  ` , then EŒ`; k can be evaluated in O.1/ time provided that the row minima of the first ` rows are already known. 3. E is a totally monotone matrix. Larmore and Schieber [77] have developed an algorithm that generalizes the SMAWK to run in linear time in this case as well. Their algorithm is also known as the LARSCH algorithm – again, as with the SMAWK algorithm, the name was derived using initials of authors of the original paper in which the algorithm was introduced in Larmore and Schieber [77]. Both SMAWK and LARSCH are important in dynamic programming speedup and are explained in the next section.

6

The SMAWK and LARSCH Algorithm

This section gives “ready to implement” descriptions of the SMAWK and LARSCH algorithms mentioned in the previous section.

6.1

The Matrix Searching Problem

The matrix searching problem is the problem of finding all row minima of a given matrix M . If n; m are the number of rows columns, respectively, of M , the problem clearly takes ‚.nm/ time in the worst case. However, if M is totally monotone the problem can be solved in O.n C m/ time using the SMAWK algorithm [3].

Advanced Techniques for Dynamic Programming Column 0

Column 1

69

Column 2

E[1]

Row 1

c[0,1]+E[0]

Row 2

c[0,2]+E[0]

c[1,2]+E[1]

Row 3

c[0,3]+E[0]

c[1,3]+E[1]

c[2,3]+E[3]

Row 4

c[0,4]+E[0]

c[1,4]+E[1]

c[2,4]+E[2]

E[2]

E[3]

c[3,4]+E[4]

E[4]

Fig. 16 The “online” protocol of the tableau. Note that once the minimum of row 4 is known, column 4 is “knowable”

SMAWK is recursive, but uses two kinds of recursion, called INTERPOLATE and REDUCE, which are described in the following. Let 1  J.i /  m be the index of the minimum element of the i th row.1 Since M is totally monotone, J.i /  J.k/ for any i < k. 1. For small cases, SMAWK uses a trivial algorithm. If m D 1, the problem is trivial. If n D 1, simply use linear search. 2. If n  2, then INTERPOLATE can be used. Simply let M 0 be the bn=2c  m submatrix of M consisting of all even indexed rows of M . Recursively, find all row minima of M 0 , and then use linear search to find the remaining minima of M in O.n C m/ time. 3. If m > n, then REDUCE can be used. The set fJ.i /g clearly has cardinality at most n. REDUCE selects a subset of the columns of M which has cardinality at most n, and which includes Column J.i / for all 1  i  n. Let M 0 be the submatrix of M consisting of all the columns selected by REDUCE. Then, recursively, find all row minima of M 0 . These will exactly be the row minima of M . The time required to select the columns of M 0 is O.n C m/.

1

Use the leftmost rule to break ties.

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W. Bein

SMAWK then operates by alternating the two kinds of recursion. If the initial matrix is n  n, then INTERPOLATE reduces to the problem on a matrix of size roughly n=2  n, and then REDUCE reduces to the problem on a matrix of size roughly n=2  n=2, and so forth. Time Complexity Let T .n/ be the time required by SMAWK to find all row minima of an n  n totally monotone matrix. Applying both INTERPOLATE and REDUCE, obtain the recurrence T .n/ D O.n/ C T .n=2/ and thus T .n/ D O.n/. By a slight generalization of the recurrence, one can show that SMAWK solves the problem for an n  m matrix in O.n C m/ time. INTERPOLATE is explained using the code below. Code for INTERPOLATE 1: fM is an n  m totally monotone matrix, where m  n.g 2: if n D 1 then 3: J.1/ D 1 4: else 5: Let M 0 be the matrix consisting of the even indexed rows of M . 6: Obtain J.i / for all even i by a recursive call to REDUCE on M 0 . 7: for all odd i in the range 1 to n do 8: if i D n then 9: Find J.n/ 2 ŒJ.n  1/; n by linear search. 10: else 11: Find J.i / 2 ŒJ.i  1/; J.i C 1 by linear search. 12: end if 13: end for 14: end if The total number of comparisons needed to find the minima of all odd rows is at most n  1, as illustrated in Fig. 17 below. Now for REDUCE. The procedure REDUCE operates by maintaining a stack, where each item on the stack is a column (actually, a column index). Initially the stack is empty, and columns are pushed onto the stack one at a time, starting with Column 1. The capacity of the stack is n, the number of rows. But before any given column is pushed onto the stack, any number (zero or more) of earlier columns are popped off the stack. A column is popped if it is determined that it is dominated, which implies that none of its entries can be the minimum of any row. In some cases, if the stack is full, a new column will not be pushed. At the end, those columns which remain on the stack form the matrix M 0 . Dominance There is a loop invariant, namely that if the stack size is at least i , and if S Œi  D j , then all entries of Column j above Row i are known to be useless, i.e., will

Advanced Techniques for Dynamic Programming

71

Fig. 17 INTERPOLATE. M 0 consists of the hatched rows, and the minima of those rows are indicated by black dots. By the monotonicity property of J , only search the interval indicated in black to find the minimum of any odd numbered row

not be the minima of any row. Formally, M Œi 0 ; S Œi  1  M Œi 0 ; j  for all 1  i0 < i. Suppose that Column k is the next column to be possibly pushed onto the stack. If the stack is empty, then S Œ1 k, and one is done. Otherwise, it must be checked to see whether the top column in the stack is dominated. Let i be the current size of the stack, and let S Œi  D k. Then, necessarily, i  j < k. Column j is dominated if M Œi; k < M Œi; j . The reason is that M Œi 0 ; j   M Œi 0 ; S Œi  1 for any i 0 < i by the loop invariant, and that M Œi 0 ; j  < M Œi 0 ; k for all i 0  i by monotonicity. If the stack is popped, then the new top is tested for being dominated. This continues until the test fails. If the stack size is currently less than n, k is pushed onto the stack; otherwise, Column k is useless and is discarded. Code for REDUCE 1: fM is an n  m totally monotone matrix.g 2: top 0 fInitialize the stack to emptyg 3: for j from 1 to m do 4: while top > 0 and M Œtop; j  < M Œtop; S Œtop do 5: top top  1 fPop the stack.g 6: end while 7: if top < n then 8: top top C 1 and then S Œtop j fPush j onto the stack.g 9: end if 10: end for 11: Call INTERPOLATE on M 0 , consisting of columns remaining on the stack. The total number of comparisons needed to execute REDUCE (other than the recursive call) is at most 2m. One see this using an amortization argument. Each column is given two credits initially. When a column is pushed onto the stack, it spends one credit, and when it is popped off the stack it spends one credit. Each of those two events requires at most one comparison. Thus, the total number of comparisons is at most 2m.

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Fig. 18 REDUCE. M 0 consists of the hatched columns. All row minima, indicated by black dots, lie within those columns, although possibly not all columns contain minima

Figure 18 shows an example of an 8  16 matrix M , where the shaded columns survive to form M 0 .

6.2

The Online Matrix Searching Problem

Define an almost lower triangular matrix to be matrix M of n rows, numbered 1 : : : n, and m columns such that the length of the rows increases as n increases. More formally, let the columns be numbered 0 : : : m  1. Then there must exist n row lengths, 1  `Œ1  `Œ2      `Œn D m, such that M Œi; j  is defined if and only if 1  i  n and 0  j < `Œi . If `Œi  D i for all i , then M is a standard lower triangular matrix. Given an almost lower triangular matrix, let J.i / be the column index of the minimum entry in the i th row of M . (Ties are broken arbitrarily). The online matrix search problem is the problem of finding all row minima of M , with the condition that a given entry M Œi; j  is available if and only if J.k/ has been computed for every k such that `Œk < j . For example, if M is a standard lower triangular matrix, M Œi; j  is available if and only if J Œk has been computed for all k  j . Note that M Œi; 0 is available initially. Assume that the computation is done by a process B which is in communication with a supervisor A. The online computation then proceeds as follows. • A grants permission for B to see Column 0. • B reports J Œ1 (which is certainly 0) to A. • A grants permission for B to see Column 1. • B reports J Œ2 to A. • A grants permission for B to see Column 1. • B reports J Œ3 to A. • etc. Note that B is unable to compute J.i / until it has seen Columns 0 through i  1, since the minimum of Row i could be in any of those columns. Conversely, one must have computed J.1/ through J.i  1/ in order to be allowed to see those columns. Thus, the order of the events in the above computation is strict.

Advanced Techniques for Dynamic Programming

6.3

73

Algorithm LARSCH

In general, it takes O.nm/ time to solve the online matrix search problem. However, if M is totally monotone, the values of J are monotone increasing, and the online matrix search problem can be solved in O.nCm/ time by using an online algorithm LARSCH, which is the online version of SMAWK, with some adaptations. Let M be the input matrix. LARSCH works using a chain of processes, each of which is in communication with its supervisor, the process above it in the chain. The supervisor of the top process is presumably an application program. Refer to the process below a process as its child. Each process in the chain works an instance of the online matrix searching problem. The top process works the instance given by the application. Each process reports results interleaved with messages received from its supervisor. Each process P works the problem on a strictly monotone almost lower triangular matrix MP , which is a submatrix of M . If Q is the child of P , then P creates a submatrix MQ of MP and passes that submatrix to Q. Of course, P never passes the entries of the matrix to Q; rather, it tells Q which entries of M it is allowed to see. For clarity assume that each process uses its own local row and column indices, but is aware of the global indices of each entry. For example, in example calculation in Table 5, M5 Œ3; 2 D M Œ15; 11. Alternating Types. In the following detailed description of LARSCH, assume that the initial matrix M is standard lower triangular. (The algorithm can easily be generalized to cover other cases.) Let P0 ; P1 ; : : : Ph , h D 2.blog2 .n C 1/c  1/, be the processes, where Pt C1 is the child of Pt . The processes are of alternating types. If t is even, say that Pt has standard type, while if t is odd, say that Pt has stretched type. LetMt be the submatrix ˘ of M visible to Pt . M0 D M . If t is odd, then Mt has nt D 2t =2 n  2t =2  1 rows and mt D 2nt columns, and the length of its i th row is 2i . If t  2 is even, the Mt has nt D nt 1 rows and mt D nt columns, and the length of its i th row is i . The choice of Mt C1 as a submatrix of Mt is ˘illustrated below. If t is even, of Mt , ie. all rows of odd then the rows of Mt C1 are rows 3; 5; 7; : : : n1 2 index other than 1. Row i of Mt C1 consists of all but the last entry of Row 2i C 1 of Mt . If t is odd, then Mt C1 has the same number of rows as Mt , but only half the columns. Since there are nt rows in Mt , the number of possible columns of Mt which contain values of J cannot exceed nt . The REDUCE procedure of LARSCH chooses exactly nt columns of Mt in a way that makes certain that all columns which contain row minima of Mt are chosen. The figure below illustrates one possible choice (Fig. 19). The shaded entries are passed to Mt C1 . Note that not every entry of a selected column is part of Mt C1 .

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Table 5 LARSCH algorithm example: INTERPOLATE and REDUCE 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 3 5 7 9 11 13 15 17 19 3 5 7 9 11 13 15 17 19

32 31 10 5 19 35 27 22 33 18 41 30 48 60 54 40 69 90 78 0 10 19 27 33 41 48 54 69 78 0 10 19 27 33 41 48 54 69 78

1 48 15 9 23 39 31 26 37 22 45 33 50 62 56 41 69 89 77 1 15 23 31 37 45 50 56 69 77 1 23 31 37 45 50 56 69 77

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

21 14 27 42 33 27 37 21 43 30 46 58 52 36 64 84 71 2

16 27 42 33 26 36 20 42 28 43 55 48 32 60 79 66 3

40 52 41 34 44 28 50 35 49 61 53 37 64 81 68 4

60 47 39 48 31 52 37 51 62 53 36 62 77 64 5

51 41 47 30 51 36 50 61 52 34 59 83 60 6

39 44 27 47 32 46 54 46 28 53 76 52 7

54 36 55 40 53 60 52 33 57 73 54 8

41 57 42 55 62 53 34 58 72 53 9

46 34 44 53 42 23 47 61 42 10

34 42 49 38 19 43 57 38 11

47 53 42 22 45 59 39 12

48 37 14 39 53 33 13

41 18 43 56 35 14

19 38 51 30 15

35 45 22 16

48 24 17

18

27 33 37 43 46 52 64 71 2

27 33 36 42 43 48 60 66 3

41 44 50 49 53 64 68 7

47 48 52 51 53 62 64 11

47 51 50 52 59 60 13

44 47 46 46 53 52 15

55 53 52 57 54 16

57 55 53 58 53

44 42 47 42

42 38 43 38

42 45 39

37 39 33

43 35

38 30

22

24

33 37 43 46 52 64 71

36 42 43 48 60 66

47 46 46 53 52

42 38 43 38

37 39 33

38 30

22

Advanced Techniques for Dynamic Programming

75

Fig. 19 Shaded entries of Mt indicate the submatrix MtC1 passed to the child process. Left: odd t . Right: even t

6.4

Standard Type Process: Pt for t Even (INTERPOLATE)

Code for a Process of Standard Type: Pt for t Even. 1: fMatrix Mt with nt rows and nt columnsg 2: for i from 1 to nt do 3: Receive permission to view Column i  1 from supervisor. 4: if i D 1 then 5: J.1/ 0 6: else if i is even and i < nt then 7: Grant permission to child to view Columns i  2 and i  1. 8: Receive J Partial from child. 9: fMin of all entries but last of Row i C 1 is in Column J Partial.g 10: Find J.i / 2 ŒJ.i  1/; J Partial by linear search. 11: else if i is even and i D nt then 12: Find J.nt / 2 ŒJ.i  1/; nt  1 by linear search. 13: else[i  3 is odd] 14: if Mt Œi; i  1 < Mt Œi; J Partial then 15: J.i / i 1 16: else 17: J.i / J Partial 18: end if 19: end if 20: Send J.i / to supervisor. fMin of Row i is in Column J.i /g 21: end for

6.5

Standard Type Process: Pt for t Odd (REDUCE)

A stack called the column stack is maintained. Let top be the number of items (which are column indices) stored in the stack, and let S Œi  be the i th entry (counting from the bottom) of the stack. That is, S Œi  is defined for all 1  i  top.

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Each time process Pt is able to view a new column, if the column stack is not empty, it pops all columns which are dominated by the new column off the stack, and then pushes the new column. The domination rule is that Column S Œi  is dominated by the new column, say Column j , if Mt Œi; j  < Mt Œi; S Œi . Of course, it is possible that j  2i , in which case Mt Œi; j  is taken to be infinity. Code for a Process of Stretched Type: Pt for t Odd. fMatrix Mt with nt rows and 2nt columnsg top 0 fInitialize the column stack to emptyg for i from 1 to nt do Receive permission to view Columns 2i  2 and 2i  1 from supervisor. Pop all columns dominated by Column 2i  2 from column stack. Push Column 2i  2 onto column stack. Pop all columns dominated by Column 2i  1 from column stack. Push Column 2i  1 onto column stack. Grant permission to child to view Column S Œi . fThe i th column on the stack, which will become Column i  1 of Mt C1 g Receive J.i / from child. fMin of Row i is in Column J.i /g Send J.i / to supervisor. end for How may a new column possibly dominate existing columns on the column stack? Lines 2, 3, and 4 of the code below are the detail of Line 5 and Line 7 of the code above, while Lines 5 and 6 are the detail of Line 6 and Line 8 of the code above. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

1: 2: 3: 4: 5: 6:

Code for Popping and Pushing the Column Stack. fColumn j is the new columng while top > 0 and 2  top > j and Mt Œtop; j  < Mt Œtop; S Œtop do top top  1 fPop off the dominated columng end while top top C 1 S Œtop j fPush the new columng

7

The Quadrangle Inequality and Binary Search Trees

Another type of speedup is based in the Knuth-Yao quadrangle inequality. This section discusses this kind of speedup and the relation with SMAWK/LARSCH speedup.

7.1

Background

Recall construction of optimal binary search trees discussed in Sect. 2.5. Gilbert and Moore [59] gave a O.n3 / time algorithm. More than a decade later, in 1971,

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it was noticed by Knuth [71] that, using a complicated amortization argument, the Bi;j can all be computed using only O.n2 / time. Around another decade later, in the early 1980s, Yao (see [96, 97]) simplified Knuth’s proof and, in the process, showed that this dynamic programming speedup worked for a large class of problems satisfying a quadrangle inequality property. Many other authors then used the Knuth-Yao technique, either implicitly or explicitly, to speed up different dynamic programming problems. For this see for example the work of Wessner [93], the work of Atallah, Kosaraju, Larmore, Miller, and Teng [9] and Bar-No and Ladner [14]. Even though both the SMAWK algorithm and the Knuth-Yao (KY) speedup (best described in [71, 96, 97]) use an implicit quadrangle inequality in their associated matrices, on second glance, they seem quite different from each other. In the SMAWK technique, the quadrangle inequality is on the entries of a given m  n input matrix, which can be any totally monotone matrix. The KY technique, by contrast, uses a quadrangle inequality in the upper-triangular n  n matrix B. That is, it uses the QI property of its result matrix to speed up the evaluation, via dynamic programming, of the entries in the same result matrix. Aggarwal and Park [2] demonstrated a relationship between the KY problem and totally-monotone matrices by building a 3-D monotone matrix based on the KY problem and then using an algorithm due to Wilber [94] to find tube minima in that 3-D matrix. They left as an open question the possibility of using SMAWK directly to solve the KY problem. Definition 4 A two dimensional upper triangular matrix A, 0  i  j  n satisfies the quadrangle inequality (QI) if for all i  i 0  j  j 0 , A.i; j / C A.i 0 ; j 0 /  A.i 0 ; j / C A.i; j 0 /:

(23)

Observation 2 A Monge matrix satisfies a quadrangle inequality, but a totally monotone matrix may not. Yao’s result (of [96]) was formulated as follows: For 0  i  j  n let w.i; j / be a given function and ( Bi;j D

0; if i D j I   w.i; j / C min Bi;t 1 C Bt;j ; if i < j:

(24)

i 0, PrŒY  t 

EŒY  ; t

where EŒY  is the expectation of Y . random Lemma 2 (Chernoff’s bounds) Let X1 ; X1 ; : : : ; Xn be independent 0/1P variables, for 1  P i  n, PrŒXi D 1 D pi , where 0 < pi < 1. Let X D niD1 Xi and  D EŒX  D niD1 p i . Then, for any ı > 0, 1. PrŒX  .1 C ı/  2. PrŒX  .1  ı/  e

2.2

eı .1Cı/1Cı ı 2 =2



.

.

One-Stage Pooling Designs

Two efficient randomized constructions will be given for d -disjunct and .d I z/disjunct matrices, respectively. The constructions are based on the transversal design.

2.2.1 A New Construction of d-Disjunct Matrices A Las Vegas algorithm will be presented next, which for given n, d and 0 < p < 1, successfully constructs a t  n d -disjunct matrix with probability at least p, with t  cd 2 .log

2 C log n/; 1p

where c  4:28 is constant. For given n, d and 0 < p < 1, define n0 D 2n. Let  be the unique positive root of 2 ln.1 C / D : 1C   3:92 is chosen to minimize the leading constant of t (see the remarks in later part). Let

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Algorithm 1 (constructing d -disjunct matrix Mt n ) 1. Construct a random q-ary matrix Mt00 n0 with each cell randomly assigned from f1; 2; : : : ; qg, independently and uniformly. 2. For any 1  i < j  n0 , let wi;j be the random variable denoting the number of rows r such that the two entries M 0 .r; i / and M 0 .r; j / are equal. Then, EŒwi;j  D  .D

t0 /: q

Create an edge between columns i and j if wi;j  .1 C /. 3. For each edge created in Step 2, remove one of its two columns arbitrarily. Let M 00 denote the resulting matrix. 4. If M 00 has less than n .D n20 / columns, exit and the algorithm fail. 5. Using the transformation in Theorem 1, turn the first n columns of M 00 into a binary matrix Mtn with t  t0 q.

q D .1 C /d; t0 D

2n  1 t0 1C d ln ; D :  1p q

Please see Algorithm 1 as the algorithm for constructing d -disjunct matrices. In Algorithm 1 , at Step 3, M 00 must be q-ary .d; 1/-disjunct since for any column i , the union of any d other columns can only cover less than d  .1 C / D d 

t0 D t0 d

entries of column i . Therefore, if the algorithm successfully returns a matrix, it must be d -disjunct. Moreover, t  t0 q D

2n  1 .1 C /2 2 d log  log e 1p

< cd 2 .log

where c D

.1C/2  log e

2 C log n/; 1p

 4:28.

2.2.2 Analysis of Algorithm 1 The analysis of the success probability and running time of Algorithm 1 will be presented next. Success Probability. First, the expectation of the number of edges created at Step 2 is estimated. Lemma 3 Let m be the random variable denoting the number of edges created at Step 2 of Algorithm 1 , then EŒm  n.1  p/.

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Proof For 1  i < j  n0 ; 1  k  t0 , at Step 2 of Algorithm 1 , define 0/1 i;j random variable Xk such that Xk D 1 if and only if M 0 .k; i / D M 0 .k; j /: i;j

Then,

i;j

i;j

i;j

wi;j D X1 C X2 C    C Xt0 : Let X i;j be the indicator random variable for the event that there is an edge between column i and column j , that is, X

i;j

 1 there is an edge between column i and column j , that is, wi;j  .1C/I D 0 otherwise.

Since wi;j is the sum of t0 -independent 0/1 random variables, the Chernoff bound, (1) in Lemma 2, implies that  PrŒX

i;j

D 1 D PrŒwi;j  .1 C / 

e .1 C /1C

 :

Notice that q D .1 C /d , and t0 D

2n  1 1C d ln :  1p

Then,  D EŒwi;j  D

1 2n  1 t0 D ln : q  1p

From ln.1 C / D

2 ; 1C

it follows that .1 C /1C D e 2 , and so e D e  ; .1 C /1C which implies that 

Thus,

e .1 C /1C



1

D .e  /  ln

2n1 1p

D

1p : 2n  1

1p ; EŒX i;j  D PrŒX i;j D 1  2n  1 P for 1  i < j  n0 . Since m D 1i n  EŒm  1  p. n Running Time. The time required by Algorithm 1 is dominated by Step 2, which is ! n0 t0 D O.d n2 ln n/; 2 by simply counting, for all pairs of columns, the number of rows at which the two columns have equal entry. In fact, an expected O.n2 ln n/ running time can be obtained by counting along the rows. For 1  i < j  n0 , let n.i; j / denote the number of equal entries between column i and column j in the same row. Initially, set n.i; j / D 0 for 1  i < j  n0 . For each row r, let Sr;1 ; Sr;2 ; : : : ; Sr;q denote the sets of column indices such that Sr;k D fi W M 0 .r; i / D kg: Clearly, the sets Sr;k , 1  k  q, can be constructed in n0 time. For each k, increase the values of n.i; j / by 1 for all i < j and i; j 2 Sr;k . The expected number of   such pairs .i; j / for each Sr;k is EŒ jS2r;k j . Since jSr;k j are identically distributed for 1  k  q, the expected running time of Step 2 is " t0  n0 C qE

jSr;1 j 2

!#! :

Notice that jSr;1 j has the binomial distribution with parameters n0 and 1=q; thus, " n0 C qE

jSr;1 j 2

!# D n0 C q

1 2 .n  n0 / D O.n2 =d /; q2 0

and so the expected running time of Step 2, which is also the expected running time of Algorithm 1 , is t0  O.n2 =d / D O.n2 ln n/. Therefore, the following theorem is established. Theorem 2 Given n, d , and 0 < p < 1, Algorithm 1 successfully constructs a t n d -disjunct matrix with probability at least p, with t < cd 2 .log

2 C log n/; 1p

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where c  4:28 is constant. The algorithm runs in expected O.n2 ln n/ time. Remarks In Algorithm 1 ,  is chosen to minimize the leading constant of t. It is required that t0 .1 C /  ; d where  D tq0 , that is, q  .1 C /d , to guarantee that matrix M 00 is .d; 1/disjunct. To guarantee that with reasonable probability M 00 has at least n columns, it is required that n0  EŒm  n; n0  where EŒm D 2 EŒX 1;2 . This implies that n0  n PrŒX 1;2 D 1  n0  : 2

Since

n0  n max n0  D n0

1 ; 2n  1

2

which can be achieved when n0 D 2n  1 or n0 D 2n, it should have that PrŒX 1;2 D 1 

1 : 2n  1

This can be guaranteed by 

that is,  ln By plugging in  D

t0 q

D

t q2

f ./ D

 

1 ; 2n  1

.1 C /1C  O.1/ C ln n: e

and q  .1 C /d , it follows that

t

Define

e .1 C /1C

.1 C /2 ln .1C/ e

1C

.1 C /2 ln .1C/ e

1C

d 2 .O.1/ C ln n/:

D

.1 C /2 : .1 C / ln.1 C /  

To minimize f ./ for  > 0, from basic calculus, f 0 ./ D 0 implies that

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ln.1 C / D

2 : 1C

It is easy to verify that this equation has one unique positive root   3:92. Also, the equation implies that .1 C /2 .1 C /2 D : .1 C / ln.1 C /   

f ./ D

2.2.3 Error-Tolerance Case Algorithm 1 is modified next, so that given n, d , z > 1 and 0 < p < 1, the modified algorithm successfully constructs a t  n .d I z/-disjunct matrix with probability at least p, with  t  cd 2 log

  2  z 2 ; C log n C 2.1 C /d z C O 1p ln n

where   3:92 and c  4:28 are constants, and the O./ notation hides dependencies on p. First a generalization of .d; 1/-disjunct matrices is given. A q-ary matrix is .d; 1I z/-disjunct if for any column c and any set D of d other columns, there exist at least z elements in c such that each of these elements does not appear in any column of D in the same row. Clearly, by applying the same transformation in Theorem 1, one can turn a t0  n q-ary .d; 1I z/-disjunct matrix into a .d I z/-disjunct matrix with n columns and at most t0 q rows. For given n, d , z > 1 and 0 < p < 1, let n0 ,  be as in Algorithm 1 . Let q D .1 C /d C

z ln

2n1 1p

; t0 D z C

2n  1 t0 1C d ln ; D :  1p q

It can be verified that by this assignment, .1 C / D and



e .1 C /1C

t0  z ; d

 D

1p : 2n  1

Please see Algorithm 2 as the algorithm for constructing .d I z/-disjunct matrices. Firstly, at Step 3 of Algorithm 2 , M 00 must be q-ary .d; 1I z/-disjunct because for any column i , any d other columns can only cover less than d .1 C/ D t0 z of its entries. Therefore, if the algorithm successfully returns a matrix, it must be .d I z/disjunct. Secondly, when z D o.d ln n/, by similar arguments, Algorithm 2 runs in time O.d n2 ln n/ in the straightforward manner, and can be improved to expected O.n2 ln n/ time by counting the pairs of equal entries along the rows. Thirdly,

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Algorithm 2 (constructing .d I z/-disjunct matrix Mt n ) Algorithm 2 works in the same way as Algorithm 1 , except that with q D .1 C /d C and t0 D z C

1C d 

z

ln

2n1 1p

2n1 . 1p

ln

t  t0 q .1 C /2 2 . log e/z2 2n  1 d log C 2.1 C /d z C  log e 1p log 2n1 1p   2   z 2 ; C log n C 2.1 C /d z C O  cd 2 log 1p ln n

D

2

where c D .1C/  log e  4:28. For the success probability, if one let m be the random variable denoting the number of edges created at Step 2, since 

e .1 C /1C

 D

1p 2n  1

still holds, the same result in Lemma 3 also holds here, that is, EŒm   n.1  p/: Therefore, the probability that there are less than n columns left at Step 4 (i.e., the failure probability of Algorithm 2 ) is at most PrŒm > n  1  p. The following theorem is established. Theorem 3 Given n, d , z > 1 and 0 < p < 1, Algorithm 2 successfully constructs a t  n .d I z/-disjunct matrix with probability at least p, with  t  cd log 2

  2  z 2 C log n C 2.1 C /d z C O ; 1p ln n

where   3:92 and c  4:28 are constants. When z D o.d ln n/, the algorithm runs in expected O.n2 ln n/ time.

2.3

Two-Stage Pooling Designs

New two-stage pooling designs, which require a number of tests asymptotically no more than a factor of log3 3 (the factor approaches log2 e  1:44 as d tends to infinity) of the information-theoretic lower bound d log.n=d /, will be presented next.

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This improves the previously best upper bound of 4.1 C o.1// times the information theoretic bound in [24] by a factor of more than 2. For a 0/1 matrix M , let C denote the set of columns of M , recall that M is d -disjunct if for any d -sized subset D of C , each column in C  D is not covered by U.D/, where U.D/ denotes the union of the columns in D. Such matrices form the basis for nonadaptive (one-stage) pooling designs. However, a d -disjunct matrix 2 with n columns requires no less than . dloglogd n / rows, which is a factor of d= log d of the information-theoretic lower bound. Instead of determining all the positives immediately, in [24] the authors relax the property of M by introducing the concept of .d; k/-resolvable matrices, which form a good two-stage group testing regimen. A 0/1 matrix M is called .d; k/-resolvable if, for any d -sized subset D of C , there are fewer than k columns in C  D that are covered by U.D/. Thus, a matrix is d -disjunct if and only if it is .d; 1/-resolvable. For a set of n items in which at most d are positives, one can construct a “trivial two-stage” pooling design based on a t  n .d; k/-resolvable matrix as follows. Define the first round tests according to the rows of the matrix. By identifying the items in a negative pool (a pool with negative test outcome) as negatives, one can restrict the positives to a set D 0 of size smaller than d C k. Then, perform an additional round of tests on each item in D 0 individually. Thus, the total number of tests of the two stages is less than t C d C k.

2.3.1 Near Optimal Two-Stage Pooling Designs Let M1 be a q-ary matrix, and let C denote the set of columns of M1 . Matrix M1 is said to be .d; 1I k/-resolvable if, for any d -sized subset D of C , there are fewer than k columns in C  D that are covered by D. Here by saying a column c is covered by D, it means that for each element of c, the element appears at least once in some column of D in the same row. By applying the transformation in Theorem 1, one can turn a t0  n q-ary .d; 1I k/-resolvable matrix into a t  n .d; k/-resolvable matrix with t  t0 q. Let M 0 be a random t0 n q-ary (where q will be specified later) matrix with each cell assigned randomly from f1; 2; : : : ; qg, independently and uniformly. For each set D of d columns and a column c … D, for each element ci (i D 1; 2; : : : ; t0 ) in c,  d the probability that ci appears in some column of D in the same row is 1 1  q1 ; thus, the probability that every element in c appears in some column of D in the d  same row, that is, c is covered by D, is Œ1  1  q1 t0 . Parameter t0 is chosen such that "   # t0 1 1 d D 1 1 ; q nd

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that is, t0 D 

log.n  d / h  d i : log 1  1  q1

Let C denote the set of columns of M 0 . For any set D of d columns of M 0 , and for each c 2 C  D, let Xc be the indicator variable such that Xc D 1 if and only if c is covered by D. Then, 1 PrŒXc D 1 D : nd Define X XD D Xc : c2C D

Then, XD is the random variable denoting the number of columns in C  D that are covered by D. Since XD is the sum of .n  d / i.i.d. 0/1 random variables and EŒXD  D 1, the Chernoff’s bound implies that the probability that D covers at least .1 C ı/ columns in C  D is eı : .1 C ı/1Cı

PrŒXD  .1 C ı/ 

Therefore, the probability that M 0 is not .d; 1I 1 C ı/-resolvable, that is, there exists some set D of d columns that covers at least .1 C ı/ columns in C  D, is at most ! n eı pD : d .1 C ı/1Cı In order to satisfy p < 1, it suffices to assign ı such that 

1Cı e

1Cı D nd ;

since which implies that ! .1 C ı/1Cı .1 C ı/1Cı n d > Dn > : eı e 1Cı d /1Cı D nd implies . 1Cı / Notice that . 1Cı e e

1Cı e

d

D n e ; thus,

1 C ı D .1 C o.1//

d ln n : ln.d ln n/

Hence, by probabilistic arguments, the existence of a t0  n q-ary .d; 1I 1 C ı/resolvable matrix with t0 and ı as specified above has been proved.

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By applying the transformation in Theorem 1, one can turn the t0  n q-ary .d; 1I 1 C ı/-resolvable matrix M 0 into a (binary) t n .d; 1 C ı/-resolvable matrix M with q log n i: h t  t0 q <  log 1  .1  q1 /d Define Cd .x/ D

x

; for x > 1:  log 1  .1  x1 /d

Choosing q to be the positive integer that minimizes Cd .x/, and let Cd D Cd .q/. Then, t  Cd log n. For d  1, Cd =d  Cd .3d /=d D

3  log 1  .1 

1 d / 3d



3 : log 3

Also it is not hard to see that when d D 1, C1 D C1 .3/ D log3 3 indeed holds. Furthermore, the following lemma estimates that q D ‚.d / and Cd ! d log e as d ! 1. Lemma 4 For d  1, let q D q.d / be the point that minimizes Cd .x/ D x for x > 1, and let Cd D Cd .q/. Then, q.d / D ‚.d /, and  logŒ1.1 1 /d  x

limd !1 Cd =d D log e. To prove Lemma 4, the following useful fact is proved first. Fact 1 Let f .y/ D ln y ln.1  y/, 0 < y < 1. Then f .y/ achieves maximum at y D 12 . Proof of Fact 1: By symmetry, it is sufficient to show that f 0 .y/ > 0 for 0 < y < 12 . Since f 0 .y/ D D

1 1 ln.1  y/  ln y y 1y 1 Œ.1  y/ ln.1  y/  y ln y ; y.1  y/

let g.y/ D .1  y/ ln.1  y/  y ln y; Next it will be showed that g.y/ > 0 for 0 < y < 12 . Rewrite 1 : g.y/ D ln.1  y/ C y ln y.1  y/ 1 For 0 < y < 12 , ln y.1y/ > ln 4 since y.1  y/ < 14 ; thus,

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g.y/ > ln.1  y/ C y ln 4: Let h.y/ D ln.1  y/ C y ln 4: 1 , h0 .y/ > 0 for 0 < y < 1  ln14 and h0 .y/ < 0 Notice that h0 .y/ D ln 4  1y 1 1 for 1  ln 4 < y < 2 . Thus, h.y/ is monotone increasing when 0 < y < 1  ln14 and monotone decreasing when 1  ln14 < y < 12 . From h.0/ D h. 12 / D 0, one can obtain that 1 h.y/ > 0 for 0 < y < : 2

Therefore, for 0 < y < 12 , g.y/ > h.y/ > 0, and so f 0 .y/ D

1 g.y/ y.1y/

> 0.



Proof of Lemma 4: Notice that if q1 satisfies .1  q11 /d D 12 , then q1 D ‚.d / since q1 ! log e as d ! 1. Moreover, d Cd .q1 / D q1 D ‚.d /: The lemma is proved by contradiction. First assume that q.d / D O.d / does not hold, that is, for any c > 0 and any d0 > 0, there exists d > d0 such that q.d / > cd . Then, since dq > c (for simplicity q is used instead of q.d /, if it is clear from the context), as c ! 1, Cd .q/ D



q  d  1  log 1  1  q q

i   log 1  1  dq

Dd

h

q d

log dq

D !.d /I here, a  b means that limc!1 ab D 1. However, this contradicts since q is the point that minimizes Cd .q/ and on the other hand Cd .q1 / D ‚.d /. On the other hand, assume that q.d / D .d / does not hold, that is, for any c > 0 and any d0 > 0, there exists d > d0 such that q.d / < cd . Write Cd .q/ D

D

q  d  1  log 1  1  q q ln 2 ): q i dq h 1  ln 1  1  q (

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q  Since 0 < 1  q1
1, as c ! 0,

d q

>

1 c

! 1, and

q i dq h 1  q1
e q ln 2 D d d 1 q 1 q

this also contradicts (here a  b means that limc!0 ab D 1). Therefore, q.d / D ‚.d /. q  Next Cd as d ! 1 is estimated. Since 1  q1 < 1e for q > 1, thus     dq d 1 d 1 1 < D e q ; q e  #    d 1 d <  log 1  e  q ;  log 1  1  q "

and

it follows that Cd .q/ D

>

q  d  1  log 1  1  q q   d  log 1  e  q

D h

d ln 2 i :  d  dq ln 1  e  q

d

ln 2 Let y D e  q , then  dq D ln y, and Cd .q/ > ln ydln.1y/ . Since ln y ln.1y/ achieves 1 maximum at y D 2 (Fact 1), Cd .q/ > d log e for q > 1, thus Cd > d log e for d  1. On the other hand, as mentioned at the beginning of the proof, as d ! 1, q1 d ! log e, and Cd .q1 / D q1 ! d log e. Therefore, as d ! 1, Cd ! d log e.  The above arguments showed existence of a t  n .d; 1 C ı/-resolvable matrix d ln n with t  Cd log n and 1 C ı D .1 C o.1// ln.d ln n/ , which implies the following theorem.

Theorem 4 Given n and d , there exists a two-stage pooling design for finding up to d positives from n items using no more than Cd log n C d C ı C 1 tests, where

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Cd D min x2N

3 x  i  log 3 d;  1 d  log 1  1  x h

d ln n for d  1, and ı D .1 C o.1// ln.d . Moreover, ln n/

lim Cd =d D log e:

d !1

2.4

Probabilistic Pooling Designs

A probabilistic pooling design identifying up to d positives from n items with high probability will be presented next. In a probabilistic group testing algorithm, one may identify a positive item as negative; such a wrongly identified item is called a false negative; a negative item which is wrongly identified as positive is called a false positive. Clearly, the algorithm correctly identifies all positives if and only if there are no false positives or false negatives. Previous works on probabilistic nonadaptive group testing algorithms can be found from, among others, [40,41,44]. Algorithm. Given n and d , first construct a t0  n random q-ary matrix M 0 with each cell randomly assigned from f1; 2; : : : ; qg independently and uniformly (where t0 and q will be specified later). Then, use the transformation in Theorem 1 to obtain a t  n 0/1 matrix M with t  t0 q. Associate the n items with the columns of M , and test the pools indicated by the rows of M . The items not in any negative pool are identified as positives. Analysis. Let D be the set of columns corresponding to the positives, then jDj  d . First, it is easy to see that no positive item will be identified as negative if there is no error in the test outcomes. For any negative item, let c denote the column associated with it, then the item is wrongly identified if and only if c is covered by U.D/ in M , or equivalently, c is covered by D in M 0 (here the same notations c and D are used for different matrices M and M 0 , to denote the corresponding columns). The probability that c is covered by D, as analyzed in Sect. 2.3, is "

  # t0 "  # t0  1 jDj 1 d 1 1  1 1 : q q

Choosing q and t0 such that "

 # t0  1p 1 d ; D 1 1 q n

that is, t0 D 

1 log n C log 1p

logŒ1  .1  q1 /d 

:

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Then, the probability that there exists some negative item wrongly identified is no more then 1 .n  jDj/Œ1  .1  /d t0  1  p; q which implies that with probability at least p, the above algorithm successfully identifies all the positives. The number of pools required is no more than q 1 /: .log n C log 1 d 1  p logŒ1  .1  q /  By choosing q to be the positive integer minimizing t  t0 q D 

Cd .x/ D obtaining

x for x > 1;  logŒ1  .1  x1 /d 

t  Cd .log n C log

1 /: 1p

Theorem 5 The above one-stage algorithm, with probability at least p, correctly 1 identifies up to d positives from n items using no more than Cd .log n C log 1p / tests. Remarks 1. The one-stage probabilistic pooling design is also transversal. This design never gets false negatives, while the probabilistic algorithms in [40, 41, 44] never get false positives. The algorithm in [44] identifies up to 9 positives from 18,918,900 items using 5,460 tests, with success probability of 98.5 %. For the method proposed here, n D 18;918;900, d D 9, and p D 0:985, by choosing q D 14, it 1 requires Cd .q/.log n C log 1p / < 408 tests, which is much fewer. 2. In contrast to the two-stage design in Sect. 2.3, this probabilistic algorithm is explicitly given and can be easily implemented in practice. In addition, one can extend this algorithm to two stage, by performing an additional round of individual tests on the candidate positives identified by the first round, so that no item will be wrongly identified. It is easy to verify that, for this extended two-stage probabilistic algorithm, by choosing the same value q, and choosing t0 such that Œ1  .1  q1 /d t0 D n1 , the expected total number of tests required is no more than Cd log n C d C 1, which is better than the deterministic two-stage design in Sect. 2.3.

2.5

Conclusion and Future Studies

New one- and two-stage pooling designs, together with new probabilistic pooling designs, are presented in this section. The approach presented works for both errorfree and error-tolerance scenarios. The following remarks end off this section: 1. The constructions of pooling designs in Sects. 2.3 and 2.4 can also be generalized to error-tolerance case, in a similar manner as in the construction of .d I z/disjunct matrices in Sect. 2.2.3. The details are omitted due to the similarities.

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2. Efficient constructions (i.e., in time polynomial in n and d ) of the two-stage designs in Sect. 2.3 are not given. Up to now, no efficient construction of twostage pooling designs using the number of tests within a constant factor of the theoretic lower bound is known. In [14], the construction requires information n time, and in [24], the authors gave existence proof as in this section. n 2d Although once such a design is found it can be used as many times as wanted, efficient construction is an important issue. 3. The two-stage pooling design presented in Sect. 2.3 uses the number of tests asymptotically within a factor of Cd =d ( log3 3 for general d , and tends to log2 e  1:44 as d ! 1) of the information theoretic bound d log.n=d /. Can two-stage algorithms do as good as fully adaptive algorithms, that is, achieve a factor of asymptotically 1 of the information theoretic bound? Or, how good could it be? 4. Last but the least, efficient (i.e., polynomial time in n and d ) deterministic constructions of d -disjunct matrices with t D O.d 2 log n/ are known [2, 50]. Regarding the leading constant within the big-O notation, the results indicate that they are considerably larger than the result given in this section (where the leading constant is approximately 4:28). An efficient randomized construction of d -disjunct matrices with t D O.d 2 log n/ and efficient decoding (in time polynomial in t) is given in [33], and an efficient deterministic construction with the same properties is obtained recently [45]. Improved constructions of disjunct matrices are interesting to investigate.

3

New Complexity Results on Nonunique Probe Selection

Given a collection of n targets and a sample S containing at most d of these targets, and a collection of m probes each of them hybridizes to a subset of the given targets, the goal is to select a subset of probes, such that all targets in S can be identified by observing the hybridization reactions between the selected probes and S . For each probe p, there is hybridization reaction between p and S if S contains at least one target that hybridizes with p, otherwise there is no hybridization reaction. The above probe selection problem has been extensively studied recently [5, 31, 51, 52, 56] due to its important applications, particularly in molecular biology. For example, one application of this identification problem is to identify viruses (targets) from a blood sample. The presence or absence of the viruses is established by observing the hybridization reactions between the blood sample and some probes; here, each probe is a short oligonucleotide of size 8–25 that can hybridize to one or more of the viruses. A probe is called unique if it hybridizes to only one target; otherwise, it is called nonunique. Identifying targets using unique probes is straightforward. However, in situations where the targets have a high degree of similarity, for instance when identifying closely related virus subtypes, finding unique probes for every target is difficult. In [54], Schliep, Torney, and Rahmann proposed a group testing method using nonunique probes to identify targets in a given sample. Since each

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nonunique probe can hybridize with more than one target, the identification problem becomes more complicated. One important issue is to select a subset from the given nonunique probes so that the hybridization results can be decoded, that is, determine the presence or absence of targets in sample S . Also, the number of selected probes is exactly the number of hybridization experiments required, and it is desirable to select as few probes as possible to reduce the experimental cost. In [35, 54], two heuristics using greedy and linear programming-based techniques, respectively, are proposed for choosing a suitable subset of nonunique probes. In [11], the computational complexities of some basic problems in nonunique probe selection are studied, in the context of the theory of NP -completeness (see Chap. 10 in [17,19] and [30]). The complexity results in [11] will be presented in this section.

3.1

Preliminaries

The nonunique probe selection problem can be formulated as follows. Given a collection of n targets t1 ; t2 ; : : : ; tn , and a collection of m nonunique probes p1 ; p2 ; : : : ; pm , a sample S is known to contain at most d of the n targets. The probe-target hybridizations can be represented by an m  n 0-1 matrix M . Mi;j D 1 indicates that probe pi hybridizes to target tj , and Mi;j D 0 indicates otherwise. The subset of probes selected corresponds to a subset of rows in M , which forms a submatrix H of M with the same number of columns. The hybridization results between the selected probes and S also can be represented as a 0-1 vector V . Vi D 1 indicates that there is hybridization reaction between pi and S , that is, pi hybridizes to at least one target in S , and Vi D 0 indicates otherwise. If there is no error in the hybridization experiments, then V is equal to the union of the columns of H that correspond to the targets in S . Here, the union of a subset of columns is simply the Boolean sum of these column vectors. In order to identify all targets in S , the submatrix H should satisfy that all unions of up to d columns in H are different; in other words, H should be dN -separable. Also, as mentioned above, it is desirable to minimize the number of rows in H . A matrix H is said to be dN -separable if all unions of up to d columns in H are different. However, the following equivalent definition is more useful in the proofs here. Let H be a t  n Boolean matrix. For each i 2 f1; 2; : : : ; tg, define Hi D fj j 1  j  n; Hi;j D 1g: For any subset S of f1; 2; : : : ; ng and any i 2 f1; 2; : : : ; tg, write  1 if Hi \ S 6D ;I Hi .S / D 0 otherwise. Two sets S1 ; S2 f1; 2; : : : ; ng are said to be separated by H if there exists an integer i , 1  i  t, such that Hi .S1 / 6D Hi .S2 /. Matrix H is said to be dN -separable if for any two different subsets S1 , S2 of f1; 2; : : : ; ng, with jS1 j  d and jS2 j  d , S1 and S2 can be separated by H .

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115

Complexity of Minimal dN -Separable Matrix

In nonunique probe selection, one natural problem of interests is to determine whether a submatrix H chosen is dN -separable and minimal. By minimal, it means that the removal of any row from H will make it no longer dN -separable. The problem can be formulated as follows. M IN -SEPARABILITY (M INIMAL SEPARABILITY ): Given a t  n Boolean matrix H and an integer d  n, determine whether it is true that (a) H is dN -separable, and (b) for any submatrix Q of H of size .t  1/  n, Q is not dN -separable.

For a given binary matrix H and a positive integer d , the problem to determine whether H is dN -separable is known to be coNP -complete ([17], Theorem 10.2.1). Here a DP -completeness proof of problem MIN-SEPARABILITY will be presented. The class DP is the collection of sets A which are the intersection of a set X 2 NP and a set Y 2 coNP . The notion of DP -completeness has been used to characterize the complexity of the “exact-solution” version of many NP -complete problems. For instance, the exact traveling salesman problem, which asks, for a given edge-weighted complete graph G and a constant K, whether the minimum weight of a traveling salesman tour of the graph G is equal to K, is DP -complete (see [47], Theorem 17.2). In addition, the “critical” version of some NP -complete problems is also known to be DP -complete. For instance, the following problem is the critical version of the 3-satisfiability problem and has been shown to be DP -complete by Papadimitriou and Wolfe [48]: M IN -3-UNSAT: Given a 3-CNF Boolean formula ' which consists of clauses C1 ; C2 ; : : : ; Cm , determine whether it is true that (a) ' is not satisfiable, and (b) for any j , 1  j  m, the formula 'j that consists of all clauses C` , ` 2 f1; 2; : : : ; mg  fj g, is satisfiable.

Although most exact-solution version of NP -complete problems have been shown to be DP -complete, many critical versions are not known to be DP complete. The problem MIN-SEPARABILITY may be viewed as a critical version of the dN -separability problem. Its DP -completeness will be proved by constructing a reduction from MIN-3-UNSAT. Theorem 6 MIN-SEPARABILITY is DP -complete. Proof Recall that DP D fX \ Y j X 2 NP; Y 2 coNP g. A problem A is DP -complete if A 2 DP and, for all B 2 DP , B Pm A. For convenience, for any e j is used to denote the .t  1/  n submatrix of H with the j th t  n matrix H , H row removed. First, to see that MIN-SEPARABILITY 2 DP , let ˚ ej X D .H; d / j H is a t  n 0/1 matrix, 1  d  n, .8j; 1  j  t/ H

is not dN -separable ; and Y D f.H; d / j H is a t  n 0/1 matrix, 1  d  n, H is dN -separableg:

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It is clear that MIN-SEPARABILITY D X \Y . It is also not hard to see that X 2 NP and Y 2 coNP . In particular, to see that X 2 NP , note that .H; d / 2 X if and only if there exist 2t subsets Sj;1 , Sj;2 of f1; 2; : : : ; ng, for j 2 f1; 2; : : : ; tg, such that, for each j , Hk .Sj;1 / D Hk .Sj;2 / for all k 2 f1; 2; : : : ; tg  fj g. Next, a reduction from MIN-3-UNSAT to MIN-SEPARABILITY will be described. Let ' be a 3-CNF Boolean formula which consists of m clauses C1 , C2 ; : : : ; Cm , over n variables x1 ; x2 ; : : : ; xn . For each j 2 f1; 2; : : : ; mg, let 'j denote the Boolean formula that consists of all clauses C` for ` 2 f1; 2; : : : ; mg  fj g. From ', a .3n C m C 1/  .2n C 2/ Boolean matrix H will be constructed, and define d D n C 1. For convenience, the columns of H are denoted by X D fxi ; xN i j 1  i  ng [ fy; zg; and denote the rows of H by T D fxi ; xN i ; ui j 1  i  ng [ fyg [ fCj j 1  j  mg: Next H is defined by specifying each row of it: 1. For each 1  i  n, let Hxi D fxi g, HxN i D fxN i g, and Hui D fxi ; xN i ; zg. 2. Hy D fyg. 3. For each 1  j  m, let HCj D fxi j xi 2 Cj g [ fxN i j xN i 2 Cj g [ fy; zg (so that jHCj j D 5). To prove the correctness of the reduction, first verify that if ' is not satisfiable, then H is dN -separable. To see this, let S1 and S2 be two subsets of X , each of size  n C 1. Case 1. S1  fzg 6D S2  fzg. Then, there exists v 2 X  fzg such that v 2 S1 S2 . Then, Hv .S1 / 6D Hv .S2 /. Case 2. S1  fzg D S2  fzg. Then, it must be true that S1 S2 D fzg. Without loss of generality, assume S2 D S1 [ fzg. Note that jS2 j  n C 1 implies jS1 j  n. Subcase 2.1. There exists an integer i such that jS1 \ fxi ; xN i gj 6D 1. First, if jS1 \fxi ; xN i gj D 0 for some i , then Hui .S1 / D 0 and Hui .S2 / D 1 (because z 2 S2 ). Next, if jS1 \ fxi ; xN i gj D 2 for some i , then it must have jS1 \ fxk ; xN k gj D 0 for some k, because jS1 j  n. Then, again Huk .S1 / D 0 6D 1 D Huk .S2 /. Subcase 2.2. jS1 \ fxi ; xN i gj D 1 for all i 2 f1; 2; : : : ; ng. Note that, in this case, y 62 S1 . Define a Boolean assignment  W fx1 ; x2 ; : : : ; xn g ! fTRUE; FALSEg by .xi / D TRUE if and only if xi 2 S1 : Since ' is not satisfiable, there exists a clause Cj that is not satisfied by . It means that Cj \ S1 D ;, and so HCj .S1 / D 0. However, HCj .S2 / D 1 since z 2 S2 . The above completes the proof that H is dN -separable. ev Next, it will be showed that if 'j is satisfiable for all j D 1; 2; : : : ; m, then H N is not d -separable for all v 2 T . First, for v 2 X  fzg, let S1 D fzg and S2 D fv; zg.

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Then, for all rows w 2 X  fz; vg, Hw .S1 / D 0 D Hw .S2 /. Also, for all other rows w 2 T  X , Hw .S1 / D Hw .S2 / D 1 since z 2 Hw . So, S1 and S2 are not separable e v. by H Next, consider the case v D ui for some i 2 f1; 2; : : : ; ng. Let S1 D fxk j 1  k  n; k 6D i g [ fyg; and S2 D S1 [ fzg. It is clear that jS1 j D n and jS2 j D n C 1. Now the following e ui . claim is made: S1 and S2 are not separable by H To prove the claim, note that the rows Hxk , HxN k , for 1  k  n, and row Hy cannot separate S1 from S2 , since S1  fzg D S2  fzg. Also, rows Huk .S1 / D Huk .S2 / D 1, for all k 2 f1; 2; : : : ; ng  fi g, because jS1 \ fxk ; xN k gj D 1 if k 6D i . In addition, for any j D 1; 2; : : : ; m, HCj .S1 / D 1 D HCj .S2 /, since y 2 S1 . It e ui cannot separate S1 from S2 . follows that H Finally, consider the case v D Cj for some j 2 f1; 2; : : : ; mg. Note that 'j is satisfiable. So, there is a Boolean assignment  W fx1 ; x2 ; : : : ; xn g ! fTRUE; FALSEg satisfying all clauses C` , except Cj . Define S1 D fxi j .xi / D TRUEg [ fxN i j .xi / D FALSEg; and S2 D S1 [ fzg. Then, similar to the argument for the case v D ui , one can verify that Hw .S1 / D Hw .S2 / for w 2 X  fzg, and for w 2 fui j 1  i  ng. In addition, for any clause C` , with ` 6D j , C` is satisfied by . It follows that C` \ S1 6D ; and e v is not dN -separable, for HC` .S1 / D 1 D HC` .S2 /. This completes the proof that H all v 2 T . Conversely, it will be showed that if ' 62 MIN-3-UNSAT, then .H; n C 1/ 62 MINSEPARABILITY. First, consider the case that ' is a satisfiable formula. Let  W fx1 ; x2 ; : : : ; xn g ! fTRUE; FALSEg be a Boolean assignment satisfying '. Define S1 D fxi j .xi / D TRUEg [ fxN i j .xi / D FALSEg; and S2 D S1 [ fzg. Then, similar to the earlier proof, one can verify that H cannot separate S1 from S2 . In particular, HCj .S1 / D 1 for all j 2 f1; 2; : : : ; mg, because  satisfies Cj and so Cj \ S1 6D ;. Thus, .H; n C 1/ 62 MIN-SEPARABILITY. Next, assume that there exists an integer j 2 f1; 2; : : : ; mg such that 'j is not e Cj is dN -separable. The proof of the claim satisfiable. The following claim is made: H is similar to the first part of the proof (for the statement that if ' is not satisfiable then H is dN -separable). Case 1. S1  fzg 6D S2  fzg. Then, there exists v 2 X  fzg such that v 2 S1 S2 . So, Hv .S1 / 6D Hv .S2 /. Case 2. S1  fzg D S2  fzg. Then, it must be true that S1 S2 D fzg, and one may assume S2 D S1 [ fzg. It must have jS2 j  n C 1 and jS1 j  n. Subcase 2.1. There exists an integer i such that jS1 \ fxi ; xN i gj 6D 1. Similar to the earlier proof, if jS1 \ fxi ; xN i gj D 0 for some i D 1; 2; : : : ; n, then Hui can be

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used to separate S1 from S2 . If jS1 \ fxi ; xN i gj D 2 for some i D 1; 2; : : : ; n, then jS1 \ fxk ; xN k gj D 0 for some k, and again Huk separates S1 from S2 . Subcase 2.2. jS1 \ fxi ; xN i gj D 1 for all i 2 f1; 2; : : : ; ng. Then, since jS1 j  n, y 62 S1 . Define a Boolean assignment  W fx1 ; x2 ; : : : ; xn g ! fTRUE; FALSEg by .xi / D TRUE if and only if xi 2 S1 . Since 'j is not satisfiable, there exists a clause C` , ` 6D j , such that .C` / D FALSE. It means that C` \ S1 D ;, and so HC` .S1 / D 0. However, HC` .S2 / D 1 since z 2 S2 . So, HC` separates S1 from e Cj is dN -separable, and hence .H; n C 1/ 62 S2 . This completes the proof that H MIN-SEPARABILITY. 

3.3

N Minimum d-Separable Submatrix

A more important problem in nonunique probe selection is to find a minimum subset of probes that can identify up to d targets in a given sample. In the matrix representation, the problem can be formulated as the following: Given a binary matrix M and a positive integer d , find a minimum dN -separable submatrix of M with the same number of columns (problem MIN-dN -SS in [17], Chap. 10). For d D 1, MIN-dN -SS has been proved to be NP -hard ([17], Theorem 10.3.2), by modifying a reduction used in the proof of the NP -completeness of the problem MINIMUM-TEST-SETS in [30]. For fixed d > 1, MIN-dN -SS is believed to be NP -hard; however, up to now, no formal proof is known. Next the decision version of MIN-dN -SS is considered. dN -SS (dN -SEPARABLE SUBMATRIX ): Given a t  n Boolean matrix M and two integers d; k > 0, determine whether there is a k  n submatrix H of M that is dN -separable.

Recall that †P2 is the complexity class of problems that are solvable in nondeterministic polynomial time with the help of an NP -complete set as an oracle. For instance, the following problem SAT2 is †P2 -complete ([19], Theorem 3.13): Given a Boolean formula ' over two disjoint sets X and Y of variables, determine whether there exists an assignment to variables in X so that the resulting formula (over variables in Y ) is a tautology. It is easy to see that dN -SS is in †P2 . It is conjectured to be †P2 -complete. Here a similar problem that is a little more general than dN -SS will be considered, and its †P2 -completeness will be proved. dN -SSRR (dN -SEPARABLE SUBMATRIX WITH RESERVED ROWS): Given a t  n Boolean matrix M and three integers d > 0; s; and k  0, determine whether there is a dN -separable .s C k/  n submatrix H of M that contains the first s rows of M and k rows from the remaining t  s bottom rows of M .

Let ' be a Boolean formula, an implicant of ' is a conjunction C of literals that implies '. The following problem is proved to be †P2 -complete by Umans [55]. SHORTEST IMPLICANT CORE: Given a DNF formula ' D T1 C T2 C    C Tm , and an integer p, determine whether ' has an implicant C that consists of p literals from the last term Tm .

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By a reduction from SHORTEST IMPLICANT CORE, one can obtain the following result. Theorem 7 dN -SSRR is †P2 -complete. Proof The problem dN -SSRR can be solved by a nondeterministic machine that guesses a .s C k/  n submatrix H of M which contains the first s rows of M and then determines whether H is dN -separable. Note that the problem of determining whether a given matrix H is dN -separable is in coNP . Thus, dN -SSRR 2 †P2 . Next, dN -SSRR is proved to be †P2 -complete by constructing a polynomial-time reduction from SHORTEST IMPLICANT CORE to it. To define the reduction, let .'; p/ be an instance of the problem SHORTEST IMPLICANT CORE, that is, let ' D T1 C T2 C    C Tm be a DNF formula over n variables x1 ; x2 ; : : : ; xn , and let p be an integer > 0. Note that each term Tj , 1  j  m, of ' is a conjunction of some literals. Also, Tj is used to denote the set of these literals. Assume that the last term Tm of ' has q literals `1 ; `2 ; : : : ; `q . Define a .3n C m C q/  .2n C 1/ Boolean matrix M as follows: 1. Let the 2n C 1 columns of M be X D fx1 , xN 1 , x2 , xN 2 ; : : : ; xn , xN n , zg and the 3n C m C q rows of M be T D fxi ; xN i ; ui j 1  i  ng [ ftj j 1  j  mg [fcj j 1  j  qg. 2. For i D 1; 2; : : : ; n, Mxi D fxi g, MxN i D fxN i g, and Mui D fxi ; xN i ; zg. 3. For j D 1; 2; : : : ; m, Mtj D fxi j xN i 2 Tj g [ fxN i j xi 2 Tj g [ fzg. (Note that Mtj \ Tj D ;). 4. The bottom q rows of M are Mcj D f`j ; zg, for j D 1; 2; : : : ; q. Let d D n C 1, s D 3n C m, and k D p, and consider the instance .M; d; s; k/ for the problem dN -SSRR. First assume that ' has an implicant C of size p that is a subset of Tm . Let H be the submatrix of M that consists of the first s D 3n C m rows plus the k D p rows Mcj for which `j 2 C . The following claim is made: H is dN -separable. That is, for any subsets S1 and S2 of fx1 ; xN2 ; : : : ; xn ; xN n ; zg of size  d , there exists a row in H that separates them. Case 1. S1  fzg 6D S2  fzg. Then, there exists v 2 X  fzg such that v 2 S1 S2 . Then, Mv .S1 / 6D Mv .S2 /, and so H separates S1 from S2 . Case 2. S1  fzg D S2  fzg. Then, it must be true that S1 S2 D fzg. Without loss of generality, assume S2 D S1 [ fzg. Note that jS2 j  n C 1 implies jS1 j  n. Subcase 2.1. There exists an integer i such that jS1 \ fxi ; xN i gj 6D 1. First, if jS1 \ fxi ; xN i gj D 0 for some i , then Mui .S1 / D 0 and Mui .S2 / D 1 (because z 2 S2 ). Next, if jS1 \fxi ; xN i gj D 2 for some i , then it must have jS1 \fxk ; xN k gj D 0 for some k, because jS1 j  n. Then, again Muk .S1 / D 0 6D 1 D Muk .S2 /. It follows that H separates S1 from S2 .

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Subcase 2.2. jS1 \ fxi ; xN i gj D 1 for all i 2 f1; 2; : : : ; ng. Define a Boolean assignment  W fx1 ; x2 ; : : : ; xn g ! fTRUE; FALSEg by .xi / D TRUE if and only if xi 2 S1 . This is further divided into two subcases: Subcase 2.2.1.  satisfies the conjunction C . Since C is an implicant of ' D T1 C T2 C    C Tm ,  must satisfy some Tj , 1  j  m. Thus, Tj S1 : For any xi 2 Tj , .xi / D TRUE, and so xi 2 S1 , and for any xN i 2 Tj , .xi / D FALSE , and so xN i 2 S1 . It follows that Mtj .S1 / D 0 since Mtj \ Tj D ;. On the other hand, Mtj .S2 / D 1 since z 2 Mtj \ S2 . So, Mtj , and hence H , separates S1 from S2 . Subcase 2.2.2.  does not satisfy C . Then, for some literal `j 2 C , .`j / D 0. Thus, `j 62 S1 , and Mcj .S1 / D 0. On the other hand, Mcj .S2 / D 1 since z 2 Mcj . Thus, Mcj , which is a row in H , separates S1 from S2 . Conversely, assume that H is a .3n C m C k/  .2n C 1/ submatrix of M that contains the first 3n C m rows of M and is dN -separable. Let C be the conjunction of literals `j for which Mcj is a row in H . Then, obviously, jC j D k. Now the following claim is made: C is an implicant of '. Let  W fx1 ; x2 ; : : : ; xn g ! fTRUE; FALSEg be a Boolean assignment that satisfies C . It will be showed that  satisfies '. Let S1 D fxi j .xi / D TRUEg [ fxN i j .xi / D FALSEg; and S2 D S1 [ fzg. Then, S1 and S2 can be separated by some row in H . Since S2 D S1 [ fzg, they are not separable by a row Mxi or MxN i , for any i D 1; 2; : : : ; n. In addition, since jS1 \fxi ; xN i gj D 1 for all i D 1; 2; : : : ; n, they cannot be separated by row Mui , for any i D 1; 2; : : : ; n. Furthermore, note that for any literal `j 2 C , .`j / D 1 and so `j 2 S1 and Mcj .S1 / D Mcj .S2 / D 1. Thus, S1 and S2 cannot be separated by any row Mcj of H . Therefore, S1 and S2 must be separable by a row Mtj , for some j D 1; 2; : : : ; m. That is, Mtj .S1 / D 0 6D 1 D Mtj .S2 /. Since Mtj contains the complements of the literals in Tj , Tj S1 . It follows that  satisfies the term Tj , and hence '. 

3.4

Conclusion

In this section, the computational complexities of problems related to nonunique probe selection are presented. The problem of verifying the minimality of a dN -separable matrix is showed to be DP -complete, and hence is intractable, unless DP D P . For the problem of finding a minimum dN -separable submatrix, it is conjectured to be †P2 -complete and, hence, is even more difficult than the minimal dN -separability problem. To support this conjecture, the problem dN -SSRR, which is a little more general than the minimum dN -separable submatrix problem, is shown to be †P2 -complete. The complexity of the original problem MIN-dN -SS remains open.

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Parameterized Complexity Results on NGT

Given an m  n binary matrix and a positive integer d , to decide whether the matrix is d -separable (dN -separable, or d -disjunct) is a basic problem in nonadaptive group testing (NGT). They are known to be coNP-complete in classical complexity theory [18]. Thus, one should not expect any polynomial time algorithm to solve any of them. However, since in most applications d n, an interesting question is whether there are efficient algorithms solving the above decision problems for small values of d . In [12] by studying the parameterized complexity of the above three problems with d as the parameter, the authors gave a negative answer to the above question. More formally, they studied the parameterized decision problems p-DISJUNCTNESS-TEST, p-SEPARABILITY-TEST, and p-SEPARABILITY -TEST defined as follows (where N denotes the set of positive integers). p-DISJUNCTNESS-TEST Instance: A binary matrix M and d 2 N . Parameter: d . Problem: Decide whether M is d -disjunct. p-SEPARABILITY-TEST Instance: A binary matrix M and d 2 N . Parameter: d . Problem: Decide whether M is d -separable. p-SEPARABILITY -TEST Instance: A binary matrix M and d 2 N . Parameter: d . Problem: Decide whether M is dN -separable. The main results obtained in [12] will be presented in this section; they are summarized in the following theorem. Theorem 8 p-DISJUNCTNESS-TEST, p-SEPARABILITY -TEST, and p-SEPARABILITY-T EST are all co-W[2]-complete. W[2] is the parameterized complexity class at the second level of the W-hierarchy, and co-W[2] is the class of all parameterized problems whose complements are in W[2]. They will be formally introduced in the sequel. Theorem 8 indicates that, given an m  n binary matrix and a positive integer d , a deterministic algorithm with running time f .d /  .mn/O.1/ (where f is an arbitrary computable function) to decide whether the matrix is d -separable (dN -separable, or d -disjunct) does not exist unless the class W[2] collapses to FPT (the class of all fixed-parameter tractable problems), which is commonly conjectured to be false.

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Preliminaries

Before proving Theorem 8, the notions of fixed-parameter tractability, relational structures, first-order logic, and the W-hierarchy of parameterized complexity classes are formally introduced.

4.1.1 Fixed-Parameter Tractability The theory of fixed-parameter tractability [16, 27] has received considerable attention in recent years, for both theoretical research and practical computation. The notations and conventions in [27] are adopted here. Let † denote a fixed finite alphabet. A parameterization of † is a polynomial time computable mapping  W † ! N . A parameterized problem (over †) is a pair .Q; / consisting of a set Q † and a parameterization  of † . An algorithm A with input alphabet † is an fpt-algorithm with respect to , if for every x 2 † the running time of A on input x is at most f ..x//jxjO.1/ , for some computable function f . A parameterized problem .Q; / is fixed-parameter tractable if there is an fpt-algorithm with respect to  that decides Q. The key point of the definition of fpt-algorithm is that the superpolynomial growth of running time is confined to the parameter .x/, which is usually known to be comparatively small. The class of all fixed-parameter tractable problems is denoted by FPT. Many NP-hard problems such as the VERTEX COVER problem [8] and the ML TYPE-CHECKING problem [37] have been shown to be fixed-parameter tractable. On the other hand, there is strong theoretical evidence that certain well-known parameterized problems, for instance the INDEPENDENT SET problem and the DOMINATING SET problem, are not fixed-parameter tractable [16]. This evidence is provided, similar to the theory of NP-completeness, via a completeness theory based on the following notion of reductions: Let .Q; / and .Q0 ;  0 / be parameterized problems over alphabets † and †0 , respectively. An fpt-reduction from .Q; / to .Q0 ;  0 / is a mapping R W † ! .†0 / such that: 1. For all x 2 † , x 2 Q if and only if R.x/ 2 Q0 . 2. R is computable by an fpt-algorithm (with respective to ). That is, there is a computable function f such that R.x/ is computable in time f ..x//jxjO.1/ . 3. There is a computable function g W N ! N such that  0 .R.x//  g..x//, for all x 2 † . In the above definition, the last requirement is to ensure that class FPT is closed under fpt-reductions, that is, if a parameterized problem .Q; / is reducible to another parameterized problem .Q0 ;  0 / and .Q0 ;  0 / 2 FPT, then .Q; / 2 FPT. 4.1.2 Relational Structures In later discussions, the conventions in descriptive complexity theory are adopted, in which instances of decision problems are viewed as structures of some vocabulary instead of languages over some finite alphabet.

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A (relational) vocabulary  is a set of relation symbols. Each relation symbol R 2  has an arity arity.R/  1. A structure A of vocabulary  consists of a set A called the universe and an interpretation RA Aari ty.R/ of each relation symbol R 2 . For a tuple aN 2 Aari ty.R/ , write RA aN (or aN 2 RA ) to denote that aN belongs to the relation RA . Here only nonempty finite vocabularies and structures with a finite universe are considered. Recall that a hypergraph is a pair H D .V; E/ consisting of a set V of vertices and a set E of hyperedges. Each hyperedge is a subset of V . Graphs are hypergraphs with hyperedges of cardinality two. The following example illustrates how to represent a hypergraph using a relational structure. Example 1 Let HG be the vocabulary consisting of the unary relation symbols VERT and EDGE and the binary relation symbol I . A hypergraph H D .V; E/ can be represented by a relational structure H of vocabulary HG as follows: • The universe of H is V [ E. • VERT H WD V and EDGE H WD E. • I H WD f.v; e/ W v 2 V; e 2 E, and v 2 eg is the incidence relation.

4.1.3 First-Order Logic First the syntax of first-order logic is briefly recalled. Let  be a vocabulary. Atomic first-order formulas of vocabulary  are of the form x D y or Rx1 : : : x` , where R 2  is `-ary (i.e., has arity `) and x; y; x1 ; : : : ; x` are variables. First-order formulas of vocabulary  are built from atomic formulas using Boolean connectives ^ (and), _ (or), : (negation), together with the existential and universal quantifiers 9 and 8. The connectives ! (implication) and $ (equivalence) are not part of the language defining first-order formulas, but they are used as abbreviations: ' ! stands for :' _ , and ' $ stands for .' ! / ^ . ! '/. A variable x is called a free variable of ' if x occurs in ' but is not in the scope of a quantifier binding x. Write '.x1 ; : : : ; x` / to indicate that all free variables of ' belong to set fx1 ; : : : ; x` g. A formula without free variables is called a sentence. Let both †0 and …0 denote the class of quantifier-free first-order formulas. For t  0, let †t C1 be the class of all formulas .9x1 : : : 9x` /', where ' 2 …t , and let …t C1 be the class of all formulas .8x1 : : : 8x` /', where ' 2 †t . For formulas of second-order logic, in addition to the individual variables, they may also contain relation variables; each of the relation variables has a prescribed arity. Here lowercase letters (e.g., x; y; z) are used to denote individual variables, and uppercase letters (e.g., X; Y; Z) are used to denote relation variables. As in [27], for convenience free relation variables are allowed to be in first-order formulas, since the crucial difference between first-order and second-order logic is not that secondorder formulas can have relation variables, but that second-order formulas can quantify over relations. Therefore, here the syntax of first-order logic is enhanced by

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including new atomic formulas of the form Xx1 : : : x` , where X is an `-ary relation variable. The meaning of formula Xx1 : : : x` is: The tuple of elements interpreting .x1 ; : : : ; x` / is contained in the relation interpreting the relation variable X . The classes such as †t and …t are also extended to include formulas with free relation variables. It is worth emphasizing again that in first-order logic quantification over relation variables is now allowed.

4.1.4 W-Hierarchy First a brief introduction to the W-hierarchy of parameterized complexity classes is given. Roughly speaking, the W-hierarchy classifies problems according to the syntactic form of their definitions, and the definitions are formalized using languages of mathematical logic. The W-hierarchy can be defined in several different ways; here the following definition based on the weighted Fagin-defined problems is adopted. Let '.X / be a first-order formula with a free relation variable X of arity s. Define p-WD' to be the following parameterized decision problem. p-WD' : Instance: A structure A and k 2 N . Parameter: k. Problem: Decide whether there is a relation S As of cardinality k such that A ˆ '.S /. Here, A ˆ '.S / stands for that structure A satisfies sentence '.S / (or, A is a model of '.S /), and S is called a solution for ' in structure A. The readers are referred to, for example, Sect. 4.2 of [27], for more detailed introduction to the semantics of first-order formulas. For a class ˆ of first-order formulas, let p-WD-ˆ be the class of all parameterized problems p-WD' with ' 2 ˆ. For t  1, define W[t]WD Œp-WD-…t fpt , which is the class of all parameterized problems that are fpt-reducible to some problems in p-WD-…t . The classes W[t], for t  1, form the W-hierarchy. Thus, the levels of W-hierarchy essentially correspond to the number of alternations between universal and existential quantifiers in the definitions of their complete problems. Problems hard for W[1] or higher class are assumed not to be fixed-parameter tractable. For instance, the I NDEPENDENT SET problem is W[1]-complete, and the DOMINATING SET problem is W[2]-complete. For a parameterized problem .Q; / over the alphabet †, let .Q; /c denote its complement, that is, the parameterized problem .† n Q; /. Let C be a parameterized complexity class. Then co-C is defined to be the class of all parameterized problems .Q; / such that .Q; /c 2 C . Clearly, FPT = co-FPT. From the definition of fpt-reductions, it is easy to see that if class C is closed under fptreductions, so is co-C . In particular, each class W[t], t  1, gives rise to a new parameterized complexity class co-W[t]. Also, it is easy to prove that if .Q; / is complete in parameterized complexity class C under fpt-reductions, then .Q; /c is complete in class co-C under fpt-reductions.

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125

Proof of Theorem 8

Now the proof of Theorem 8 is ready to presented. For a binary matrix M , let RM be the set consisting of all rows in M , and let CM be the set consisting of all columns in M . Relational Structure for a Binary Matrix. Let BM be the vocabulary consisting of the unary relation symbols ROW and COLUMN and the binary relation symbol I. Then, the binary matrix M can be represented by a structure M of vocabulary BM , where the universe of M is RM [ CM , and the interpretations for the relation symbols in BM are as follows: • ROWM WD RM . • COLUMNM WD CM . • IM WD f.r; c/ W r 2 RM ; c 2 CM , and M.r; c/ D 1g, which is the incidence relation. The proof of Theorem 8 is partitioned into the following six lemmas: Lemma 5 p-DISJUNCTNESS-TEST 2 co-W[2]. Proof Consider the following complement problem of p-DISJUNCTNESS-TEST. p-NONDISJUNCTNESS-TEST Instance: A binary matrix M and d 2 N . Parameter: d . Problem: Decide whether M is NOT d -disjunct. A …2 formula nondisj(X) with a free relation variable X of arity 2 will be defined, and p-NONDISJUNCTNESS-TEST will be showed to be equal to p-WDnond i sj.X / (see Sect. 4.1.4 for the definition of problem p-WD' ). This implies that pNONDISJUNCTNESS-TEST is in p-WD-…2 , therefore is in W[2]. A binary matrix is not d -disjunct if and only if there is a set D of d columns and another column c … D such that U.D/ covers c. The idea here is assuming that the solution S to X is of the form f.ci ; c/ W ci 2 Dg. Therefore, X should be a binary relation variable, and the solution S to X should have cardinality d . Define 1 WD 8c1 8c2 .Xc1 c2 ! .COLUMNc1 ^ COLUMNc2 ^ .c1 ¤ c2 ///;

and define 2 WD 8c3 8c4 8c5 8c6 ..Xc3 c4 ^ Xc5 c6 / ! .c4 D c6 //:

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Here 1 and 2 are to guarantee that the solution S to X of cardinality d has the form f.c1 ; c/; : : : ; .cd ; c/g, where c 2 CM , ci 2 CM , and ci ¤ c, for 1  i  d . Define nondisj 0 .X / WD 8r9c7 9c8 .ROW r ! .Xc7 c8 ^ .I rc8 ! I rc7 ///: Here nond i sj 0 .X / is to guarantee that the solution S to X satisfies that the union of columns in fci W .ci ; c/ 2 S g covers c (so that M is not d -disjunct). Finally, define nondisj.X / WD 1 ^ 2 ^ nond i sj 0 .X /; which is equivalent to a …2 formula. 2 From the above definition, clearly if there exists a relation S CM of cardinality d such that M ˆ nond i sj.S /, then M is not d -disjunct. On the other hand, if M is not d -disjunct, then there exist a subset D of d columns c1 ; : : : ; cd and another column c … D such that c is covered by U.D/. It is not hard to verify that the relation S D f.c1 ; c/; : : : ; .cd ; c/g satisfies M ˆ nond i sj.S /. Therefore, M is not d -disjunct if and only if there exists a relation S of cardinality d such that M ˆ nond i sj.S /. That is, p-NONDISJUNCTNESS-TEST is p-WDnond i sj.X /. Thus, p-NONDISJUNCTNESS-TEST 2 W[2], and so p-DISJUNCTNESS-TEST 2 co-W[2].  Lemma 6 p-SEPARABILITY-TEST 2 co-W[2]. Proof Consider the following complement problem of p-SEPARABILITY-TEST. p-NONSEPARABILITY-TEST Instance: A binary matrix M and d 2 N . Parameter: d . Problem: Decide whether M is NOT d -separable. A formula nonsep.Y / with a free relation variable Y of arity 2 will be defined, and p-NONSEPARABILITY-TEST will be shown to be equal to p-WDnonsep.Y / . A binary matrix is not d -separable if and only if there exist two distinct subsets D1 and D2 , each contains d columns such that U.D1 / D U.D2 /. Assume that D1 D fc11 ; c12 ; : : : ; c1d g and D2 D fc21 ; c22 ; : : : ; c2d g. The idea is to assume that the solution S to Y is of the form f.c11 ; c21 /; .c12 ; c22 /; : : : ; .c1d ; c2d /g, and so Y should be a binary relation variable, and the solution S to Y should have cardinality d . Define the formula nonsep.Y / such that it satisfies the following: There exists a 2 relation S CM of cardinality d such that M ˆ nonsep.S / if and only if M is not d -separable. Define 3 WD 8c1 8c2 .Yc1 c2 ! .COLUMNc1 ^ COLUMNc2//; 4 WD 8c3 8c4 8c5 8c6 ..Yc3 c4 ^ Yc5 c6 / ! ..c3 D c5 / $ .c4 D c6 ///; and 5 WD 9c7 9c8 8c9 ..Yc7 c8 ^ :Yc9 c7 //:

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2 Here 3 is to guarantee that the relation variable Y CM ; 4 is to build a bijection between the first component of elements in S and the second component of elements in S , which guarantees that the two subsets fc1i W 9c s:t: .c1i ; c/ 2 S g and fc2j W 9c s:t: .c; c2j / 2 S g (which intend to be D1 and D2 , respectively) have the same cardinality; 5 is to guarantee that the two subsets fc1i W 9c s:t: .c1i ; c/ 2 S g and fc2j W 9c s:t: .c; c2j / 2 S g are distinct from each other. Define

nonsep 0 .Y / WD 8r.ROW r ! ..9c10 9c11 Yc10 c11 ^ I rc10 / $ .9c12 9c13 Yc12 c13 ^ I rc13 ///; which is to guarantee that the solution S to Y satisfies that the union of columns in fc1i W 9c s:t: .c1i ; c/ 2 S g is equal to the union of columns in fc2j W 9c s:t: .c; c2j / 2 S g. From basic logic computation, it is not hard to verify that nonsep 0 .Y / is a …2 formula with free relation variable Y . Finally, define nonsep.Y / WD 3 ^ 4 ^ 5 ^ nonsep 0 .Y /: 2 From the above definition of nonsep.X /, if a relation S CM of cardinality d satisfies that M ˆ nonsep.S /, then the two subsets fc1i W 9c s:t: .c1i ; c/ 2 S g and fc2j W 9c s:t: .c; c2j / 2 S g both contain d columns of M and are distinct from each other; moreover, their unions are identical. This implies that M is not d -separable. On the other hand, if M is not d -separable, then there exist two distinct subsets D1 and D2 , each contains d columns such that U.D1 / D U.D2 /. Assume that D1 D fc11 ; : : : ; c1d g and D2 D fc21 ; : : : ; c2d g. It is not hard to verify that the relation

S D f.c11 ; c21 /; .c12 ; c22 /; : : : ; .c1d ; c2d /g satisfies M ˆ nonsep.S /. 2 of cardinality d such that M ˆ From above, there exists a relation S CM nonsep.S / if and only if M is not d -separable; therefore, p-NONSEPARABILITYTEST is p-WDnonsep.Y / . 3 and 4 are …1 formulas; 5 is a †2 formula; nonsep 0 .Y / is a …2 formula; therefore, nonsep.Y / D 3 ^ 4 ^ 5 ^ nonsep 0 .Y / is equivalent to a †3 formula, which implies that p-NONSEPARABILITY-TEST is in p-WD-†3 . Here the following fact is applied: p-WD-†3 p-WD-…2 (more generally, p-WD-†t C1 p-WD-…t , for t  1. The main idea to prove this conclusion is not complicated; the reader is referred to, e.g., Proposition 5.4 in [27] for the proof). Thus, p-NONSEPARABILITY-TEST 2 W[2], and so p-SEPARABILITY-TEST 2 co-W[2].  Lemma 7 p-SEPARABILITY -TEST 2 co-W[2].

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Proof Since W[2] is closed under fpt-reductions, so is co-W[2]. It will be showed next that p-SEPARABILITY -TEST is fpt-reducible to p-SEPARABILITY-TEST. Since the latter is in co-W[2] (Lemma 6), this implies that p-SEPARABILITY TEST 2 co-W[2]. The reduction can be obtained immediately from the following fact (Lemma 2.1.6 in [17]): A binary matrix M 0 containing a zero column is d -separable if and only if the matrix M obtained by removing this zero column from M 0 is dN -separable. Let .M; d / be an instance of p-SEPARABILITY -TEST, where M is a binary matrix and the parameter d is a positive integer. Map .M; d / to .M 0 ; d /, where M 0 is obtained by adding a zero column to M . From the above lemma, .M; d / 2 pSEPARABILITY -TEST if and only if .M 0 ; d / 2 p-SEPARABILITY-TEST. It is easy to see that this is an fpt-reduction from p-SEPARABILITY -TEST to p-SEPARABILITY-TEST. Lemma 8 p-DISJUNCTNESS-TEST is co-W[2]-complete. Proof A hitting set in a hypergraph H D .V; E/ is a set T of vertices that intersects each hyperedge, that is, T \ e ¤ ; for all e 2 E. The classical HITTING-SET problem is to find a hitting set of a given cardinality k in a given hypergraph H, which is known to be NP-complete. The following parameterized hitting set problem is W[2]-complete (see, e.g., Theorem 7.14 in [27]). p-HITTING-SET Instance: A hypergraph H and k 2 N . Parameter: k. Problem: Decide whether H has a hitting set of k vertices. An ftp-reduction from p-HITTING-SET to p-NONDISJUNCTNESS-TEST will be given, based on an idea similar to that in [18]. Let .H; k/ with H D .V; E/ be an instance of p-HITTING-SET, where V D f1; : : : ; ng, E D fe1 ; : : : ; em g, and each ei , 1  i  m, is a subset of V . Define d D k, and define an .n C m/  .n C 1/ binary matrix M with rows Ri as follows (here each row is represented as a subset of the set of all columns f1; 2; : : : ; n C 1g, in the most natural way): Ri D fi g; RnCj D ej [ fn C 1g;

i D 1; : : : ; nI j D 1; : : : ; m:

First, assume that H has a hitting set T V of size k. Consider the subset S1 D T of columns of M . Since T is a hitting set of H, U.S1 / covers column n C 1. Notice that jS1 j D d and column n C 1 is not in S1 , M is not d -disjunct. Conversely, assume that M is not d -disjunct. Then, there exist a subset S1 of d columns in f1; 2; : : : ; n C 1g and another column c … S1 such that U.S1 / covers c. From the way the first n rows of matrix M are defined, c can only be column n C 1. Thus, column n C 1 is not in S1 . Set T D S1 , then jT j D k, and T is a subset of V . Since U.S1 / covers column n C 1, it is easy to see that T is a hitting set of H.

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It is not hard to verify that the above is an fpt-reduction. Therefore, p-NONDISJUNCTNESS-TEST is W[2]-complete, and so p-DISJUNCTNESS-TEST is co-W[2]-complete.  Lemma 9 p-SEPARABILITY -TEST is co-W[2]-complete. Proof To prove the lemma, an ftp-reduction from p-HITTING-SET to p-NONSEPARABILITY -TEST will be given. For an instance .H; k/ of p-HITTING-SET, define matrix M in the same way as in the proof of Lemma 8, and define d D k C 1. Next the correctness of this reduction will be shown. First, assume that H has a hitting set T V of size k. Consider the following two subsets of columns in M : S1 D T and S2 D T [ fn C 1g. Then, for 1  i  n, it is obvious that U.S1 /i D U.S2 /i ; for n < i  n C m, since T is a hitting set of H, U.S1 /i D 1 D U.S2 /i . Notice that jS1 j; jS2 j  d and S1 ¤ S2 , M is not dN -separable. Conversely, assume that M is not dN -separable. Then, there exist two subsets S1 and S2 of columns in f1; 2; : : : ; n C 1g such that jS1 j; jS2 j  d , S1 ¤ S2 , and U.S1 / D U.S2 /. Since U.S1 /i D U.S2 /i for 1  i  n, it follows that S1 \ f1; : : : ; ng D S2 \ f1; : : : ; ng: To have S1 ¤ S2 , column n C 1 must belong to exactly one of S1 and S2 . Without loss of generality, assume that n C 1 … S1 and n C 1 2 S2 . Set T D S1 , then jT j D jS1 j D jS2 j  1  d  1 D k; and T is a subset of V . From U.S1 /i D U.S2 /i D 1 for n < i  n C m, T is a hitting set of H. Therefore, p-NONSEPARABILITY -TEST is W[2]-complete, and so p-SEPARABILITY -TEST is co-W[2]-complete. Lemma 10 p-SEPARABILITY-TEST is co-W[2]-complete. Proof Since as proved before in Lemma 7 that p-SEPARABILITY -TEST is fpt-reducible to p-SEPARABILITY-TEST, and in Lemma 9 that p-SEPARABILITY TEST is co-W[2]-complete, p-SEPARABILITY-TEST is co-W[2]-hard. Together with Lemma 6 that p-SEPARABILITY-TEST 2 co-W[2], it is obtained that p-SEPARABILITY-TEST is co-W[2]-complete.

4.3

Discussion

In this section, the parameterized complexity is established for the following three basic problems in pooling design: Given an m  n binary matrix and a positive integer d , to decide whether the matrix is d -separable (dN -separable, or

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d -disjunct). It is showed that these problems are co-W[2]-complete; thus, do not admit algorithms with running time f .d /  .mn/O.1/ for any computable function f . The best known algorithms for the above general problems are all in a bruteforce manner. It is interesting to investigate that whether these problems admit better algorithms. For instance, are there algorithms with running time no.d / O.m/ to solve these problems when d is small compared to n?

5

Upper Bounds on the Minimum Number of Rows of Disjunct Matrices

A 0-1 matrix is d -disjunct if no column is covered by the union of any d other columns, where the union means the bitwise Boolean sum of these d column vectors. In other words, a 0-1 matrix is called d -disjunct if for any column C and any d other columns, there exists at least one row such that the row has value 1 at column C and value 0 at all d other columns. The same structure is also called cover-free family [25, 29, 53] in combinatorics and superimposed code [22, 23, 34] in information theory. It is called a d -disjunct matrix in group testing [17, 32, 39]. A 0-1 matrix is .d I z/-disjunct [22,39] if for any column C and any d other columns, there exist at least z rows such that each of them has value 1 at column C and value 0 at all the other d columns. Thus, d -disjunct is .d I 1/-disjunct. Besides other applications, d -disjunct and .d I z/-disjunct matrices form the basis for errorfree and error-tolerant nonadaptive group testing algorithms, respectively. These algorithms have applications in many practical areas such as DNA library screening [3, 6, 17, 43] and multi-access communications [57]. Let t.d; n/ denote the minimum number of rows required by a d -disjunct matrix with n columns. The bounds on t.d; n/ have been extensively studied in the fields of combinatorics, information theory, and group testing, under different terminologies. 2 For lower bounds, it is known that t.d; n/ D . d loglogd n / [21, 29, 53]. In particular, 2

d .1 C o.1// log n, which is D’yachkov and Rykov [21] proved that t.d; n/  2 log d the best lower bound so far. For upper bounds on t.d; n/, it is known that t.d; n/ D O.d 2 log n/ [2, 22, 32, 33, 45, 50]. In [22], Dyachkov, Rykov, and Rashad obtained the following asymptotic upper bound on t.d; n/ with a rather involved proof, which is currently the best.

Theorem 9 (Dyachkov, Rykov, and Rashad [22]) For d constant and n ! 1, t.d; n/ 

d Œ1 C o.1/ log n; Ad

where  Ad D max

max

0p1 0P 1

  1p p : .1  P / log.1  p d /Cd P log C.1  P / log P 1P

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1 d log e

131

as d ! 1.

For .d I z/-disjunct matrices, let t.d; nI z/ denote the minimum number of rows required by a .d I z/-disjunct matrix with n columns. For given d and z, D’yachkov, Rykov, and Rashad [22] studied the value of limn!1 logt n , among others, and they 2

proved that t.d; nI z/  cΠd loglogd n C .z  1/d  where c is a constant. In [13], by using q-ary .d; 1/-disjunct matrices [17, 20] and the probabilistic method (see, e.g., [1]), the authors gave a very short proof for the currently best upper bound on t.d; n/. In contrast to the previous result in [22] (Theorem 9), which is an asymptotic upper bound, the upper bound on t.d; n/ in [13] does not contain the asymptotic term o.1/. Also, the method in [13] is generalized to obtain a new upper bound on t.d; nI z/. These results will be presented in this section.

5.1

Upper Bounds on t.d; n/

Theorem 10 For n > d  1, t.d; n/ 

where Bd D maxq>1

  d  log 1 1 q1 q

d C1 log n; Bd

. Moreover, Bd !

1 d log e

as d ! 1.

Before proving the above theorem, the concept of q-ary .d; 1/-disjunct matrix will be first introduced: A matrix is called q-ary .d; 1/-disjunct if it is q-ary, and for any column C and any set D of d other columns, there exists an element in C such that the element does not appear in any column of D in the same row. As described in [17, 20], one can transform a q-ary .d; 1/-disjunct matrix M to a (binary) d -disjunct matrix M 0 as follows. Replace each row Ri of M by several rows indexed with entries of Ri . For each entry x of Ri , the row with index x is obtained from Ri by turning all x’s into 1’s and all others into 0’s. The following fact is useful in later proof, which is the same as Theorem 1 only using different notations: Fact 2 (Theorem 3.6.1 in [17]) A t  n q-ary .d; 1/-disjunct matrix M yields a t 0  n d -disjunct matrix M 0 with t 0  tq. Now it is time to present the proof of Theorem 10. Proof of Theorem 10: Given n > d  1, first construct a random t  n q-ary (q > 1) matrix M with each entry assigned randomly and uniformly from f1; 2; : : : ; qg, where q and t will be specified later. For each column C and a set D of d other columns, for each element ci (i D 1; 2; : : : ; t) of C , the probability that ci appears in some column of D in the same row is 1  .1  q1 /d . Thus, the probability that every element of C appears in some column of D in the same row is Œ1  .1  q1 /d t .

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M is not .d; 1/-disjunct if and only if there exist a column C and a set D of d other columns such that the above holds. Therefore, the probability that M is not .d; 1/-disjunct is no more than ! n 1 .n  d / Œ1  .1  /d t : d q What follows next is to try to minimize tq, the number of rows of the d -disjunct matrix M 0 as in Fact 2, under the condition that q and t satisfy nd C1 Œ1  .1 

1 d t /   1: q

(1)

Notice that Eq. (1) implies ! n 1 .n  d / Œ1  .1  /d t < 1I d q thus, the probability that M is .d; 1/-disjunct is greater than zero. Therefore, by probabilistic argument, Eq. (1) implies the existence of a t  n q-ary .d; 1/-disjunct matrix, and so a d -disjunct matrix with n columns and at most tq rows. To satisfy Eq. (1), let .d C 1/ log n tD :  logŒ1  .1  q1 /d  Define Bd .q/ D

 logŒ1  .1  q1 /d 

then tq D

q

;

.d C 1/ log n : Bd .q/

Let q0 be the point that maximizes Bd .q/, and let Bd D Bd .q0 / (one can estimate that q0 D ‚.d / and Bd D ‚. d1 /, since the proof here can stand alone without this observation; they are put into Lemma 11) in later part. By assigning q D q0 , it follows that t.d; n/  .tq/jqDq0 D

.d C 1/ log n .d C 1/ log n D : Bd .q0 / Bd

Next estimate Bd as d ! 1. Since .1  q1 /q
1,

    dq d 1 d 1 1 < D e q ; q e

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and  logŒ1  .1 

d 1 d /  <  log.1  e  q /: q

It follows that

Bd .q/ D

 logŒ1  .1  q1 /d  q d

 log.1  e  q / < q D

d d 1 Πln.1  e  q /: d ln 2 q

d

Let x D e  q , then  dq D ln x, and Bd .q/
1. Thus, Bd < lnd2 for d  1. On the other hand, when q satisfies .1  q1 /d D 12 , as d ! 1, it is easy to see that dq ! ln12 , and Bd .q/ D q1 ! lnd2 . Therefore, as 1  d ! 1, Bd ! lnd2 D d log e. Lemma 11 Given d  1, let q0 D q0 .d / be the point that maximizes Bd .q/ D  logŒ1.1 q1 /d  q ‚. d1 /.

for q > 1. Then, as d ! 1, q0 .d / D ‚.d /, and Bd D Bd .q0 / D

Proof Notice that if q1 satisfies .1  q11 /d D 12 , then q1 D ‚.d /, since qd1 ! ln12 as d ! 1. Moreover, Bd .q1 / D q11 D ‚. d1 /. The lemma is proved by contradiction. First assume that q0 D O.d / does not hold, that is, for any c > 0 and any d0 > 0, there exists d > d0 such that q0 .d / > cd . Then, since qd0 > c, as c ! 1,

Bd .q0 /d D  D

 logŒ1  .1 

1 d /  q0

q0  logŒ1  .1  q0 log qd0 q0 d

D o.1/;

d q0 /

d

d

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here a  b means that limc!1

a b

D 1. Thus, Bd .q0 / D

o.1/ : d

However, this is a contradiction since q0 is the maximum point of Bd .q/ and Bd .q1 / D ‚. d1 / with .1  q11 /d D 12 . On the other hand, assume that q0 D .d / does not hold, that is, for any c > 0 and any d0 > 0, there exists d > d0 such that q0 .d / < cd . Then, Bd .q0 /d D

D Since 0 < .1 

1 q0 / q0


1, as c ! 0, Œ.1 

1 d /  q0

 logŒ1  .1 

d q0

>

1 c

d:

! 1, and

1 q0 qd d /  0 < e q0 ! 0: q0

Thus,

Bd .q0 /d 

Œ.1 

d

1 q0 q0 q0 / 

q0 ln 2

d

D

d 1 d 1 Œ.1  /q0  q0 ln 2 q0 q0


1

d C1 z log n C log log n C O.1/; Bd Bd

 logŒ1.1 q1 /d  . q

Moreover, Bd !

1 d log e

as d ! 1.

A q-ary matrix is called .d; 1I z/-disjunct if for any column C and any set D of d other columns, there exists at least z elements in C such that each of these elements does not appear in any column of D in the same row. Clearly, by using the same method mentioned above, one can transform a t  n q-ary .d; 1I z/-disjunct matrix to a .d I z/-disjunct matrix with n columns and at most tq rows. Proof of Theorem 14: For given n; d , and z, similarly construct a random t  n q-ary (q > 1) matrix M with each entry assigned randomly and uniformly from f1; 2; : : : ; qg; q and t will be specified later. For each column C and a set D of d other columns, for each element ci of C , the probability that ci appears in some column of D in the same row is 1  .1  q1 /d . Thus, the probability that there exist t  z C 1 elements of C such that each of them appears in some column of D in the same row is at most ! ! t t 1 1 d t zC1 D Œ1  .1  /d t zC1 : Œ1  .1  /  q q z1 t zC1 M is not .d; 1I z/-disjunct if and only if there exists a column C and a set D of d other columns such that the above holds. Therefore, the probability that M is not .d; 1I z/-disjunct is no more than n .n  d / d

!

! t 1 Œ1  .1  /d t zC1 : q z1

The goal is to minimize tq, the number of rows of the corresponding .d I z/disjunct matrix, under the condition that nd C1 t z Œ1  .1 

1 d t z /   1: q

(2)

Notice that Eq. (2) implies n .n  d / d

!

! t 1 Œ1  .1  /d t zC1 < 1: z1 q

Thus, the probability that M is .d; 1I z/-disjunct is greater than zero, which similarly implies the existence of a t  n q-ary .d; 1I z/-disjunct matrix, and a .d I z/-disjunct matrix with n columns and at most tq rows.

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Let q0 be the point that maximizes Bd .q/ D

 logŒ1  .1  q1 /d  q

:

Assign q D q0 . To satisfy Eq. (2), which is equivalent to .d C 1/ log n C z log t  .t  z/ logŒ1  .1  let tD

1 d / ; q0

.d C 1/ log n C z C t1 :  logŒ1  .1  q10 /d 

Then, t1 should satisfy (

.d C 1/ log n z log C z C t1  logŒ1  .1  q10 /d  Let t1 D

)  t1 logŒ1  .1 

1 d /  q0

(3)

z log log n C t2 :  logŒ1  .1  q10 /d 

From Eq. (3), t2 should satisfy that

z  logŒ1  .1 

( 1 d /  q0

log

.d C 1/  logŒ1  .1  1 C log n

1 d /  q0

z log log n C z C t2  logŒ1  .1  q10 /d 

!)  t2 (4)

For d and z constants (thus, q0 is also constant), as n ! 1, the minimum value of t2 satisfying Eq. (4) is t2 D

z  logŒ1  .1 

1 d /  q0

log

.d C 1/  logŒ1  .1 

1 d /  q0

D O.1/:

Thus, tD

.d C 1/ log n z log log n C C O.1/  logŒ1  .1  q10 /d   logŒ1  .1  q10 /d 

satisfies Eq. (2) (where the constant term z in t is absorbed in O(1)). Therefore, the number of rows of the corresponding .d I z/-disjunct matrix is at most

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137

d C1 z log n C log log n C O.1/; Bd Bd

where Bd D Bd .q0 / D max q>1

Also, Bd ! as proved in Theorem 10.

6

 logŒ1  .1  q1 /d  q

:

1 as d ! 1; d log e 

Transformation from Error-Tolerant Separable Matrices to Error-Tolerant Disjunct Matrices

Let M be a 0/1 matrix. For any set S of columns of M , U.S / will denote the union of the row indices of 1-entries of all columns in S . When S is the singleton set fC g, by abusing the notation, U.S / is simply written as C: Matrix M is called d -separable if for any two distinct d -sets S and S 0 of columns, U.S / ¤ U.S 0 /. M is called dN -separable if the restrictions jS j D d and jS 0 j D d above are changed to jS j  d and jS 0 j  d , respectively. Finally, M is called d -disjunct if for any d -set S of columns and any column C not in S , C is not contained in U.S /. These three properties of 0/1 matrices have been widely studied in the literature of nonadaptive group testing designs (pooling designs), which have applications in DNA screening [17, 25, 32, 38, 39]. It has long been known that d -disjunctness implies dN -separability which in turn implies d -separability [17, Chap. 2]. Recently, Chen and Hwang [7] found a way to construct a disjunct matrix from a separable matrix to complete the cycle of implications. Theorem 12 (Chen and Hwang [7]) Suppose M is a 2d -separable matrix. Then one can construct a d -disjunct matrix by adding at most one row to M . The notion of d -separability, dN -separability, and d -disjunctness has their errortolerant versions. A 0/1 matrix M is called .d I z/-separable if jU.S /4U.S 0/j  z for any two d -sets of columns of M . It is .dN I z/-separable if the restriction of d -sets is changed to two sets each with at most d elements. Finally, M is .d I z/-disjunct if for any d -set S of columns and any column C not in S , jC n U.S /j  z. Note that the variable z represents some redundancy to tolerate errors [17]. For z D 1, the error-tolerant version is reduced to the original version. In [17], Du and Hwang attempted to extend Theorem 12 to its error-tolerant version, as stated in the following theorem: Theorem 13 ([17], Theorem 2.7.6) Suppose M is a .2d I z/-separable matrix. Then one can obtain a .d I z/-disjunct matrix by adding at most z rows to M .

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By Theorem 13, Du and Hwang obtained the following corollary: Corollary 1 ([17], Theorem 2.7.7) A .d I 2z/-separable matrix can be obtained from a .2d I z/-separable matrix by adding at most z rows. Unfortunately, Theorem 13 is incorrect; thus, Corollary 1 is also incorrect, as seen by the following counterexample. Let 0

1 B0 M1 D B @0 0

0 1 0 0

0 0 1 0

1 0 0C C 0A 1 :

It is easily verified that M1 is .2I 2/-separable. It will be showed next adding two rows to M1 cannot produce a .1I 2/-disjunct matrix. Let C1 ; C2 ; C3 ; and C4 denote the four columns of M1 . Suppose setting C D Ci and S D fCj g, i ¤ j . Then two rows are needed such that each containing Ci but not Cj . One such row is already provided by M1 . So one .1; 0/-pair is needed in a new row. Since this is required for each pair of .i; j / with i ¤ j , there are 4  3 D 12 choices of .i; j / pair, and each such pair needs a .1; 0/-pair in a new row, or equivalently, the new rows should provide 12 such .1; 0/-pairs. However, one new row can provide at most four .1; 0/-pairs (achieved by a row with two 1-entries and two 0-entries). So two new rows are not sufficient to provide the 12 .1; 0/-pairs required by the .1I 2/-disjunctness property. In [9], the authors gave a correct version of Theorem 13 and obtained a more rigorous statement of Theorem 12. Their results will be presented in this section.

6.1

Main Results

Lemma 12 ([17], Lemma 2.1.1) Suppose M is a d -separable matrix with n columns where d < n, then it is k-separable for every positive integer k  d . Note that the condition d < n in Lemma 12 is necessary as seen by the following example: Let 0 1 110 M2 D @ 1 0 1 A 011

:

M2 is trivially 3-separable. However, it is not 2-separable, as the union of any pair of its columns is identical. Now Lemma 12 is generalized to an error-tolerant version. Lemma 13 If a matrix M with n columns is .d I z/-separable for d < n, then it is .kI z/-separable for every positive integer k  d .

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Proof It suffices to prove M is .d 1I z/-separable. Assume that M is not .d 1I z/separable. Then there exist two distinct sets S and S 0 each consisting of d  1 columns of M such that jU.S /4U.S 0/j < z. If jS n S 0 j D jS 0 n S j  2, then there must exist a pair of columns .Cx ; Cy / such that Cx 2 S n S 0 and Cy 2 S 0 n S . It is easy to see that jU.S [ fCy g/4U.S 0 [ fCx g/j  jU.S [ fCy g/4U.S 0 /j  jU.S /4U.S 0 /j: This violates the .d I z/-separability of M , as desired. Now consider the case of jS n S 0 j D jS 0 n S j D 1. It is obvious that jS [ S 0 j D d . Thanks to d < n, one can take a column C of M which is in neither S nor S 0 . It is easily seen that jU.S [ fC g/4U.S 0 [ fC g/j  jU.S /4U.S 0/j < z: This contradicts the .d I z/-separability of M , completing the proof.



Now it is ready to give a correct version of Theorem 13. Theorem 14 Suppose M is a .2d I z/-separable matrix with n columns where n  2d C 1. Then one can obtain a .d I dz=2e/-disjunct matrix by adding at most dz=2e rows to M . Proof Suppose M is not .d I dz=2e/-disjunct. Then there exist a column C and a set S of d other columns such that jC n U.S /j < dz=2e. By adding at most dz=2e rows to M such that each row has a 1-entry at column C and 0-entries at all columns in S , one can obtain jC n U.S /j  dz=2e. Of course, there may exist another pair .C 0 ; S 0 / where C 0 is a column and S 0 is a set of d columns other than C 0 , such that jC 0 n U.S 0 /j < dz=2e in M . Then break it up by using those dz=2e rows in the same fashion. Here what one needs to show is that this procedure is not self-conflicting, that is, there does not exist two pairs .C; S / and .C 0 ; S 0 / such that jC n U.S /j < dz=2e, yet on the other hand C 2 S 0 while jC 0 n U.S 0 /j < dz=2e. Suppose to the contrary that there exist two pairs .C; S / and .C 0 ; S 0 / in M as described above with jS j D jS 0 j D d . Define S0 D fC 0 g [ S [ S 0 ; S1 D S0 n fC g; and S2 D S0 n fC 0 g: Let s D jS0 j, then s  2d C 1 and jS1 j D jS2 j D s  1  2d . Note that S1 ¤ S2 , but they have the same cardinality which is less than 2d C 1. Next it will be showed that the symmetric difference of U.S1 / and U.S2 / is less than z, thus violating the assumption of .2d I z/-separability. Since the only column in S1 but not in S2 is C 0 and jC 0 n U.S 0 /j < dz=2e, it follows that jU.S2 / n U.S1 /j < dz=2e: (5) Similarly, one can obtain

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jU.S1 / n U.S2 /j < dz=2e:

(6)

Equation (5) along with Eq. (6) gives jU.S /4U.S 0/j < z, implying that M is not .s  1I z/-separable. This contradicts with Lemma 13, and so the theorem is proved.  Corollary 2 Suppose M is a 2d -separable matrix with n columns where n2d C1. Then one can obtain a d -disjunct matrix by adding at most one row to M . Proof It follows from Theorem 14 by setting z D 1: Corollary 2 is a more rigorous version of Theorem 12. The following example shows the necessity of the extra condition n  2d C 1 in Corollary 2. Let 0

110 B1 0 1 M3 D B @0 1 1 000

1 1 0C C 0A 1 :

Then M3 is trivially 4-separable, but it can be easily verified that no row can be added to M3 to make it 2-disjunct. Similarly, any matrix with 2d columns is trivially .2d I z/-separable, and one does not expect that adding dz=2e rows to an arbitrary matrix with 2d columns would make it .d I dz=2e/-disjunct. To see a specific counterexample, note that M1 is trivially a .4I 4/-separable matrix, but adding two rows does not make it a .2I 2/-disjunct matrix – It is even not .1I 2/disjunct as indicated at the end of Sect. 1. Corollary 3 Suppose M is a .2d I z/-separable matrix with n columns where n  2d C 1. Then, for any positive integer k  dz=2e, one can obtain a .d I k/disjunct matrix by adding at most k rows to M . Proof The proof of Theorem 14 shows that there does not exist two pairs .C; S / and .C 0 ; S 0 / such that jC n U.S /j < dz=2e, yet on the other hand, C 2 S 0 while jC 0 n U.S 0 /j < dz=2e. In fact, the term dz=2e can be replaced by any positive integer k which satisfies the symmetric difference of U.S1 / and U.S2 / is less than z. Therefore, for any k  dz=2e, one can obtain a .d I k/-disjunct matrix by adding at most k rows to M in the same fashion.  The following equivalence relation is given in [17] without giving a proof. Now a proof is given, and a stronger result is obtained by using the equivalence relation. Lemma 14 ([17], Lemma 2.7.5) A matrix M is .dN I z/-separable if and only if it is .d I z/-separable and .d  1I z/-disjunct.

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Proof Suppose M is .dN I z/-separable but not .d  1I z/-disjunct; in other words, there exists a set S of d 1 columns other than a column C such that jC nU.S /j  z. Then it is easy to see that jU.S [ fC g/4U.S /j D jU.S [ fC g/ n U.S /j  z; a contradiction to .dN I z/-separability. Thus, M is .d  1I z/-disjunct and .d I z/separable trivially. Let M be .d I z/-separable and .d  1I z/-disjunct. It suffices to show that jU.X /4U.Y /j  z for any two sets X , Y of at most d columns. If jX j D jY j  d , then jU.X /4U.Y /j  z by .d I z/-separability and Lemma 13. Assume jX j < jY j  d , then there exists a column Cy 2 Y but not in X . By .d  1I z/-disjunctness, it must have jCy n U.X /j  z; hence, jU.X /4U.Y /j  z. It concludes the proof.  By Lemmas 14 and 13, Corollary 3 is extended to a stronger version. Corollary 4 Suppose M is a .2d I z/-separable matrix with n columns where n  N 1I k/2d C 1. Then, for any positive integer k  dz=2e, one can obtain a .d C separable matrix by adding at most k rows to M .

6.2

Concluding Remarks

The following remarks demonstrate the optimality of the results presented in this section. 1. The constraint k  dz=2e in Corollary 3 is necessary to make the number of rows added to be independent of n and d . To see a specific example, consider that M is an .ndz=2e/  n matrix such that each column has dz=2e 1-entries and any 2 columns have no intersection. Then, M is .2d I z/-separable. Since every column has only dz=2e 1-entries, to make M .d I k/-disjunct by adding rows, the rows added must form a .d I k  dz=2e/-disjunct submatrix when k > dz=2e. In this case, the minimum number of rows required would depend on n; d , and k  dz=2e. 2. Let N be a 0/1 matrix of constant row sum 1 and constant column sum z, and let M be obtained from N by adding one zero column. It is easy to verify that M is .2d I z/-separable. Since there is a zero column in M , one cannot obtain from M a .d I k/-disjunct matrix by adding less than k rows. This shows that the bound on the number of additional rows given in Corollary 3 is optimal in this sense.

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Conclusion

Some recent algorithmic, complexity, and mathematical results on nonadaptive group testing (and on pooling design) are presented in this monograph. New construction of disjunct matrices with even reduced number of rows remains interesting to investigate. The complexity of the problem MIN-dN -SS introduced in Sect. 3.3 remains unsolved. On the bounds of the minimum number t.d; n/ of rows of d -disjunct matrices with n columns, closing the gap between O.d 2 log n/ and 2 . d loglogd n / remains as a major open problem in extremal combinatorics.

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43. H.Q. Ngo, D.Z. Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, in DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, vol. 55 (American Mathematical Society, Providence, 2000), pp. 171–182 44. H.Q. Ngo, D.Z. Du, New constructions of non-adaptive and error-tolerance pooling designs. Discret. Math. 243, 161–170 (2002) 45. H.Q. Ngo, E. Porat, A. Rudra, Efficiently decodable error-correcting list disjunct matrices and applications, in Proceedings of the 38th International Colloquium on Automata, Languages and Programming, Zurich, Switzerland, 2011, pp. 557–568 46. M. Olson, L. Hood, C. Cantor, D. Botstein, A common language for physical mapping of the human genome. Science 245, 1434–1435 (1989) 47. C.H. Papadimitriou, Computational Complexity (Addison-Wesley, New York, 1994) 48. C.H. Papadimitriou, D. Wolfe, The complexity of facets resolved, J. Comput. Syst. Sci. 37, 2–13 (1988) 49. H. Park, W. Wu, Z. Liu, X. Wu, H.G. Zhao, DNA screening, pooling design and simplicial complex. J. Comb. Optim. 7, 389–394 (2003) 50. E. Porat, A. Rothschild, Explicit non-adaptive combinatorial group testing schemes, in Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Reykjavik, Iceland, 2008, pp. 748–759 51. S. Rahmann, Rapid large-scale oligonucleotide selection for microarrays, in Proceedings of the 1st IEEE Computer Society Conference on Bioinformatics (CSB’ 02), Stanford, CA, USA, 2002, pp. 54–63 52. S. Rahmann, Fast and sensitive probe selection for DNA chips using jumps in matching statistics, in Proceedings of the 2nd IEEE Computer Society Bioinformatics Conference (CSB’ 03), Stanford, CA, USA, 2003, pp. 57–64 53. M. Ruszink´o, On the upper bound of the size of the r-cover-free families. J. Comb. Theory Ser. A 66, 302–310 (1994) 54. A. Schliep, D.C. Torney, S. Rahmann, Group testing with DNA chips: generating designs and decoding experiments, in Proceedings of the 2nd IEEE Computer Society Bioinformatics Conference (CSB’ 03), Stanford, CA, USA, 2003, pp. 84–93 55. C. Umans, The minimum equivalent DNF problem and shortest implicants, in Proceedings of 39th IEEE Symposium on Foundation of Computer Science, Palo Alto, CA, USA, 1998, pp. 556–563 56. X. Wang, B. Seed, Selection of oligonucleotide probes for protein coding sequences. Bioinformatics 19, 796–802 (2003) 57. J.K. Wolf, Born again group testing: multiaccess communications. IEEE Trans. Inf. Theory IT-31, 185–191 (1985)

Advances in Scheduling Problems Ming Liu and Feifeng Zheng

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Emergence of psd Delivery Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Literature Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Single-Machine Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preliminary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Makespan, Maximum Lateness, Maximum Tardiness, and Number of Tardy Jobs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Total Weighted Completion Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Total Weighted Discounted Completion Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Total Weighted Tardiness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Total Completion Time with Release Dates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Approximation for Total Completion Time with Release Dates. . . . . . . . . . . . . . . . . . . . . . 3 Parallel Identical Machine Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Makespan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Total Completion Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146 146 147 148 150 150 151 152 155 155 158 159 161 161 168 168 168 169

M. Liu () School of Economics & Management, Tongji University, Shanghai, People’s Republic of China e-mail: [email protected] F. Zheng Glorious Sun School of Business and Management, Donghua University, Shanghai, People’s Republic of China e-mail: [email protected] 145 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 77, © Springer Science+Business Media New York 2013

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Abstract

This chapter reviews a research focus of scheduling in manufacturing facilities over the last decade. Manufacturing environment has a great influence on the processing times of jobs. The chapter reviews basic concept about past-sequencedependent delivery times and covers recently results on scheduling with such constraint.

1

Introduction

Scheduling is a form of decision making that plays a crucial role in manufacturing and service industries. Scheduling is a key component of combinatorial optimization. In the current competitive environment, effective scheduling has become a necessity for survival in the marketplace. Companies have to meet shipping dates that have been committed to customers, as failure to do so may result in a significant loss of goodwill. They also have to schedule activities or jobs in such a way as to use the resources available in an efficient manner. Scheduling deals with the allocation of scarce resources to tasks over time. It is a decision-making process with the goal of optimizing one or more objectives. The resources and tasks in an organization can take many forms. The resources may be machines in a workshop, runways at an airport, crews at a construction site, processing units in a computing environment, and so on. The tasks may be operations in a production process, takeoffs and landings at an airport, stages in a construction project, executions of computer programs, and so on. Each task may have a certain priority level, an earliest possible starting time, and a due date. The objective may be the minimization of the completion time of the last task, and another may be the minimization of the number of tasks completed after their respective due dates [12]. This chapter focuses on a cutting-edge topic in machine scheduling problem, that is, scheduling with past-sequence-dependent (abbr. psd) job delivery times. The topic is important because of the theoretical insights it provides and also of its practical implications.

1.1

Emergence of psd Delivery Time

In many industries, manufacturing environment has a great influence on the processing times of jobs. A growing body of evidence shows that the manufacturing environment or waiting time may have an adverse effect on the total processing time of a job before its delivery to the customer. In some situations, the waiting time-induced adverse effect does not impede the job’s suitability to be processed. Moreover, it can be eliminated after the main processing of the job with an extra

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time caused. Such an extra time for eliminating adverse effect between the main processing and the delivery of job is viewed as a psd delivery time. Note that the treatment of the adverse effect for a job does not occupy any machine and has no relation with the schedule of the job’s main processing. For analysis convenience, it is generally assumed that the psd delivery time of a job is proportional to the job’s waiting time. One special case is that the psd delivery time is a simple linear function of the wait time. One application with psd delivery time is in electronic manufacturing industry. An electronic component may be exposed to certain electromagnetic and/or radioactive fields while waiting in the machine’s preprocessing area, and regulatory authorities require the component to be “treated” (e.g., in a chemical solution capable of removing/neutralizing certain effects of electromagnetic/radioactive fields) for an amount of time proportional to the job’s exposure time to these fields. The treatment is performed immediately after the component has been processed on the machine to ensure a guaranteed delivery to the customer [6]. Such a postprocessing operation is usually called the job “tail” or the job “delivery time.” Unlike the traditional assumption of a job-specific constant delivery time in the scheduling literature [4], it is assumed that the psd delivery time of a job is proportional to the job’s waiting time.

1.2

Literature Review

It has been widely studied in recent decades for one scheduling scenario such that the waiting time of a job has an adverse effect on the job’s main processing. The scenario with waiting time-induced adverse effect is mainly partitioned into three categories. In the first category, each job has a so-called deteriorating processing time such that the processing time of a job increases in its waiting time. The concept of deterioration is introduced by Browne and Yechiali [2] and it describes a kind of adverse effect of waiting time. The second category originated from [5], in which the adverse effect must be removed prior to the main processing of the job by performing a setup operation. The authors introduced the concept of past-sequence-dependent setup time to describe the adverse effect. The third and newest category is from [6], in which the waiting time-induced adverse effect does not impede the suitability to be processed by one machine, and the adverse effect shall be removed prior to delivering the job to the customer. They established a model to incorporate the adverse effect of waiting into a postprocessing operation by introducing the concept of past-sequence-dependent delivery time. The treatment of the adverse effect of each job, that is, the job’s psd delivery time, must follow immediately the main processing of the job. This chapter focuses on this category.

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Notation

In all the following scheduling problems considered, the number of jobs and that of machines are assumed to be finite. Generally, the number of jobs is denoted by n and the number of machines by m (see [12]). The following notations are associated with job Jj : Processing time (pij ). pij represents the main processing time of job Jj on machine Mi . It is simply denoted as pj if the main processing time of Jj is uniform on all machines or if there is only one machine allowed to process job Jj . In the rest of this chapter, sometimes the processing of Jj means its main processing on a machine. Due date (dj ). dj represents the committed date at which job Jj is shipped on board or arrived at the customer. There may incur a penalty if the completion time of a job is strictly later than its due date. Weight (wj ). wj represents some priority or cost for satisfying job Jj . Its specific meaning depends on manufacturing settings. Delivery time (qj ). Literally, qj means the time a completed job needs to be delivered to a customer. In this chapter, delivery time usually represents a postprocessing of a job after its completion on the machine. (Note that this postprocessing operation is not done on the machine). One scheduling problem is usually described by a triplet ˛jˇj (refer to [1]). The ˛ field describes the machine environment such as single machine or parallel machines, the ˇ field provides details on job-processing characteristics and constraints, and the  field describes the minimization objectives such as the makespan and the total completion times. Below, possible settings in each of the three fields are given in this chapter. First, there are two possible machine environments specified in the ˛ field: Single machine (1). There is a single machine for job processing in the manufacturing system. Identical machines in parallel (Pm). There are m identical parallel machines in the system. Each job requires only one operation and may be processed on any one of the m machines. Secondly, there may be multiple entries on job-processing restrictions and constraints specified in the ˇ field, including: Past-sequence-dependent delivery time (qpsd ). qpsd represents the time required for eliminating waiting time-induced adverse effect after the main processing of job. (Note that this part of processing is not done on the machine). Release date (rj ). Job Jj becomes available for processing at its release date rj . If the symbol does not appear in the ˇ field, it implies rj D 0 for any job Jj and all the jobs can be started at the beginning. Preemption (prmp). prmp represents that it is unnecessary to keep a job on a machine, once started, until completion. The scheduler is allowed to interrupt the processing of a job at any time point and arrange another job on the machine instead. In this chapter, the focus is on the resumption case such that if a preempted job is afterward rescheduled on the machine (or another machine in the case of parallel machines), it is started from where it was preempted but not from the start.

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Precedence constraint (prec). prec means that there exist processing precedence constraints among jobs such that one or more jobs are required to be completed before the processing of some other job. In this chapter, three specific forms of precedence constraint are considered: a chain in which each job has at most one predecessor and at most one successor, an intree in which each job has at most one successor, and an outtree in which each job has at most one predecessor. Before introducing the minimization objectives in the  field, some fundamental notations are given. In a processing schedule, let Cj be the completion time of job Jj , that is, the completion time of the processing of Jj on a machine plus the job’s psd delivery time. The lateness of job Jj is defined as Lj D Cj  dj ; which is positive or negative when Jj is completed later or earlier than its due date, respectively. The tardiness of job Jj is defined as Tj D maxfCj  dj ; 0g D maxfLj ; 0g: The difference between the tardiness and the lateness lies in that the former is never negative, that is, the jobs completed earlier than its due date are counted out when counting Tj . The unit penalty of job Jj is defined as  Uj D

1; if Lj > 0I 0; otherwise:

The lateness, tardiness, and unit penalty are the three basic due date-related parameters considered in the literature. The possible objective functions specified in  field are: Makespan (Cmax ). The makespan, defined as maxfC1 ; : : : ; Cn g, is equivalent to the completion time of the last job to leave the system. A minimum makespan usually implies a high utilization of the machine(s). Maximum lateness (Lmax ). The maximum lateness Lmax is defined as maxfL1 ; : : : ; Ln g. It measures the worst violation of due date among completed jobs. Maximum tardiness (Tmax ). The maximum tardiness Tmax is defined as maxfT1 ; : : : ; Tn g. It also measures the worst violation of due date among completed jobs but omits the jobs with negative Plateness. Total weighted completion time ( wj Cj ). The sum of the weighted completion times of the n jobs gives an indication of the total holding or inventory costs P incurred by the schedule. For the case ofPrj D 0, the total completion times Cj is equivalent to total flow time, and wj Cj is equivalent to total weighted flow time. P Discounted total weighted completion time ( wj .1  e rCj /). This is a more general cost function than total weighted completion time. The costs are discounted at a rate r 2 .0; 1/ per unit time. That is, if job Jj is not completed by time t, an additional cost wj re rt dt is incurred over period Œt; t C dt. (Note that

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R Cj

wj re rt dt D wj .1  e rCj / where Cj is a variable). The value of r is usually less than 0.1. P Total weighted tardiness ( wj Tj ). This objective is also a more general cost function than total weighted P completion time. Number of tardy jobs ( Uj ). This is not only an objective of academic interest but also a measure in common use in practice due to its easy operation. All the above objective functions belong to so-called regular performance measures which are nondecreasing functions of jobs’ completion times, that is, C1 ; : : : ; Cn . 0

2

Single-Machine Model

This section considers the following single-machine scheduling models related to minimization objectives. For completion time-related objectives, it includes makespan, that is, the maximum completion time, total completion time, and total weighted completion time. For due date-related objectives, it includes maximum lateness and number of tardy jobs. All the objective functions considered in this section are regular, which means the objective function is a nondecreasing function of job’s completion time. For most models considered in this section, jobs are all released at the beginning, and thus there is no advantage in adopting preemptions. For these models, it can be shown that an optimal schedule is a nonpreemptive one, and there are no idle time segments between the processing of any two jobs. However, if jobs are with different release dates in nonpreemptive environment, it may be advantageous to contain unforced idleness. This section is mainly based on [6, 10, 11].

2.1

Preliminary

Consider a single machine to process a set of n jobs which are available at time zero. Let pj , dj denote, respectively, the processing time and due date of job Jj for j D 1; : : : ; n. Given a nonpreemptive schedule , let JŒj  , pŒj  , and dŒj  denote the job occupying the j -th position in , its processing time, and its due date, respectively. Denote by SŒj  ./ or simply SŒj  the starting time of job JŒj  in . In the environment with psd delivery time, the processing of JŒj  must be followed immediately by its psd delivery time qŒj  . This chapter considers the case of simple linear function for qŒj  such that qŒj  is proportional to the waiting time or starting time of job JŒj  . (Readers may extend the discussion to the case of linear or some other function in application necessities). Consequently, qŒj  is formulated as qŒj  D SŒj  ;

j D 1; : : : ; n;

(1)

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where   0 is a normalizing constant. Observe that in a single-machine environment where the machine is always available for processing and all the n jobs are available at time zero, the starting time SŒj  in a schedule without (redundant) idle time satisfies SŒ1 D 0; jP 1 (2) SŒj  D pŒi  ; j D 2; : : : ; n: i D1

Let CŒj  denote the completion time of job JŒj  in a schedule, that is, the completion time of the processing of JŒj  on the machine plus the job’s psd delivery time. Therefore, CŒ1 D pŒ1 C qŒ1 D pŒ1 ; CŒj  D SŒj  C pŒj  C qŒj  D

jP 1 i D1

(3)

pŒi  C pŒj  C qŒj 

D .1 C  /

jP 1 i D1

pŒi  C pŒj  ;

j D 2; : : : ; n:

pŒi  C pŒj  ;

j D 1; : : : n;

Accordingly, CŒj  D .1 C  /

j 1 X

(4)

i D1

where

0 P i D1

2.2

pŒi  D 0.

Makespan, Maximum Lateness, Maximum Tardiness, and Number of Tardy Jobs

For convenience, let P D

Pn

j D1 pj .

Theorem 1 Problem 1jqpsd jCmax can be solved in O.n/ time. Proof It is clear from (Eq. 4) that CŒj  > CŒj 1 , and therefore Cmax D CŒn D P C 

n1 X

pŒi  D P C .P  pŒn / D .1 C  /P  pŒn :

(5)

i D1

Since P is a constant, Cmax is minimized when pŒn is maximal. Consequently, any sequence with pŒn D maxj D1;:::;n fpj g is optimal for the problem. Since the longest job can be identified in O.n/ time, the problem can be solved in O.n/ time as well. 

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P Below it is shown that 1jqpsd jLmax , 1jqpsd jTmax , and 1jqpsd j Uj problems can be reduced to the corresponding problems without psd delivery times by appropriate transformations for pj and dj . Specifically, in accordance with expression (Eq. 5), Pj LŒj  D CŒj   dŒj  D i D1 .1 C  /pŒi   .dŒj  C pŒj  / and TŒj  D maxfLŒj  ; 0g; therefore, the substitutions of pj0 D .1 C  /pj and dj0 D dj C pj , j D 1; : : : ; n, reduce the above Lj , Tj expressions to the corresponding expressions without P psd delivery times. Consequently, the 1jqpsd jLmax P,n1jqpsd jTmax , and 1jqpsd j Uj problems reduce to 1jjLmax , 1jjTmax , and 1jj j D1 Uj problems, respectively. As a result, 1jqpsd jLmax and 1jqpsd jTmax can be solved in O.n log n/ time P by implementing the earliest due date (EDD) sequence on due dates dj0 , and 1jj Uj can be solved in O.n log n/ time by implementing Algorithm 3.3.1 in [12] with processing times pj0 and due dates dj0 .j D 1; : : : ; n/.

2.3

Total Weighted Completion Time

Consider the single-machine scheduling problem to minimize the total P weighted completion time with psd delivery time. Denote the model as 1jqpsd j wj Cj . It is shown that shortest weighted processing time (SWPT) rule is optimal for the model. Theorem 2 For 1jqpsd j

P

wj Cj , SWPT rule is optimal.

Proof By formula (Eq. 4), n P j D1

wŒj  CŒj  D

n P j D1

wŒj  .1 C  /

D .1 C  /

n P j D1

wŒj 

j P

i D1 j P

i D1

! pŒj   pŒj 

pŒj   

n P

(6) wŒj  pŒj  :

j

On the right-hand side of the above second equation, the first item .1 C  / j n n P P P wŒj  pŒj  is minimized by SWPT rule, and the second item  wŒj  pŒj  is

j D1

i D1

j D1

a constant which has no relation with the processing sequence. This completes the proof.  Corollary 1 For 1jqpsd j

P

Cj , SPT rule is optimal.

For an alternative proof of this corollary, readers can refer to [6].

2.3.1 Total Weighted Completion Time with Precedence Constraints The problem of single-machine scheduling to minimize total weighted completion time with arbitrary precedence constraints is NP-hard, due to the NP-hardiness of 1jprecjCmax (without psd delivery time). So, the focus has now been shifted to several special cases and exploit polynomial time algorithms for the cases.

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Consider the simplest form of precedence constraints (i.e., precedence constraints that take the form of parallel chains). This problem can be solved by a relatively simple and efficient (polynomial time) algorithm. The algorithm is based on some fundamental properties of scheduling with precedence constraints. Consider the case with two job chains. The first chain consists of jobs J1 ; : : : ; Jk and the second chain consists of jobs JkC1 ; : : : ; Jn . The precedence constraints in the two chains are as follows: J1 ! J2 !    ! Jk and JkC1 ! JkC2 !    ! Jn : In the above chains, Ji precedes Ji C1 for i  i < k and k C 1  i < n. The following lemma is based on the nonpreemption assumption such that if the first job of one chain is started for processing on the machine, then the whole chain shall be processed till its end without preemption. The lemma answers the question, with the purpose to minimize the total weighted completion time, which one of the two chains should he process first? Pk

Lemma 1 Lemma 3.1.3 [12] If

j D1 wj

Pk

j D1

pj

Pn

wj

> . Pn

j DkC1 pj

: 

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Consider the preemptive case other than that in Lemma 1 such that the processing of a chain may be preempted and then it is not necessary to go through each chain at a time. In other words, in the preemptive case, a scheduler may process some jobs of one chain (while adhering to the precedence constraints), switch over to another chain, and later on return to the first chain. A basic concept related to a chain of jobs is present. Given a chain J1 ! J2 !    ! Jk , let l  be the smallest index that satisfies ( Pl ) Pl  j D1 wj j D1 wj D max : Pl  Pl l2f1;:::;kg j D1 pj j D1 pj The ratio on the left-hand side of the above equation is called the -factor of the chain and it is denoted by .J1 ; : : : ; Jk /. Job Jl  is referred to as the job that determines the -factor of the chain. For the objective to minimize the total weighted completion time, the following lemma holds in the case of multiple chains. Lemma 2 Lemma 3.1.3 [12] If job Jl  determines .J1 ; : : : ; Jk /, then there exists an optimal sequence that processes jobs J1 ; : : : ; Jl  one after another without any interruption by jobs from other chains. Proof See the proof of Lemma 3.1.3 [12].



The result of this lemma is intuitive. If one scheduler had decided to start processing a string of jobs, it makes sense to continue processing the string on the machine until job Jl  is completed without processing any other job in between. Given the form of chains for precedence constraints, the above two lemmas are the basis for constructing a simple algorithm that minimizes the total weighted completion time. Algorithm 3.1.4 [12]: Whenever the machine is freed, select among the remaining chains the one with the highest -factor. Process this chain without interruption up to and including the job that determines its -factor.

Consider other chains with more general precedence constraints than the parallel chains mentioned above. Some notations on directed tree are introduced, in which each node represents a job and each directed arc represents a precedence constraint between the corresponding two nodes. If there is only one job without successors (or predecessors) in a tree, then it is called an intree (or outtree). In an intree, the single job with no successors is called the root and is located at level 1; all jobs immediately preceding the root are at level 2; all those immediately preceding the jobs at level 2 are located at level 3, and so on. All the jobs without predecessors in the intree are referred to in graph theory terminology as leaves. In an outtree, the definitions are on the contrary way. The single job without predecessors is called the root and is located at level 1; all the jobs immediately following the root are located at level 2, and so on. The jobs without successors in an outtree are called leaves.

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In both intree and outtree, the jobs with no predecessors are generally referred to as starting jobs. There is a single starting job in an outtree while there may be several starting jobs in anP intree. P For 1jintreej wj Cj and 1jouttreej wj Cj , there are decomposition algorithms (see [13]), which can be directly applied to the P corresponding scenarios withPpsd delivery times, that is, 1j; qpsd ; intreej wj Cj and 1jqpsd ; outtreej wj Cj .

2.4

Total Weighted Discounted Completion Time

P Consider the objective of total weighted discounted completion time P wj .1  e rCj /, where r (0 < r < 1) is the discount factor. The problem 1jqpsd j wj .1  e rCj / gives rise to a different priority rule, that is, scheduling jobs in nonincreasing wj erpj order of . This rule is referred to as the Modified weighted discounted 1er.1C /pj shortest processing time (MWDSPT) rule. Theorem 3 MWDSPT rule is optimal for 1jqpsd j

P

wj .1  e rCj /.

Proof By contradiction. Assume otherwise that there exists another optimal schedule S , in which job Jj immediately precedes job Jk , while wj e rpj wk e rpk > : 1  e r.1C /pk 1  e r.1C /pj Let t be the time at which job Jj starts its processing in S . An adjacent pairwise interchange between the two jobs results in a new schedule S 0 . It is observed that for each of the other jobs except Jj and Jk , its starting time and end time in the two sequences are the same. So, the only difference between S and S 0 related to the objective value is due to jobs Jj and Jk . Together with formula (Eq. 3), the total contribution of the two jobs to the objective under S is wj .1  e r.t Cpj Cqj / / C wk .1  e r.t Cpj Cpk Cqk / / D wj .1  e r.t Cpj C t / / C wk .1  e r.t Cpj Cpk C.t Cpj // /:

(7)

The total contribution of jobs Jk and Jj to the objective under S 0 is obtained by interchanging the j s and ks in formula (Eq. 7). By algebraic calculation, it can be shown that the objective value under S 0 is less than that under S , implying a contradiction. This completes the proof. 

2.5

Total Weighted Tardiness

P It is well known that problem 1jj wj Tj is NP-hard in theP strong sense, which is a P special case of 1jqpsd j wj Tj . This implies that 1jqpsd j wj Tj is also NP-hard

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in the strong sense. In this subsection, the main focus is on a polynomially solvable special case. Given a job instance, if any two jobs Jj ; Jk with pj  pk satisfy dj  dk and wj  wk , then it is said the job instance has agreeable due dates and agreeable weights. Theorem 4 If a job instance has agreeable due dates Pand agreeable weights, then Earliest Due Date (EDD) rule is optimal for 1jqpsd j wj Tj . Proof Suppose otherwise in an optimal schedule S , there exist two jobs Jj and Jk with pj  pk such that Jk is scheduled before Jj . By agreeable due dates and weights, dj  dk and wj  wk is obtained. Performing a position interchange on jobs Jk and Jj in S while keeping the processing order of all the other jobs unchanged, a new schedule S 0 is obtained. It suffices to prove that the total weighted tardiness under S 0 is no more than that of S . Let A and P .A/ be the set of jobs between Jj and Jk and its total processing time in S . Let t be the time at which job Jk starts its processing under S . Jk is immediately followed by A and then Jj . Under the new schedule S 0 , however, Jj will start its processing at time t, and it is immediately followed by A and then Jk . Notice that the total weighted tardiness of all the jobs except Jj and Jk under S 0 is less than or equal to that under S due to pj  pk . The rest of the proof is to show that the total weighted tardiness of Jj and Jk under S 0 is no more than that under S . The total weighted tardiness of Jj and Jk under S is wk Tk C wj Tj D wk maxft C pk C  t  dk ; 0g Cwj maxft C pk C P .A/ C pj C .t C pk C P .A//  dj ; 0g: While under S 0 , the total contribution of Jj and Jk is wj Tj0 C wk Tk0 D wj maxft C pj C  t  dj ; 0g Cwk maxft C pj C P .A/ C pk C .t C pj C P .A//  dk ; 0g: Define  D .wj Tj0 C wk Tk0 /  .wk Tk C wj Tj /. Depending on , two cases are discussed. Case 1: t C pk C  t  dk  0. Case 1.1: t C pj C  t  dj  0.  D Œwj .t C pj C  t  dj / Cwk .t C pj C P .A/ C pk C .t C pj C P .A//  dk / Œwk .t C pk C  t  dk / Cwj .t C pk C P .A/ C pj C .t C pk C P .A//  dj /

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D .1 C  /Œwk .pj C P .A//  wj .pk C P .A// 0 where the above inequality is due to wj  wk and pj  pk . Case 1.2: t C pj C  t  dj < 0.  D wk Œt C pj C P .A/ C pk C .t C pj C P .A//  dk  Œwk .t C pk C  t  dk / Cwj .t C pk C P .A/ C pj C .t C pk C P .A//  dj / D wk .1 C  /.pj C P .A// wj .t C pk C P .A/ C pj C .t C pk C P .A//  dj /  wk Œ.1 C  /.pj C P .A// .t C pk C P .A/ C pj C .t C pk C P .A//  dj / D wk Œ.pj  pk /  .t C pk C  t  dj /  wk .t C pk C  t  dj /  wk .t C pk C  t  dk / 0 where the first inequality is due to wj  wk and t C pk C P .A/ C pj C .t C pk C P .A//  dj  t C pk C  t  dk  0 by the condition of Case 1, the second and third inequalities are due to pj  pk and dj  dk respectively, and the last inequality is due to case condition t C pk C  t  dk  0. Case 2: t C pk C  t  dk < 0. Case 2.1: t C pj C P .A/ C pk C .t C pj C P .A//  dk  0. Case 2.1.1: t C pj C  t  dj  0.  D Œwj .t C pj C  t  dj / Cwk .t C pj C P .A/ C pk C .t C pj C P .A//  dk / wj .t C pk C P .A/ C pj C .t C pk C P .A//  dj / D wk .t C pj C P .A/ C pk C .t C pj C P .A//  dk / wj .1 C  /.pk C P .A// < wk .1 C  /.pj C P .A//  wj .1 C  /.pk C P .A// 0

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where the first inequality is due to case condition t C pk C  t  dk < 0 and the second inequality is due to wj  wk and pj  pk . Case 2.1.2:

t C pj C  t  dj < 0.  D wk .t C pj C P .A/ C pk C .t C pj C P .A//  dk / wj .t C pk C P .A/ C pj C .t C pk C P .A//  dj /  wj Œ.pj  dk /  .pk  dj / D wj Œ.pj  pk / C .dj  dk /  0;

where the first inequality is due to wk  wj and the second inequality is due to pj  pk and dj  dk . Case 2.2: t C pj C P .A/ C pk C .t C pj C P .A//  dk < 0. Case 2.2.1: t C pj C  t  dj  0.  D wj .t C pj C  t  dj ; 0/ wj Œt C pk C P .A/ C pj C .t C pk C P .A//  dj  D wj .1 C  /.pk C P .A// 0 Case 2.2.2:

t C pj C  t  dj < 0.

 D 0  wj maxft C pk C P .A/ C pj C .t C pk C P .A//  dj ; 0g  0: This completes the proof.

2.6



Total Completion Time with Release Dates

P It is already P known that 1jrj j Cj is NP-hard (see [9]), and therefore 1jrj ; qpsd j Cj is NP-hard. The focus is to exploit some polynomially solvable cases of the latter problem. Remember that in Sect. 2.3, Corollary 1 shows that SPT rule Pproduces an optimal schedule for the model without release dates, that is, 1jqpsd j Cj . This subsection deals P with the model with release dates and preemption, that is, 1jqpsd ; prmp; rj j Cj . In the preemptive case, one job on processing may be interrupted at some time and resumed from where it was preempted at a later time. It is proved that a modified shortest remaining processing time (MSRPT) rule is optimal for the model. Notice that in the preemption model, on the case where one job Jj has started processing for more than one time, Sj is defined as the first starting time of the job. The value of psd delivery time depends on Jj ’s first starting time, that is, qj D Sj . The following lemma shows one property in an optimal schedule.

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Lemma 3 In an optimal schedule for 1jqpsd ; prmp; rj j processing at its release date.

159

P

Cj , each job starts

Proof By contradiction. Suppose otherwise in an optimal schedule , job Jj starts processing at time Sj with Sj > rj . Construct a new preemptive schedule  0 by inserting processing Jj at time rj with a time length of zero and keep the processing schedule of all jobs in  unchanged. Since the value of Sj decreases from Sj in  to rj in  0 , together with formula (Eq. 4), the psd delivery time and the completion time of Jj in  0 are strictly less than that in . The completion times of all other jobs are the same in  and  0 . Hence, the total completion time in  0 is less than that in , contradicting that  is an optimal schedule.  Algorithm MSRPT: (At any time Principle 1 is superior to Principle 2) Principle 1: At the release date of each job, process it with a time length of zero. Principle 2: If the machine is idle or a job arrives at time t , schedule a job by SRPT (shortest remaining processing time) rule. Ties are broken arbitrarily. If there is no available job at the time, wait until a new job arrives. Note that the remaining processing time includes psd delivery time.

Theorem 5 Algorithm MSRPT is optimal for 1jqpsd ; prmp; rj j

P

Cj .

Proof First, give a sketch of the proof. By Lemma 3, the psd delivery time of each Jj satisfies qj D Sj D  rj , and thus it has no relation P with the schedule of the processing of Jj . The problem reduces to 1jprmp; rj j Cj . By the argument of position interchange on the processing of any two consecutive jobs, it can be proved that MSRPT produces an optimal schedule. The theorem follows. 

2.7

Approximation for Total Completion Time with Release Dates

For the model with release dates, if preemption is allowed, the previous subsection shows that there exists an optimal algorithm. However, Pif preemption is not allowed, the corresponding problem, denoted by 1jq ; r j Cj , is NP-hard due to the psd j P NP-hardness of 1jrj j Cj . Due to the NP-hardness result, the approximation P algorithm is developed for problem 1jqpsd ; rj j Cj in this subsection. The main idea of the proposed approximation algorithm below is to transform a preemptive schedule into a corresponding non-preemptive schedule. The total relaxation of jobs’ completion times during the transformation is bounded by a constant factor. P Given an instance of problem 1jqpsd ; prmp; rj j Cj and a preemptive schedule P which is produced by MSRPT algorithm, let Cj .P / be the completion time of job Jj in P . The following algorithm, named CONVERT, reveals how to convert schedule P into a feasible nonpreemptive schedule N .

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Algorithm CONVERT: Input: Preemptive schedule P Output: Nonpreemptive schedule N 1. Form a list L of jobs J1 ; J2 ; : : : ; Jn in the increasing order of their completion times Cj .P /, j D 1; 2; : : : ; n. 2. Produce a nonpreemptive and feasible schedule of the n jobs in the same completion order as in L, with the constraint that each job starts processing not earlier than its release date.

P Theorem 6 Given an optimal preemptive schedule P for 1jqpsd ; prmp; rj j Cj . Algorithm CONVERT produces in O.n/ time a nonpreemptive schedule N in which Cj .N /  2.1 C  /Cj .P / for each job Jj . Proof By the description of Algorithm CONVERT, it works as follows. Consider an arbitrary job Jj in P . Let Sjl .P / be the last starting time of Jj and kj the length of time for its last processing. That is, the last processing of Jj is within time interval ŒSjl .P /; Sjl .P / C kj  in schedule P . Remove all the previous processing time segments of Ji in a total length of pj kj before time Sjl .P /, and insert pj kj extra units of time at time Sjl .P /Ckj for processing Jj continuously within interval ŒSjl .P /; Sjl .P / C pj . Note that this time interval might overlap with the processing time intervals of either preceding or succeeding jobs. Hence, in the nonpreemptive schedule N , the starting time of job Jj or Sj .N / may be postponed to another time later than Sjl .P / with the assurance that there is no unforced idle time between the processing of any two jobs. It is observed that in the worst case of N , all the jobs completed before Jj are started on or after time Sjl .P /. Together with the fact that the jobs completed before Jj in P are the same as that in N , the following is obtained: Sj .N /  Sjl .P / C

P kWCk .P / 0, A is a .1 C /-approximation algorithm running in polynomial time in both input size and 1=. Without loss of generality, it is assumed that 0 <   1. For the special case with m D 1, that is, 1jqpsd jCmax , it is already known that SPT rule produces an optimal schedule. It immediately leads to the following lemma. Lemma 4 In an optimal solution of P mjqpsd jCmax , all the jobs on each machine Mi (i 2 f1; : : : ; mg) are sequenced by SPT rule. According to the above lemma, index the n jobs according to SPT rule. Similar to the establishment of FPTAS in [7, 8], introduce variables xj (j D 1; : : : ; n) such that xj D i if job Jj is assigned to machine Mi (i 2 f1; : : : ; mg). Let vector x D .x1 ; x2 ; : : : ; xn / be one assignment of all the n jobs and X D fxg be the vector set containing all the solutions to the problem considered. Note that the only concern is nondelay schedule here since an optimal solution must be such a schedule. For each vector x, adopt function fji .x/ and gji .x/, respectively, to denote the processing workload (excluding psd delivery time) and the makespan (including psd delivery time) of machine Mi immediately after scheduling job Jj . Let hj .x/ D maxi D1;:::;m fgji .x/g. So hn .x/ is our desired value for any given x. Define the following initial and recursive functions on X : Initial function: f0i .x/ D 0;

i D 1; : : : ; mI

g0i .x/

i D 1; : : : ; mI

D 0;

h0 .x/ D 0: Recursive function for j D 1; ::; n:

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fji .x/ D fji1 .x/ C pj ;

i D xj I

fji .x/

otherwise.

D

fji1 .x/;

gji .x/ D .1 C  /fji1 .x/ C pj ;

i D xj I

gji .x/ D gji 1 .x/;

otherwise.

hj .x/ D maxi D1;:::;m fgji .x/g: Therefore, problem P mjqpsd jCmax reduces to minimize hn .x/;

x 2 X:

Since the proposed algorithm A contains procedure Partition.A; e; ı/ which was proposed in [7, 8], restate the procedure for comprehension completion. In Partition.A; e; ı/, parameter A  X , e is a nonnegative integer function on X , and 0 < ı  1. The procedure partitions A into k.e/  1 disjoint subsets which satisfy some specific properties. The value of k.e/ is to be determined in the procedure. Procedure Partition.A; e; ı/: Step 1. Arrange vectors x 2 A in the order x .1/ ; x .2/ : : : ; x .jAj/ such that 0  e.x .1/ /  e.x .2/ /  : : :  e.x .jAj/ /. Step 2.1. Assign vectors x .1/ ; x .2/ : : : ; x .i1 / to set Ae1 such that e.x .i1 / /  .1 C ı/e.x .1/ / and e.x .i1 C1/ / > .1 C ı/e.x .1/ /. If such i1 does not exist, take Aek.e/ D Ae1 D A and stop. Step 2.2. Assign vectors x .i1 C1/ ; x .i1 C2/ : : : ; x .i2 / to set Ae2 such that e.x .i2 / /  .1 C ı/ e.x .i1 C1/ / and e.x .i2 C1/ / > .1 C ı/e.x .i1 C1/ /. If such i2 does not exist, take Aek.e/ D Ae2 D A  Ae1 and stop. Step 2.3. Recursively, construct vector set Aei for i  3 as in Step 2.2 until x .jAj/ is included in Aek.e/ for some integer k.e/.

Procedure Partition.A; e; ı/ runs in O.jAj log jAj/ time since Step 1 consumes O.jAj log jAj/ time while the rest of the steps for vector partition takes only O.jAj/ time. Further, we use two properties of Partition.A; e; ı/ proposed in [7, 8] in the development of our FPTAS and the establishment of its time complexity. Property 1 je.x/  e.x 0 /j (j D 1; : : : ; k.e/). Property 2 k 

log e.x .jAj/ / ı



ı minfe.x/; e.x 0 /g for any fx; x 0 g

2

Aej

C 2 for 0 < ı  1 and 1  e.x .jAj/ /.

It is now ready to present the FPTAS A for P mjqpsd jCmax below. Algorithm A : Step 1. (Initialization) Reindex the jobs in the nondecreasing order of pj . Set Y0 D f.0; 0; : : : ; 0/g (which means no job is assigned initially) and j D 1. „ ƒ‚ … n

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Step 2.1. (Generate Y1 ; : : : ; Yn ) For set Yj 1 , construct Yj0 by adding xj 2 f1; : : : ; mg in position j of each vector from Yj 1 . Calculate the following recursive function for each x 2 Yj0 : fji .x/ D fji1 .x/ C pj ; i D xj I fji .x/ D fji1 .x/; gji .x/

D .1 C

gji .x/

D

/fji1 .x/ C

otherwise. pj ;

gji 1 .x/;

i D xj I otherwise.

hj .x/ D maxiD1;:::;m fgji .x/g: Step 2.2. If j D n, take Yn D Yn0 and go to Step 3.  and perform the following operations: Step 2.3. Otherwise, if j < n, set ı D 2n 0 i Step 2.3.1. Call Partition.Yj ; fj ; ı/ (i D 1; : : : ; m) to divide set Yj0 into k.f i / disjoint fi

fi

fi

subsets Y1 ; Y2 ; : : : ; Yk.f i / . Step 2.3.2. Call Partition.Yj0 ; hj ; ı/ to divide set Yj0 into k.h/ disjoint subsets h Y1h ; Y2h ; : : : ; Yk.h/ . T fm T h f1 T Step 2.3.3. Divide set Yj0 into disjoint subsets Ya1 am b D Ya1    Yam Yb for a1 D 1; 2; : : : ; k.f 1 /;    ; am D 1; 2; : : : ; k.f m /; b D 1; 2; : : : ; k.h/. Step 2.3.4. For each nonempty subset Ya1 am b , choose a vector x .a1 am b/ such that hj .x .a1 am b/ / D minfhj .x/jx 2 Ya1 am b g. Step 2.3.5. Set Yj WD fx .a1 am b/ ja1 D 1; 2; : : : ; k.f 1 /;    ; am D 1; 2; : : : ; k.f m /; T fm T h f1 T b D 1; 2; : : : ; k.h/, and Ya1    Yam Yb ¤ ;g; j D j C 1, and go to Step 2.1. 0 Step 3. (Solution) Select vector x 2 Yn such that hn .x 0 / D minfhn .x/jx 2 Yn g.

The purpose of Step 2.3 is to reduce the solution space in the current iteration which will be used to generate the solution space in the next iteration. In each iteration, such reduction guarantees that the function values of omitted solutions are within .1 C ı/ times of the values of remained solutions. Theorem 9 For P mjqpsd jCmax , algorithm A finds a solution x 0 such that hn .x 0 /  .1 C /hn .x  /. Proof Let vector x  D .x1 ; : : : ; xn / be an optimal solution for P mjqpsd jCmax . Suppose there is a partial optimal solution .x1 ; : : : ; xj ; 0; : : : ; 0/ 2 Ya1 am b  Yj0 for some j and a1 ; : : : ; am ; b. By Step 2 of algorithm A , such j always exists, for instance, j D 1. By Step 2.3.4 of algorithm A , no matter whether or not .x1 ; : : : ; xj ; 0; : : : ; 0/ is chosen as x .a1 am b/ , it is obtained by Property 1 that jfji .x1 ; : : : ; xj ; 0; : : : ; 0/  fji .x .a1 am b/ /j  ıfji .x1 ; : : : ; xj ; 0; : : : ; 0/; i D 1; : : : ; m; and jhj .x1 ; : : : ; xj ; 0; : : : ; 0/  hj .x .a1 am b/ /j  ıhj .x1 ; : : : ; xj ; 0; : : : ; 0/;

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which means hj .x .a1 am b/ /  .1 C ı/hj .x1 ; : : : ; xj ; 0; : : : ; 0/:

(8)

Now establish the relation between solutions obtained by A and a partial optimal solution in the next iteration. Consider two vectors which assign j C 1 jobs, namely, .x1 ; : : : ; xj ; xjC1 ; 0; .a a b/

.a a b/

: : : ; 0) and xO .a1 am b/ D .x1 1 m ; : : : ; xj 1 m ; xjC1 ; 0; : : : ; 0/. (xO is chosen as a bridge connecting a partial optimal solution and the solutions produced by A in the next iteration). Without loss of generality, assume xjC1 D u. It follows that jgju C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/  gju C1 .xO .a1 am b/ /j D j.1 C  /fju .x1 ; : : : ; xj ; 0; : : : ; 0/ C pj  .1 C  /fju .x .a1 am b/ /  pj j D .1 C  /jfju .x1 ; : : : ; xj ; 0; : : : ; 0/  fju .x .a1 am b/ /j  .1 C  /ıfju .x1 ; : : : ; xj ; 0; : : : ; 0/  ıgju C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/: Therefore, gju C1 .xO .a1 am b/ /  .1 C ı/gju C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/: According to the above analysis, the following is obtained: gji C1 .xO .a1 am b/ /  .1 C ı/gji C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/:;

i D 1; : : : ; m:

Combining inequality (Eq. 9) and the definition of hj .x/, jhj C1 .xO .a1 am b/ /  hj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/j D j maxi D1;:::;m fgji C1 .xO .a1 am b/ /g  maxi D1;:::;m fgji C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/gj  ıgji C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/  ı maxi D1;:::;m fgji C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/g D ıhj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/:

(9)

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Consequently, hj C1 .xO .a1 am b/ /  .1 C ı/hj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/:

(10)

Set ıj D ı in this, namely, j th, iteration. Then the following inequalities follows directly by inequalities (Eq. 8)–(Eq. 10): hj .x .a1 am b/ /  .1 C ıj /hj .x1 ; : : : ; xj ; 0; : : : ; 0/: gji C1 .xO .a1 am b/ /  .1 C ıj /gji C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/;

(11) (12)

i D 1; : : : ; m:

hj C1 .xO .a1 am b/ /  .1 C ıj /hj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/:

(13)

Without loss of generality, assume that xO .a1 am b/ 2 Yc1 cm d  Yj0 C1 and algorithm A chooses x .c1 cm d / instead of xO .a1 am b/ in its .j C 1/th iteration. By Property 1 and inequalities (Eq. 12) and (Eq. 13), the following are is obtained: jgji C1 .xO .a1 am b0 / /  gji C1 .x .c1 cm d / /j  ıgji C1 .xO .a1 am b/ /  ı.1 C ıj /gji C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/; i D 1; : : : ; m and

jhj C1 .xO .a1 am b/ /  hj C1 .x .c1 cm d / /j  ıhj C1 .xO .a1 am b/ /

(14)

 ı.1 C ıj /hj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/: Together with inequalities (Eq. 13) and (Eq. 14), jhj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/  hj C1 .x .c1 cm d / /j  jhj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/  hj C1 .xO .a1 am b/ /j Cjhj C1 .xO .a1 am b/ /  hj C1 .x .c1 cm d / /j  .ıj C ı.1 C ıj //hj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/ D .ı C .1 C ı/ıj /hj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/: Set ıj C1 D ı C .1 C ı/ıj in this .j C 1/th iteration. It follows that hj C1 .x .c1 cm d / /  .1 C ıj C1 /hj C1 .x1 ; : : : ; xj ; xjC1 ; 0; : : : ; 0/: Repeat the above analysis for j C 2; : : : ; n. There exists an x 0 2 Yn such that

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hn .x 0 /  .1 C ın /hn .x  /;

(15)

where ın is by the following induction: ın D ı C .1 C ı/ın1 D ı C .1 C ı/.ı C .1 C ı/ın2 / D  Dı

nj P1

.1 C ı/i C .1 C ı/nj ıj :

i D0

Remember that it has been set ıj D ı. By the above formula, ın achieves its maximal value when j D 1. That is, ın  ı Dı

n2 P i D0

.1 C ı/i C .1 C ı/n1 ı

n1 P i D0

.1 C ı/i

D .1 C ı/n  1 D

n P i D1

n.n1/.ni C1/ i ı: iŠ

 in the above formula on the right-hand side of By Step 2.3, substituting ı D 2n the third equation, the following is obtained:

ın  

n P i D1 n P i D1



n.n1/.ni C1/ iŠ 1 iŠ

Pn



  i 2n

  i 2

i D1

 1 i 2

 : Together with inequality (Eq. 15), hn .x 0 /  .1 C /hn .x  /: By Step 3 of the algorithm, vector x 0 is chosen such that hn .x 0 /  hn .x 0 /  .1 C /hn .x  /: 

This completes the proof. 2

log L log n/ time. Theorem 10 Algorithm A requires O. m.mC1/n 

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Proof It suffices to consider the most time-consuming operation of iteration j in Step 2, that is, Partition. The time complexity of procedure Partition.Yj0 ; fji ; ı/ as well as Partition.Yj0 ; hj ; ı/ is O.Yj0 log Yj0 /. In Step 2, by the construction of Yj0 C1, P jYj0 C1 j  mjYj j  m  k.f 1 /    k.f m /  k.h/ is obtained. Define L D niD1 pi . By Property 2, for each i 2 f1; : : : ; mg, k.f i / 

2n log L C 2: 

k.h/ 

2n log L C 2: 

Similarly,

mn 0 0 Therefore, jYj0 j  2mn  log L C 2 D O.  log L/ and then jYj j log jYj j D mn O.  log L log n/. There are at most n iterations, in each of which procedure Partition is called m C 1 times. So the time complexity of algorithm A is 2 log L log n/. The theorem follows.  O. m.mC1/n 

3.2

Total Completion Time

P For problem P mjqpsd j Cj , earliest completion time (ECT) rule is equal to SPT rule. Below, a fundamental property for an ECT schedule is presented. P Property 3 Among all feasible schedules for P mjqpsd j Cj , an ECT schedule has the most number of completed jobs at any time point during processing. The above property directly deduces the following theorem. Theorem 11 ECT (or SPT) rule is optimal for P mjqpsd j

4

P

Cj .

Conclusion

This chapter has provided an overview of scheduling with past-sequence-dependent delivery times. It has described the major results on single and identical parallel machine settings. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 71101106, 71172189, and 71090404/71090400.

Cross-References Complexity Issues on PTAS Greedy Approximation Algorithms

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Recommended Reading 1. A. Borodin, R. El-yaniv, Online Computation and Competitive Analysis (Cambridge University Press, Cambridge, 1998) 2. S. Browne, U. Yechiali, Scheduling deteriorating jobs on a single processor. Oper. Res. 38, 495–498 (1990) 3. P. Brucker, Scheduling Algorithms, 5th edn. (Springer, Berlin, 2006) 4. R.M. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic machine scheduling: a survey. Ann. Discret. Math. 5, 287–326 (1979) 5. C. Koulamas, G.J. Kyparisis, Single-machine scheduling problems with past-sequencedependent setup times. Eur. J. Oper. Res. 187, 1045–1049 (2008) 6. C. Koulamas, G.J. Kyparisis, Single-machine scheduling problems with past-sequencedependent delivery times. Int. J. Prod. Econ. 126, 264–266 (2010) 7. M.Y. Kovalyov, W. Kubiak, A fully polynomial approximation scheme for minimizing makespan of deteriorating jobs. J. Heuristics 3, 287–297 (1998) 8. M.Y. Kovalyov, W. Kubiak, A fully polynomial approximation scheme for the weighted earliness-tardiness problem. Oper. Res. 47, 757–761 (1999) 9. J.K. Lenstra, A.H.G. Rinnooy Kan, P. Brucker, Complexity of machine scheduling problems. Ann. Discret. Math. 1, 343–362 (1977) 10. M. Liu, F. Zheng, C. Chu, Y. Xu, New results on single-machine scheduling; past-sequencedependent delivery time. Theor. Comput. Sci. 438, 55–61 (2012) 11. M. Liu, F. Zheng, C. Chu, Y. Xu, Single-machine scheduling with past-sequence-dependent delivery times and release times. Inf. Process. Lett. 112, 835–838 (2012) 12. M. Pinedo, Scheduling: Theory, Algorithms, and Systems, 2nd edn. (Prentice Hall, New York, 2005) 13. F.B. Sidney, Decomposition algorithms for single-machine sequencing with precedence relations and deferral costs. Oper. Res. 23, 283–298 (1975)

Algebrization and Randomization Methods Liang Ding and Bin Fu

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Notations and Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Rudimentary Knowledge of Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Schwartz-Zippel Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 k-Path Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Brief History of the k-Path Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 O  .2k / Time Randomized Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hamiltonian Path Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A Brief History of TSP and Hamiltonian Path Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Overall Idea of Bj¨orklund’s Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Notations and Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Cycle Cover Cancelation in Characteristic Two. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Determinants and Inclusion-Exclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Hamiltonicity in Undirected Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Minimum Common String Partition Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A Brief History of MCSP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 An O.2n nO.1/ / Time Algorithm for General Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Algebraic FPT Algorithms for Channel Scheduling Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A Brief History of LTDR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Algebraic FPT Algorithms for LTDR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Complexity of Testing Monomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Notations and Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Q 3SAT and Related NP-Complete Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PQ 7.3 Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Complexity for Coefficient of Monomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Hardness of Approximation of Integration and Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Inapproximation of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Inapproximation of Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172 173 173 174 176 177 178 178 185 185 186 187 188 190 195 201 202 203 204 205 206 208 208 210 211 212 213 213 215

B. Fu () • L. Ding Department of Computer Science, The University of Texas-Pan American, Edinburg, TX, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected]

171 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 80, © Springer Science+Business Media New York 2013

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9 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Abstract

Over the past decades, people have dedicated great efforts to solve some NPcomplete problems. Hamiltonian path problem is one of classic problems which baffle people many years. In recent years, some algebrization and randomization techniques have been developed and applied in the design of algorithms, which provides a credible alternative to traditional methods. Then the testing monomials theories were developed based on above techniques, which have been found applications in solving some critical problems in graph theory, networking, bioinformatics, etc. We give a self-contained description of the examples of successful applications by combining algebrization techniques and randomization techniques.

1

Introduction

This chapter presents examples to show some important algebrization and randomization techniques that have been effectively applied in the design of algorithms for some classic or burgeoning problems. These techniques were initialized to solve the classic path and packing problem. Then they are found applications in wide range areas such as graph theory, networking, bioinformatics, etc. Through exploiting the techniques of algebrization and randomization and combining them, some classic intractable problems such as k-path problem and Hamiltonian path problem, which baffle people for a long time, have got breakthroughs. Our aim is to make readers recognize that the new developed algebrization and randomization methods are very powerful tools for designing algorithms. The chapter is organized as follows. Section 2 provides the basic notations and definitions adopted in the rest of this chapter. Also the proof of some well-known results are showed to make the chapter self-contained. The senior readers may skip this section. In the following sections, several examples which are based on the algebrization and randomization techniques are presented. The organization for each section is similar: the beginning paragraphs give a brief glance of the history of the problem, and then the details of the algorithms and the proof the correctness are showed next. Section 3 presents an O  .2k / time algorithm for the k-path problem. The k-path problem, which is also known as the longest path problem, has been studied for decades. With the development of bioinformatics, the k-path problem has found many applications. Combining algebrization and randomization, a big step improvement is achieved to solve the k-path problem. In Sect. 4, the similar technique is used to solve the famous Hamiltonian path problem. The Hamiltonian path problem is an important graph problem that has

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been studied extensively. It is presented as an example for many standard textbooks about complexity and algorithm. For a general graph with n vertices, the fastest algorithm to solve the Hamiltonian path problem runs in O  .2n / time. And this runtime went on about half a century since 1950s. This section shows an O  .1:657n/ time algorithm to solve the Hamiltonian path problem for the general direct graphs. The theory talked about in Sect. 7, which is named testing monomial theory, comes from the methods used in previous two sections. Because of the important applications, it is necessary to investigate the testing monomial theory and understand how the theory relates to critical problems in complexity. This section works on the question which ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its sum-product expansion. It discusses the hardness of testing monomial for some special polynomials and gives algorithms to solve it. Sections 5 and 6 are devoted to discussions on the applications of the testing monomial theory. Two problems, which both are newly burgeoning problems, are presented separately in two sections. Section 5 shows an O.2n nO.1/ / time algorithm for minimum common string partition problem (MCSP). Since the applications in genome sequencing of computational biology, the MCSP problem has drawn a lot of attentions. In Sect. 6, algebraic fixed parameter tractable algorithms are presented for Channel Scheduling Problem. The Channel Scheduling Problem focuses on developing efficient scheduling algorithms form the client point of view with the objective of minimizing the access time and number of channel switches for the wireless data broadcast. By solving these two problems, the readers can see the practical applicability of the testing monomial theory. Also it reflects the power of the algebrization and randomization. As most chapters of this book talk about the approximation algorithms for the combinatorial optimization problems, the reader can see the discussion about the parameterized complexity of combinatorial optimization problems in Chapter Connections Between Continuous and Discrete Extremum Problems, Generalized Systems, and Variational Inequalities. This chapter contains some recent technologies that combine algebrization and randomization to develop parameterized algorithms and exact algorithms for some NP-hard problems. These methods also bring the new development of complexity theory. We give a self-contained description for all the results in the chapter.

2

Preliminaries

This section reviews some basic notations and theorems which will be adopted throughout the chapter.

2.1

Basic Notations and Definitions

In mathematics, polynomial is one of the most basic algebra concepts. It can be described as an expression constituted by a sum of powers in one or more

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variables multiplied by coefficients. It sounds simple, but it hides many profound and complicated computation problems. The concept of polynomial runs through all the chapter. Let P 2 fZ; Z2 ;    ; ZN g, N > 2. For variables x1 ;    ; xn , let PŒx1 ;    ; xn  denote the communicative ring of all the n-variate polynomials with j j coefficients from P. For 1  i1      ik  n,  D xi11    xikk is called a Pk monomial. The degree of , denoted by deg./, is sD1 js .  is multilinear, if j1 D    D jk D 1, i.e.,  is linear in all its variables xi1 ;    ; xik . For any given integer c  1,  is called a c-monomial, if 1  j1 ;    ; jk < c. An arithmetic circuit, or circuit for short, is a direct acyclic graph with C gates of unbounded fan-ins,  gates of two fan-ins, and all terminals corresponding to variables. The size, denoted by s.n/, of a circuit with n variables is the number of gates in it. A circuit is called a formula, if the fan-out of every gate is at most one, i.e., the underlying direct acyclic graph is a tree. A polynomial p.x1 ;    ; xn / can be represented by a circuit easily. By definition, any polynomial p.x1 ;    ; xn / can be expressed as a sum of a list of monomials, called the sum-product expansion. The degree of the polynomial is the largest degree of its monomials in the expansion.

2.2

Rudimentary Knowledge of Algebra

Senior readers who are familiar with algebra may skip this section. Most of notations which are used in the following sections coincide with standard notations. However, to make the chapter self-contained, it is necessary to introduce some of them which are infrequently seen in combinatorial optimization. Rank and characteristic are very important concepts which appear throughout this chapter. The definitions are given as follows. Definition 1 Let G be a group and a be an element of G, e is the identity of G, and the rank of the element a is defined as the smallest positive integer n which makes an D e. If n does not exist, a has infinite rank. The rank of a is usually represented as jaj. Definition 2 The characteristic of a ring R, often denoted C har.R/, is defined to be the smallest number of times to get the additive identity element 0 by summarizing the multiplicative identity element 1 of R; the ring R is said to have characteristic zero if this repeated sum never reaches the additive identity. Theorem 1 ([25]) If R is a ring with identity and without zero divisors, then all of the elements except the additive identity element 0 have the same characteristic which is either 0 or a prime p. In abstract algebra, a field F is a commutative ring whose nonzero elements form a group under multiplication. Thus, each nonzero element a of F has inverse element a1 . It is easy to see that there is no zero divisor in F . Therefore, the above

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theorem holds for a field F . Similar to a ring, a field also has its characteristic. Because of Theorem 1, the characteristic of a finite field F must be a prime number. The main methods based on algebrization and randomization are established over some special algebraic structures. Two special algebraic space GF .p n / and Zk2 are frequently used here. GF .p n / represents the Galois fields (or finite field) with p n elements. The Galois fields are classified by size. There is exactly one finite field up to isomorphism of size p k for each prime p and positive integer k. Specifically, the Galois fields or the finite field has the following properties [44]: first, the Galois fields can be classified by the number of elements, so-called the order, which has the form p n . Here, p is the characteristic of the field which is a prime number, and n is a positive integer. Second, for every prime number p and positive integer n, there exists a finite field with p n elements. Third, any two finite fields with the same number of elements are isomorphic. That is, under some renaming of the elements of one of these, both its addition and multiplication tables become identical to the corresponding tables of the other one. For more proofs of the properties, readers can refer to Jacobson’s book [44]. For example, the elements of GF .22 /, which has characteristic two, can be represented as the four polynomials 0; 1; x; 1 C x over GF .2/, where computations are done modulo x 2 C x C 1, such as x 3 D x  x 2 D x  .1 C x/ D x C x 2 D 1 and x 2 D x  x D .x 2 C 1/  x D x 3 C x D 1 C x in GF .22 /. Given the operation with componentwise addition modulo 2 over binary k-vectors, Zk2 is the group of all these binary k-vectors. If W0 is used to denote the all-zeros vector of Zk2 , then for each v 2 Zk2 , v2 D W0 is achieved. In linear algebra, determinant and permanent are two fundamental concepts over square matrix which has many applications. Given an n-by-n matrix A D .ai;j /, the determinant of A is defined as det.A/ D

X  2Sn

sgn./

n Y

ai; .i /

i D1

Here, the sum is computed over all permutations  of the f1; 2;    ; ng. For each permutation , sgn./ denotes the signature of . It is positive 1 for even  and negative 1 for odd . Evenness or oddness of  can be defined as follows: the permutation is even(odd) if the new sequence can be obtained by an even(odd) number of switches of numbers. The permanent is a function of the square matrix similar to the determinant. The permanent of an n-by-n matrix A is perm.A/ D

n XY

ai; .i /

 2Sn i D1

Over a ring of characteristic two, the well-known fact for a square matrix A is that the determinant is identical to the permanent. det.A/ D perm.A/

(1)

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Schwartz-Zippel Lemma

The Schwartz-Zippel lemma is a well-known tool that is commonly used in probabilistic polynomial identity testing. It was developed independently by Schwartz and Zippel in around 1980 [67, 77]. And it is the theoretical basis of the techniques which will be presented in the following sections. Lemma 1 ([65]) Let Q.x1 ;    ; xn / 2 FŒx1 ;    ; xn  be a multivariate polynomial of total degree d . Fix any finite set S 2 F, and let r1 ;    ; rn be chosen independently and uniformly at random from S. Then PrŒQ.r1 ;    ; rn / D 0jQ.x1 ;    ; xn / ¤ 0 

d jSj

(2)

Proof The idea derives from single variable case: a single variable polynomial of degree d has no more than d roots. It seems intuitively that a similar statement would hold for multivariate polynomials. The proof is by induction on the number of variables n. As was mentioned before, for base case n D 1, the probability that d Q.r1 / D 0 is at most jSj . Assume the lemma holds for a multivariate polynomial with up to n  1 variables. Consider the polynomial Q.x1 ;    ; xn / by factoring out the variable x1 : Q.x1 ;    ; xn / D

k X

x1i Qi .x2 ;    ; xn /

i D0

where k  d is the largest exponent of x1 in Q. Since P is not identical to 0, the coefficient of x1k Qi .x2 ;    ; xn / is not equal to zero by our choice. It is easy to see that the total degree of Qk .x2 ;    ; xn / is at most d  k, by induction hypothesis: PrŒQk .r2 ;    ; rn / D 0 

d k jSj

Then consider the case that Qk .r2 ;    ; rn / ¤ 0, Q.x1 ;    ; xn / becomes a univariate polynomial: q.x1 / D Q.x1 ; r2 ;    ; rn / D

k X

x1i Qi .r2 ;    ; rn /

i D0

The polynomial q.x1 / has degree k, so the base case implies that PrŒQ.r1 ;    ; rn / D 0jQk .r2 ;    ; rn / ¤ 0 

k jSj

Using events A and B to denote Q.r1 ;    ; rn / D 0 and Qk .r2 ;    ; rn / D 0, respectively, by probability theory,

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PrŒA D PrŒA \ B C PrŒA \ B c  D PrŒBPrŒAjB C PrŒB c PrŒAjB c   PrŒB C PrŒAjB c  

k d k C jSj jSj

D

d jSj

So the induction completes.

3



k-Path Problem

In graph theory, given a graph G D .V; E/, a simple path of G is a path that does not have any repeated vertices. The longest path problem is the problem of finding a simple path of maximum length in the given graph G. The decision version of the longest path problem is called the k-path problem. The graph G and an integer k are given as input; the aim of the k-path problem is to determine if graph G contains a simple path of length at least k. Unlike the shortest path problem, even with negative edge weights, as long as the graph G does not have negative-weight cycles, the shortest paths between two vertices can be found in polynomial time. Finding the longest simple path between two vertices is NP-complete even if all edge weights are equal to 1. The NPcompleteness of the decision problem can be easily proved by a reduction from the Hamiltonian path problem. Theorem 2 The k-path problem is NP-complete. Proof For a given graph G D .V; E/ and an integer k, a certificate P can be easily verified in polynomial time, so the k-path problem is in class NP. It is known that Hamiltonian path is NP-complete [23]. The next is to prove Hamiltonian path p k-path, which shows that the k-path problem is NP-hard. Given a instance of Hamiltonian path G1 D .V1 ; E1 /, it can be transformed to a k-path instance by simple asking whether there exists a simple path of length jV1 j in G1 . Clearly, if G1 has a Hamiltonian path, since it traverses all possible vertices, this Hamiltonian path is the simple path of G1 with length jV1 j. Conversely, if there is simple path of length jV1 j in the graph G1 , it is a path that visits all the vertices of G1 without repetition, and this is a Hamiltonian path by definition.  The longest path problem can be transformed into the shortest path problem by utilizing the dual nature of optimizations. Given a longest path problem with input graph G, G can be converted into the shortest simple path problem on graph H which is same as the original graph G but with edge weights negated. It is

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known that the shortest path problem can be solved in polynomial time for graph without negative-weight cycle. This approach cannot be used to solve H because any positive-weight cycle in G produces a negative-weight cycle in H . Thus, the shortest path problem on a graph with negative-weight cycle is also NP-complete. However, the longest path problem is polynomial-time solvable on acyclic graphs. Since the acyclic graph has no cycles, these is no negative-weight cycle on H . Thus, any shortest path algorithm can be used to solve the shortest problem on H .

3.1

A Brief History of the k-Path Problem

For a graph G D .V; E/ with the number of nodes jV j D n and an integer k, the naive algorithm of the k-path problem enumerates all sequences of length k C 1 with time bound O.nk /. When k D n, it has been known very early that the problem can be solved in O  .2k / time using dynamic programming method [8, 38, 51]. In 1985, B. Monien first improved the time bound to O  .kŠ/. Thus, the problem can be solved in polynomial time when k  log n= log log n. After about 10 years, Papadimitriou and Yannakakis [66] conjectured that the .log n/-path problem can be solved in polynomial time. Almost in the meantime, Alon, Yuster, and Zwick [2] verified the conjecture. They gave a randomized algorithm running in O  ..2e/k /  O  .5:44k / time and a deterministic O  .c k / time algorithm, where c is a large constant. With the development of bioinformatics, the k-path problem has found applications both in biological subnetwork matchings [53] and in detecting signaling pathways in protein interaction networks [68]. Probably driven by the demand of applications, faster algorithms appeared after 2006. J. Kneis et al. [56] provided a O  .4k / randomized algorithms and O  .c k / deterministic algorithm with c D 16. The deterministic complexity was further reduced by Chen et al. [19], who presented more efficient color-coding schemes and gave an O  .12:8k / time deterministic algorithm. The space complexity of the algorithm is O.k log k n C m/ which is a remarkable improvement over the exponential ln k space complexity of the colorcoding approach. Very recently, a big step was taken by Koutis [61] and Williams [74]. Koutis gave a randomized algorithm for k-path that runs in O  .23k=2 /  O  .2:83k / time. By improving Koutis’s approach, Williams presented a O  .2k / time randomized algorithm in 2008. They used very typical approaches which combine algebrization methods with randomization methods to obtain more efficient algorithms.

3.2

O  .2k / Time Randomized Algorithm

For this section, the main techniques are from the works of Williams [74] and Koutis [61]. Our objective is to make their works more readable for the inexperienced. Also an introduction of this line of research will make readers have clear mentality. The general idea is to reduce the k-path problem to the

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multilinear monomial testing for the sum-product expansion of a polynomial which is constructed by the input graph. Improving Koutis’s result [61], Williams [74] developed an O.2k nO.1/ / time randomized algorithm such that it outputs yes with high probability if there is a multilinear monomial and always outputs no if there is no multilinear monomial. According to above idea, the first problem is how to transform the given graph G for the k-path problem into a polynomial. Assume graph G has vertex set v1 ; v2 ;    ; vn . Let A be the adjacency matrix of G, and let x1 ; x2 ;    ; xn be !  variables. Define a matrix BŒi; j  D AŒi; j xi . Let 1 T be the row n-dimension !   vector of all 1’s and x be the column n-dimension vector defined by ! x Œi  D xi . So the graph G can be transformed into the k-walk polynomial Pk .x1 ;    ; xk / D !  T k1 ! 1 B   x . The k-walk polynomial has a good property that it reflects all the walk of length k in it sum-product expansion. Then the following claim can be proved. Define the polynomial X

Pk .x1 ;    ; xk / D

xi1    xik

i1 ; ;ik is a walk in G

It is easy to see that the graph G has a k-path vi1 ;    ; vik iff Pk .x1 ;    ; xk / has a multilinear monomial of xi1    xik of degree k in its sum-product expansion (see an example below). So the k-path problem for G can be reduced to the problem that detects whether Pk .x1 ;    ; xk / has multilinear monomial in its sum-product expansion. It is trivial to see whether Pk .x1 ;    ; xk / has a multilinear monomial when its sum-product expansion is already known. Unfortunately, the sum-product expression is essentially a troublesome thing which is infeasible to realize. That is because a polynomial may often have exponentially many monomials in its expansion. Lemma 2 There is a polynomial-time algorithm such that given an undirected graph G and a parameter k, it outputs a polynomial size circuit S for a multivariate polynomial C.:/ such that G has a k-path if and only if the sum-product expansion of C.:/ contains a multilinear monomial. Proof For each vertex vi 2 V , define a polynomial pk;i as follows: p1;i D xi ; 0 pkC1;i D xi @

X

.vi ;vj /2E

A polynomial for G is defined as

1 pk;j A ; k > 1:

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p.G; k/ D

n X

pk;i :

i D1

Obviously, p.G; k/ can be represented by an arithmetic circuit. It is easy to see that the graph G has a k-path vi1    vik iff p.G; k/ has a monomial of xi1    xik in its sum-product expansion.  Example 1 Consider the following simple graph G1 with three nodes: v2

v1 v3

The adjacency matrix A1 of G1 and the matrix B1 Œi; j  D A1 Œi; j xi are given below: 0 1 0 1 010 0 x1 0 B1 D @ x2 0 x2 A A1 D @ 1 0 1 A 010 0 x3 0 !   Then the 3-walk polynomial P3 .x1 ; x2 ; x3 / D 1 T  B12  ! x D 2x1 x2 x3 C x12 x2 C 2 2 2 x1 x2 C x2 x3 C x2 x3 . Clearly, x1 x2 x3 is a multilinear monomial in P3 .x1 ; x2 ; x3 /, so there is a corresponding 3-path v1 ! v2 ! v3 in G1 . It is easy to see that Pk can be represented by a circuit of size O.k.m C n// where m is the number of edges in G. Let F D GF .23Clog k / and W0 be the all-zeros vector of Zk2 . Williams’s randomized algorithm is introduced as follows:

Algorithm 1 Williams’s Algorithm Input: a graph G with vertex set v1 ; v2 ;    ; vn and a parameter k Output: If G does not have a k-path, the algorithm always outputs false; if G has a k-path, the algorithm outputs true with probability no less than 1=5. Steps: 1: Calculate Pk as described above. 2: Implement Pk with a circuit C . 3: Pick n uniform random vectors z1 ; z2 ;    ; zn from Zk2 . 4: For each multiplication gate gi in C , pick a uniform random wi 2 F n 0. Insert a new gate that multiplies the output of gi with wi and feeds the output to those gates that read the output of gi . 5: Let C 0 be the new arithmetic circuit and P 0 be the new polynomial represented by C 0 . Output true iff P 0 .W0 C z1 ;    ; W0 C zn / ¤ 0. End of Algorithm

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The basic idea of Williams’s algorithm is to substitute variables of Pk with random group algebra elements of Zk2 such that in the sum-product expansion of Pk , all the non-multilinear monomials in Pk become zero and some multilinear monomials survive. And in order to avoid making the coefficients of all multilinear monomials zero, the original scalar-free multiplication circuit is augmented by adding random scalar multiplications over a field large enough that the remaining multilinear monomial evaluates to nonzero with decent probability. Let s.n/ be the size of the arithmetic circuit used to represent Pk . Then Theorem 3 gives theoretical support for Williams’s algorithm. In above algorithm, readers need to pay attention to two points. First, all the k additions of F .Zk2 / have the P and multiplications are over a ring F .Z2 /. Elements k form g2Zk ag g, where each ag 2 F . The addition of F .Z2 / is defined in a point2 wise manner: 0 @

X

1

0

X

ag g A C @

g2Zk2

1 bg g A D

g2Zk2

X

.ag C bg /g

g2Zk2

The multiplication of two vectors .a1 ;    ; ak /T and .b1 ;    ; bk /T in F .Zk2 / is equal to .c1 ;    ; ck /T with ci D ai C bi .mod 2/ for i D 1; 2;    ; k. The multiplication has the form of a convolution: 0 @

X g2Zk2

1 0 ag g A  @

X

1 bg g A D

g2Zk2

X

.ag av /.gv/ D

g;v2Zk2

X g2Zk2

0 @

X

1 ah bhg A g

h2Zk2

Example 2 Let u1 ; u2 2 GF .22 /ŒZ32 , where 0 1 0 1 1 0 u1 D @ 0 A C x @ 0 A ; 1 0

0 1 0 1 0 1 1 1 0 u2 D @ 0 A C @ 0 A C @ 1 A : 1 1 0

Then 20 1 0 13 20 1 0 1 0 13 0 1 0 1 1 u1 C u2 D 4@ 0 A C x @ 0 A5 C 4@ 0 A C @ 0 A C @ 1 A5 0 1 0 1 1 0 1 0 1 1 1 D .1 C x/ @ 0 A C @ 1 A 1

1

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20 1 0 13 20 1 0 1 0 13 0 1 0 1 1 u1  u2 D 4@ 0 A C x @ 0 A5  4@ 0 A C @ 0 A C @ 1 A5 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 1 D .1 C x/ @ 0 A C x @ 1 A C .1 C x/ @ 0 A C @ 1 A 0

0

1

1

Second point is that in the evaluation of the k-path polynomial Pk , the algorithm corresponds to picking random yi;j;c in F for c D 1;    ; k  1, i; j D 1;    ; n, !   letting Bc Œi; j Dyi;j;c BŒi; j , then evaluating Pk0 .x1 ;    ; xn /D 1 T  Bk1    B1  ! x on the appropriate vectors. Before proving the correctness of Williams’s algorithm, a lemma which was proved by Koutis [61] and then further developed by Williams [74] over any field of characteristic two needs to be shown first. Lemma 3 If z1 ; z2 ;    ; zi 2 Zk2 are linearly dependent over GF .2/, then Qi k k j D1 .W0 C zj / D 0 in F ŒZ2 . Otherwise, if z1 ; z2 ;    ; zi 2 Z2 are linearly Qi P independent over GF .2/, then j D1 .W0 C zj / D z2Zk z. 2

Proof If z1 ; z2 ;    ; zi are linearly dependent, there is a nonempty subset T of the vectors that sum to the all-zeros vector. In F ŒZk2 , this is equivalent to Y

zj D W0 :

j 2T

Let S  T be arbitrary. By multiplying both sides by Y j 2S

zj D

Y

Q

j 2.ST / zj ,

zj :

j 2.ST /

P Q Q Thus, j 2T .W0 C zj / D S T . j 2S zj / D 0 mod 2, since each product Q appears twice in the sum. Since F is characteristic two, ij D1 .W0 C zj / D 0 over F ŒZk2 . Therefore, linearly dependent vectors lead to a cancelation Q of monomials. On the other hand, when z1 ; z2 ;    ; zi are linearly independent, ij D1 .W0 C zj / is just the sum over Q all vectors in the span of z1 ; z2 ;    ; zi , since each vector in the span is of the form j 2S zj for some S  Œi , and there is a unique way to generate each vector in the span.  Lemma 4 Let v1 ;    ; vn be n random vectors in Z2 . Then with probability at least 1 4 , they are linearly independent.

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k Proof Pk For the first m vectors v1 ;    ; vk , there are 2 vectors that can be expressed as i D1 ai vk , where ai 2 Z2 for i D 1;    ; m. Thus, for a random n-dimensional P k vector vmC1 from Z2 , with probability 22n , vmC1 cannot be expressed as kiD1 ai vk for some ai 2 Z2 for i D 1;    ; m. The probability that v1 ;    ; vn linearly independent is

      n  1 2 2n1 1Y 1 1 n 1 n  1 n  1 2 2 2 2 2 k

(3)

kD2

D

1 : 4

(4) 

Example 3 Let z1 ; z2 ; z3 ; z4 2 Z32 be four vectors: 0 1 0 z1 D @ 1 A 0

0 1 1 z2 D @ 0 A 0

0 1 0 z3 D @ 0 A 1

0 1 1 z4 D @ 1 A 0

Since z1  z2  z4 D W0 , z1 ; z2 ; zQ 3 ; z4 are linearly dependent. To coincide with above proof, let T D fz1 ; z2 ; z3 g. Then j 2T .W0 C zj / D .W0 C z1 /  .W0 C z2 /  .W0 C z4 / D W0 C z1 C z2 C z4 C z1  z2 C z1  z4 C z2  z4 C z1  z2  z4 D 0. The above equation equals to zero because z1  z2  z4 D W0 , z1 D z2  z4 , z2 D z1  z4 , z4 D z1  z2 in the sum-product and F is characteristic two. On the other hand, it isQeasy to see that z1 ; z2 ; z3 are linearly independent. So it can be easily verified that 3j D1 .W0 C zj / is the sum over all vectors in the span of z1 ; z2 ; z3 . Theorem 3 ([74]) Given a graph G with vertex set v1 ; v2 ;    ; vn and a parameter k  n, Williams’s algorithm is a randomized O  .2k nO.1/ / time algorithm such that if there is a k-path in G, it outputs true with probability no less than 1=5 and always outputs false if there is no k-path in G. Proof By the observation of Koutis [61] that for any zi 2 Zk2 , .W0 C zi /2 D W02 C 2vi C v2i D W0 C 0 C W0 D 0 mod 2: Thus, since F has characteristic two, all squares in P equal to 0 in P 0 .W0 C z1 ;    ; W0 C zn /. It illustrates that if P .x1 ;    ; xn / does not have a multilinear monomial, then for any choice of vi , P 0 .W0 C z1 ;    ; W0 C zn / D 0 over F ŒZk2 . Without loss of generality, assume that every multilinear monomial of P has degree at least k, and at least one multilinear monomial has degree exactly k. If not, let k 0 < k be the minimum degree of a multilinear monomial in P . By trying

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all j D 1;    ; k and multiplying the final output of the circuit for P by j new variables xnC1 ;    ; xnCj , a polynomial P j is obtained to be fed to the randomized algorithm. Note that when j D k  k 0 , our assumption holds. By above assumption, every multilinear monomial in the sum-product expansion of P has the form c  xi1    xik00 , where k 00  k and c 2 Z. For each such monomial, there is a corresponding collection of multilinear monomial P 0 , each of Q 0 the form w1    wk 0 1 kj D1 .W0 C zij /, where the sequence w1 ;    ; wk 0 1 is distinct for every monomial in the collection (as the sequences of multiplication gates g1 ;    ; gk 0 1 are distinct). Since there are no scalar multiplication in the arithmetic circuit, these monomials do not have leading coefficients. Then it can be proved that if the sum-product expansion of P .x1 ;    ; xn / has a multilinear monomial, for random choices of wi ’s and zi ’s, P 0 .W0 C z1 ;    ; W0 C zn / ¤ 0 with probability at least 1=5. 0 Since k 00 > k vectors are P linearly dependent, by Lemma 3, P .W0 Cz1 ;    ; W0 C zn / equals to either 0 or c z2Zk for some c 2 F . The vectors zl1 ;    ; zlk chosen for 2 the variables in a multilinear monomial P are linearly independent with probability at least 1=4, because the probability that a random k  k matrix over GF .2/ has full rank is at least 1=4 according to Lemma 4. If S is used to represent the set of those multilinear monomials in P which correspond to k linearly independent vectors in P 0 .W0 C z1 ;    ; W0 C zn /, then P for some ci 2 F , the coefficient c D i ci 2 F mentioned in the last paragraph corresponds to the i th multilinear monomial in S . Here, each coefficient ci comes from a sum of products of k  1 elements wi;1 ;    ; wi;k1 corresponding to some multiplication gates gi;1 ;    ; gi;k1 . Note that in the k-path case, each ci is a sum of products P of the form wi;1 wi;2    wi;k1 . Imagining the wi ’s as variables, the sum Q D i ci is a degree-k polynomial over F . Assuming S ¤ ;, since each monomial in Q has coefficient 1, Q is not identical to zero. Then by SchwartzZippel Lemma 1, randomly assigning values to the variables of Q results in an evaluation of 0 2 F with probability at most k=jF j D 1=23 . Since S ¤ ; with probability at least 1=4, and the coefficient c ¤ 0 with probability at least 1  1=23 D 7=8, the overall probability of success is at least 1=4  7=8 > 1=5. In the remaining paragraph, the runtime of the algorithm is discussed. However, since all the additions and multiplications takes place over F ŒZk2 , it is necessary to account for the cost of each arithmetic. Consider thePsum-product expansion of P 0 .W0 C z1 ;    ; W0 C zn /. It will be converted into Ci .:/vi , where each vi is a k-dimensional vector over Z2 , and Ci .:/ is a circuit for a multivariate polynomial. According P to the construction of in Lemma 2, each multiplication gate is for .mi vj /. CP i .:/vi /, where mi is a 0 monomial. It takes O.2k / steps to convert it into the format P  Ci .:/vi . Assume that the final evaluation has the format Ci .:/vi . The size of each Ci .:/ can be maintained in the size O.nO.1/ / as the original size of circuit P .:/ is O.nO.1/ /. The existing algorithm needs to be used for the polynomial identity, which takes O.nO.1/ / time for each Ci .:/, to test if each Ci .:/ is zero by Lemma 1. Therefore, the total time is O.2k nO.1/ /. 

Algebrization and Randomization Methods

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185

Hamiltonian Path Problem

In graph theory, a Hamiltonian path is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian cycle is a cycle in an undirected cycle which visits each vertex exactly once and also returns to the starting vertex. For example, every cycle graph and complete graph with more than two vertices is Hamiltonian. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the lcosian game, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. The Hamiltonian path problem and the Hamiltonian cycle problem are two classic problems of determining whether there exists a Hamiltonian path and a Hamiltonian cycle, respectively, for a given directed or undirected graph. For a given graph G, the Hamiltonian path problem is equivalent to the Hamiltonian cycle problem by adding a new vertex and connecting it to all the vertices of G. Both problems are in Karp’s original list [51] of NP-complete problems. For the specific proof of NP-completeness of the problems, please refer to standard algorithm books (e.g., [23]). Actually, the NP-hardness of Hamiltonian path problem has been used in the proof of Theorem 2. Through setting the distance between two cities to a finite constant if they are adjacent and infinity otherwise, the Hamiltonian path problem is a special case of the traveling salesman problem (TSP). The TSP was first formulated as a mathematical problem in 1930s. It is one of the most intensively studied problems in computational mathematics. Given a list of cities and their pairwise distances, the TSP asks for a shortest possible tour that visits each city exactly once. Even though the problem is computational difficult, a large amount of heuristics and exact methods are given to solve different instances of it [3]. Furthermore, the TSP has many applications in planning, logistics, and the manufacture of microchips. It is also equivalent to some bioinformatics problems such as DNA sequencing after slightly modified.

4.1

A Brief History of TSP and Hamiltonian Path Problem

The TSP was defined formally in the 1800s and was first studied by K. Menger and other mathematicians during the 1930s. The naive solution of the TSP would be try all permutations and see which one is cheapest using brute force search. The running time for this approach is in O.nŠ/, the factorial of the number of cities. So this approach becomes impractical even for only 20 cities. In almost 200 years from 1930s, plenty of algorithms including heuristics, approximations, and exact algorithms have been developed. Since the topic mainly focus on the Hamiltonian path detection problem in this section, we only introduce the exact algorithms for briefness. Later in the 1950s and 1960s, the TSP was reduced to an integer linear program by G. Dantzig et al., and the cutting plane method was developed to solve it. In the same period, Bellman [7, 8] and Karp [38] provided the dynamic programming algorithm for the TSP which runs in O.n2 2n / time. Given a complete

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graph with n vertices and edge weights l W E ! RC , the algorithm is based on defining ws;t .X / for s; t 2 X  V as the weight of the lightest path from s to t in the induced graph GŒX  visiting all vertices in X exactly once. Here, ws;t .X / obeys the following simple recursive equation:  minu2X nfs;t g ws;u .X n ftg/ C l.ut/ jX j > 2 ws;t .X / D l.st/ jX j  2 The dynamic programming solution requires exponential space. Using inclusionexclusion, the problem can be solved in time within a polynomial factor of 2n and polynomial space [5, 52, 57]. Then great progress was made by Grotschel et al., which uses cutting planes and branch and bound to solve instances with up to 2,392 cities. After that, Applegate, Bixby, Chvatal, and Cook developed the program Concorde that has been used in many recent record solutions. Except for the improvements for the general graph, there are some commendable results for restricted graph classes. Broersma et al. [13] described how to solve the Hamiltonian path problem for claw-free graphs in O  .1:682n / time. Iwama and Nakashima [43] showed that the TSP in cubic graphs admits an O  .1:251n/ time algorithm by improving slightly on Eppstein [28]. In graphs of maximum degree 4, Gebauer [35] provided an algorithm to count the Hamiltonian cycles in O  .1:715n/ time. Bj¨orklund et al. [12] presented that the traveling salesman problem in bounded degree graphs can be solved in time O..2  /n /, where  > 0 depends only on the degree bound but not on the number of cities, n. In 2005, by transforming a TSP instance into an instance of microchip layout problem, Cook and others computed an optimal tour for an instance which contains 33,810 cities. Though for the instances with tens of thousands of cities, the TSP can be solved. There is still no algorithm which breaks the time bound O.2k / for the general graph. So Woeginger [75] asked in open problem that whether there exists an exact algorithm which runs in time O  .c n / for the TSP and the Hamiltonian path problem for some c < 2. After 7 years, this problem has been solved very recently by Bjorklund [9], who improved the bounds to O.1:414n/ and O.1:657n/ for the bipartite graph and the general graph, respectively.

4.2

¨ Overall Idea of Bjorklund’s Approach

About 50 years ago, the O  .2n / time bound for detecting Hamiltonian path problem has been established. No exact algorithm which was developed to beat the bound until half a century later. Bj¨orklund [9] presented a Monte Carlo algorithm for n-vertex undirected graph running in O  .1:657n/ time, which is a breakthrough for this fundamental and classic problem. This crucial contribution was inspired by the algebraic sieving method for k-path problem [61, 74] which is introduced in Sect. 3. The basic idea of Bj¨orklund’s approach is similar to the inclusion-exclusion algorithm which can be summarized as first counting all closed n-walks through

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s and then canceling out crossing n-walks. However, comparing with inclusionexclusion algorithm, Bj¨orklund’s approach has some limitations. It only works for undirected graph, it is randomized, and also it cannot count the number of solutions. The same as Williams’s algorithm [74] for the k-path, Bj¨orklund’s approach evaluates multivariate polynomials over fields of characteristic two. The difference between them is that instead of sieving the k-path among all the walks, Bj¨orklund’s approach sieves the Hamiltonian cycles among all the cycle covers. A cycle cover in a directed n-vertex graph is a set of n arcs such that every vertex is the origin of one arc and the end of another. The arcs together describe disjoint cycles covering all vertices of the graph. In particular, the cycle covers contain the Hamiltonian cycle. The main contribution of Bj¨orklund’s work is that it not only further develops Koutis-William’s ideas but also introduces determinants to count weighted cycle covers to extend the ideas of using determinants to count perfect matchings [10]. However, unlike Koutis-William who construct small arithmetic circuits, Bj¨orklund depends on the efficient algorithms for computing a matrix determinant numerically. In the following paragraphs of this section, a concrete introduction about Bj¨orklund’s appoach is given, which further shows the tremendous ability of algebrization and randomization for the algorithm design.

4.3

Notations and Definitions

This section first introduces some particular notations and definitions for Bj¨orklund’s algorithm. Efforts were made to explain the details of the methods more clearly; however, for letting readers simply refer to the original papers, most of the notations are consistent with Bj¨orklund’s. A cycle cover can be formally defined as follows. Definition 3 For a direct graph D D .V; E/, a cycle cover is a subset C  E such that for every vertex v 2 V , there is exactly one arc av1 2 C starting in v and exactly one arc av2 2 C ending in v. Let cc.D/ be the family of all cycle covers of D and let hc.D/ be the set of Hamiltonian cycle covers. It is easy to see that hc.D/  cc.D/. Because among all the cycle covers of G, the Hamiltonian cycle covers are some particular cycle covers which consist of only one big cycle passing through all vertices of D. Let cc.D/ n hc.D/ be set of nonHamiltonian cycle covers, i.e., cc.D/= hc.D/ contains all the cycle covers which have more than one small cycles. A Hamiltonian cycle cover of a direct graph can be simply written as a vertex order .v0 ; v1 ;    ; vn1 / (cf. Fig. 1). Given a function f W X ! Y , f is surjective (or onto) if each element of Y can be obtained by applying f to some element of X . Let f 1 W Y ! 2X be the preimage of f , then f 1 .b/ D fa 2 X W f .a/ D bg. Then an important definition for labeled cycle cover sum will be given next. The name stems from the fact that every arc in cycle cover is labeled by a nonempty

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Fig. 1 The direct graph with a non-Hamiltonian cycle cover Œ.A; E; C /; .B; D; F / and a Hamiltonian cycle .A; E; F; B; D; C /

subset of a set of labels. It is important because the Hamiltonian path detection problem can be reduced to labeled cycle cover sum calculating problem. Definition 4 Suppose D D .V; E/ is a directed graph, L is a label set, and f is a function on a ring R which takes an edge a in G and the a subset of labels in L as inputs. The labeled cycle cover sum for D is defined as ƒ.D; L; f / D

X

X Y

f .a; g 1 .a//

(5)

C 2cc.D/ gWLC a2C

In above definition, all the cycles covers are summarized in the outer sum, and the inner sum is over all surjective function g from the label set L to the cycle cover set C with the two head arrow  representing a surjective function.

4.4

Cycle Cover Cancelation in Characteristic Two

In Sect. 4.2, it has been mentioned that the basic idea is first counting too much and then canceling out every false contribution. Following this idea, since the cycle covers of the graph contains all the Hamiltonian paths, the objective is to cancel out all the non-Hamiltonian paths and hence keep the Hamiltonian cycle covers. Thus, this section first describes how the non-Hamiltonian cycle covers are canceled out in the labeled cycle cover sum over the ring R with characteristic two and then discusses how to keep the Hamiltonian cycle covers. Before presenting the lemma, two definitions are necessary. Definition 5 • A direct graph G D .V; E/ is bidirected if for every arc uv 2 E, it has an arc vu 2 E in the opposite direction. • For an arbitrarily chosen special vertex s,f : A  2L n f;g ! R is an s-oriented mirror function if f .uv; Z/ D f .vu; Z/ for all Z and all u ¤ s and v ¤ s. The next lemma proves that how the non-Hamiltonian cycle covers vanish, which also imply that the nonexistence of false positives in Bj¨orklund’s randomized algorithm for detecting the Hamiltonian path.

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Fig. 2 Considering Fig. 1 as a bidirected graph, this figure shows a cycle cover Œ.AEC /; .BDF / which contains a labeled cycle C D .BDF / with the arcs along the cycle C is to pair with the cycle cover in Fig. 3

Lemma 5 Given a bidirected graph D D .V; E/, a finite set L, and special vertex s 2 V , let f be an s-oriented mirror function over a ring R of characteristic two. Then ƒ.D; L; f / D

X

X

Y

f .a; g 1 .a//

(6)

H 2hc.D/ gWLH a2H

Proof To cancel out all the labeled non-Hamiltonian cycle covers in the labeled cycle cover sum, a mapping M is created to partition all the non-Hamiltonian cycle covers into dual pairs such that both cycle covers in each pair contribute the same term to the sum. Since the ring R is of characteristic two, all these terms cancel. Let C 2 cc.D/ and C be the first cycle of C not passing through s. Since all the non-Hamiltonian cycle cover consists of at least two cycles and all cycles are vertex disjoint, there must exist one such C. There are two cases. In the general case, consider any fixed order of the cycles. Let C 0 D C except for the cycle C which is reversed in C 0 , i.e., every arc uv 2 C is replaced by the arc in the opposite direction vu in C 0 . Since G is bidirected, this arc must exist. In the special case when C contains only two arcs, C 0 is identical to C . The function g 01 is identical to g 1 on C n C and is defined by g 01 .uv/ D g 1 .vu/ for all arcs uv 2 C. Then the mapping is constructed by M.C; g/ D .C 0 ; g 0 /. And clearly, .C; g/ ¤ M.C; g/ and .C; g/ D M.M.C; g//. Therefore, the mapping M uniquely pairs up the labeled non-Hamiltonian cycle covers. Figures 2 and 3 give an example. From the known conditions of the lemma, f is an s-oriented mirror function. Thus, f .uv; Z/ D f .vu; Z/ for all arcs uv not incident to s and all Z 2 2L n ;. Also since F is a Q ring of characteristic two Qand .C; g/ and M.C; g/ have same labels, the equality a2C f .a; g 1 .a// D a2C 0 f .a; g 0 1 .a// satisfies. Therefore, the addition of the above tow products from .C; g/ and M.C; g/ equals to 0.  The above lemma shows how the non-Hamiltonian cycle covers cancel out in labeled cycle cover sum if the function f meets some requirements. However, in order to detect the Hamiltonian cycle covers, at least one Hamiltonian cycle cover reserves is still needed. The technique used here is very similar to the

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Fig. 3 This Figure is achieved by applying mapping M to Fig. 2. It has a cycle cover Œ.AEC /; .BFD/ which contains a labeled cycle C D .BFD/. The cycle .BDF / in Fig. 2 and the cycle .BFD/ in this figure keep the same labeling and hence pair up to cancel out the label cycle cover sum

one used in the k-path problem. The labeled cycle cover sum are transformed into a multivariate polynomial such that it would have monomial with nonzero coefficient if Hamiltonian cycle exists. Moreover, it would not have monomials with nonzero coefficient resulting from non-Hamiltonian cycle covers, as a consequence of Lemma 5. Then Freivalds’s fingerprint technique is employed to detect if the multivariate polynomial is identical to zero or not. If it is zero, the original direct graph does not contain Hamiltonian cycles, whereas if it is not zero, the original direct graph contains at least one Hamiltonian cycle. A careful reader could discover that for the same Hamiltonian cycle in a bidirected graph, it has two opposite directions. If the two directions of the Hamiltonian cycle contribute the same terms to the sum, they would vanish in the labeled cycle cover sum. That is why the s-oriented mirror function is defined early in this section. In particular for the special vertex s and any su; us 2 E, the above transformation ensures that f .su; X / and f .us; X / will not share variables. The details of the transformation will be discussed in Sect. 4.6.

4.5

Determinants and Inclusion-Exclusion

In this section, the readers would know how to compute labeled cycle cover sum efficiently. This technique can be found in Bj¨orklund’s recent work [10]. In Sect. 2.2, it has been mentioned that for a matrix A over a ring of characteristic two, the result det.A/ D perm.A/ holds. Moreover, given a weighted direct graph D D .V; E/, let the weights of the edges be defined by a function w W A ! R and define a jV j  jV j matrix with rows and columns representing the vertices V  Ai;j D

w.ij / 0

W ij 2 E : otherwise,

Algebrization and Randomization Methods

191

then X

perm.A/ D

Y

w.a/:

(7)

C 2cc.D/ a2C

The above Eq. (7) holds because of two aspects. First, a cycle cover of the direct graph D is a collection of vertex-disjoint directed cycles that covers all nodes of D. Hence each vertex i in D has a unique successor .i / in the cycle cover and  is a permutation on f1; 2;    ; ng where n is the number of vertices in D. Then every cycle cover of D corresponds to a permutation  on f1; 2; P   ; ng. Thus, Q according to the definition of permanent in Sect. 2.2, perm.A/  C 2cc.D/ a2C w.a/. On the other hand, any permutation  on f1; 2;    ; ng corresponds to a possible cycle cover in which there is an edge from vertex i to vertex .i / for each i in G. If there exists an edge created by the permutation  which is not a right edge of D, the cycle cover created by the permutation  is not a right cycle cover of D. Thus, the weight function w will P Q make the contribution from this permutation zero. Thus, perm.A/  C 2cc.D/ a2C w.a/. The remaining paragraphs describe how to compute labeled cycle cover sum through a sum of an exponential number of determinants. To this end the matrices Mf .Z/i;j are defined as for every Z  L  Mf .Z/i;j D

f .ij; Z/ 0

W ij 2 E; Z ¤ ; : otherwise.

(8)

Then the following polynomial p.f; r/ is introduced to compute an associated labeled cycle cover sum in characteristic two: p.f; r/ D

X Y L

det

X

! r

jZj

Mf .Z/

(9)

ZY

It is easy to see that p.f; r/ consists of a sum of an exponential number of determinants in which r is indeterminate whose aim is to control the total rank of the subsets used as labels in the labeled cycle covers. The following lemma shows that the labeled cycle cover sum is identical to the coefficient of the monomial r jLj in p.f; r/. For a polynomial q.r/ in an indeterminate r, Œr l q.r/ is the coefficient of monomial r l in q.r/. Lemma 6 For a directed graph D, a set L of labels, and any f W A  2L n ; ! GF .2k /, Œr jLj p.f; r/ D ƒ.D; L; f /:

(10)

Proof Since the determinant is equivalent to the permanent in rings of characteristic two (Eq. 1) and the relationship between permanent and cycle covers,

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p.f; r/ D

X

perm

Y L

D

X

! r jZj Mf .Z/

ZY

X

Y

X

Y L C 2cc.D/ a2C

ZY

X

! r

jZj

f .a; Z/ :

After calculating the inner product, p.f; r/ D

X

X

X

Y

r jq.a/j f .a; q.a//:

Y L C 2cc.D/ qWC !2Y nf;g a2C

Changing the order of summation,

p.f; r/ D

X

X

X

C 2cc.D/ qWC !2L nf;g

S

Y

a2C q.a/Y Y L

! r

jq.a/j

f .a; q.a// :

(11)

a2C

Claim 1 In Eq. (11), all the terms from the functions q W C ! 2L n f;g satisfied S a2C q.a/  L cancel in rings of characteristic two. S From a2C q.a/ SL, the union of q.a/ over all the elements of S C does not cover all of L, i.e., L n a2C q.a/ ¤ ;. Then for each subset T  L n a2C q.a/, there is a term derived from T (let Y D SL n T ) in the innermost summation. It is easy to see that for all the subsetsSof Ln a2C q.a/, the terms derived from them are equal. And in total there are 2jLn a2C q.a/j equal terms. Since the ring characteristic is two, these terms cancel. Because of Claim 1, in Eq. (11),

p.f; r/ D

X

X

P

r

a2C

jq.a/j

C 2cc.D/ qWC !2L nf;g S a2C q.a/DL

Y

! f .a; q.a// :

(12)

a2C

S P Since a2C q.a/ D L and a2C jq.a/j D jLj implies 8a ¤ b W q.a/ \ q.b/ D ;, the coefficient of r jLj is Œr jLj p.f; r/ D

X

X

Y

C 2cc.D/

qWC !2L nf;g S a2C q.a/DL 8a¤bWq.a/\q.b/D;

a2C

! f .a; q.a// :

Therefore, after inverting the function q, Œr jLj p.f; r/ identifies the labeled cycle cover sum in Definition 4. 

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Based on the above lemma, the labeled cycle cover sum can be computed by calculating the coefficient of a polynomial. This is a typical algebrization which transforms a graph theoretical problem into a algebraic problem. The algebrization enables a relatively efficient algorithm for computing the labeled cycle cover sum. Before presenting the details of the algorithm, some definitions and a Lemma are given to compute the labeled cycle cover sum. These results are developed by Bj¨orklund et al. in [11]. For more information, please refer to their paper [11] and Yates’ fast zeta transform [76]. Definition 6 Let N be a set of n elements with n  1 and let us assume that N D f1; 2; : : : ; ng. The set of all subsets of N is denoted by 2N . Assume that R is an arbitrary ring. Let f be a function that associates with every subset S  N an element f .S / of the ring R. The M¨obius transform of f is the function fO that associates with every X  N the ring element fO.X / D

X

f .S /:

(13)

S X

Given the M¨obius transform fO, the original function f may be recovered via the M¨obius inversion formula f .S / D

X

.1/jS nX j fO.X /:

(14)

X S

The fast M¨obius transform [54, 76] is the following algorithm for computing the M¨obius transform in O.n2n / ring operations. Lemma 7 There is an O.n2n / time algorithm to compute fO.X / for all X  N . Proof To compute fO given f, let initially fO0 .X / D f .X / for all X  N and then iterate for all j D 1; 2;    ; n and X  N as follows: ( fOj .X / D

fOj 1 .X / fOj 1 .X n fj g/ C fOj 1 .X /

W if j … X; W if j 2 X:

(15)

It is straightforward to verify by induction on j that this recurrence gives fOn .X / D fO.X / for all X  N in O.n2 2n / ring operations. Let Nj D f1; 2;    ; j g and N0 D ;. Using induction on j , we can show that P fOj .X / D Xj Nj \X f .Xj [ Yj /, where Yj D X  Nj .

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O O P It is trivial for j D 0 since f0 .X / D f .X /. Assume that fj .X / D Xj Nj \X f .Xj [ Yj /. P It is easy to see that fOj C1 .X / D Xj C1 Nj C1 \X f .Xj C1 [ Yj C1 / from the O definition of fj C1 .X / and the hypothesis of induction.  Then the following lemma gives the details about computing the labeled cycle cover sum for a direct graph. Lemma 8 The labeled cycle cover sum ƒ.D; L; f / for a function f with codomain GF .2k / on a directed graph G on n vertices, and with 2k > jLjn, can be computed in O..jLj C n/O.1/ 2jLj / arithmetic operations over GF .2k /, where ! is the square matrix multiplication exponent. Proof The labeled cycle cover sum ƒ.D; L; f / can be computed by means of Lemma 6. It would be recognized that the runtime of the algorithm is exponential in the number of labels but polynomial in the size of the input graph. According to Lemma 6, for polynomial p.f; r/, the aim is to achieve the coefficient of r jLj . It is easy to see from Eq. (12) that p.f; r/ is a polynomial of r with maximum degree jLjn. Thus, the polynomial for jLjn choices of r is evaluated, and interpolation is used to solve for the sought coefficient. The condition 2k > jLjn is required to make sure the interpolation has enough different points. Here, a generator g of the multiplicative group in GF .2k / is use to generate the interpolation point r D g 0 ; g 1 ;    ; gjLjn . To do interpolation, for instance, the O.jLj2 n2 / time Lagrange interpolation is one of the choices. From the above analysis, it can be achieved that the second part of the runtime bound. However, the much more time-consuming procedures areP matrixes tabulating jZj and determinant calculating in formula (Eq. 9). Let T .Y / D Mf .Z/: ZY r jLj tabulating T .Y / for all Y  L can be done in O.jLj2 P / time using Lemma 7. Moreover, for the determinant calculating of p.f; r/ D Y L det.T .Y //, it can O.1/ jLj be done in O.n 2 / as the determinant of an n  n matrix can be computed in O.nO.1/ / time using any standard method such as Gauss elimination to convert the matrix of the determinant into a right triangular one. Since there are jLjn values of r, the total runtime for the determinant calculating is bounded by O..jLjCn/O.1/ 2jLj /. Therefore, by summing up all the time required by the matrix tabulating, the determinant calculating, and the Lagrange interpolation, the labeled cycle cover sum can be computed in O..jLj C n/O.1/ 2jLj / time.  The complexity of the above lemma can be made as low as O..jLj2 n C jLjn1C! /2jLj C jLj2 n2 / as mentioned in [9] by using the algorithms of Bunch and Hopcroft [14] for computing determinant and the Coppersmith-Winograd square matrix multiplication exponent [22], where ! is the square matrix multiplication exponent.

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Fig. 4 A bipartite graph contains two halves. To reduce the Hamiltonian path to the labeled cycle cover sum, the left half U is used as a bidirected graph and the right half V is a label set

4.6

Hamiltonicity in Undirected Graphs

In this section, an O  .1:657n/ time algorithm is introduced to detect Hamiltonian path for direct graphs. For the direct graphs G D .V; E/, the overall idea is to reduce Hamiltonian path to labeled cycle cover sum. In Bj¨orklund et al.’s paper, they first apply the reduction to bipartite graphs. It gives an O  .1:414n/ time algorithm for bipartite graphs. Definition 7 A bipartite graph is a graph whose vertices can be divided into two disjoint sets V1 and V2 such that every edge connects a vertex in V1 to one in V2 . For the bipartite graph, a smaller bidirected graph D is constructed on one of the halves, and the other half is used as labels in a labeled cycle cover sum on D. (cf. Fig. 4). For the general direct graph, the core idea is same as the bipartite case. However, there are some slight differences. Compared with the bipartite case, the general graphs do not have the bipartite features. Hence, it seems impossible to partition the vertices into two equal parts of which the Hamiltonian cycle alternate vertices from one part to the other part. Then it is difficult to know which subset of the vertices could serve as labels. To solve this problem, a uniformly randomly chosen partition V D V1 [V2 with jV1 j D jV2 j can be exploited. That is because the number of transitions along a fixed Hamiltonian cycle from a vertex in one part to a vertex in the other part is n=2 in expectation. Note that in the bipartite case it is n. So for random partition of the general graph, there still is a good chance that the transitions will happen along the fixed Hamiltonian cycle. The vertices in V1 in the bipartite case were handled at a polynomial-time cost, whereas the vertices in V2 case at a price of a factor 2 each in the runtime. In the same vein, the vertices in V1 followed by a vertex in V2 along a fixed Hamiltonian cycle in the general case will be computationally cheap. The alternation would not happen each time along the Hamiltonian cycle. Therefore, arcs need to be distinguished according to the partition.

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The arcs connecting adjacent vertex pairs vi , vi C1 along a Hamiltonian cycle H are called unlabeled by V2 if both vi and vi C1 belong to V1 . The remaining arcs are referred to as labeled by V2 . Then the arcs of H can be partitioned into two parts of which L.H / and U.H / represent the set of the labeled arcs and the set of the unlabeled arcs, respectively. Define hcVm2 .G/ as the subset of hc.G/ of Hamiltonian cycles H which have precisely m arcs unlabeled by V2 along H . A vertex s is fixed: two variables xuv and xvu are introduced for every edge uv 2 E such that u 2 V2 or v 2 V2 (or both). xuv D xvu works except when u D s or v D s. For a complete bidirected graph D D .V1 ; F /, F is the set of arcs connecting every two points u and v in V1 such that there exists a path from u to v only through points in V2 in G or edge uv 2 GŒV1 . V2 contains some of the labels. In addition to V2 , a set Lm contains m extra labels is added to handle arcs unlabeled by V2 . For each edge uv in GŒV1  and every element d 2 Lm , new variables xuv;d , xvu;d are introduced and set to xuv;d D xvu;d except when u D s or v D s. See Fig. 5 for an example. For two vertices u; v 2 V and a nonempty subset X  V , Pu;v .X / is defined as the family of all simple paths in G from u to v passing through exactly the vertices in X (in addition to u and v). For uv 2 F and ;  X  V2 , f .uv; X / D

X

Y

xwz :

P 2Pu;v .X / wz2P

For every arc uv 2 F such that uv is an edge in GŒV1 , and every d 2 Lm , (16) f .uv; d / D xuv;d : In all other points, f is set to zero. For simplicity, the general graphs with even number of vertices is considered.

Algorithm 2 Bj¨orklund’s Algorithm for Hamiltonian Path Input: A graph G D .V1 ; V2 ; E/ with vertex set v1 ; v2 ;    ; vn Output: If G does not have a Hamiltonian path, the algorithm always outputs “No”; if G has a Hamiltonian path, the algorithm outputs “Yes” with probability no less than 1  .1=2/n . Steps: 1: Uniformly yield a partition V1 [ V2 D V at random, with jV1 j D jV2 j D n=2. Select k large enough such that 2k > cn for some c > 1. 2: Let m D 0. 3: Transform the input graph G into a symbolic labeled cycle cover sum ƒ.D; V2 [ Lm ; f / with D and f defined as above and the label set L D V2 [ Lm . 4: Pick a random point p from the field GF .2k / and evaluate ƒ.D; V2 [Lm ; f / in p over GF .2k / using the method from Lemmas 8 and 9. 5: Let m D m C 1. Repeat steps 35 until m D mmax D 0:205n. 6: Repeat steps 25 for r D nc 20:024n runs with constant c to be selected later. 7: If any result shows that ƒ.D; V2 ; f / is a nonzero polynomial, output “Yes”; otherwise, output “No.” End of Algorithm

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197

Hamiltonian undirected graph G = (V, E ) V1

V = V 1 ∪ V2

v1 x(v5v6,d1)v5 x(v6v7,d2)v6 v7 U(H) Unlabeled

v10 x(v10v11,d3) v11

|V1| = |V2|

V2

x(v1v2)

v2 x(v2v3) v3 x(v3v4) v4

x(v4v5) x(v7v8) x(v9v10)

v8 x(v8v9) v9

x(v11v12)

v12

L(H) Labeled

x(v12v13) v13 x(v13v14,d4)

v15

x(v14v15)

v16

v14

x(v15v16) x(v16v17)

v17

x(v17v18) v18 Lm= {d1,d2,d3,d4}

Fig. 5 The undirected graph G D .V; E/ with a Hamiltonian path is uniformly randomly partitioned into two equal parts V1 and V2 . Each part has nine vertices. The red variables show the labeled arcs, whereas the blue variables represent the unlabeled arcs. Here, for clearness of the figure, one unlabeled arc which is from v18 to v1 is omit. Thus, the extra label set Lm has five elements handling arcs unlabeled by V2 . This figure is very similar to the bipartite graph except that some unlabeled

The above algorithm has two loops. Since the partition yield randomly in step 1, the inner loop on m is to make sure all possible numbers of unlabeled arcs by V2 can be evaluated. The algorithm runs for r D nO.1/ 20:024n times to guarantee that the probability of the false negatives is exponentially small in n. Then the following lemma shows the relationship between labeled cycle cover sum ƒ.D; V2 [ Lm ; f / and the Hamiltonian cycles of G. And Lemma 9 ensures the correctness of the algorithm for the general graphs. Lm , and f defined above, Lemma 9 With G; D; VP 2 ; U; V; L; m; P Q Q 1. ƒ.D; V2 [Lm ; f / D H 2hc m .G/ .  WU .H /!Lm uv2U .H / xuv; .uv/ /. uv2L.H / xuv / V2 with  one to one. 2. ƒ.D; V2 [ Lm ; f / is the zero polynomial iff hcVm2 .G/ D ;. Proof (i) Since D is bidirected and f is an s-oriented mirror function, by Lemma 5 ƒ.D; V2 [ Lm ; f / D

X

X

Y

f .a; g 1 .a//:

H 2hc.D/ gWV2 [Lm H a2H

By omitting the vertices which belong to V2 , it is easy to see that there is a mapping from a Hamiltonian cycle H 2 hc.G/ to a Hamiltonian cycle

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in D. Observing that the above equation is over all the Hamiltonian cycles in D, detecting the Hamiltonicity of G can be done in the backward way, to expand the Hamiltonian cycles in D into Hamiltonian cycles of G. Since previous works were first done on the Hamiltonian cycles in D, it is necessary to extend the definition of labeled and unlabeled arc which were defined before for Hamiltonian cycles in G only. For a Hamiltonian cycle H 2 hc.D/ labeled by the function g W V2 [ Lm  H , an arc uv 2 H is labeled by V2 if g 1 .uv/  V2 and unlabeled by V2 if g1 .uv/ 2 Lm . HL and HU are used to represent the labeled and unlabeled set of arcs, respectively. According to the definition of the function f , the arcs of a Hamiltonian cycle in D in the labeled cycle cover sum either are labeled by an element of Lm or a nonempty subset of V2 , since these are the only subsets of the labels for which f is nonzero. Moreover, the definition of f at Eq. (16) tells us that every arc unlabeled by V2 along a Hamiltonian cycle consumes exactly one of the m labels in Lm , and all labels are used. Thus, only the Hamiltonian cycles in D with exactly m arcs unlabeled by V2 leave a nonzero contribution. Note that there is another restriction that a cycle H leaves a nonzero result only if the m arcs unlabeled by V2 along the cycle are also edges in G. Considering all the above restrictions on the inner summation in above formula, X X ƒ.D; V2 [ Lm ; f / D ƒHU .Lm /ƒHL .V2 / H 2hc.D/ HU [HL DH HU \HL D; jHU jDm 8a2HU Wa2E

with 0

X

ƒHU .Lm / D @

Y

1 f .a; .a//A

 WHU !Lm a2HU

and 0 ƒHL .V2 / D @

X

Y

1 f .a; g 1 .a//A :

gWV2 HL a2HL

Through analysis above, it is easy to see that the summation in ƒHU .Lm / is over all functions  which are one to one. Replacing f by its definition, the inner expressions is expanded to 0 ƒHU .Lm / D @

X

Y

 WHU !Lm uv2HU

and

1 xuv; .uv/ A

Algebrization and Randomization Methods

0 ƒHL .V2 / D @

X

199

Y

X

Y

1 xwz A :

gWV2 HL uv2HL P 2Pu;v .g 1 .uv// wz2P

Since every vertex in V2 is mapped to by precisely one arc in F on a Hamiltonian cycle H in D, for arcs a1 ; a2 2 HL and a1 ¤ a2 , g 1 .a1 / \ g 1 .a2 / D ;. Thus, for every arc uv in the labeled set HL , ƒHL .V2 / exhausts all the possible simple paths from u to v passing through the vertices in X with ;  X  V2 . Then through picking one simple path for each labeled arc in HL , the combination of all the simple paths grouped with all the unlabeled arcs actually forms a Hamiltonian cycle in G. Therefore, rewriting the expression as a sum of Hamiltonian cycles in G, the result establishes as claimed:

ƒ.D; V2 [ Lm ; f / D

X H 2hcVm .G/ 2

0 @

X

Y

10 xuv; .uv/ A @

 WU .H /!Lm uv2U .H /

Y

1 xuv A

uv2L.H /

(ii) If the graph G has no Hamiltonian cycles, the sum is clearly zero from above equation. Otherwise, if G has Hamiltonian cycles, then from the inner summation of the above equation, it is easy to see that each Hamiltonian cycle contributes a set of mŠ different monomials per orientation of the cycle (one for each permutation of ). Each edge of the cycle contributes a variable for the monomials. Since two different Hamiltonian cycles each has an edge and the other has not, monomials resulting from different Hamiltonian cycles are different. By the definition of s-oriented mirror function, the variables associated to the oppositely directed arcs incident to s are different. Thus, the monomials formed by oppositely oriented Hamiltonian cycles are unique in the sum.  Lemma 10 There is an O.n/ time algorithm to randomly partition a set of (even) n elements into two equal parts with n=2 elements each. Proof Let V be a set of n D 2m elements. Let V1 be empty in the beginning. Select a random element from V , put it into V1 , and remove the selected element from V . Repeat the above step until V1 contains m elements. It is easy to see equal subset of size m of V has the same probability to be generated. Let U be a subset m of m elements in V . With probability 2m , the first selected is selected from U . Assume that the first i elements are all selected from U for some i < m. With mi probability 2mi , the i C 1-th element is selected from U . Thus, the probability that U is generated by the algorithm is 1 1 m m1   D 2m : 2m 2m  1 mC1 m



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We need to following probability amplification lemma for the randomized algorithm for Hamiltionian cycle problem. Lemma 11 is a standard method in the randomized algorithm design. Lemma 11 Let A be an randomized algorithm for the Hamiltonian cycle problem such that for every G D .V; E/ without Hamiltonian cycle, it always returns “No” (zero false positive), and for every G D .V; E/ with a Hamiltonian cycle, 1 it returns “Yes” with probability at least ˛.n/ with ˛.n/ > 1. Then there is another 0 O.1/ algorithm A by repeating A n ˛.n/ times such that for every G D .V; E/ without Hamiltonian cycle, it always returns “No,” and for every G D .V; E/ with a Hamiltonian cycle, it returns “Yes” with probability at least 1  21n . Proof Let r D nc ˛.n/ with constant c to be determined later. The algorithm A is repeated r times. If one of the r times returns “Yes,” then the algorithm A0 returns “Yes.” Otherwise, the algorithm A0 returns “No.” It is easy to see that A0 has a zero false positive. If the input graph G D .V; E/ has a Hamiltonian path, then 1 r with probability at most .1  ˛.n/ / , all of the r times execution of A returns “No.” 1 r Therefore, with probability at least 1  .1  ˛.n/ /  1  21n , at least one of the r times of A executions returns “Yes” if constant c is selected to be large enough.  Theorem 4 ([10]) There is a Monte Carlo algorithm detecting whether an undirected graph on n vertices is Hamiltonian or not running in O  .1:657n/ time, with no false positives and false negatives with probability exponentially small in n. Proof From Lemma 9, the readers can realize the practicability of Bj¨orklund’s algorithm for the general graphs. The only thing left is to analyze the runtime and the probability of false negatives of our run results. For the runtime, from the definition of f .uv; X / for ;  X  V2 , to evaluate the labeled cycle cover sum ƒ.D; V2 [ Lm ; f /, the function f needs to be tabulated for all subsets of V2 . This can be achieved by running a variant of the Bellman-HeldKarp recursion. Formally, let fO W .V  V /  2V2 ! GF .2k / be defined for u ¤ v and ;  X  V2 by X

fO.uv; X / D

Y

xwz

P 2Pu;v .X / wz2P

then f .uv; X / D fO.uv; X / for uv 2 F and ;  X  V2 . For jX j > 1 the recursion fO.uv; X / D

X

xuw fO.wv; X n w/

w2X;uv2E

can be used to tabulate f .uv; X / for all ;  X  V2 . Since V2 has n=2 vertices, the tabulation for f can be done in O  .2n=2 / time. After tabulation, ƒ.D; V2 [ Lm ; f / can be evaluated by Lemma 8. Then the runtime of the evaluation is close relevant

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201

to the size of the label set V2 [ Lm . Since the size of V2 is fixed to n=2, the only need is the analysis of the maximum size of Lm . Let G D .V; E/ be a Hamiltonian undirected graph and V1 [ V2 D V with jV1 j D jV2 j. We will show m. 1  m /2mn1 .n=21/2 2 ‚ m 2 np Pr.jhcVm2 .G/j > 0/  m1 n .n=2/ . n /2m 4n 2n1:5

! :

(17)

To obtain the inequality in (Eq. 17), it is necessary to count the number of possibility that one fixed Hamiltonian cycle H D .v0 ; v1 ; c    ; vn1 / has exactly m arcs unlabeled by V2 and moreover has v0 2 V1 and vn1 2 V2 . For such a H , there are exactly n=2  m indices i such that xi 2 V1 and xi C1 2 V2 , and just as many indices i where xi 2 V2 and xi C1 2 V1 . Take Fig. 5 as an example, since m D 5, there are n=2  m D 4 vertices .v1 ; v7 ; v11 ; v14 / which satisfy xi 2 V1 and xi C1 2 V2 . And similarly there are 4 vertices .v4 ; v9 ; v12 ; v17 / which satisfy xi 2 V2 and xi C1 2 V1 . The ordered list of these transition indices i1 ; i2 ;    ; in2m uniquely describes the partition. In other words, any such list with 0  i1 ; 8j W ij < ij C1 , and in2m D n  1 corresponds to a unique partition. Set i0 D 1 and define the positive integers dj D ij ij 1 for all 0 < j  n2m. Then d1 ; d2 ;    ; dn2m1 describes a partition of the vertices in V1 in n=2  m groups. Analogously, d2 ; d4 ;    ; dn2m describes a partition of the vertices in V2 in n=2  m groups. The number of ways to write a positive integer as a sum of k positive integers is .p1 /. Thus, by multiplying k1 the number of ways to partition the vertices in V1 with the number of ways for n V2 , and dividing with the total number of balanced partition .n=2 /, the inequality in Eq. (17) is achieved. The second bound is derived by replacing the binomial coefficients p with their n factorial definition and using Stirling’s approximation for nŠ 2 ‚..n=e/ 2 n/. P max mmax m Since Pr.. m jhc .G/j D 0/  1  Pr.jhc j > 0//, to bring the V2 mD0 V2 probability of false negatives down to exp..n//, the number of runs r D nO.1/ Pr1 .jhcVm2max j > 0/ is needed by Lemma 11. Solving for the local minimum of the total runtime, using the inequality in (Eq. 17) to bound the probability, mmax D 0:205n and r D nO.1/ 20:024n runs. Therefore, the total runtime is bounded by O  .1:657n/. 

5

Minimum Common String Partition Problem

In this section, an application of the multilinear testing problem is presented to illustrate the practicability of algebrization. By means of multilinear testing for polynomials, a new exponential algorithm was recently developed by Fu et al. [33] to solve the minimum common string partition problem (MCSP). The problem MCSP has attracted a lot of attention, partly because of its applications in computational biology, text processing, and data compression. More specifically, MCSP has a close connection with the genome rearrangement problems such as edit

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L. Ding and B. Fu

distance and sorting by reversals. In fact, MCSP was initially studied exactly as a problem on “edit distance with moves” [70]. Definition 8 Given a pair of strings S and S 0 , the common string partition problem is to find partitions for S D S1    Sk and S 0 D S10    Sk0 such that S10 ;    ; Sk0 is a rearrangement of S1 ;    ; Sk and k is the least. Naturally, in the minimum common string Ppartition problem, two strings X and Y of length n are given over an alphabet . Let each symbol appear the same number of times in X and Y . This condition is certainly sufficient and necessary for us to compute a common string partition for X and Y . For example, two strings X D beadcdb and Y D abdebc have a common partition .< b; e; ab; c; db >; < ab; db; e; b; c >/. There are several variants of MCSP. Restricted version is called where each letter occurs at most d times in each input string as d MCSP. When the input strings are alphabet of size c, the corresponding problem is called MCSPc .

5.1

A Brief History of MCSP

The problem d -MCSP has been well studied. 2-MCSP (and therefore MCSP) is NPhard and APX-hard [37]. Several approximation algorithms have been proposed for MCSP problem [37, 59]. Chen et al. [18] studied the problem of computing signed reversal distance with duplicates (SRDD). They introduced the signed minimum common partition problem as a tool for dealing with SRDD and observed that for any two related signed strings X and Y , the size of a minimum common partition and the minimum number of reversal operations needed to transform X and Y are within a factor of 2. Kolman and Walen [60] devised an O.d 2 /-approximation algorithm running in O.n/ time for SRDD. Chrobak et al. [21] analyzed the greedy algorithm for MCSP; they showed that for 2-MCSP the approximation ratio is exactly 3, for 4-MCSP the approximation ratio is .log n/, and for the general MCSP, the approximation ratio is between .n0:43 / and O.n0:67 /. Kaplan and Shafrir [50] improved the lower bound to .n0:46 / when the input strings are over an alphabet of size O.log n/. Kolman [58] described a simple modification of the greedy algorithm; the approximation ratio of the modified algorithm is O.p 2 / for p-MCSP. Most recently, Goldstein and Lewenstein presented an greedy algorithm for MCSP [36]. In the framework of parameterized complexity, Damaschke first solved MCSP by an FPT algorithm with respect to parameters k (size of the optimum solution), r (the repetition number), and t (the distance ratio, depending on the shortest block in the optimum solution) [24]. Jiang et al. showed that both d -MCSP and M CSP c admit FPT algorithms when d and c are (constant) parameters [47]. However, it remains open whether MCSP admits any FPT algorithm parameterized only on k.

Algebrization and Randomization Methods

5.2

203

An O.2n nO.1/ / Time Algorithm for General Cases

This section shows that algebrization, especially multilinear monomial testing, is a very useful tool by presenting an exact solution for the MCSP problem. The overall idea is to convert it into a detection of a multilinear monomial in the sum-product expansion of a multivariate polynomial. Definition 9 Let S D a1 a2    an be a string of length n; • For an integer i 2 Œ1; n, define S Œi  D ai . • For an interval Œi; j   Œ1; n, define S Œi; j  to be the substring ai ai C1    aj of S . The aim is to create a polynomial in which a multilinear monomial is used to encode a solution for the partition problem. Then the following theorem explains how to construct polynomial according to two given strings S and S 0 and gives the proof of the correctness of encoding. Theorem 5 ([33]) There is an O.2n nO.1/ / time algorithm for the minimum common string partition problem. Proof Let S and S 0 be two input strings of length n. Let each position i of S 0 have a unique variable xi toPrepresent it. Define P Œa; b D xa xaC1    xb . For each interval Œs; t, define Gs;t D S 0 Œs 0 ;t 0 DS Œs;t  P Œs 0 ; t 0 . For each i  n, define a polynomial Fi;1 D G1;i . It represents all the substrings of S 0 that match S Œ1; i . This is formally stated by Claim 2 below. Claim 2 S 0 Œu; v is a substring of S 0 with S 0 Œu; v D S Œ1; i  if and only if there is a multilinear monomial P Œu; v in Fi;1 . Let F0;t D 1 for each t  1. For each i  n, define

Fi;t C1 D

X

Gj;i  Fj 1;t :

(18)

1j i

This polynomial has the property for encoding some partial common partition. It is stated in Claim 3. Claim 3 Let i be an arbitrary integer parameter in the range Œ1; n. Then there is a partition S Œi1 ; j1 ;    ; S Œit ; jt  for S Œ1; i  such that S Œir ; jr  D S 0 Œir0 ; jr0  for some disjoint intervals Œi10 ; j10 ;    ; Œit0 ; jt0  that are subset of Œ1; n if and only if there Q is a multilinear monomial Q in the sum of product expansion of Fi;t with Q D trD1 P Œir0 ; jr0 . Proof Prove by induction. For the case t D 1, it follows from Claim 2. Assume that the claim is true for t.

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Considering the case t C 1, the following two cases for the induction are needed to be analyzed: • Assume that there is a partition S Œi1 ; j1 ;    ; S Œit C1 ; jt C1  for S Œ1; i  such that 1 D i1  j1 D i2  1 < j2 D    jt D it C1  1 < jt C1 D i and S Œir ; jr  D S 0 Œir0 ; jr0  for r D 1;    ; t C 1 with some disjoint intervals Œi10 ; j10 ;    ; Œit0C1 ; jt0C1 . By our inductive hypothesis, there is a multilinear Qt 0 0 monomial Q0 D rD1 P Œir ; jr  that is in the sum of product expansion of Fjt ;t . By the definition of Git C1 ;jt C1 , Git C1 ;jt C1 contains the multilinear monomial P Œit0C1 ; jt0C1  in its sum of product expansion. By the definition of Fi;t C1 in Eq. (18), multilinear monomial Q D Q0  P Œit0C1 ; jt0C1  in the sum of expansion of Fi;t C1 . • Assume that there is multilinear monomial Q in Fi;t C1 . By the definition of Fi;t C1 in Eq. (Eq. 18), there is an integer j and an interval Œit0C1 ; jt0C1  such that S Œj; i  D S 0 Œit0C1 ; jt0C1 , and there is a multilinear monomial Q0 with Q D P Œit0C1 ; jt0C1   Q0 . By the inductive hypothesis, there is a partition S Œi1 ; j1 ;    ; S Œit ; jt  for 0 0 0 S Œ1; j  1 such that for each 1  r  t, QSt Œir ; jr 0 D 0S Œir ; jr  for some disjoint 0 0 0 0 0 intervals Œi1 ; j1 ;    ; Œit ; jt , and Q D rD1 P Œir ; jr . Since Q is a multilinear monomial, there is no intersection between Œit0C1 ; jt0C1  and [trD1 Œir0 ; jr0 .  By Claim 3, it is easy to see that there is a partition of k segments if and only if there is a multilinear monomial x1 x2    xn in the sum of product expansion of Fn;k . Then there is an O.2n nO.1/ / time algorithm to find the multilinear monomial by evaluating Fn;k from bottom-up. During the evaluation, only the multilinear monomials are kept in Fi;t . Since the total number of variables is n, the total number of multilinear monomials is at most 2n . Note that Gj;i has at most O.n/ monomials. Therefore, each multiplication Gj;i Fj 1;t takes O.2n nO.1/ / time. The arithmetic expression for Fn;k involves nO.1/ C and  operations. Therefore, the total time for generating all the multilinear monomials in the sum of product expansion of Fn;k is O.2n nO.1/ /. Testing the existence of multilinear monomial x1 x2    xn in the sum of product expansion over Fn;k gives the existence of a common partition of at most k blocks. Tracing how the multilinear monomial x1 x2    xn is formed with a bottom-up approach to find all multilinear monomial in the sum of product expansion of Fn;k shows how a common partition of at most k blocks is computed. This can be seen in the proof of Claim 3. 

6

Algebraic FPT Algorithms for Channel Scheduling Problem

In this section, the algorithms for scheduling data retrieval in multichannel wireless data broadcast environments will be introduced. These algorithms are derived from the multilinear testing theory. Moreover, they also use algebraic method and randomized method to minimize the total time for channel scheduling. Generally, the Channel Scheduling Problem, which is also named the least time data retrieving (LTDR)

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problem, is belong to wireless data broadcast. In many applications, such as stock quotes, flight schedules, and traffic news [1], the LTDR problem appears when people may want to download multiple data items at a time. There are two major performance concerns with respect to wireless data broadcast, one is tuning time and the other is access time. The tuning time, defined as the amount of time a client spends on listening to the channels, is a performance criterium to evaluate the energy consumption, while the access time, defined as the time duration from the moment a client submits a query to the moment that all the required data are downloaded, reflects the response delay of queries. The LTDR problem is the problem of finding a schedule to download a set of required data, which are cyclically broadcasted via multiple broadcast channels, such that all the required data can be downloaded in a short time. The formal definition of the LTDR problem is as follows. Definition 10 Given a query data set S and a broadcast program of k channels, find a valid schedule to download all the data items in S in some order so that the total access time is minimized. In the following paragraphs, the related works about LTDR is first introduced. Then the algebraic FPT algorithms for LTDR will be presented in the next section. These works can be found in the recently accepted paper by Gao, Lu, Wu, and Fu [34].

6.1

A Brief History of LTDR

As it is talked about before, access time is one of the major factors with respect to the performance of the wireless data broadcast. Many works have been done to minimize the access time [42, 46, 71], etc. The LTDR problem, which is from the client point of view, is one of the optimization problems in the field of wireless data broadcast. Under the assumption that the indices of all required data are already obtained, the LTDR problem focuses on developing efficient scheduling algorithms from the client point of view with the objective of minimizing the access time. Compared to the massive amount of works for scheduling the data retrieval process at the client side, there has been little work done on scheduling the data retrieval process at the client side. When there is only one broadcast channel, or the client requests single data item at a time, the data retrieval scheduling problem is straightforward. But in many cases, the clients may want to download multiple data per request [39–42, 48, 62, 71]. Juran et al. and Hurson et al. presented several data retrieval heuristics in [42, 48] to reduce the access time and the number of channel switchings. In [71], Shi et al. investigated how to schedule the downloading process at the client side by using multiple parallel processes. It is worthy to mention that the studies in [42, 48, 71] assume the data are partitioned over multiple channels without replications. This assumption simplifies the algorithm design, but there are many data allocation scheduling may result in the replicative data appearing on channels [1, 30, 41].

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Here, the LTDR problem assumes that the data are allowed replicatively. Compared with the model studied in [42, 48, 71], this is more generalized assumption which fits all wireless data broadcast programs. Under this assumption, Du et al. give the complexity analysis and some approximation algorithms of the LTDR problem in their recently work [27].

6.2

Algebraic FPT Algorithms for LTDR

In this section, a fixed parameter tractable algorithm with computational time O.2t .nkh/O.1/ / for the LTDR problem is introduced. Here, k is the number of channels, t is the least number of programs to be downloaded in a set of programs S , n is the maximal time slots, and h is the maximal number of channel switches. A problem with input size n and parameter k has a fixed parameter tractable (FPT) algorithm if there is an algorithm that runs in an O.f .k/nO.1/ / time. Looking for FPT algorithms for NP-hard problems has been widely studied in the last two decades. Although it is unknown if the LTDR problem is NP-hard, it looks a challenging problem and hard to find the optimal solution. Specifically, the LTDR problem can be described as the clients want to download a set S of m programs from k channels by a receiver. Each channel Ci is partitioned into many discrete time slots, and a program pi takes integer si;j of time slots in channel j . If the receiver switches from channel i at time t to channel j , it can start downloading the program in channel j at time t C 2. As described before, the target is to minimize the total time and the number of switches among those channels. Definition 11 A subset program S 0 D fxi1 ;    ; xia g  S is .i; t; h/-downloadable if the receiver can download all programs in S 0 in time interval Œ1; t, the total number of channel switches is at most h, and a program xij is downloaded from channel i in the final time phase. The following lemma transform the channel switches allowed LTDR problem to the monomial testing problem. Lemma 12 There is polynomial-time algorithm such that it constructs a circuit for a polynomial Fi;t;h;a such that for each .i; t; h/-downloadable subset S 0 D fpi1 ;    ; pia g  S , the sum of product expansion of Fi;t;h;a contains a multilinear monomial .xi1 ;    ; xia /Y , where each variable xi represents program pi for i D 1;    ; m, and Y is a multilinear monomial with only y variables. Proof For each program pi 2 S , let variable xi represent it. A recursive way is used to define the polynomial Fi;t;h;a : 1. Fi;t;0;0 D P 0. 2. Fi;t;0;1 D xj . pj 2S; pj is entirely in the time interval Œt1 ;t2  of channel i

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P 3. FP D yi;t;hC1;bC1;1 . t1 0, and if f is not satisfiable, we have A.g/ D 0 since A.:/ is a r.n/-approximation and r.n/  1. We can know if the coefficient of x1    xn in the sum of product expansion of g is positive in polynomial time. Thus, .3; 3/-SAT is solvable in polynomial time. Since .3; 3/-SAT is NP-complete, we have P D NP. 

9

Conclusion

The combination of randomization and polynomial brings efficient algorithm for the classical problems k-path and Hamiltonian path. The polynomial method has applications in the areas of wireless networking and bioinformatics. It can be seen that testing the existence of multilinear monomials in the sum-product expansion of a polynomial is NP-complete and computing the coefficient of a multilinear monomial in the sum-product expansion is #P-hard. These results shows the polynomial methods links to the central areas of computational complexity theory. This chapter explains some methods that are related to algorithm and computational complexity. It is expected to see the further development of the theory in the future. Acknowledgements The authors would like to thank Dr. Dingzhu Du for his encouragements when writing this book chapter. We would also like to thank Zhixiang Chen, Yang Liu, and Robert Schweller for a wonderful experience of collaboration working in this area. Finally, we would like

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to thank the reviewers for their helpful comments for an earlier version of this chapter. Bin Fu is supported in part by the National Science Foundation Early Career Award CCF-0845376.

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Algorithmic Aspects of Domination in Graphs Gerard Jennhwa Chang

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Definitions and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Graph Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Variations of Domination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Special Classes of Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Properties of Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Labeling Algorithm for Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dynamic Programming for Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Primal–Dual Approach for Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Power Domination in Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Tree Related Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Interval Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Interval Orderings of Interval Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Domatic Numbers of Interval Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Weighted Independent Domination in Interval Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Weighted Domination in Interval Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Weighted Total Domination in Interval Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Weighted Connected Domination in Interval Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chordal Graphs and NP-Completeness Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Perfect Elimination Orderings of Chordal Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 NP-Completeness for Domination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Independent Domination in Chordal Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Strongly Chordal Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Strong Elimination Orderings of Strongly Chordal Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Weighted Domination in Strongly Chordal Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Domatic Partition in Strongly Chordal Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Permutation Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cocomparability Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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G.J. Chang Department of Mathematics, National Taiwan University, Taipei, Taiwan e-mail: [email protected]

221 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 26, © Springer Science+Business Media New York 2013

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9 Distance-Hereditary Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Hangings of Distance-Hereditary Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Weighted Connected Domination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271 271 272 274 275

Abstract

Domination in graph theory has many applications in the real world such as location problems. A dominating set of a graph G D .V; E/ is a subset D of V such that every vertex not in D is adjacent to at least one vertex in D. The domination problem is to determine the domination number .G/ of a graph G that is the minimum size of a dominating set of G. Although many theoretic theorems for domination and its variations have been established for a long time, the first algorithmic result on this topic was given by Cockayne, Goodman, and Hedetniemi in 1975. They gave a linear-time algorithm for the domination problem in trees by using a labeling method. On the other hand, at about the same time, Garey and John constructed the first (unpublished) proof that the domination problem is NP-complete. Since then, many algorithmic results are studied for variations of the domination problem in different classes of graphs. This chapter is to survey the development on this line during the past 36 years. Polynomial-time algorithms using labeling method, dynamic programming method, and primal–dual method are surveyed on trees, interval graphs, strongly chordal graphs, permutation graphs, cocomparability graphs, and distance-hereditary graphs. NP-completeness results on domination are also discussed.

1

Introduction

Graph theory was founded by Euler [88] in 1736 as a generalization of the solution to the famous problem of the K¨onigsberg bridges. From 1736 to 1936, the same concept as graph, but under different names, was used in various scientific fields as models of real-world problems; see the historic book by Biggs et al. [22]. This chapter intends to survey the domination problem in graph theory from an algorithmic point of view. Domination in graph theory is a natural model for many location problems in operations research. As an example, consider the following fire station problem. Suppose a county has decided to build some fire stations, which must serve all of the towns in the county. The fire stations are to be located in some towns so that every town either has a fire station or is a neighbor of a town which has a fire station. To save money, the county wants to build the minimum number of fire stations satisfying the above requirements. Domination has many other applications in the real world. The recent book by Haynes et al. [112] illustrates many interesting examples, including dominating

Algorithmic Aspects of Domination in Graphs Fig. 1 King domination on a 3  5 Chessboard (a) Chessboard (b) Chinese chessboard

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a

b K K

K

K

K K

queens, sets of representatives, school bus routing, computer communication networks, .r; d /-configurations, radio stations, social network theory, land surveying, and kernels of games. Among them, the classical problems of covering chessboards by the minimum number of chess pieces are important in stimulating the study of domination, which commenced in the early 1970s. These problems certainly date back to De Jaenisch [83] and have been mentioned in the literature frequently since that time. A simple example is to determine the minimum number of kings dominating the entire chessboard. The answer to an m  n chessboard is d m3 ed n3 e. In the Chinese chess game, a king only dominates the four neighbor cells which have common sides with the cell the king lies in. In this case, the Chinese king domination problem for an m  n chessboard is harder. Figure 1 shows optimal solutions to both cases for a 3  5 board. The above problems can be abstracted into the concept of domination in terms of graphs as follows. A dominating set of a graph G D .V; E/ is a subset D of V such that every vertex not in D is adjacent to at least one vertex in D. The domination number .G/ of a graph G is the minimum size of a dominating set of G. For the fire station problem, consider the graph G having all towns of the county as its vertices and a town is adjacent to its neighbor towns. The fire station problem is just the domination problem, as .G/ is the minimum number of fire stations needed. For the king domination problem on an m  n chessboard, consider the king’s graph G whose vertices correspond to the mn squares in the chessboard, and two vertices are adjacent if and only if their corresponding squares have a common point. For the Chinese king domination problem, the vertex set is the same, but two vertices are adjacent if and only if their corresponding squares have a common side. Figure 2 shows the corresponding graphs for the king and the Chinese king domination problems on a 3  5 chessboard. The king domination problem is just the domination problem, as .G/ is the minimum number of kings needed. Black vertices in the graph form a minimum dominating set. Although many theoretic theorems for the domination problem have been established for a long time, the first algorithmic result on this topic was given by Cockayne et al. [56] in 1975. They gave a linear-time algorithm for the domination problem in trees by using a labeling method. On the other hand, at about the same time, Garey and Johnson (see [99]) constructed the first (unpublished) proof that the domination problem is NP-complete for general graphs. Since then, many

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a

b

Fig. 2 King’s graphs for the chess and the Chinese chess (a) For chess; (b) For Chinese chess

algorithmic results are studied for variations of the domination problem in different classes of graphs. The purpose of this chapter is to survey these results. This chapter is organized as follows. Section 2 gives basic definitions and notation. In particular, the classes of graphs surveyed in this chapter are introduced. Among them, trees and interval graphs are two basic classes in the study of domination. While a tree can be viewed as many paths starting from a center with different branches, an interval graph is a “thick path” in the sense that a vertex of a path is replaced by a group of vertices tied together. Section 3 investigates different approaches for domination in trees, including labeling method, dynamic programming method, and primal–dual approach. These techniques are used not only for trees but also for many other classes of graphs in the study of domination as well as many other optimization problems. Section 4 is for domination in interval graphs. It is in general not clear to which classes of graphs the results in trees and interval graphs can be extended. For some classes of graphs, the domination problem becomes NP-complete, while for some classes it is polynomially solvable. Section 5 surveys NP-completeness results on domination, for which chordal graphs play an important role. The remaining sections are for classes of graphs in which the domination problem is solvable, including strongly chordal graphs, permutation graphs, cocomparability graphs, and distance-hereditary graphs.

2

Definitions and Notation

2.1

Graph Terminology

A graph is an ordered pair G D .V; E/ consisting of a finite nonempty set V of vertices and a set E of 2-subsets of V , whose elements are called edges. Sometimes V .G/ is used for the vertex set and E.G/ for the edge set of a graph G. A graph is trivial if it contains only one vertex. For any edge e D fu; vg, it is said that vertices u and v are adjacent and that vertex u (respectively, v) and edge e are incident. Two distinct edges are adjacent if they contain a common vertex. It is convenient to henceforth denote an edge by uv rather than fu; vg. Notice that uv and vu represent the same edge in a graph.

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Fig. 3 A graph G with six vertices and seven edges

u

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Two graphs G D .V; E/ and H D .U; F / are isomorphic if there exists a bijection f from V to U such that uv 2 E if and only if f .u/f .v/ 2 F . Two isomorphic graphs are essentially the same as one can be obtained from the other by renaming vertices. It is often useful to express a graph G diagrammatically. To do this, each vertex is represented by a point (or a small circle) in the plane and each edge by a curve joining the points (or small circles) corresponding to the two vertices incident to the edge. It is convenient to refer to such diagram of a graph as the graph itself. In Fig. 3, a graph G with vertex set V D fu, v, w, x, y, zg and edge set E D fuw, ux, vw, wx, xy, wz, xzg is shown. Suppose A and B are two sets of vertices. The neighborhood NA .B/ of B in A is the set of vertices in A that are adjacent to some vertex in B, that is, NA .B/ D fu 2 A W uv 2 E for some v 2 Bg: The closed neighborhood NA ŒB of B in A is NA .B/ [ B. For simplicity, NA .v/ stands for NA .fvg/, NA Œv for NA Œfvg, N.B/ for NV .B/, N ŒB for NV ŒB, N.v/ for NV .fvg/, and N Œv for NV Œfvg. The notion u  v stands for u 2 N Œv. The degree deg.v/ of a vertex v is the size of N.v/ or, equivalently, the number of edges incident to v. An isolated vertex is a vertex of degree zero. A leaf (or end vertex) is a vertex of degree one. The minimum degree of a graph G is denoted by ı.G/ and the maximum degree by .G/. A graph G is r-regular if ı.G/ D .G/ D r. A 3-regular graph is also called a cubic graph. A graph G 0 D .V 0 ; E 0 / is a subgraph of another graph G D .V; E/ if V 0  V and E 0  E. In the case of V 0 D V , G 0 is called a spanning subgraph of G. For any nonempty subset S of V , the (vertex) induced subgraph GŒS  is the graph with vertex set S and edge set EŒS  D fuv 2 E W u 2 S and v 2 S g:

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A graph is H -free if it does not contain H as an induced subgraph. The deletion of S from G D .V; E/, denoted by G  S , is the graph GŒV n S . G  v is a short notation for G  fvg when v is a vertex in G. The deletion of a subset F of E from G D .V; E/ is the graph G F D .V; E nF /. G e is a short notation for G feg if e is an edge of G. The complement of a graph G D .V; E/ is the graph G D .V; E/, where E D fuv 62 E W u; v 2 V and u ¤ vg: Suppose G1 D .V1 ; E1 / and G2 D .V2 ; E2 / are two graphs with V1 \ V2 D ;. The union of G1 and G2 is the graph G1 [ G2 D .V1 [ V2 ; E1 [ E2 /. The join of G1 and G2 is the graph G1 C G2 D .V1 [ V2 ; EC /, where EC D E1 [ E2 [ fuv W u 2 V1 and v 2 V2 g: The Cartesian product of G1 and G2 is the graph G1 G2 D .V1  V2 ; E /, where V1  V2 D f.v1 ; v2 / W v1 2 V1 and v2 2 V2 g; E D f.u1 ; u2 /.v1 ; v2 / W .u1 D v1 ; u2 v2 2 E2 / or .u1 v1 2 E1 ; u2 D v2 /g: In a graph G D .V; E/, a clique is a set of pairwise adjacent vertices in V . An i -clique is a clique of size i . A 2-clique is just an edge. A 3-clique is called a triangle. A stable (or independent) set is a set of pairwise nonadjacent vertices in V . For two vertices x and y of a graph, an x-y walk is a sequence x D x0 , x1 , : : :, xn D y such that xi 1 xi 2 E for 1  i  n, where n is called the length of the walk. In a walk x0 , x1 , : : :, xn , a chord is an edge xi xj with ji  j j  2. A trail (path) is a walk in which all edges (vertices) are distinct. A cycle is an x-x walk in which all vertices are distinct except the first vertex is equal to the last. A graph is acyclic if it does not contain any cycle. A graph is connected if for any two vertices x and y, there exists an x-y walk. A graph is disconnected if it is not connected. A (connected) component of a graph is a maximal subgraph which is connected. A cut-vertex is a vertex whose deletion from the graph results in a disconnected graph. A block of a graph is a maximal connected graph which has no cut-vertices. The distance d.x; y/ from a vertex x to another vertex y is the minimum length of an x-y path; and d.x; y/ D 1 when there is no x-y path. Digraphs or directed graphs can be defined similar to graphs except that an edge .u; v/ is now an ordered pair rather than a 2-subset. All terms in graphs can be defined for digraphs with suitable modifications by taking into account the directions of edges. An orientation of a graph G D .V; E/ is a digraph .V; E 0 / such that for each edge fu; vg of E, exactly one of .u; v/ and .v; u/ is in E 0 and the edges in E 0 all come from this way.

Algorithmic Aspects of Domination in Graphs

2.2

227

Variations of Domination

Due to different requirements in the applications, people have studied many variations of the domination problem. For instance, in the queen’s domination problem, one may ask the additional property that two queens do not dominate each other or any two queens must dominate each other. The following are most commonly studied variants of domination. Recall that a dominating set of a graph G D .V; E/ is a subset D of V such that every vertex not in D is adjacent to at least one vertex in D. This is equivalent to that N Œx \ D ¤ ; for all x 2 V or [y2D N Œy D V . A dominating set D of a graph G D .V; E/ is independent, connected, total, or perfect (efficient) if GŒD has no edge, GŒD is connected, GŒD has no isolated vertex, or jN Œv \ Dj D 1 for any v 2 V n D. An independent (respectively, connected or total) perfect dominating set is a perfect dominating set which is also independent (respectively, connected or total). A dominating clique (respectively, cycle) is a dominating set which is also a clique (respectively, cycle). For a fixed positive integer k, a k-dominating set of G is a subset D of V such that for every vertex v in V , there exists some vertex u in D with d.u; v/  k. An edge dominating set of G D .V; E/ is a subset F of E such that every edge in E n F is adjacent to some edge in F . For the above variations of domination, the corresponding independent, connected, total, perfect, independent perfect, connected perfect, total perfect, clique, cycle, and k- and edge domination numbers are denoted by i .G/, c .G/, t .G/, per .G/, iper .G/, cper .G/, tper .G/, cl .G/, cy .G/, k .G/, and e .G/, respectively. A dominating set D of a graph G D .V; E/ corresponds to a dominating P function which is a function f W V ! f0; 1g such that u2N Œv f .u/  1 for P any v 2 V . The weight of f is w.f / D f .v/. Then .G/ is equal to the v2V minimum weight of a dominating function of G. Several variations of domination are defined in terms of functions as follows. A signed dominating function is a P function f W V ! fC1; 1g such that u2N Œv f .u/  1. The signed domination number s .G/ of G is the minimum weight of a signed dominating function. A Roman dominating function is a function f W V ! f0; 1; 2g such that any vertex v with f .v/ D 0 is adjacent to some vertex u with f .u/ D 2. The Roman domination number Rom .G/ of G is the minimum weight of a Roman dominating function. A set-valued variation of domination is as follows. For a fixed positive integer k, a k-rainbow dominating function is a function f W V ! 2f1;2;:::;kg such that P f .v/ D ; implies [u2N.v/ f .u/ D f1; 2; : : : ; kg. The weight of f is w.f / D v2V jf .v/j. The k-rainbow domination number krain .G/ of G is the minimum weight of a krainbow dominating function. Notice that 1-rainbow domination is the same as the ordinary domination. A quite different variation motivated from the power system monitoring is as follows. For a positive integer k, suppose D is a vertex subset of a graph G D .V; E/. The following two observation rules are applied iteratively:

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Fig. 4 A graph G of 13 vertices and 19 edges

v1

v4

v6

v9

v11

v7 v2

v12 v5

v3

v10

v8

v13

• Observation rule 1 (OR1). A vertex in D observes itself and all of its neighbors. • Observation rule 2 (OR2). If an observed vertex is adjacent to at most k unobserved vertices, then these vertices become observed as well. The set D is a k-power dominating set of G if all vertices of the graph are observed after repeatedly applying the above two observation rules. Alternatively, let S 0 .D/ D N ŒD and Si C1 .D/ D [fN Œv W v 2 Si .D/; jN.v/nSi .D/j  kg for i  0. Notice that S0 .D/  S1 .D/  S2 .D/     and there is a t such that Si .D/ D St .D/ for i  t. Using this notation, D is a k-power dominating set if and only if St .D/ D V . The k-power domination number kpow .G/ of G is the minimum size of a k-power dominating set. For the case of k D 1, 1-power domination is called power domination. Figure 4 shows a graph G of 13 vertices and 19 edges, whose values of  .G/ for variations of domination and the corresponding optimal sets/functions are given below. .G/ i .G/ c .G/ t .G/ per .G/ iper .G/ cper .G/ tper .G/ cl .G/ cy .G/ 2 .G/ k .G/ e .G/ s .G/

D 3, D 3, D 6, D 5, D 4, D 1, D 10, D 6, D 1, D 8, D 2, D 1, D 3, D 5,

D  D fv2 ; v8 ; v9 g; D  D fv2 ; v8 ; v9 g; D  D fv2 ; v4 ; v5 ; v7 ; v10 ; v12 g; D  D fv2 ; v4 ; v7 ; v10 ; v12 g; D  D fv2 ; v3 ; v8 ; v9 g; infeasible; D  D fv1 ; v2 ; v4 ; v5 ; v6 ; v7 ; v9 ; v10 ; v11 ; v12 g; D  D fv1 ; v2 ; v4 ; v5 ; v12 ; v13 g; infeasible; D  D fv2 ; v4 ; v6 ; v9 ; v12 ; v10 ; v7 ; v5 g; D  D fv6 ; v7 g; D  D fv7 g for k  3; D  D fv2 v4 ; v9 v12 ; v7 v8 g; f  .vi / D 1 for i D 1; 6; 8; 11 and f  .vi / D C1 for other i ;

Algorithmic Aspects of Domination in Graphs

Rom .G/ D 6, 1rain .G/D 3, 2rain .G/D 6, 3rain .G/D 8, 4rain .G/D 10, 5rain .G/D 12,

krain .G/D 13, kpow .G/D 1,

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f  .vi / D 2 for i D 2; 8; 9 and f  .vi / D 0 for other i ; f  .vi / D f1g for i D 2; 8; 9 and f  .vi / D ; for other i ; f  .vi / D f1; 2g for i D 2; 8; 9 and f  .vi / D ; for other i ; f  .vi / D f1g for i D 3; 6; 8; 13, f  .v1 / D f  .v10 / D f2g, f  .v5 / D f  .v11 / D f3g and f  .vi / D ; for other i ; f  .vi / D f1g for i D 3; 6; 8; 13, f  .v1 / D f2g; f  .v10 / D f2; 4g, f  .v5 / D f3; 4g; f  .v11 / D f3g and f  .vi / D ; for other i ; f  .vi / D f1g for i D 3; 6; 8; 13, f  .v1 / D f2g; f  .v10 / D f2; 4; 5g, f  .v5 / D f3; 4; 5g; f  .v11 / D f3g and f  .vi / D ; for other i ; f  .vi / D f1g for all i for k  6; D  D fv2 g for k  1.

The (vertex-)weighted versions of all of the above vertex-subset variations of domination can also be considered. Now, every vertex v has a weight w.v/ of real number. The problem is to find a dominating set D in a suitable variation such that w.D/ D

X v2D

w.v/

is as small as possible. Denote this minimum value by  .G; w/, where  stands for a variation of the domination problem. When w.v/ D 1 for all vertices v, the weighted cases become the cardinality cases. For some variations of domination, the vertex weights may assume to be nonnegative as the following lemma shows. Lemma 1 Suppose G D .V; E/ is a graph in which every vertex is associated with a weight w.v/ of real number. If w0 .v/ D maxfw.v/; 0g for all vertices v 2 V , then for any  2 f;; c; t; k; kpowg we have  .G; w/ D  .G; w0 / C

X w.v/ 0 are integers. Output. A minimum R-dominating set D of T . Method. D ; for i D 1 to n  1 do let vj be the parent of vi ; if (avi  bvi ) then bvj minfbvj ; bvi C 1g; else if (avi D 0) then D D [ fvi g; bvj 1; end if; else if (0 < avi < bvi ) then avj minfavj ; avi  1g; bvj minfbvj ; bvi C 1g; end if; end for; if (avn < bvn ) then D D [ fvn g. The labeling algorithm is also used for many variations of domination. For instance, Mitchell and Hedetniemi [162] and Yannakakis and Gavril [196] gave labeling algorithms for the edge domination problem in trees. Laskar et al. [149] gave a labeling algorithm for the total domination problem in trees. Next, an example of labeling algorithm using strong tree orderings is demonstrated. Chang et al. [47] investigated the k-rainbow domination problem on trees. For technical reasons, they in fact dealed with a more general problem. A krainbow assignment is a mapping L that assigns each vertex v a label L.v/ D .av ; bv / with av ; bv 2 f0; 1; : : : ; kg. A k-L-rainbow dominating function is a function f W V .G/ ! 2f1;2;:::;kg such that for every vertex v in G the following conditions hold:

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(L1) jf .v/j  av . (L2) j [u2N.v/ f .u/j  bv whenever f .v/ D ;. The k-L-rainbow domination number kLrain .G/ of G is the minimum weight of a k-L-rainbow dominating function. A k-L-rainbow dominating function f of G is optimal if w.f / D kLrain .G/. Notice that k-rainbow domination is the same as k-L-rainbow domination if L.v/ D .0; k/ for each v 2 V .G/. Theorem 4 Suppose v is a leaf adjacent to u in a graph G with a k-rainbow assignment L. Let G 0 D G  v and L0 be the restriction of L on V .G 0 /, except that when av > 0 we let bu0 D maxf0; bu  av g. Then the following hold: .1/ If av > 0, then kLrain .G/ D kL0 rain .G 0 / C av . .2/ If av D 0 and au  bv , then kLrain .G/ D kL0 rain .G 0 /. Theorem 5 Suppose N.u/ D fz; v1 ; v2 ; : : : ; vs g such that v1 ; v2 ; : : : ; vs are leaves in a graph G with a k-rainbow assignment L. Assume avi D 0 for 1  i  s and bv1  bv2  : : :  bvs > au . Let b  D minfbvi C i  1 W 1  i  s C 1g D bvi  C i   1, where bvsC1 D au and i  is chosen as small as possible. If G 0 D G  fv1 ; v2 ; : : : ; vs g and L0 is the restriction of L on V .G 0 / with modifications that au0 D bvi  and bu0 D maxf0; bu  i  C 1g, then kLrain .G/ D kL0 rain .G 0 / C i   1. Remark that for the case when the component of G containing u is a star, the vertex z does not exist. There is in fact no vertex vsC1 . The assignment of bvsC1 D au is for the purpose of convenience. In the case of i  D s C 1, it just means that au0 is the same as au . The theorems above then give the following linear-time algorithm for the k-Lrainbow domination problem in trees. Algorithm RainbowDomTreeL. Find the k-L-domination number of a tree. Input. A tree T D .V; E/ in which each vertex v is labeled by L.v/ D .av ; bv /. Output. The minimum k-L-rainbow dominating number r of T . Method. r 0; get a breadth first search ordering x1 ; x2 ; : : : ; xn for the tree T rooted at x1 ; for j D 1 to n do sj 0; fnumber of children xi with axi D 0 and bxi > axj g for j D n to 2 step by 1 do s sj ; v xj ; if s > 0 then fapply Theorem 5g let u D v and z; v1 ; v2 ; : : : ; vs ; b  ; i  be as described in Theorem 5; r r C i   1; au bvi  ; bu maxf0; bu  i  C 1g; end if; else fapply Theorem 4g let u D xj 0 be the parent of v; if av > 0 then f bu maxf0; bu  av g; r r C av g;

Algorithmic Aspects of Domination in Graphs

v1

v2

237

v3

v4

v5 Fig. 6 A tree T with a unique minimum independent dominating set

else if au < bv then sj 0 sj 0 C 1; end else; end do; if ax1 > 0 then r r C ax1 ; else if bx1 > 0 then r

r C 1.

As these algorithms suggest, the labeling algorithm may only work for problems whose solutions have “local property.” For an example of problem without local property, the independent domination problem in the tree T of Fig. 6 is considered. The only minimum independent dominating set of T is fv1 ; v3 g. If the tree ordering Œv1 ; v2 ; v4 ; v3 ; v5  is given, the algorithm must be clever enough to put the leaf v1 into the solution at the first iteration. If another tree ordering Œv5 ; v4 ; v3 ; v2 ; v1  is given, the algorithm must be clever enough not to put the leaf v5 at the first iteration. So, the algorithm must be one that not only looks at a leaf and its only neighbor but also has some idea about the whole structure of the tree. This is the meaning that the solution does not have a “local property.”

3.3

Dynamic Programming for Trees

Dynamic programming is a powerful method for solving many discrete optimization problems; see the books by Bellman and Dreyfus [14], Dreyfus and Law [85], and Nemhauser [167]. The main idea of the dynamic programming approach for domination is to turn the “bottom-up” labeling method into “top-down.” Now a specific vertex u is chosen from G. A minimum dominating set D of G either contains or does not contain u. So it is useful to consider the following two domination problems which are the ordinary domination problem with boundary conditions:  1 .G; u/ D minfjDj W D is a dominating set of G and u 2 Dg:  0 .G; u/ D minfjDj W D is a dominating set of G and u … Dg: Lemma 2 .G/ D minf 1 .G; u/,  0 .G; u/g for any graph G with a specific vertex u.

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u

v

G

H

Fig. 7 The composition of two trees G and H

Suppose H is another graph with a specific vertex v. Let I be the graph with the specific vertex u, which is obtained from the disjoint union of G and H by joining a new edge uv; see Fig. 7. The aim is to use  1 .G; u/;  0 .G; u/;  1 .H; v/, and  0 .H; v/ to find  1 .I; u/ and 0  .I; u/. Suppose D is a dominating set of I with u 2 I . Then D D D 0 [ D 00 , where D 0 is a dominating set of G with v 2 D 0 and D 00 is a subset of V .H / which dominates V .H /  fvg. There are two cases. In the case of v 2 D 00 , D 00 is a dominating set of H . On the other hand, if v … D 00 , then D 00 is a dominating set of H  v. In order to cover the latter case, the following new problem is introduced:  00 .G; u/ D minfjDj W D is a dominating set of G  ug:

Note that  00 .G; u/   0 .G; u/, since a dominating set D of G with u 62 D is also a dominating set of G  u. Theorem 6 Suppose G and H are graphs with specific vertices u and v, respectively. Let I be the graph with the specific vertex u, which is obtained from the disjoint union of G and H by joining a new edge uv. Then the following statements hold: .1/  1 .I; u/ D  1 .G; u/ C minf 1 .H; v/;  00 .H; v/g: .2/  0 .I; u/ D minf 0 .G; u/ C  0 .H; v/;  00 .G; u/ C  1 .H; v/g: .3/  00 .I; u/ D  00 .G; u/ C .H / D  00 .G; u/ C minf 1 .H; v/;  0 .H; v/g:

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239

Proof (1) This follows from the fact that D is a dominating set of I with u 2 D if and only if D D D 0 [ D 00 , where D 0 is a dominating set of G with u 2 D 0 and D 00 is a dominating set of H with v 2 D 00 or a dominating set of H  v. (2) This follows from the fact that D is a dominating set of I with u … D if and only if D D D 0 [ D 00 , where either D 0 is a dominating set of G with u … D 0 and D 00 is a dominating set of H with v 62 D 00 or D 0 is a dominating set of G  u and D 00 is a dominating set of H with v 2 D 00 . (3) This follows from the fact that D is a dominating set of I  u if and only if D D D 0 [ D 00 , where D 0 is a dominating set of G  u and D 00 is a dominating set of H .  The lemma and the theorem above then give the following dynamic programming algorithm for the domination problem in trees. Algorithm DomTreeD. Determine the domination number of a tree. Input. A tree T with a tree ordering Œv1 ; v2 ; : : : ; vn . Output. The domination number .T / of T . Method. for i D 1 to n do  1 .vi / 1I  0 .vi / 1I  00 .v/ 0I end do; for i D 1 to n  1 do let vj be the parent of vi ;  1 .vj /  1 .vj / C minf 1 .vi /;  00 .vi /gI 0  .vj / minf 0 .vj / C  0 .vi /;  00 .vj / C  1 .vi /gI 00  .vj /  00 .vj / C minf 1 .vi /;  0 .vi /gI end do; .T / minf 1 .vn /;  0 .vn /g. The advantage of the dynamic programming method is that it also works for problems whose solutions have no local property. As an example, Beyer et al. [21] solved the independent domination problem by this method. Moreover, the method can be used to solve the vertex-edge-weighted cases. The following derivation for the vertex-edge-weighted domination in trees is slightly different from that given by Natarajan and White [166]: Define  1 .G; u; w; w/;  0 .G; u; w; w/, and  00 .G; u; w; w/ in the same way as 1  .G; u/,  0 .G; u/, and  00 .G; u/, except that jDj is replaced by w.D/. Lemma 3 .G; w; w/ D minf 1 .G; u; w; w/,  0 .G; u; w; w/g for any graph G with a specific vertex u. Theorem 7 Suppose G and H are graphs with specific vertices u and v, respectively. Let I be the graph with the specific vertex u, which is obtained from the

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disjoint union of G and H by joining a new edge uv. The following statements hold: .1/  1 .I; u; w; w/ D  1 .G; u; w; w/ C minf.H; v; w; w/; w.uv/ C  00 .H; v; w; w/g: .2/  0 .I; u; w; w/ D minf 0 .G; u; w; w/ C .H; w; w/;  00 .G; u; w; w/ C w.uv/ C  1 .H; v; w; w/g: .3/  00 .I; u; w; w/ D  00 .G; u; w; w/ C .H; w; w/: The lemma and the theorem above then give the following algorithm for the vertex-edge-weighted domination problem in trees. Algorithm VEWDomTreeD. Determine the vertex-edge-weighted domination number of a tree. Input. A tree T with a tree ordering Œv1 ; v2 ; : : : ; vn , and each vertex v has a weight w.v/ and each edge e has a weight w.e/. Output. The vertex-edge-weighted domination number .T; w; w/ of T . Method. for i D 1 to n do .vi /  1 .vi / w.vi /I 0  .vi / 1I  00 .vi / 0I end do; for i D 1 to n  1 do let vj be the parent of vi ;  1 .vj /  1 .vj /C minf.vi /; w.uv/ C  00 .vi /gI 0  .vj / minf 0 .vj / C .vi /;  00 .vj / C w.uv/ C  1 .vi /gI 00  .vj /  00 .vj / C .vi /I .vj / minf 1 .vj /;  0 .vj /gI end do; .T; w; w/ .vn /. The dynamic programming method is also used in several papers for solving variations of the domination problems in trees; see [12, 96, 114, 122, 184, 201].

3.4

Primal–Dual Approach for Trees

The most beautiful method used in domination may be the primal–dual approach. In this method, besides the original domination problem, the following dual problem is also considered. In a graph G D .V; E/, a 2-stable set is a subset S  V in which every two distinct vertices u and v have distance d.u; v/ > 2. The 2-stability number ˛2 .G/ of G is the maximum size of a 2-stable set in G. It is easy to see the following inequality. Weak Duality Inequality: ˛2 .G/  .G/ for any graph G. Note that the above inequality can be strict, as shown by the n-cycle Cn that ˛2 .Cn / D b n3 c but .Cn / D d n3 e.

Algorithmic Aspects of Domination in Graphs

241

For a tree T , an algorithm which outputs a dominating set D  and a 2-stable set S with jD  j  jS  j is designed. Then 

jS  j  ˛2 .T /  .T /  jD  j  jS  j; which implies that all inequalities are equalities. Consequently, D  is a minimum dominating set, S  is a maximum 2-stable, and the strong duality equality ˛2 .T / D .T / holds. The algorithm starts from a leaf v adjacent to u. It also uses the idea as in the labeling algorithm that u is more powerful than v since N Œv  N Œu. Instead of choosing v, u is put into D  . Besides, v is also put into S  . More precisely, the algorithm is as follows. Algorithm DomTreePD. Find a minimum dominating set and a maximum 2-stable set of a tree. Input. A tree T with a tree ordering Œv1 ; v2 ; : : : ; vn . Output. A minimum dominating set D  and a maximum 2-stable set S  of T. Method. D ; S ; for i D 1 to n do let vj be the parent of vi ; (assume vj D vn for vi D vn ) if (N Œvi  \ D  D ) then D D  [ fvj g;  S S  [ fvi g; end if: end do. To verify the algorithm, it is sufficient to prove that D  is a dominating set, S  is a 2-stable set, and jD  j  jS  j. D  is clearly a dominating set as the if-then statement does. Suppose S  is not a 2-stable set, that is, there exist vi and vi 0 in S  such that i < i 0 but dT .vi ; vi 0 /  2. Let Ti D T Œfvi ; vi C1 ; : : : ; vn g. Then Ti contains vi , vj , and vi 0 . Since d.vi ; vi 0 /  2, the unique vi -vi 0 path in T (and also in T 0 ) is either vi ; vi 0 or vi ; vj ; vi 0 . In any case, vj 2 N Œvi 0 . Thus, at the end of iteration i , D  contains vj . When the algorithm processes vi 0 , N Œvi 0  \ D  ¤  which causes that S  does not contain vi 0 , a contradiction. jD  j  jS  j follows from that when vj is added into D  , which may or may not already be in D  , a new vertex vi is always added into S  . Theorem 8 Algorithm DomTreePD gives a minimum dominating set D  and a maximum 2-stable set S  of a tree T with jD  j D jS  j in linear time. Theorem 9 (Strong Duality) ˛2 .T / D .T / for any tree T .

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The primal–dual approach was in fact used by Farber [90] and Kolen [143] for the weighted domination problem in strongly chordal graphs. It was also used by Cheston et al. [55] for upper fraction domination in trees.

3.5

Power Domination in Trees

Power domination is a most different variation of domination. The observation rules make it so different from the ordinary domination as well as other variations. Haynes et al. [111] gave a linear-time algorithm for the power domination problem in trees. This subsection demonstrates a neat linear-time algorithm offered by Guo et al. [103]. Algorithm PowerDomTree. Find a minimum power dominating set of a tree. Input. A tree T rooted at vertex r. Output. A minimum power dominating set D of T. Method. sort the non-leaf vertices of T in a list L according to a post-order traversal of T ; D ;; while L ¤ frg do v the first vertex in L; L L n fvg; if v has at least two unobserved children then D D [ fvg; exhaustively apply the two observation rules to T ; end if; end while; if r is unobserved then D D [ frg. Theorem 10 Algorithm PowerDomTree gives a minimum power dominating set of a tree in linear time. Proof First is to prove that the output D of the algorithm is a power dominating set. The proof is based on an induction on the depth of the vertices u in T , denoted by depth.u/. Note that the proof works top-down, whereas the algorithm works bottomup. For depth.u/ D 0, it is clear that u, which is r, is observed due to the second “if”-condition of the algorithm. Suppose that all vertices u with depth.u/ < k with k > 0 are observed. Consider a vertex u with depth.u/ D k, which is a child of the vertex v. If v 2 D, then u is observed; otherwise, due to the induction hypothesis, v is observed. Moreover, if depth.v/  1, then the parent of v is observed as well. Because of the first “if”-condition of the algorithm, v is not in D only if v has at most one unobserved child during the “while”-loop of the algorithm processing v. If vertex u is this only unobserved child, then it gets observed by applying OR2 to v, because v itself and all its neighbors (including the parent of v and its children)

Algorithmic Aspects of Domination in Graphs

243

with the only exception of u are observed by the vertices in D. In summary, it is concluded that all vertices in T are observed by D.  Next is to prove the optimality of D by showing a more general statement: Claim Given a tree T D .V; E/ rooted at r, Algorithm PowerDomTree outputs a power dominating set D such that jD \ Vu j  jD 0 \ Vu j for any minimum power dominating set D 0 of T and any tree vertex u, where Vu denotes the vertex set of the subtree of T rooted at u. Proof of the Claim. Let Tu D .Vu ; Eu / denote the subtree of T rooted at vertex u. Let ` denote the depth of T , that is, ` WD maxv2V depth.v/. For each vertex u 2 V , define du WD jD \ Vu j and du0 WD jD 0 \ Vu j. The claim is to be proved by an induction on the depth of tree vertices, starting with the maximum depth ` and proceeding to 0. Since vertices u with depth.u/ D ` are leaves and the algorithm adds no leaf to D, it is the case that du D 0 and so du  du0 for all u with depth.u/ D `. Suppose that du  du0 holds for all u with depth.u/ > k with k < `. Consider a vertex u with depth.u/ D k. Let Cu denote the set of children of u. Then, for all v 2 Cu , by the induction hypothesis, dv  dv0 . In order to show du  du0 , one only has to consider the case that (a) u 2 D,

(b) u … D 0 ,

and (c)

P v2Cu

dv D

P v2Cu

dv0 .

(1)

In all other cases, du  du0 always holds. In the following, it will be shown that this case does not apply. Assume that u ¤ r. The argument works also for u D r. From (c) and that dv  dv0 for all v 2 Cu (induction hypothesis), dv D dv0 for all v 2 Cu . Moreover, (a) is true only if u has two unobserved children v1 and v2 during the “while”-loop of the algorithm processing u (the first “if”-condition). In other words, this means that vertices v1 and v2 are not observed by the vertices in .D \ Vu / n fug. In the following, it is shown that u has to be included in D 0 in order for D 0 to be a valid power dominating set. To this end, the following statement is needed: For all x 2 D 0 \ Vvi with 1  i  2 there is some x 0 2 W.vi ;x/ \ D,

(2)

where W.vi ;x/ denotes the path between vi and x (including vi and x). Without loss of generality, consider only i D 1. Assume that statement (2) is not true for a vertex x 2 D 0 \ Vv1 . Since x … D, one can infer that dx < dx0 . Since no vertex from W.v1 ;x/ is in D and, by induction hypothesis, dy  dy0 for all y 2 Vv1 , it follows that dx 0 < dx0 0 for all vertices x 0 2 W.v1 ;x/ , in particular, dv1 < dv01 . This contradicts the fact that dvi D dv0i for all children vi of u. Thus, statement (2) is true.

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By statement (Eq. 2), for each vertex x 2 D 0 \ Vvi (i 2 f1; 2g), there exists a vertex x 0 2 D on the path W.vi ;x/ . Thus, if vertex vi is observed by a vertex x 2 D 0 \ Vvi , then there is a vertex x 0 2 D \ W.vi ;x/ that observes vi . However, (a) in (Eq. 1) is true only if v1 and v2 are unobserved by the vertices in D \ Vv1 and D \ Vv2 . Altogether, this implies that v1 and v2 cannot be observed by the vertices in D 0 \ Vv1 and D 0 \ Vv2 . Since D 0 is a power dominating set of T , vertex u is taken into D 0 ; otherwise, u has two unobserved neighbors v1 and v2 . OR2 can never be 0 applied to u whatever Pvertices from P V n V0u are in D . It concludes that the case that 0 u 2 D; u … D , and v2Cu dv D v2Cu dv does not apply. This completes the proof of the claim.  Concerning the running time, the following explains how to implement the exhaustive applications of OR1 and OR2. Observe that if OR2 is applicable to a vertex u during the bottom-up process, then all vertices in Tu can be observed at the current stage of the bottom-up process. In particular, after adding vertex u to D, all vertices in Tu can be observed. Thus, exhaustive applications of OR1 and OR2 to the vertices of Tu can be implemented as pruning Tu from T which can be done in constant time. Next, consider the vertices in V n Vu . After adding u to D, the only possible application of OR1 is that u observes its parent v. Moreover, the applicability of OR2 to the vertices x lying on the path from u to r is checked, in the order of their appearance, the first v, and the last r. As long as OR2 is applicable to a vertex x on this path, x is deleted from the list L that is defined in the first line of the algorithm and prune Tx from T . The linear running time of the algorithm is then easy to see: The vertices to which OR2 is applied are removed from the list L immediately after the application of OR2 and are never processed by the instruction in the first line inside the “while”-loop of the algorithm. With proper data structures such as integer counters storing the number of observed neighbors of a vertex, the applications of the observation rules to a single vertex can be done in constant time. Post-order traversal of a rooted tree is clearly doable in linear time. 

3.6

Tree Related Classes

Besides the methods demonstrated in the previous subsections, the “transformation method” sometimes is also used in the study of domination. Roughly speaking, the method transforms the domination problem in certain graphs to another well-known problem, which is solvable. As this method depends on the variation of domination and the type of graphs, it will be mentioned only when it is used in the problem surveyed in this chapter. There are some classes of graphs which have treelike structures, including block graphs, (partial) k-trees, and cacti. A block graph is a graph whose blocks are complete graphs. For results on variations of domination in block graphs, see [39, 43, 126, 130, 202]. A cactus is a graph whose blocks are cycles. Hedetniemi [118] gave a linear-time algorithm for the domination problem in cacti.

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245

For a positive integer k, k-trees are defined recursively as follows: (a) A complete graph of k C 1 vertices is a k-tree; (b) the graph obtained from a k-tree by adding a new vertex adjacent to a clique of k vertices is a k-tree. Partial k-trees are subgraphs of k-trees. For results on variations of domination in k-trees and partial k-trees, see [3, 66, 95, 172, 180, 189, 190].

4

Interval Graphs

4.1

Interval Orderings of Interval Graphs

Recall that an interval graph is the intersection graph of a family of intervals in the real line. In many papers, people design algorithms or prove theorems for interval graphs by using the interval models directly. This very often is accomplished by ordering the intervals according to some nondecreasing order of their right (or left) endpoints. For instance, suppose G D .V; E/ is an interval graph with an interval model fIi D Œai ; bi  W 1  i  ng; where b1  b2      bn . One can solve the domination problem for G by using exactly the same primal–dual algorithm for trees except replacing the 4th line of Algorithm DomTreePD by let j be the largest index such that vj 2 N Œvi . In order to prove that the revised algorithm works for interval graphs, it is only necessary to show that S  is a 2-stable set. Suppose to the contrary that S  contains two vertices vi and vi 0 with i < i 0 and d.vi ; vi 0 /  2, say there is a vertex vk 2 N Œvi \N Œvi 0 . Consider the largest indexed vertex vj of N Œvi  as chosen in iteration i of the algorithm. Since vk 2 N Œvi , k  j . Consider two cases: Case 1. j  i 0 . In this case, k  j  i 0 . Then bk  bj  bi 0 . Since vk 2 N Œvi 0 , Ik intersects Ii 0 and so ai 0  bk . Therefore, ai 0  bj  bi 0 and so Ij intersects Ii 0 . Case 2. i 0 < j . In this case, i < i 0 < j . Then bi  bi 0  bj . Since vj 2 N Œvi , Ii intersects Ij and so aj  bi . Therefore, aj  bi 0  bj and so Ij intersects Ii 0 . In any case, vj 2 N Œvi 0 . As vj is put into D  in iteration i , when the algorithm processes vi 0 , N Œvi 0  \ D  ¤ ; so that S  does not contain vi 0 , a contradiction. Therefore, S  is a 2-stable set. As one can see, the arguments in Cases 1 and 2 are quite similar. This is also true in many other proofs for interval graphs. One may expect that a unified property can be applied. This is in fact the so-called interval ordering in the following

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theorem (see [175]). Notice that once property (IO) below holds, the conclusion vj 2 N Œvi 0  above follows immediately. Theorem 11 G D .V; E/ is an interval graph if and only if G has an interval ordering which is an ordering Œv1 ; v2 ; : : : ; vn  of V satisfying i < j < k and vi vk 2 E imply vj vk 2 E.

.IO/

Proof .)/ Suppose G is the intersection graph of fIi D Œai ; bi  W 1  i  ng: Without loss of generality, assume that b1  b2      bn . Suppose i < j < k and vi vk 2 E. Then bi  bj  bk . Since vi vk 2 E, it is the case that Ii \ Ik ¤ ; which implies that ak  bi . Thus, ak  bj  bk and so Ij \ Ik ¤ ;, that is, vj vk 2 E. .(/ On the other hand, suppose (IO) holds. For any vi 2 V , let i  be the minimum index such that vi  2 N Œvi  and let interval Ii D Œi  ; i . If vi vk 2 E with i < k, then k   i < k and so Ii \ Ik ¤ ;. If Ii \ Ik ¤ ; with i < k, say j 2 Ii \ Ik , then k   j  i < k. Since vk  vk 2 E, by (IO), vi vk 2 E. Therefore, G is an interval graph with fIi D Œi  ; i  W 1  i  ng as an interval model.



The problem of recognizing interval graphs and giving their interval ordering is a fundamental problem. Several linear-time recognition algorithms for interval graphs have been developed in the literature; see Booth and Leuker [26], Korte and Mohring [144], Hsu and his coauthors [123–125,181], and Habib et al. [104], among many others. Note that a linear-time algorithm may not be easily implementable, and so there are still some efforts to search for new (simple) interval graph recognition algorithms and new ways to reconstruct an interval representation or just the interval ordering of a given interval graph. The 6-sweep LBFS algorithm by Corneil et al. [70] is a such one. It is believed that a 3-sweep LBFS algorithm is possible.

4.2

Domatic Numbers of Interval Graphs

Another interesting usage of the interval ordering is the following rewriting for the result on the domatic numbers of interval graphs obtained by Bertossi [24]. He transformed the domatic number problem on interval graphs to a network flow problem as follows. This is a typical example of the transformation method. First, Bertossi’s method is described as follows. Suppose G D .V; E/ is an interval graph, whose vertex set V D f1; 2; : : : ; ng, with an interval model fŒai ; bi  W 1  i  ng. Without loss of generality, assume that

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4

Fig. 8 The construction of H for an interval graph of domatic number 2

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3 2

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no two intervals share a common endpoint and a1 < a2 <    < an . Two “dummy” vertices 0 and n C 1 are added to the graph with b0 < a1 and bn < anC1 . Then an acyclic directed network H is constructed as follows. The vertex set of H is f0; 1; 2; : : : ; n; n C 1g, and there is a directed edge .i; j / in H if and only if j 2 P .i / [ Q.i /, where P .i / D fk W ai < ak < bi < bk g and Q.i / D fk W bi < ak and there is no h with bi < ah < bh < ak g: Figure 8 shows an example of H . Bertossi then proved that any path from vertex 0 to vertex nC1 in H corresponds to a proper dominating set of G and vice versa. His arguments have a flaw. In fact this statement is not true as 0; 2; 3; 5; 6 is a path in the directed network H in Fig. 8, but its corresponding dominating set f2; 3; 5g has a proper subset f2g that is also a dominating set. To be precise, he only showed that (P1) a 0-.n C 1/ path in H corresponds to a dominating set of G, and (P2) a proper dominating set of G corresponds to a 0-.n C 1/ path in H . Besides, the entire arguments can be treated in terms of the interval ordering as follows. Now assume that Œ0; 1; 2; : : : ; n; n C 1 is an interval ordering of the graph G D .V; E/ with two isolated vertices 0 and n C 1 added. Then construct a directed network H 0 with vertex set f0; 1; 2; : : : ; n; n C 1g and edge set fij W i < j and .i < h < j imply ih 2 E or hj 2 E/g: Notice that H 0 is not the same as H . However, statements (P1) and (P2) remain true if H is replaced by H 0 . Also, there is a simpler proof using property (IO), which is different from their original proof using the endpoints of the intervals. The argument is as follows:

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First, a 0-.n C 1/ path P W 0 D i0 ; i1 ; : : : ; ir ; irC1 D n C 1 in H certainly corresponds to a dominating set D D fi0 ; i1 ; : : : ; ir ; irC1 g of G by the definition of the edge set of H 0 . This proves (P1). Conversely, for a proper dominating set D of G, consider the corresponding path P . For any 0  s  r, suppose is < h < isC1 . Since D is a dominating set of G, there exists some ij 2 D such that hij 2 E. Case 1. j  s. Then is h 2 E by (IO). Case 2. j D s C 1. Then hisC1 2 E. Case 3. j > s C 1. If N ŒisC1   N Œij , then D n fvsC1 g is a dominating set of G, violating that D is a proper dominating set. So, there is some vertex k 2 N ŒisC1  n N Œij . The case of isC1 < ij < k or h < k < ij implies k 2 N Œij  by (IO), a contraction. The case of k < h < isC1 implies hisC1 2 E. In any case, is isC1 is an arc in H . This proves (P2). Primal–dual approaches were also used in [156,187] to solve the domatic number problem in interval graphs. Manacher and Mankus [158] made it possible to get an O.n/ algorithm for the problem. Peng and Chang [168] used the primal–dual method to get a linear-time algorithm for the problem in strongly chordal graphs; see Sect. 6.3.

4.3

Weighted Independent Domination in Interval Graphs

There are many algorithms for variants of domination in interval graphs; see [16, 19, 48, 50, 53, 174, 175]. Among them, Ramalingam and Pandu Rangan [175] gave a unified approach to the weighted independent domination, the weighted domination, the weighted total domination, and the weighted connected domination problems in interval graphs by using the interval orderings. Their algorithms are demonstrated in this and the following subsections. Now, suppose G D .V; E/ is an interval graph with vertex set V D f1; 2; : : : ; ng, where Œ1; 2; : : : ; n is an interval ordering of G. Assume that each vertex is associated with a real number as its weight. Notice that except for independent domination, according to Lemma 1, assume that the weights are nonnegative. Consider the following notation:

Vi D f1; 2; : : : ; i g and Gi denotes the subgraph GŒVi  induced byVi : V0 is the empty set: low.i / D minimum element in N Œi : maxlow.i / D maxflow.j / W low.i /  j  i g: Li D fmaxlow.i /; maxlow.i / C 1; : : : ; i g: Mi D fj W j > i and j is adjacent to i g:

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For any family X of sets of vertices, minfX g denotes a minimum weighted set in X . If X is the empty set, then minfX g denotes a set of infinite weight. Notice that the vertices in the set f1; 2; : : : ; low.i /  1g are not adjacent to i and the vertices in flow.i /; low.i / C 1; : : : ; i g are adjacent to i . The vertices in Li form a maximal clique in the graph Gi . Let j be the vertex such that low.i /  j  i and maxlow.i / D low.j /. Then, low.i /  low.j /  j  i . It can easily be seen that N Œj  is a subset of Li [ Mi . Furthermore, in Gi , j is adjacent only to the vertices in Li . Having all of these, it is ready to establish the solutions to weighted independent domination problem in interval graphs. Let IDi denote an independent dominating set of the graph Gi , and let MIDi denote the minimum weighted IDi . Notice that, in Gi , the set Li is a maximal clique and that there is a vertex in Li which is not adjacent to any vertex in Vi n Li . Hence, any IDi contains exactly one vertex j in Li . Furthermore, it is necessary and sufficient that IDi n fj g dominates Vlow.j /1 and contains no vertex adjacent to j . Hence, IDi n fj g is an independent dominating set of Glow.j /1 . In other words, a set is an IDi if and only if it is of the form IDlow.j /1 [ fj g for some j in Li . These give the following lemma: Lemma 4 (a) MID0 D ;. (b) For 1  i  n, MIDi D minfMIDlow.j /1 [ fj g W j 2 Li g: A linear-time algorithm for the weighted independent domination problem in interval graphs then follows. The detailed description is omitted as it is easy. Similarly, in the following three subsections, only recursive formulas for variations of domination in interval graphs are presented.

4.4

Weighted Domination in Interval Graphs

Let Di denote a subset of V that dominates Vi . Unlike independent domination, it is not necessary to restrict Di as a subset of Vi . Let MDi denote a minimum weighted Di . Since there is a vertex in Li whose neighbors are all in Li [ Mi , the set Di contains some vertex j in Li [ Mi . It is necessary and sufficient that the remaining set Di n fj g dominates Vlow.j /1 since j dominates all vertices in Vi n Vlow.j /1 and no vertex in Vlow.j /1 . (Note that if j 2 Li [ Mi , then low.j /  i ). In other words, a set is a Di if and only if it is of the form Dlow.j /1 [ fj g for some j in Li [ Mi . These give the following lemma: Lemma 5 (a) MD0 D ;. (b) For 1  i  n, MDi D minfMDlow.j /1 [ fj g W j 2 Li [ Mi g:

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Weighted Total Domination in Interval Graphs

Let TDi denote a subset of V that totally dominates Vi and let MTDi be a minimum weighted TDi . Let PDi denote a subset of V that totally dominates fi g [ Vlow.i /1 , and let MPDi be a minimum weighted PDi . As in domination, TDi also includes some vertex j in Li [ Mi . If j 2 Li , then it is necessary and sufficient that the set TDi n fj g totally dominates Vlow.j /1 [ fj g. If j 2 Mi , then it is necessary and sufficient that the set TDi n fj g totally dominates Vlow.j /1 . Similarly, any PDi includes some vertex j adjacent to i . By the definition, j  low.i /. Hence, it is necessary and sufficient that the PDi n fj g totally dominates Vminflow.i /1;low.j /1g . These give the following lemma: Lemma 6 (a) MTD0 D ;. (b) For 1  i  n,   [ MTDi D min fMPDj [ fj g W j 2 Li g fMTDlow.j /1 [ fj g W j 2 Mi g : MPDi D minfMTDminflow.j /1;low.i /1g [ fj g W j 2 N.i /g: Note that in the original paper by Ramalingam and Pandu Rangan [175], there is a typo that uses j 2 N Œi  rather than j 2 N.i / in the formula for MPDi .

4.6

Weighted Connected Domination in Interval Graphs

Let CDi denote a connected dominating set of Gi that contains the vertex i , and let MCDi denote a minimum weighted CDi . If low.i / D 1, then MCDi is fi g since all vertices have nonnegative weights. If low.i / > 1, then any CDi contains vertices other than i and hence some vertex adjacent to i in Gi . Let j be the maximum vertex in CDi n fi g. Assume that low.j / < low.i /. Otherwise j is removed to get a CDi of the same or lower weight. If low.j / < low.i /, then any other vertex of Gj adjacent to i is also adjacent to j . Thus, it is necessary and sufficient that CDi n fi g is a CDj . These give the following lemma: Lemma 7 (a) If low.i / D 1, then MCDi D fi g. (b) For low.i / > 1, MCDi D minfMIDj [ fi g W j 2 N Œi  and j < i and low.j / < low.i /g: (c) minfMCDn W i 2 Ln g is a minimum weighted connected dominating set of the graph G.

Algorithmic Aspects of Domination in Graphs

5

Chordal Graphs and NP-Completeness Results

5.1

Perfect Elimination Orderings of Chordal Graphs

251

It was seen in previous sections that the domination problem is well solved for trees and interval graphs. People then try to generalize the results for general graphs. However, the NP-completeness theory raised by Cook suggests that this is in general quite impossible. Garey and Johnson, in an unpublished paper (see [99]), pointed out that the dominating problem is NP-complete by transforming the vertex cover problem to it. Vertex Cover Instance: A graph G D .V; E/ and a positive integer k  jV j. Question: Is there a subset C  V of size at most k such that each edge xy of G has either x 2 C or y 2 C ? Their idea can in fact be modified to prove many NP-complete results for the domination problem and its variations in many classes of graphs. Among these classes, the class of chordal graphs is most interesting in the study of many graph optimization problems. Chordal graphs are raised in the theory of perfect graphs; see [101]. It contains trees, interval graphs, directed path graphs, split graphs, undirected path graphs, etc., as subclasses. Recall that a graph is chordal if every cycle of length at least four has a chord. The following property is an important characterization of chordal graphs. The proof presented here is from Theorem 4.3 in [102], except that the maximum cardinality search is not introduced explicitly. Theorem 12 A graph G D .V; E/ is chordal if and only if it has a perfect elimination ordering which is an ordering Œv1 ; v2 ; : : : ; vn  of V such that i < j < k and vi vj ; vi vk 2 E imply vj vk 2 E:

.PEO/

Proof .)/ For any ordering  D Œv1 ; v2 ; : : : ; vn , define the vector d./ D .dn ; dn1 ; : : : ; d1 /; where each ds D jft W t > s and vt is adjacent to vs gj. Choose an ordering  such that d./ is lexicographically largest. Suppose p < q < r and vr 2 N.vp / n N.vq /. Consider the ordering  0 obtained from  by interchanging vp and vq . Then ds0 D ds for all s > q and jft W t > q and vt 2 N.vp /gj D dq0  dq D jft W t > q and vt 2 N.vq /gj: However, r is a vertex such that r > q and vr 2 N.vp / n N.vq /. Then, there exists some s > q such that vs 2 N.vq / n N.vp /. This gives

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p < q < r; vr 2 N.vp / n N.vq / imply vs 2 N.vq / n N.vp / for some s > q: . / Next, . / is used to prove the following claim: Claim There does not exist any chordless path P W vi1 ; vi2 ; : : : ; vix with x  3 and iy < ix < i1 for 1 < y < x. [Notice that the claim implies (PEO) as vk ; vi ; vj is a chordless path with i < j < k whenever vj vk 62 E.] Suppose to the contrary that such a path P exists. Choose one with a largest ix . Since i2 < ix < i1 and vi1 2 N.vi2 / n N.vix /, by . /, there exists some ixC1 > ix such that vixC1 2 N.vix / n N.vi2 /. Let z be the minimum index such that z  2 and vixC1 vz 2 E. Note that z exists and z  3. For the case when vi1 vixC1 62 E, P 0 W vi1 ; vi2 ; : : : ; viz1 ; viz ; vixC1 (or its inverse) is a chordless path of length at least three with iy < ixC1 < i1 (or iy < i1 < ixC1 ) for 1 < y  z. In this case, ix < ixC1 is a contradiction to the choice of P . For the case when vi1 vixC1 2 E, P 0 together with the edge vi1 vixC1 form a chordless cycle of length at least four, a contradiction to the fact that G is chordal. .(/ On the other hand, suppose (PEO) holds. For any cycle of length at least four, choose the vertex of the cycle with the least index. By (PEO), the two neighbors of this vertex in the cycle are adjacent. 

5.2

NP-Completeness for Domination

This section first demonstrates Garey and Johnson’s proof that the domination problem is NP-complete. The proof is adapted for split graphs, which are special chordal graphs. A split graph is a graph whose vertex set is the disjoint union of a clique C and a stable set S . Notice that a split graph is chordal as the ordering with the vertices in S first and the vertices in C next gives a perfect elimination ordering. Theorem 13 The domination problem is NP-complete for split graphs. Proof The proof is given by transforming the vertex cover problem in general graphs to the domination problem in split graphs. Given a graph G D .V; E/, construct the graph G 0 D .V 0 ; E 0 / with vertex set V 0 D V [ E and edge set E 0 D fv1 v2 W v1 ¤ v2 in V g [ fve W v 2 eg: Notice that G 0 is a split graph whose vertex set V 0 is the disjoint union of the clique V and the stable set E (Fig. 9). If G has a vertex cover C of size at most k, then C is a dominating set of G 0 of size at most k, by the definition of G 0 . On the other hand, suppose G 0 has a dominating set D of size at most k. If D contains any e 2 E, say e D uv, then

Algorithmic Aspects of Domination in Graphs Fig. 9 A transformation to a split graph

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f c

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g d

d G

G⬘

replace e with u to get a new dominating set of size at most k. In this way, assume that D is a subset of V . It is then clear that D is a vertex cover of G of size at most k. Since the vertex cover problem is NP-complete, the domination problem is also NP-complete for split graphs.  Note that the dominating set of G 0 in the proof above in fact induces a connected subgraph. Corollary 1 The total and the connected domination problems are NP-complete for split graphs. In fact, the proof can be modified to get Theorem 14 The domination problem is NP-complete for bipartite graphs. Proof The vertex cover problem in general graphs is transformed to the domination problem in bipartite graphs as follows. Given a graph G D .V; E/, construct the graph G 0 D .V 0 ; E 0 / with vertex set V 0 D fx; yg [ V [ E and edge set E 0 D fxyg [ fyv W v 2 V g [ fve W v 2 eg: Notice that G 0 is a bipartite graph whose vertex set V 0 is the disjoint union of two stable sets fxg [ V and fyg [ E (Fig. 10). If G has a vertex cover C of size at most k, then fyg [ C is a dominating set of G 0 of size at most k C 1. On the other hand, suppose G 0 has a dominating set D of

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Fig. 10 A transformation to a bipartite graph

a

a e

e b

b f

f x

c

y c

g d

g d

G

G⬘

size at most k C 1. Since NG 0 Œx D fx; yg, D must contain x or y. One may assume that D contains y but not x, as .D n fxg/ [ fyg is also a dominating set of size at most k C 1. Since y 2 D, if D contains any e 2 E, say e D uv, one can replace e with u to get a new dominating set of size at most k C 1. In this way, one may assume that D n fyg is a subset of V . It is then clear that D n fyg is a vertex cover of G of size at most k. Since the vertex cover problem is NP-complete, the domination problem is NP-complete for bipartite graphs.  Corollary 2 The total and the connected domination problems are NP-complete for bipartite graphs. There are many other NP-complete results for variations of domination; see [12, 15, 25, 66, 76, 80, 96, 99, 114, 126, 131, 132, 165, 201, 202]. Most of the proofs are more or less similar to the above two. Only very few are proved by using different methods. As an example, Booth and Johnson [25] proved that the domination problem is NP-complete for undirected path graphs, which is another subclass of chordal graphs, by reducing the three-dimensional matching problem to it. Theorem 15 The domination problem is NP-complete for undirected path graphs. Proof Consider an instance of the three-dimensional matching problem, in which there are three disjoint sets W , X , and Y each of size q and a subset M D fmi D .wr ; xs ; yt / W wr 2 W; xs 2 X and yt 2 Y for 1  i  pg of W  X  Y having size p. The problem is to find a subset M 0 of M having size exactly q such that each wr 2 W , xs 2 X , and yt 2 Y occurs in precisely one triple of M 0 .

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{Ai , Bi , Ci : 1 ≤ i ≤ p}

{Ai , Bi , Ci , Di } {Ai , Bi , Di , Fi }

{Ci , Di , Gi }

{R r } {Ai : wr ∈ mi }

{Ss } {Bi : xs ∈ mi }

{Tt } {Ci : yt ∈ mi }

for wr ∈ W

for xs ∈ X

for yt ∈ Y

{Ai , Bi , Ei } {Ai , Ei , Hi }

tree T

{Bi , Ei , Ii }

for mi ∈ M Fig. 11 A transformation to an undirected path graph

Given an instance of the three-dimensional matching problem, construct a tree T having 6p C 3q C 1 vertices from which an undirected path graph G is obtained. The vertices of the tree, which are represented by sets, are explained below. For each triple mi 2 M , there are six vertices that depend only upon the triple itself and not upon the elements within the triple: fAi ; Bi ; Ci ; Di g fAi ; Bi ; Di ; Fi g fCi ; Di ; Gi g fAi ; Bi ; Ei g fAi ; Ei ; Hi g fBi ; Ei ; Ii g

for 1  i  p:

These six vertices form the subtree of T corresponding to mi , which is illustrated in Fig. 11. Next, there is a vertex for each element of W , X , and Y that depends upon the triples of M to which each respective element belongs: fRr g [ fAi W wr 2 mi g

for wr 2 W;

fSs g [ fBi W xs 2 mi g

for xs 2 X;

fTt g [ fCi W yt 2 mi g

for yt 2 Y:

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Finally, fAi ; Bi ; Ci W 1  i  pg is the last vertex of T . The arrangement of these vertices in the tree T is shown in Fig. 11. This then results in an undirected path graph G with vertex set fAi ; Bi ; Ci ; Di ; Ei ; Fi ; Gi ; Hi ; Ii W 1  i  pg [ fRj ; Sj ; Tj W 1  j  qg of size 9p C 3q, where the undirected path in T corresponding to a vertex v of G consists of those vertices (sets) containing v in the tree T . The theorem then follows from the claim that G has a dominating set of size 2p C q if and only if the three-dimensional matching problem has a solution. Suppose D is a dominating set of G of size 2p C q. Observe that for any i , the only way to dominate the vertex set fAi ; Bi ; Ci ; Ci ; Di ; Ei ; Fi ; Gi ; Hi ; Ii g corresponding to mi with two vertices is to choose Di and Ei and that any larger dominating set might just as well consist of Ai , Bi , and Ci , since none of the other possible vertices dominate any vertex outside of the set. Consequently, D consists of Ai , Bi , and Ci for tmi ’s, and Di and Ei for p  t other mi ’s, and at least maxf3.q  t/; 0g Rr ; Ss ; Tt . Then, 2p C q D jDj  3t C 2.p  t/ C 3.q  t/ D 2p C 3q  2t and so t  q. Picking q triples mi for which Ai , Bi , and Ci are in D form a matching M 0 of size q. Conversely, suppose the three-dimensional matching problem has a solution M 0 of size q. Let D D fAi ; Bi ; Ci W mi 2 M 0 g [ fDi ; Ei W mi 2 M n M 0 g: It is straightforward to check that D is a dominating set of G of size 3q C2.pq/ D 2p C q. 

5.3

Independent Domination in Chordal Graphs

Farber [91] showed a surprising result that the independent domination problem is solvable by using a linear programming method. On the other hand, it is known [64] that the weighted independent domination problem is NP-complete. The proof has the same spirit of the proof of Theorem 13. Theorem 16 The weighted independent domination problem is NP-complete for chordal graphs. Proof The vertex cover problem in general graphs is transformed to the weighted independent domination problem in chordal graphs as follows. Given a graph G D .V; E/, construct the following chordal graph G 0 D .V 0 ; E 0 / with

Algorithmic Aspects of Domination in Graphs Fig. 12 A transformation to a weighted chordal graph

257 a⬙

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G

vertex set V 0 D fv00 ; v0 ; v W v 2 V g [ E and edge set E 0 D fv00 v0 ; v0 v W v 2 V g [ fve W v 2 eg [ fe1 e2 W e1 ¤ e2 in Eg: The weight of each e 2 E is 2jV j C 1 and the weight of each vertex in V 0 is 1 (Fig. 12). If G has a vertex cover C of size at most k, then fv00 ; v W v 2 C g[fv0 W v 2 V nC g is an independent dominating set of G 0 with weight at most jV j C k. On the other hand, suppose G 0 has an independent dominating set D of weight at most jV j C k. As k  jV j, D contains no elements in E. Let C D D \ V . It is clear that C is a vertex cover of G. Also, for each v 2 V , the set D contains exactly one vertex in fv00 ; v0 g. Thus, C is of size at most k. Since the vertex cover problem is NP-complete, the weighted independent domination problem is NP-complete for chordal graphs.  Farber’s algorithm for the independent domination problem in chordal graphs is by means of a linear programming method. Suppose G D .V; E/ is a chordal graph with a perfect elimination ordering Œv1 ; v2 ; : : : ; vn  and vertex weights w1 ; w2 ; : : : ; wn of real numbers. Write i  j for vi 2 N Œvj ; ej for i  j and i  j; i< ej for i  j and i  j: i> It follows from the definition of a perfect elimination ordering that each ej g Cj D fvi W i > is a clique. Thus, a set S of vertices is independent if and only if each Cj contains at most one vertex of S . Also, S is a dominating set if and only if for each j , S contains at least one vertex vi with i  j . Consider the following linear problem:

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P1 .G; w/ W Minimize subject to

Pn

i D1

wi xi ;

i j

xi  1 for each j;

ie >j

xi  1 for each j;

P P

xi  0

for each i:

It follows from the above comments that there is a one-to-one correspondence between independent dominating sets of G and feasible 0–1 solutions to P1 .G; w/. Moreover, an optimal 0–1 solution to P1 .G; w/ corresponds to a minimum weighted independent dominating set in G. Notice that a set of vertices of G is an independent dominating set in G if and only if it is a maximal independent set in G. Consequently, there exist independent dominating sets, and P1 .G; w/ is a feasible linear program. It will follow from the algorithm presented below that if every vertex of G has a weight in f1; 0; 1; 2; : : :g, then P1 .G; w/ has an optimal 0–1 solution and that the following dual program has an integer optimal solution: D1 .G; w/ W Maximize

Pn

j D1 .yj

 zj /; P subject to j i yj  je

>j , and Œv1 ; v2 ; : : : ; vn  is a perfect elimination ordering. Hence, ke 0, yk  wi , and ke 0, then yk  wi . Lemma 12 At the end of stage 2, for each j , if g.j / D 0, then xk D 1 for some ke >j . Proof It is clear from the instructions in stage two that, for each i , if xi D 1 at any time during the algorithm, then xi D 1 at the end of the algorithm. Suppose g.j / D 0. If xj was 2 just prior to scanning vj in stage two, then xj was assigned the value of 1 when vj was scanned, and hence, xj D 1 at the end of the algorithm. In this case, choose k D j as desired. Otherwise, xj was 0 prior to scanning vj . In that case, xj D 0 by virtue of the fact that xk D 1 for some previously scanned neighbor vk of vj , that is, xk D 1 for some ke >j .  Proof of Theorem 17. It is easy to see that Algorithm IndDomChordal halts after O.jV j C jEj/ operations. Next is to show that the final values of x1 ; x2 ; : : : ; xn and y1 ; y2 ; : : : ; yn ; z1 ; z2 ; : : : zn are feasible solutions to P1 .G; w/ and D1 .G; w/, respectively, and then to verify that these solutions satisfy the conditions of complementary slackness. It then follows that they are optimal. (i) Feasibility of dual solution: Clearly zj  0 for each j . By Lemmas 8 and 9, yj  0 and f .j /  0 for each j . (ii) Feasibility of primal solution: The instructions in stage two guarantee that if xi D xj D 1, then vi vj 62 E, and if xi D 0, then xj D 1 for some j  i . Since a 0–1 primal solution is feasible if and only if the set fvj W xj D 1g is an independent dominating set in G, it suffices to show that each xi is either 0 ei such that g.j / D 0. or 1. According to Lemma 11, for each i , there is a j < ej , ke According to Lemma 12, there is a ke >j such that xk D 1. Since i > >j , and Œv1 ; v2 ; : : : ; vn  is a perfect elimination ordering, it follows that i  k. If i D k, then xi D 1; otherwise xi D 0. (iii) Complementary slackness: If xi > 0, P then g.i / DP 0 by the choice of xi , and hence, f .i / D 0, by Lemma 10. Thus, j i yj  je 0, then g.j P / D 0 by the choice of zj , and so xk D 1 for some ke >j , by Lemma 12. Hence, ie x  1. Equality follows from the fact that the primal i >j solution is feasible. Suppose yj > 0 but xi D xk D 1 for i  j and k  j . Since xi D xk D 1, g.i / D g.k/ D 0, and so j  minfi; kg by Lemma 8, Lemma 10, and the choice

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ej and ke of yj . Thus, i > >j , which implies i  k since Œv1 ; v2 ; : : : ; vn  is a perfect elimination ordering. On the other P hand, vi vk 62 E since xi D xk D 1. Consequently i D k. Hence, if yj > 0, then i j xi  1. Equality follows from the fact that the primal solution is feasible. 

6

Strongly Chordal Graphs

6.1

Strong Elimination Orderings of Strongly Chordal Graphs

Strongly chordal graphs were introduced by several people [44,93,120] in the study of domination. In particular, most variations of the domination problem are solvable in this class of graphs. There are many equivalent ways to define them. This section adapts the notation from Farber’s paper [93]. A graph G D .V; E/ is strongly chordal if it admits a strong elimination ordering which is an ordering Œv1 ; v2 ; : : : ; vn  of V such that the following two conditions hold: (a) If i < j < k and vi vj ; vi vk 2 E, then vj vk 2 E. (b) If i < j < k < ` and vi vk ; vi v` ; vj vk 2 E, then vj v` 2 E. Notice that an ordering satisfying condition (a) is a perfect elimination ordering. Hence, every strong elimination ordering is a perfect elimination ordering, and every strongly chordal graph is chordal. The notion i  j stands for vi 2 N Œvj . The following equivalent definition of the strong elimination ordering is more convenient in many arguments for strongly chordal graph. Lemma 13 An ordering Œv1 ; v2 ; : : : ; vn  of the vertices of G is a strong elimination ordering of G if and only if i  j; k  `; i  k; i  ` and j  k imply j  `:

.SEO/

Proof Suppose Œv1 ; v2 ; : : : ; vn  is a strong elimination ordering of G. Suppose that i  j , k  `, i  k, i  `, and j  k. If i D j or k D ` or j D `, then clearly j  `. Suppose, on the other hand, that i < j , k < `, and j ¤ `. By symmetry, assume that i  k. Now consider three cases: Case 1. Suppose j D k. Then i < j < ` and vi vj ; vi v` 2 E, whence vj v` 2 E, by (a) in the definition of a strong elimination ordering. Case 2. Suppose j < k. Then i < j < k < ` and vi vk ; vi v` ; vj vk 2 E, whence vj v` 2 E, by (b) in the definition of a strong elimination ordering. Case 3. Suppose k < j . If i D k, then vk v` 2 E. Otherwise, i < k < ` and vi vk ; vi v` 2 E, whence vk v` 2 E. Consequently, vk v` ; vk vj 2 E and either k < ` < j or k < j < `. In either case, vj v` 2 E. This completes the proof of necessity. The proof of sufficiency is trivial and thus omitted. 

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Weighted Domination in Strongly Chordal Graphs

There are quite a few algorithms designed for variants of the domination problem in strongly chordal graphs; see [35, 38, 44, 45, 93, 120, 145, 194]. This section presents the linear algorithm, given by Farber [93], for locating a minimum weighted dominating set in a strongly chordal graph. Let G D .V; E/ be a strongly chordal graph with a strong elimination ordering Œv1 ; v2 ; : : : ; vn  and vertex weights w1 ; w2 ; : : : ; wn . According to Lemma 1, assume that these vertex weights are nonnegative. Consider the following linear problem: P2 .G; w/ W Minimize subject to

Pn

i D1

wi xi ;

i j

xi  1 for each j;

P

xi  0

for each i:

By definition, a set S of vertices of G is a dominating set if and only if, for each j , S contains some vertex vi such that i  j . Consequently, there is a one-to-one correspondence between feasible 0–1 solutions to P2 .G; w/ and dominating sets in G. Moreover, an optimal 0–1 solution to P2 .G; w/ corresponds to a minimum weighted dominating set in G. It follows from the algorithm below that P2 .G; w/ has an optimal 0–1 solution and that the following dual program has an optimal solution: P D2 .G; w/ W Maximize nj D1 yj ; P subject to j i yj  wi for each i; yj  0

for each j:

The algorithm presented below solves the linear programs P2 .G; w/ and D2 .G; w/. To simplify the presentation of the algorithm, define a function h and a family of sets. For each i , let h.i / D wi 

X j i

yj and Ti D fj W i  j and yj > 0g:

Note that h.i / is the slack in the dual constraint associated with vertex vi , and Ti is the set of constraints in P2 .G; w/ containing xi which must be at equality to satisfy the conditions of complementary slackness. The algorithm has two stages. Stage 1 finds a feasible solution to D2 .G; w/ by scanning the vertices in the order v1 ; v2 ; : : : ; vn ; and stage 2 uses this solution to find a feasible 0–1 solution to P2 .G; w/ by scanning the vertices in the order vn ; vn1 ; : : : ; v1 . This algorithm uses a set T to assure that the conditions of complementary slackness are satisfied. Initially, T D f1; 2; : : : ; ng, each yj D 0, and each xi D 0. When vj is scanned in stage 1, yj increases as much as possible without violating the dual constraint. In stage 2, if h.i / D 0 and Ti  T when vi is scanned, then let xi D 1 and replace T by T n Ti . Otherwise xi remains 0. A more formal description of the algorithm follows.

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Algorithm WDomSC. Determine .G; w/ for a strongly chordal graph G with nonnegative vertex weights w1 ; w2 ; : : : ; wn . Input. A strongly chordal graph G with a strong elimination ordering Œv1 , v2 , : : : , vn  and nonnegative vertex weights w1 ; w2 ; : : : ; wn . Output. Optimal solutions to P2 .G; w/ and D2 .G; w/. Method. T f1; 2; : : : ; ng; each yj 0; each xi 0; Stage 1: for j D 1 to n do yj minfh.k/ W k  j g; Stage 2: for i D n to 1 by 1 do if (h.i / D 0 and Ti  T ) then xi 1; T T n Ti ; end if. The algorithm is verified as follows. Theorem 18 Algorithm WDomSC finds a minimum weighted dominating set of a strongly chordal graph with nonnegative vertex weights in linear time provided that a strong elimination ordering is given. Proof It is easy to see that the algorithm halts after O.jV j C jEj/ operations. In order to show that the final values of x1 ; x2 ; : : : ; xn and y1 ; y2 ; : : : ; yn ; z1 ; z2 ; : : : zn are optimal solutions to P2 .G; w/ and D2 .G; w/, respectively, it suffices to show that these solutions are feasible and that they satisfy the conditions of complementary slackness. (i) Feasibility of dual solution: The instructions in stage 1 guarantee that h.i /  0 for each i and yj  0 for each j . (ii) Feasibility of primal solution: Clearly, each xi is either 0 or 1. Thus, it suffices to show that for each j , xi D 1 for some i  j . By the choice of yj , there is a k  j such that h.k/ D 0 and max Tk  j . If xk D 1, then the claim holds. Otherwise, by the algorithm, Tk was not contained in T when vk was scanned in stage 2. Since, in stage 2, the vertices are scanned in the order vn ; vn1 ; : : : ; v1 , there is some ` > k such that x` D 1 and T` \ Tk ¤ ;. Let i 2 T` \ Tk . Then i  j since max Tk  j . Thus, i  j , k < `, i  k, i  `, and j  k, whence `  j by Lemma 13, since Œv1 ; v2 ; : : : ; vn  is a strong elimination ordering. Hence, `  j and x` D 1. Consequently, the primal solution is feasible. (iii) P Complementary slackness: If xi > 0, then x1 D 1 and so h.i / D 0, that is, j i yj D wi . Suppose yj > 0. PIt is clear from the instructions that if xi D xk D 1, then Ti \ Tk D ;. Thus, i j xi  1. Equality follows from the feasibility of the primal solution. 

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Farber in fact also gave an algorithm for the weighted independent domination problem with arbitrary real weights for strongly chordal graphs. The approach is similar to that for chordal graphs (with restricted weights) in Sect. 5.2. The only difference is that the function g.i / is replaced by a set Si D fj W i  j and yj > 0g which is similar to Ti in this section. The development is similar to that in Sect. 5.2 and thus omitted.

6.3

Domatic Partition in Strongly Chordal Graphs

Peng and Chang [169] gave an elegant algorithm for the domatic partition problem in strongly chordal graphs. Their algorithm uses a primal–dual approach. Suppose Œv1 ; v2 ; : : : ; vn  is a strong elimination ordering of G D .V; E/ with the minimum degree ı.G/. Choose a vertex x of degree ı.G/. As any dominating set Di in a domatic partition of G contains at least one vertex vi in N Œx and two distinct Di have different corresponding vi , it is easy to see the following inequality. Weak Duality Inequality: d.G/  ı.G/ C 1. Their algorithm maintains ı.G/ C 1 disjoint sets. Initially, these sets are empty. The algorithm scans the vertices in the reverse order of the strong elimination ordering. A vertex is included in a set when it is scanned. When the algorithm terminates, these ı.G/ C 1 sets are dominating sets. A vertex v is completely dominated if v is dominated by all of these ı.G/ C 1 dominating sets. Algorithm DomaticSC. Determine a domatic partition of a strongly chordal graph G of size ı.G/ C 1. Input. A strongly chordal graph G D .V; E/ with a strong elimination ordering Œv1 , v2 , : : : , vn . Output. A partition of V into ı.G/ C 1 disjoint dominating sets of G. Method. Si ; for 1  i  ı.G/ C 1; for i D n to 1 step 1 do find the largest k  i such that vk is not completely dominated; let S` be a set that does not dominate vk ; S` fvi g [ S` ; if no such set exists then include vi to an arbitrary S` ; end do. Before proving the correctness of the algorithm, two lemmas are needed. Lemma 14 Assume S`  fvi C1 ; vi C2 ; : : : ; vn g and k  i , where 1  i  n. If S` does not dominate vk , then S` does not dominate vj for all j  k with j  i .

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Proof Suppose to the contrary that S` has a vertex vp dominating vj , that is, i < p and p  j . Then i < p, j  k, i  j , i  k, and p  j imply p  k by (SEO), which contradicts that S` does not dominate vk .  Let r.v/ D jfx 2 N Œv W x is not in any of the ı.G/ C 1 setsgj and ndom.v/ be the number of sets that do not dominate v during the execution of Algorithm DomaticSC. Lemma 15 Algorithm DomaticSC maintains the following invariant: r.vj /  ndom.vj / for all j 2 f1; 2; : : : ; ng: Proof The lemma is proved by induction. Initially, r.vj / D deg.vj / C 1  ı.G/ C 1 D ndom.vj / for all vj 2 V . During iteration i , only values of r.vj / and ndom.vj /, where j  i , may be altered when vi is included in a set S` . Notice that the algorithm determines the largest index k  i such that vk is not completely dominated. It then finds a set S` that does not dominate vk (S` is chosen arbitrarily when vk does not exist). For any j  i with j  k, by Lemma 14, vj was not dominated by S` . Therefore, r.vj / and ndom.vj / are decremented by one after vi is included in S` . On the other hand, for any j  i with j > k (or nonexistence of such vk ), by the choice of the vertex vk in the algorithm, vertex vj is completely dominated, that is, ndom.vj / D 0. Thus, the invariant is maintained.  Theorem 19 Algorithm DomaticSC partitions the vertex set of a strongly chordal graph G D .V; E/ into d.G/ D ı.G/ C 1 disjoint dominating sets in linear time provided that a strong elimination ordering is given. Proof Upon termination of the algorithm, r.vj / D 0 for all j 2 f1; 2; : : : ; ng. According to Lemma 15, ndom.vj / D 0 for all vj in V . That is, these ı.G/ C 1 sets are dominating sets of G. The strong duality equality d.G/ D ı.G/ C 1 then follows from the weak duality inequality. To implement that algorithm efficiently, each vertex vi is associated with a variable ndom.i / and an array L.i / of size ı.G/ C 1. Initially, ndom.i / D ı.G/ C 1 and the values of entries in Li are all zero. Thus, for each vertex it takes O.di / time to test ndom.i / to determine vk , where di is the degree of vi . It then takes

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O.ı.G/ C 1/ time to decide which set vi should go. Finally, for each vj 2 N Œvi , it takes O.1/ time to update ndom.j / and Lj . Therefore, the algorithm takes

O

7

Xn i D1

! .di Cı.G/C1/ D O.jV jCjEj/ time:



Permutation Graphs

Given a permutation  D ..1/; .2/; : : : ; .n// on the set In D f1; 2; : : : ; ng, the permutation graph of  is the graph G./ D .In ; E.// with E./ D fj k W .j  k/. 1 .j /   1 .k// < 0g: Note that  1 .j / is the position of j in the permutation . Figure 13 illustrates a permutation and its corresponding permutation graph. If a line between each integer i and its position in  is drown, then n lines are created, each with an associated integer. In this way, two vertices j and k are adjacent in G./ if and only if their corresponding lines cross. That is, G./ is the intersection graph of these n lines. Notice that an independent set in G./ corresponds to an increasing subsequence of , and a clique in G./ corresponds to a decreasing subsequence of . Permutation graphs were first introduced by Pnueli et al. in [87, 171]. Since that time quite a few polynomial-time algorithms have been constructed on permutation graphs. For example, Atallah et al. [8], Brandst¨adt and Kratsch [31]; Farber and Keil [94]; and Tsai and Hsu [191] have constructed polynomial domination and independent domination algorithms, Brandst¨adt and Kratsch [31] and Colbourn and Stewart [78] have constructed polynomial connected domination algorithms, and Brandst¨adt and Kratsch [31] and Corneil and Stewart [78] have constructed polynomial total domination algorithms. This section presents a simple O.n2 /-time algorithm, due to Brandst¨adt and Kratsch [32] for finding a minimum weighted independent dominating set in a permutation graph. Assume that the defining permutation  of the permutation graph is given as part of the input. Spinrad [186] has shown that  can be constructed in O.n2 / time, given the graph G. This algorithm takes advantage of the observation that a set is an independent dominating set if and only if it is a maximal independent set. Since maximal independent sets in permutation graphs correspond to maximal increasing subsequences in , all that is necessary is to search for such a sequence in  of minimum weight. In particular, it determines, for every j , 1  j < n, the minimum weight i .j; w/ of an independent dominating set in the subsequence .1/; .2/; : : : ; .j /, which contains .j / as the rightmost element. Let w.j / denote the weight of vertex j .

Algorithmic Aspects of Domination in Graphs Fig. 13 A permutation and its corresponding permutation graph

1

π = (2

2

3

267

3

6

4

4

5

1

6

8

7

7

2

1

3

4

6

7

5

8

8

5)

G(π)

Algorithm WIndDomPer. Solve the weighted independent domination problem for permutation graphs. Input. A permutation graph G with its corresponding permutation  on the set f1; 2; : : : ; ng and vertex weights w.1/; w.2/; : : : ; w.n/ of real numbers. Output. The weighted independent domination number i .G; w/ of G. Method.

for j D 1 to n do p.j / 0; i .j / D w..j //; end do; for k D j  1 to 1 step 1 do if (.j / > .k/ and p.j / D 0) then i .j / w..j // C i .k/; p.j / .k/; end if; else if .j / > .k/ > p.j / > 0 then i .j / minfi .j /; w..j // C i .k/g; p.j / .k/; end if end do; m p.n/; i .G; w/ i .n/; for j D n  1 to 1 step 1 do if .j / > m then i .G; w/ minfi .G; w/; i .j /g; m .j /; end if.

The algorithm is illustrated by the permutation graph G./ in Fig. 13, where  D .2 3 6 4 1 8 7 5/ and all weights are equal to 1: i .j / D .1 2 3 3 1 2 2 2/, and thus, the minimum size of an independent dominating set is 2, for example, the set f1; 5g.

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Cocomparability Graphs

A graph G D .V; E/ is a comparability graph if G has an orientation H D .V; F / such that xy; yz 2 F imply xz 2 F . In other words, G has a comparability ordering which is an ordering Œv1 ; v2 ; : : : ; vn  of V satisfying i < j < k and vi vj ; vj vk 2 E imply vi vk 2 E:

.CO/

A graph G D .V; E/ is a cocomparability graph if its complement G is a comparability graph or, equivalently, G has a cocomparability ordering which is an ordering Œv1 ; v2 ; : : : ; vn  of V satisfying i < j < k and vi vk 2 E imply vi vj 2 E or vj vk 2 E:

.CCO/

There is an O.n2:376 /-time recognition algorithm for comparability graphs and thus for cocomparability graphs [186]. This has been improved by McConnell and Spinrad who gave an O.n C m/-time algorithm constructing an orientation of any given graph G such that the orientation is a transitive orientation of G if and only if G has a transitive orientation [160]. Unfortunately, the best algorithm for testing whether the orientation is indeed transitive has running time O.n2:376 /. The class of cocomparability graphs is a well-studied superclass of the classes of interval graphs, permutation graphs, and trapezoid graphs. Domination problems on cocomparability graphs were considered for the first time by Kratsch and Stewart [148]. They obtained polynomial-time algorithms for the domination/total domination/connected domination and the weighted independent domination problems in cocomparability graphs. These algorithms are designed by dynamic programming using cocomparability orderings. Breu and Kirkpatrick [34] (see [4]) improved this by giving O.nm2 /-time algorithms for the domination and the total domination problems and an O.n2:376 /-time algorithm for the weighted independent domination problem in cocomparability graphs. On the other hand, the weighted domination, weighted total domination, and weighted connected domination problems are NP-complete in cocom‘parability graphs [49]. Also, the problem “Given a cocomparability graph G, does G have a dominating clique?” is NP-complete [148]. An O.n3 /-time algorithm computing a minimum cardinality connected dominating set of a connected cocomparability graph has been given in [148]. The following is the algorithm for this problem given by Breu and Kirkpatrick [34] (see also [4]). Let Œv1 ; v2 ; : : : ; vn  be a cocomparability ordering of a cocomparability graph G D .V; E/. For vertices u; w 2 V , write u < w if u appears before w in the ordering, that is, u D vi and w D vj implies i < j . For i  j the set fvk W i  k  j g is denoted by Œvi ; vj . Then vi vj 2 E implies that every vertex vk with i < k < j is adjacent to vi or to vj ; thus, fvi ; vj g dominates Œvi ; vj . This can be generalized as follows: Let S  V where GŒS  is connected. Then S dominates

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269

Œmin.S /; max.S / where min.S / (respectively, max.S /) is the vertex of S with the smallest (respectively, largest) index in the ordering. The following theorem and lemma are given in [148]. Theorem 20 Any connected cocomparability graph G has a minimum connected dominating set S such that the induced subgraph GŒS  is a chordless path p1 ; p2 ; : : : ; pk . Lemma 16 Suppose S  V is a minimum connected dominating set of a cocomparability graph G D .V; E/ with a cocomparability ordering Œv1 ; v2 ; : : : ; vn . If GŒS  is a chordless path p1 ; p2 ; : : : ; pk , then every vertex x < min.S / is dominated by fp1 ; p2 g and every vertex y > max.S / is dominated by fpk1 ; pk g.

The following approach enables an elegant way of locating a chordless path of minimum size that dominates the cocomparability graph. A source vertex is a vertex vi such that vk vi 2 E for all k < i , and a sink vertex is a vertex vj such that vj vk 2 E for all k > j . Then Œv1 ; v2 ; : : : ; vn  is a canonical cocomparability ordering if Œv1 ; v2 ; : : : ; vr , 1  r < n, are the source vertices and vs ; vsC1 ; : : : ; vn , 1 < s  n, are the sink vertices. Note that every cocomparability graph G has a canonical cocomparability ordering. Furthermore, given any cocomparability ordering, a canonical one can be computed in time O.n C m/. From now on, assume that Œv1 ; v2 ; : : : ; vn  is a canonical cocomparability ordering. Since the source vertices of G form a clique, any source vertex vi dominates Œv1 ; vi . Analogously, since the sink vertices of G form a clique, any sink vertex vj dominates Œvj ; vn . Therefore, the vertex set of every path between a source vertex and a sink vertex is dominating. The following theorem given in [34] enlights the key property. Theorem 21 Every connected cocomparability graph G D .V; E/ satisfying c .G/  3 has a minimum connected dominating set which is the vertex set of a shortest path between a source and a sink vertex of G. Proof Let Œv1 ; v2 ; : : : ; vn  be a canonical cocomparability ordering of G. According to Theorem 20, there is a minimum connected dominating set S of G such that GŒS  is a chordless path P W p1 ; p2 ; : : : ; pk , k  3. Construct below a chordless path P 00 between a source vertex and a sink vertex of G that has the same number of vertices as the path P . Let p1 D vi and p2 D vj . First observe that p2 D vj cannot be a source vertex, otherwise N Œp2  Œv1 ; vj  implying that fp2 ; p3 ; : : : ; pk g is also a connected dominating set of G, a contradiction. If p1 is a source vertex, then P starts at a source vertex. In this case, proceed with the path P 0 D P (possibly) rearranging pk1 ; pk . Suppose p1 D vi is not a source vertex. Then there is a source vertex u of G with up1 62 E. Since Œv1 ; v2 ; : : : ; vn  is a canonical cocomparability ordering and since

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p1 and p2 are not source vertices, u < p1 and u < p2 . Since vi vj 2 E and by Lemma 16, fvi ; vj g dominates Œ1; maxfvi ; vj g. Consequently up2 2 E. Consider the set S 0 D fu; p2 ; : : : ; pk g. Since fu; p2 g dominates Œv1 ; vj , S 0 is a dominating set. Since P W p1 ; p2 ; : : : ; pk is a chordless path, t  3 implies pt u 62 E. Thus, S 0 induces the chordless path P 0 W u; p2 ; : : : ; pk . Similarly, starting from P 0 the vertex pk can be replaced, if necessary. Vertex pk1 is not a sink vertex. If pk is a sink vertex, then S 00 D S 0 and P 00 D P 0 . Otherwise, replace pk by a sink vertex v satisfying vpk 62 E to obtain S 00 and P 00 . GŒS 00  induces a chordless path between a sink and a source vertex. The vertex set of any path between a sink and a source vertex is a dominating set. By construction S 00 is a minimum connected dominating set. Consequently S 00 is the vertex set of a shortest path between a source vertex and a sink vertex of G.  According to Theorem 21, when c .G/  3, computing a minimum connected dominating set of a connected cocomparability graph G reduces to computing a shortest path between a source and a sink vertex of G. Algorithm ConDomCC. Solve the connected domination problem in cocomparability graphs. Input: A connected cocomparability graph G D .V; E/ and a canonical cocomparability ordering Œv1 ; v2 ; : : : ; vn  of G. Output: A minimum connected dominating set of G. Methods. 1. Check whether G has a minimum connected dominating set D of size at most 2. If so, output D and stop. 2. Construct a new graph G 0 by adding two new vertices s and t to G such that s is adjacent exactly to the source vertices of G and t is adjacent exactly to the sink vertices of G. 3. Compute a shortest path P W s; p1 ; p2 ; : : : ; pk ; t between s and t in G 0 by the breadth-first search. 4. Output fp1 ; p2 ; : : : ; pk g. The correctness of Algorithm ConDomCC follows immediately from Theorem 21. The “almost linear” running time of the algorithm follows from the well-known fact that breadth-first search is a linear-time procedure. Theorem 22 For any connected cocomparability graph G D .V; E/ with a canonical cocomparability ordering Œv1 ; v2 ; : : : ; vn , Algorithm ConDomCC outputs a minimum connected dominating set of G in O.nm/ time. In fact, all parts of the algorithm except checking for a connected dominating set of size two can be done in time O.n C m/. It is clearly unsatisfactory that the straightforward test for a connected dominating set of size two dominates the overall running time. The crux is that there are even permutation graphs for which each minimum connected dominating set of size two contains neither a source nor a sink vertex (see [127, 142]). It seems that minimum

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271

dominating sets of this type cannot be found by a shortest path approach. It is an open question whether step 1 of Algorithm ConDomCC can be implemented in a more efficient way. Notice that the O.n/-time algorithms computing a minimum connected dominating set for permutation graphs [127] and trapezoid graphs [142] both rely on Theorem 21. Corneil et al. have done a lot of research on asteroidal triple-free graphs, usually called AT-free graphs [72, 75]. They are defined as those graphs not containing an asteroidal triple, that is, a set of three vertices such that between any two of the vertices, there is a path avoiding the neighborhood of the third. AT-free graphs form a superclass of the cocomparability graphs. They are a “large class of graphs” with nice structural properties, and some of them are related to domination. One of the major theorems on the structure of AT-free graphs states that every connected AT-free graph has a dominating pair, that is, a pair of vertices u; v such that the vertex set of each path between u and v is a dominating set. An O.n C m/ algorithm computing a dominating pair for a given connected ATfree graph has been presented in [75]. This can be used to obtain an O.n C m/ algorithm computing a dominating path for connected AT-free graphs (see also [73]). An O.n3 / algorithm computing a minimum connected dominating set for connected AT-free graphs is given in [10]. An O.n C m/ algorithm computing a minimum connected dominating set in connected AT-free graphs with diameter greater than three is given in [75].

9

Distance-Hereditary Graphs

9.1

Hangings of Distance-Hereditary Graphs

A graph is distance hereditary if every two vertices have the same distance in every connected induced subgraph. Distance-hereditary graphs were introduced by Howorka [121]. The characterization and recognition of distance-hereditary graphs have been studied in [11, 81, 82, 105, 121]. Distance-hereditary graphs are parity graphs [36] and include all cographs [68, 77]. The hanging hu of a connected graph G D .V; E/ at a vertex u 2 V is the collection of sets L0 .u/, L1 .u/, . . . , Lt .u/ (or L0 , L1 , . . . , Lt if there is no ambiguity), where t D maxv2V dG .u; v/ and Li .u/ D fv 2 V W dG .u; v/ D i g for 0  i  t. For any 1  i  t and any vertex v 2 Li , let N 0 .v/ D N.v/ \ Li 1 . A vertex v 2 Li with 1  i  t is said to have a minimal neighborhood in Li 1 if N 0 .x/ is not a proper subset of N 0 .v/ for any x 2 Li . Such a vertex v certainly exists.

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Theorem 23 ([11, 81, 82]) A connected graph G D .V; E/ is distance hereditary if and only if for every hanging hu D .L0 ; L1 ; : : : ; Lt / of G and every pair of vertices x, y 2 Li .1  i  t/ that are in the same component of G  Li 1 , we have N 0 .x/ D N 0 .y/. Theorem 24 ([11]) Suppose hu D .L0 ; L1 ; : : : ; Lt / is a hanging of a connected distance-hereditary graph at u. For any two vertices x; y 2 Li with i  1, N 0 .x/ \ N 0 .y/ D ; or N 0 .x/  N 0 .y/ or N 0 .x/ N 0 .y/. Theorem 25 (Fact 3.4 in [105]) Suppose hu D .L0 ; L1 ; : : : ; Lt / is a hanging of a connected distance-hereditary graph at u. If vertex v 2 Li with 1  i  t has a minimal neighborhood in Li 1 , then NV nN 0 .v/ .x/ D NV nN 0 .v/ .y/ for every pair of vertices x and y in N 0 .v/.

9.2

Weighted Connected Domination

D’Atri and Moscarini [81] gave O.jV jjEj/ algorithms for connected domination and Steiner tree problems in distance-hereditary graphs. Brandst¨adt and Dragan [29] presented a linear-time algorithm for the connected r-domination and Steiner tree problems in distance-hereditary graphs. This section presents a linear-time algorithm given by Yeh and Chang [200] for finding a minimum weighted connected dominating set of a connected distancehereditary graph G D .V; E/ in which each vertex v has a weight w.v/ that is a real number. According to Lemma 1, assume that the vertex weights are nonnegative. Lemma 17 Suppose hu D fL0 ; L1 ; : : : ; Lt g is a hanging of a connected distancehereditary graph at u. For any connected dominating set D and v 2 Li with 2  i  t, D \ N 0 .v/ ¤ ;. Proof Choose a vertex y in D that dominates v. Then y 2 Li 1 [ Li [ Li C1 . If y 2 Li 1 , then y 2 D \ N 0 .v/. So, assume that y 2 Li [ Li C1 . Choose a vertex x 2 D \ .L0 [ L1 / and an x-y path P W x D v1 ; v2 ; : : : ; vm D y using vertices only in D. Let j be the smallest index such that fvj ; vj C1 ; : : : ; vm g  Li [ Li C1 [ : : : [ Lt . Then vj 2 Li , vj 1 2 N 0 .vj /, and v and vj are in the same component of G Li 1 . By Theorem 23, N 0 .v/ D N 0 .vj / and so vj 1 2 D\N 0 .v/. In any case, D \ N 0 .v/ ¤ ;.  Theorem 26 Suppose G D .V; E/ is a connected distance-hereditary graph with a nonnegative weight function w on its vertices. Let hu D fL0 , L1 ,. . . , Lt g be a hanging at a vertex u of minimum weight. Consider the set A D fN 0 .v/: v 2 Li with 2  i  t and v has a minimal neighborhood in Li 1 g. For each N 0 .v/ in A,

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choose one vertex v in N 0 .v/ of minimum weight, and let D be the set of all such v . Then D or D [ fug or some fvg with v 2 V is a minimum weighted connected dominating set of G. Proof For any x 2 Li with 2  i  t, by Theorem 24, N 0 .x/ includes some N 0 .v/ in A. This gives Claim 1. Claim 1 For any x 2 Li with 2  i  t, x 2 N ŒLi 1 \ D. Claim 2 D [ fug is a connected dominating set of G. Proof of Claim 2. By Claim 1 and N Œu D L1 [ fug, D [ fug is a dominating set of G. Also, by Claim 1, for any vertex x in D [ fug, there exists an x-u path using vertices only in D [ fug, that is, GŒD [ fug is connected.  Suppose M is a minimum weighted connected dominating set of G. According to Lemma 17, M \ N 0 .v/ ¤ ; for each N 0 .v/ 2 A, say v 2 M \ N 0 .v/. Since any two sets in A are disjoint, jM j  jAj D jDj. Case 1. jM j D 1. The theorem is obvious in this case. Case 2. jM j > jDj. In this case, there is at least one vertex x in M that is not a v . Then w.M / 

X v

w.v / C w.x/ 

X v

w.v / C w.u/ D w.D [ fug/:

This together with Claim 2 gives that D [ fug is a minimum weighted connected dominating set of G.  Case 3. jM j D jDj  2. Since A contains pairwise P Pdisjoint sets, M D fv W 0  N .v/ 2 Ag. Then w.M / D v w.v /  v w.v / D w.D/. For any two vertices x  and y  in D, x  and y  are in M . Since GŒM  is connected, there is an x  -y  path in GŒM :    x  D v 0 ; v1 ; : : : ; vn D y :

For any 1  i  n, since vi and v are both in N 0 .vi / 2 A, by Theorem 25, i     NV nN 0 .vi / .vi / D NV nN 0 .vi / .vi /. But v i 1 2 NV nN 0 .vi / .vi /. Therefore, vi 1 2      NV nN 0 .vi / .vi / and vi 2 NV nN 0 .vi 1 / .vi 1 /. Also, that vi 1 and vi 1 are both in  N 0 .vi 1 / 2 A implies that NV nN 0 .vi 1 / .vi 1 / D NV nN 0 .vi 1 / .v i 1 /. Then vi 2    NV nN 0 .vi 1 / .vi 1 /. This proves that vi 1 is adjacent to vi for 1  i  n, and then x  D v0 ; v1 ; : : : ; vn D y  is an x  -y  path in GŒD, that is, GŒD is connected. For any x in V , since M is a dominating set, x 2 N Œv  for some N 0 .v/ 2 A. Note that v and v are both in N 0 .v/. According to Theorem 25, NV nN 0 .v/ .v / D NV nN 0 .v/ .v /. In the case of x 62 N 0 .v/, x 2 N Œv  implies x 2 N Œv , that is, D

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dominates x. In the case of x 2 N 0 .v/, NV nN 0 .v/ .v / D NV nN 0 .v/ .x/. Since GŒD is connected and jDj  2, v is adjacent to some y  2 D n N 0 .v/. Then x is also adjacent to y  , that is, D dominates x. In any case, D is a dominating set. Therefore, D is a minimum weighted connected dominating set of G.  Lemma 1 and Theorem 26 together give an efficient algorithm for the weighted connected domination problem in distance-hereditary graphs as follows. To implement the algorithm efficiently, the set A is not actually found. Instead, the following step is performed for each 2  i  t. Sort the vertices in Li such that jN 0 .x1 /j  jN 0 .x2 /j      jN 0 .xj /j: Then process N 0 .xk / for k from 1 to j . At iteration k, if N 0 .xk / \ D D ;, then N 0 .xk / is in A, and choose a vertex of minimum weight to put it into D; otherwise, N 0 .xk / 62 A and do nothing. Algorithm WConDomDH. Find a minimum weighted connected dominating set of a connected distance-hereditary graph. Input: A connected distance-hereditary graph G D .V; E/ and a weight w.v/ of real number for each v 2 V . Output: A minimum weighted connected dominating set D of graph G. Method. D ;; let V 0 D fv 2 V W w.v/ < 0g; w.v/ 0 for each v 2 V 0 ; let u be a vertex of minimum weight in V ; determine the hanging hu D .L0 ; L1 ; : : : ; Lt / of G at u; for i D 2 to t do let Li D fx1 ; x2 ; : : : ; xj g; sort Li such that jN 0 .xi1 /j  jN 0 .xi2 /j  : : :  jN 0 .xij /j; for k D 1 to j do if N 0 .xik / \ D D ; then D D [ fyg where y is a vertex of minimum weight in N 0 .xik /; end do; if not (L1  N ŒD and GŒD is connected) then D D [ fug; for v 2 V that dominates V do if w.v/ < w.D/ then D fvg; D D [ V 0. Theorem 27 Algorithm WConDomDH gives a minimum weighted connected dominating set of a connected distance-hereditary graph in linear time. Proof The correctness of the algorithm follows from Lemma 1 and Theorem 26. For each i , sort Li by using a bucket sort. Then the algorithm runs in O.jV j C jEj/ time. 

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Algorithms and Metaheuristics for Combinatorial Matrices Ilias S. Kotsireas

Contents 1 2 3 4

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries and Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclotomy Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . String Sorting Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Periodic Autocorrelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Aperiodic Autocorrelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A General String Sorting Algorithmic Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A General GAs Algorithmic Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Simulated Annealing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A General SA Algorithmic Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Particle Swarm Optimization (PSO). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Ant Colony Optimization (ACO). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Combinatorial Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

284 284 288 289 290 292 293 294 295 295 296 297 300 303 304 304 305 305

Abstract

Combinatorial matrices are matrices that satisfy certain combinatorial properties and typically give rise to extremely challenging search problems with thousands of variables. In this chapter we present a survey of some recent algorithms to search for some kinds of combinatorial matrices, with an emphasis to algorithms within the realm of optimization and metaheuristics. It is to be noted that for most kinds of combinatorial matrices there are several known infinite classes

I.S. Kotsireas Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada e-mail: [email protected]

283 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 13, © Springer Science+Business Media New York 2013

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in the literature, but these infinite classes do not suffice to cover the entire spectra of possible orders of these matrices, therefore it is necessary to resort to computational and meta-heuristic algorithms.

1

Introduction

The search for combinatorial matrices and their associated designs has been a fertile testing ground for algorithm designers for decades. These extremely challenging combinatorial problems have been tackled with algorithms using concepts and techniques from discrete mathematics, number theory, linear algebra, group theory, optimization, and metaheuristics. This chapter will survey some recent algorithms to search for combinatorial matrices, with an emphasis to algorithms within the realm of optimization and metaheuristics. The hope is that this chapter will arouse the interest of algorithm designers from various areas in order to invent new formalisms and new algorithms to search efficiently for combinatorial matrices. There are several open problems in the broad area of combinatorial matrices, and it is clear that new ideas are required to solve them. The existing algorithms continue to yield new results, especially in conjunction with the use of parallel programming techniques, but will still eventually reach a point of saturation, beyond which they will probably not be of much use. Therefore, cross-fertilization of research areas is necessary for producing new results in the search for new combinatorial matrices, at both the theoretical and the practical level. Combinatorial matrices are defined as matrices that possess some combinatorial properties, such as prescribed row/column sums and special structure described in terms of circulant matrices. The research area of combinatorial matrices has been developed in a systematic manner over the last 50 years in the four books [10, 11, 48, 88]. It is worthwhile to point out that the unifying concept of combinatorial matrices includes well-known and widely studied categories of matrices, such as adjacency and incidence matrices of graphs, Hadamard matrices, and doubly stochastic matrices. In addition, there is a number of more recent books that contain useful information and new developments in the area of combinatorial matrices and the closely related area of design theory. The book [52] (see also the update [53]) sheds considerable light in the cocyclic (i.e., group cohomological) aspects of certain kinds of combinatorial matrices and is the only systematic book-size treatment of this topic. The book [94] features a very readable exposition of design theory with emphasis on bent functions and coding theory aspects. The book [19] offers an alternative algebraic foundation of design theory. The book [54] places its emphasis on symmetric designs and their properties.

2

Preliminaries and Notations

A wide class of combinatorial matrices (or the relevant sequences) can be defined via the concepts of the periodic and aperiodic (or nonperiodic) autocorrelation functions associated to a finite sequence. These two concepts have been invented

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and studied widely within the engineering community, and their importance has been recognized in the combinatorics community as well, especially in connection with several kinds of combinatorial matrices. All sequences in this chapter will be finite and will have elements either from f1; C1g (binary sequences) or from f1; 0; C1g (ternary sequences). Definition 1 The periodic autocorrelation function associated to a finite sequence A D Œa1 ; : : : ; an  of length n is defined as PA .i / D

n X

ak akCi ; i D 0; : : : ; n  1;

kD1

where k C i is taken modulo n, when k C i > n. Definition 2 The aperiodic (or nonperiodic) autocorrelation function of a sequence Œa1 ; : : : ; an  of length n is defined as NA .i / D

ni X

ak akCi ; i D 0; : : : ; n  1;

kD1

where k C i is taken modulo n, when k C i > n. Definitions 1 and 2 are best clarified with an example. Example 1 For a finite sequence of length n D 7, A D Œa1 ; : : : ; a7 , we have PA .0/ PA .1/ PA .2/ PA .3/ PA .4/ PA .5/ PA .6/

D D D D D D D

NA .0/ D NA .1/ D NA .2/ D NA .3/ D NA .4/ D NA .5/ D NA .6/ D

a12 C a2 2 C a3 2 C a4 2 C a5 2 C a6 2 C a7 2 a1 a2 C a2 a3 C a3 a4 C a4 a5 C a5 a6 C a6 a7 C a7 a1 a1 a3 C a2 a4 C a3 a5 C a4 a6 C a5 a7 C a6 a1 C a7 a2 a1 a4 C a2 a5 C a3 a6 C a4 a7 C a5 a1 C a6 a2 C a7 a3 a1 a4 C a2 a5 C a3 a6 C a4 a7 C a5 a1 C a6 a2 C a7 a3 a1 a3 C a2 a4 C a3 a5 C a4 a6 C a5 a7 C a6 a1 C a7 a2 a1 a2 C a2 a3 C a3 a4 C a4 a5 C a5 a6 C a6 a7 C a7 a1 a1 2 C a2 2 C a3 2 C a4 2 C a5 2 C a6 2 C a7 2 a1 a2 C a2 a3 C a3 a4 C a4 a5 C a5 a6 C a6 a7 a1 a3 C a2 a4 C a3 a5 C a4 a6 C a5 a7 a1 a4 C a2 a5 C a3 a6 C a4 a7 a1 a5 C a2 a6 C a3 a7 a1 a6 C a2 a7 a1 a7 :

Definition 3 Two finite sequences Œa1 ; : : : ; an  and Œb1 ; : : : ; bn  of length n each are said to have constant periodic autocorrelation if there is a constant c such that

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PA .i / C PB .i / D c;

i D 1; : : : ; n  1:

Definition 4 Two sequences Œa1 ; : : : ; an  and Œb1 ; : : : ; bn  of length n each are said to have constant aperiodic autocorrelation if there is a constant c such that NA .i / C NB .i / D c;

i D 1; : : : ; n  1:

Note that the index i D 0Pis omitted from Definitions 3 and 4, because of the property PA .0/ D NA .0/ D nj D1 aj2 . Also note that when writing out binary and ternary sequences, it is customary in the combinatorial literature to denote 1 by , zero by 0, and C1 by C, and this is the convention adopted in the current chapter. Also note that Definitions 3 and 4 can be extended to more than two sequences. Definitions 3 and 4 are best illustrated with an example. Example 2 The following binary sequences with n D 26 have constant aperiodic autocorrelation with c D 0: C C C C  C C   C  C  C   C  C C C   C CC C C C C  C C   C  C C C C C  C    C C   The following ternary sequences with n D 30 have constant aperiodic autocorrelation with c D 0: C C      C   0000 C 0 C  C C C C   C  C  C C C C      C   C C   0  0000   C C  C  C   The following properties of the periodic and aperiodic autocorrelation functions can be observed (and verified) in Examples 1 and 2: Property 1 (Symmetry) PA .i / D PA .n  i /; i D 0; 1; : : : ; n: Property 2 (Complementarity) PA .i / D NA .i / C NA .n  i /; i D 0; 1; : : : ; n: Property 3 (Second Elementary Symmetric Function) PA .1/ C    C PA .n  1/ D 2 e2 .a1 ; : : : ; an / NA .1/ C    C NA .n  1/ D e2 .a1 ; : : : ; an /

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where e2 .a1 ; : : : ; an / D

X

287

ai aj is the second elementary symmetric function

1i 2n C 2 or PSDB .s/ > 2n C 2, then the corresponding A or B candidate sequence can be safely discarded from the search. 4.1.2 A Subtle Point: Integer PSD Values There is a subtle point regarding the computation of the PSD values that occur in string sorting algorithms, as well as other filtering-based algorithms to search for

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sequences using the PSD criterion. The subtlety comes from the fact that although in the vast majority of cases the PSD values are (positive) floating point numbers, there exist cases when the PSD values are (positive) integers. A particular case when this phenomenon occurs is described in the following result proved in [64]. n Lemma 1 Let n be an odd integer such that n  0 .mod 3/ and let m D . Let 3  2    2 i ! D e n D cos 2 Ci sin the principal n-th root of unity. Let Œa ; : : : ; an  be 1 n n a sequence with elements from f1; 0; C1g. Then we have that DFT.Œa1 ; : : : ; an ; m/ can be evaluated explicitly in closed form and PSD.Œa1 ; : : : ; an ; m/ is a nonnegative integer. The explicit evaluations are given by  1 1 DFT.Œa1 ; : : : ; an ; m/ D A1  A2  A3 C 2 2 

! p p 3 3 A2  A3 i 2 2

PSD.Œa1 ; : : : ; an ; m/ D A21 C A22 C A23  A1 A2  A1 A3  A2 A3 where A1 D

m1 X i D0

a3i C1 ; A2 D

m1 X i D0

a3i C2 ; A3 D

m1 X

a3i C3 :

i D0

It is important to carefully account for the phenomenon described by Lemma 1 in any implementation of string sorting algorithms in order not to miss any solutions due to numerical round-off error. One way of doing this is to omit the n=3-th value from the construction of the string.

4.1.3 Bracelets and Necklaces The symmetry property of periodic autocorrelation and PSD values under cyclic shifting must also be taken into account in order to avoid redundant computations for sequences that have the same periodic autocorrelation and PSD values. It turns out that the right combinatorial structures to deal with this symmetry property are bracelets and necklaces. The papers [89] and [90] and subsequent work by J. Sawada and his collaborators provide constant amortized time (CAT) algorithms to generate bracelets and necklaces, as well as extremely efficient C implementations of these algorithms.

4.2

Aperiodic Autocorrelation

The restrictions imposed by the constancy of periodic autocorrelation are described via the concept of a certain infinite class of functions associated to a finite sequence. We illustrate the main idea in the context of Turyn-type sequences, following [57]. Let n be an even positive integer and suppose that four binary sequences X D Œx1 ; : : : ; xn  and Y D Œy1 ; : : : ; yn , Z D Œz1 ; : : : ; zn  and W D Œw1 ; : : : ; wn1  of

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lengths n; n; n; n  1 respectively, have constant aperiodic autocorrelation function in the sense that NX .s/ C NY .s/ C 2NZ .s/ C 2NW .s/ D 0; for s D 1; : : : ; n  1:

(3)

The sequences X; Y; Z; W satisfying (3) are called Turyn type. We associate to a binary sequence X of length n, the periodic (of period 2) nonnegative function fX ./ D NX .0/ C 2

n1 X

NX .j / cos j:

j D1

It can be proved (see [57]) that property (Eq. 3) implies that fX ./ C fY ./ C 2fZ ./ C 2fW ./ D 6n  2; 8:

(4)

4.2.1 Aperiodic Version of the PSD Criterion Equation (4) can be used to derive the analog (an aperiodic version) of the PSD criterion in the aperiodic case. If for some value of  it turns out that fX ./ > 6n2 or fY ./ > 6n  2, or fZ ./ > 3n  1 or fW ./ > 3n  1, then the corresponding X; Y; Z; W candidate sequence can be safely discarded from the search. In addition, if for some value of  it turns out that fX ./ C fY ./ > 6n  2, then the corresponding pair .X; Y / of candidates sequences can be safely discarded from the search. Similar conditions can be derived for other partial sums of the four values fX ./, fY ./, fZ ./, fW ./ for arbitrary (but fixed) values of . The preprocessing of the sequences based on this aperiodic version of the PSD criterion is often carried  j ; j D 1; : : : ; 500 . out via a finite set of  values, such as 500

4.3

A General String Sorting Algorithmic Scheme

Based on the PSD criterion (periodic and aperiodic versions), a general string sorting algorithmic scheme can be defined as follows. INPUT: lengths and types (binary/ternary) of two sequences A, B and the autocorrelation (periodic/aperiodic) property satisfied, value of PSD constant (or its aperiodic analog) OUTPUT: sequences satisfying the input requirements (if any are found) • Preprocess each one of the two sequences separately using the applicable version of the PSD criterion. In particular, discard all candidate sequences that violate the applicable version of the PSD criterion. If the set of all possible candidate sequences is too large to preprocess in its entirety, then use randomized methods to generate a representative part of this set. • Encode candidate A-sequences by a string, namely, the concatenation of the integer parts of its PSD values, or the f ./ values, for a small set of  values.

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Encode candidate B-sequences by a similar string, but taking the difference of the PSD (or f ./ value) from the PSD constant (or its aperiodic analog). • Sort the files containing the string encodings corresponding to each Asequence and B-sequences separately. When the files containing the string encodings are too large to be sorted directly, this phase involves a distributed sorting algorithm. • Look for identical strings in the two sorted files and output the corresponding solutions. String sorting algorithms have been used in [?, 64] to compute several new weighing matrices of various weights constructed from two circulant cores, using the PSD criterion for periodic autocorrelation. The ternary sequences with n D 30 in Example 2 have been computed by the author using a C implementation of a string sorting algorithm and the aperiodic version of the PSD criterion.

5

Genetic Algorithms

Genetic algorithms (GAs) form a powerful metaheuristic method that exploits algorithmic analogs of concepts from Darwin’s theory of evolution to design efficient search methods. The theory of genetic algorithms is presented in a extremely readable way in the classical book [44]. Among the more recent systematic treatments of GAs, one can consult [87]. Genetic algorithms have been applied successfully to tackle a wide range of difficult problems across many application areas. A central concept in the theory of GAs is the “objective function.” A judicious choice of an objective function is necessary in order to be able to apply GAs to solve a specific problem. Another important ingredient in the theory of GAs is the encoding of the solution space as a set of binary vectors. We will exemplify below how to formalize a problem of searching for combinatorial matrices as a GA problem, using binary sequences with constant periodic autocorrelation 2 as a case study, and following the exposition in [59]. Let n be an odd positive integer and suppose that two binary sequences A D Œa1 ; : : : ; an  and B D Œb1 ; : : : ; bn , both of length n, have constant periodic autocorrelation function 2, as in (1). Since the m D n1 Eq. (1) must be satisfied 2 simultaneously, one can consider the following two types of objective functions arising naturally: OF1 D j PA .1/ C PB .1/ C 2 j C : : : C j PA .m/ C PB .m/ C 2 j and OF2 D .PA .1/ C PB .1/ C 2/2 C : : : C .PA .m/ C PB .m/ C 2/2 : There are two potentially important differences between the objective functions OF1 OF2 . First, OF1 is not a continuous function, while OF2 is. Second, OF1 takes on smaller values than OF2 . The GA will seek to minimize OF1 and OF2 , i.e., to find 2n f˙1g values of the 2n variables a1 ; : : : ; an ; b1 ; : : : ; bn such that OF1 and OF2

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attain the value 0. Evidently, a set of 2n f˙1g values with this property is also a solution of the original problem of finding two binary sequences with constant periodic autocorrelation function 2. The GAs requirement of encoding the solution space as a set of binary vectors is inherent in the problem of searching for binary sequences with constant periodic/aperiodic autocorrelation function.

5.1

A General GAs Algorithmic Scheme

INPUT: k binary sequences A1 ; : : : ; Ak , all of length n, and the autocorrelation (periodic/aperiodic) property satisfied OUTPUT: k-tuples of sequences satisfying the input requirements (if any are found) • Specify an initial randomly chosen population (first generation) of binary sequences of length n. • Encode the required periodic/aperiodic autocorrelation property in the form of OF1 or OF2 . • Set current generation to be the first generation. • Evolve the current generation to the next generation, using a set of genetic operators. Commonly used such operators include reproduction, crossover, and mutation. • Examine the next generation to see whether it contains k binary sequences with the required periodic/aperiodic autocorrelation property. If yes, then output this solution and exit. If no, then set current generation to be the next generation and iterate the previous step. Note that popular termination criteria for GAs include a predetermined number of generations to evolve or a predetermined amount of execution time. There are several papers that use GAs to solve design-theoretic and combinatorial problems. In [1] the authors devise a GA to search for cocyclic Hadamard matrices. In [2] the author explains how to use genetic algorithms to solve three designtheoretic problems of graph-theoretic flavor. In [50] the authors apply GAs to construct D-optimal designs. In [68] the authors use so-called competent GAs to search efficiently for weighing matrices. In [17] the authors use SGA (simple genetic algorithm) to construct Hadamard matrices of various orders. The thesis [84] is concerned with the application of GAs to construct D-optimal experimental designs. The thesis [46] contains GAs to search for normal and near-Yang sequences.

6

Simulated Annealing

Simulated annealing (SA) was presented in [58] as a new approach to approximate solutions of difficult optimization problems. This approach is inspired from statistical mechanics and is motivated by an analogy to the behavior of physical systems in the presence of heat. In the words of Kirkpatrick et al.,

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there is a deep and useful connection between statistical mechanics (the behavior of system with many degrees of freedom in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters).

In particular, they found that the Metropolis algorithm was well suited to approximate the observed behavior of solids when subjected to an annealing process. Recall that a local optimization method starts with some initial solution and, at each iteration, searches its neighbors for a better solution until no better solution can be found. A weakness of the local optimization method is that the program may get trapped at some local minima instead of detecting a global minimum. Simulated annealing addresses this issue by using randomization to improve on the local optimization search process, and in particular, occasional uphill moves (i.e., less than optimal solutions) are accepted with the hope that by doing so, a better solution will be obtainable later on. There are several parameters in the simulated annealing process which can significantly impact the actual performance, and there do not seem to be general rules on how to choose these parameters for the specific problem at hand. In the annealing process, the physical system being optimized is first “melted” at a high temperature. The temperature is then decreased slowly until the system “freezes,” and no further physical changes occur. When the system’s structure is “frozen,” this correspond to a minimum energy configuration. The physical analogy for simulated annealing is often described using the annealing method for growing a crystal. The rate at which the temperature is reduced is vitally important to the annealing process. If the cooling is done too quickly, widespread irregularities will form and the trapped energy level will be higher than what one would find in a perfectly structured crystal. It can be shown that the states of this physical system correspond to the solutions of an optimization problem. The energy of a state in the physical world corresponds to the cost of a solution in the optimization world. The minimum energy configuration that is obtained when a system is frozen corresponds to an optimal solution in an optimization problem.

6.1

A General SA Algorithmic Scheme

The basic ingredients needed to formulate a problem in a manner amenable to SA algorithms are: 1. A finite set S . This is the set of all possible solutions. 2. An objective function OF defined on S . 3. The set S  is the set of global minima (i.e., the desired solutions) using the OF . It is assumed that S   S , a proper subset of S . 4. For each i 2 S , we define a set S.i /  S  fi g to be the set of neighbors of i . 5. A nonincreasing function ˛.t/ called the cooling schedule, which controls how the temperature is lowered.

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With the previous terminology in place, a general SA algorithmic scheme looks like: Randomly select initial solution s0 ; Select an initial temperature t0 > 0; Select a temperature reduction function ˛; Repeat Repeat Randomly select a neighbor, s of s0 ; ı = OF(s)  OF(s0 ); if(ı < 0) then s0 D s; else generate random x uniformly in the range .0; 1/; if(x < e ı=t ) then s0 = s; Until iteration count = nrep Set t D ˛.t/; Until stopping condition = true s0 is the approximation to the optimal solution Recall that the goal of simulated annealing is to improve on the local search strategies by allowing some uphill moves in a controlled manner. The above SA scheme accepts a solution at each iteration if it is determined to be better than the current “best” solution. However, we also see how a less than optimal solution can be accepted and that it is accepted with probability e ı=t . Also note that for the purposes of the OF required by SA, one can use OF1 and OF2 as defined previously for GAs. Commonly used cooling schedules include ˛.t/ D ˛t, where ˛ < 1 and t ˛.t/ D where ˇ is small. 1 C ˇt The papers [12, 55, 70, 72, 77] use SA techniques to tackle design theory problems.

7

Particle Swarm Optimization (PSO)

Particle swarm Optimization (PSO) is a population-based metaheuristic algorithm. It was introduced by R.C. Eberhart and J. Kennedy in 1995 [35] primarily for solving numerical optimization problems, as an alternative to evolutionary algorithms (EAs) [3]. Its verified efficiency in challenging optimization problems as well as its easy implementation rapidly placed PSO in a salient position among the stateof-the-art algorithms. Today, PSO counts a vast number of applications in diverse scientific fields [4, 5, 78], as well as an extensive bibliography [14, 37, 56, 83], and published scientific software [100].

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Putting it formally, consider the n-dimensional global optimization problem: min f .x/:

x2X Rn

PSO probes the search space by using a swarm, namely, a set of search points: S D fx1 ; x2 ; : : : ; xN g;

xi 2 X;

i 2 I D f1; 2; : : : ; N g:

Each search point of S constitutes a particle, i.e., a search vector: xi D .xi1 ; xi 2 ; : : : ; xi n /> 2 X;

i 2 I;

which is randomly initialized in X and allowed to iteratively move within it. Its motion is based on stochastically adapted position shifts, called velocity, while it retains in memory the best position it has ever visited. These quantities are denoted as vi D .vi1 ; vi 2 ; : : : ; vi n /> ; pi D .pi1 ; pi 2 ; : : : ; pi n /> ; i 2 I; respectively. Thus, if t denotes the current iteration of the algorithm, it holds that pi .t/ D arg

min

q2f0;1;:::;t g

˚   f xi .q/ ;

i 2 I:

The best positions constitute a sort of experience for the particles, guiding them towards the most promising regions of the search space, i.e., regions that posses lower function values. In order to avoid the premature convergence of the particles on local minimizers, the concept of neighborhood was introduced [96]. A neighborhood of the i -th particle is defined as a set, NBi;s D fj1 ; j2 ; : : : ; js g  I;

i 2 NBi;s ;

and consists of the indices of all the particles with which it can exchange information. Then, the neighborhood’s best position pgi D arg min ff .pj /g j 2NBi;s

(5)

is used along with pi to update the i -th particle’s velocity at each iteration. The parameter s defines the neighborhood size. In the special case where s D N , the whole swarm constitutes the neighborhood. This case defines the so-called global PSO model (denoted as gbest), while strictly smaller neighborhoods correspond to the local PSO model (denoted as lbest). The schemes that are used for determining the particles that constitute each neighborhood are called neighborhood topologies, and they can have a crucial impact on PSO’s performance. A popular scheme is the ring topology, where each particle assumes as neighbors of the particles with its

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adjacent indices, assuming that indices recycle after N . Based on the definitions above, the iterative scheme of PSO is defined as follows [15]: h

i vij .t C 1/ D  vij .t/ C c1 R1 pij .t/  xij .t/ C c2 R2 pgi ;j .t/  xij .t/ ; (6) xij .t C 1/ D xij .t/ C vij .t C 1/;

(7)

where i D 1; 2; : : : ; N ; j D 1; 2; : : : ; n; the parameter  is the constriction coefficient; acceleration constants c1 and c2 are called the cognitive and social parameter, respectively; and R1 and R2 , are random variables uniformly distributed in the range Œ0; 1. It shall be noted that a different value of R1 and R2 is sampled for each i and j in (6) at each iteration. Also, the best position of each particle is updated at each iteration, as follows: ( pi .t C 1/ D

xi .t C 1/;

    if f xi .t C 1/ < f pi .t/ ;

pi .t/;

otherwise;

i 2 I:

(8)

The PSO variant described above was introduced by M. Clerc and J. Kennedy in [15], and it has gained increasing popularity, especially in interdisciplinary applications. This can be attributed to its simplicity, its efficiency, and its theoretical background. Based on the stability analysis [15], the parameter set  D 0:729, c1 D c2 D 2:05, was determined as a satisfactory setting that produces a balanced convergence speed of the algorithm. Nevertheless, alternative settings have been introduced in relevant works [98]. Pseudocode of PSO is reported in Algorithm 1. Unified particle swarm optimization (UPSO) was proposed as a scheme that harnesses the lbest and gbest PSO models, combining their exploration and exploitation properties [79, 82]. Let Gi .t C 1/ be the gbest velocity update of xi , while Li .t C 1/ be the corresponding lbest velocity update. Then, from Eq. (6), it holds that h

i Gij .t C 1/ D  vij .t/ C c1 R1 pij .t/  xij .t/ C c2 R2 pg;j .t/  xij .t/ ; (9) h

i Lij .t C 1/ D  vij .t/ C c1 R1 pij .t/  xij .t/ C c2 R2 pgi ;j .t/  xij .t/ ; (10) where g denotes the overall best particle. The aggregation of these search directions defines the main UPSO scheme [79]: Uij .t C 1/ D u Gij .t C 1/ C .1  u/ Lij .t C 1/;

(11)

xij .t C 1/ D xij .t/ C Uij .t C 1/:

(12)

The parameter u 2 Œ0; 1 is called the unification factor and determines the trade off between the two directions. Obviously, the standard gbest PSO is obtained by setting u D 1 in (11), while u D 0 corresponds to the standard lbest PSO. All intermediate

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values of u 2 .0; 1/ define composite UPSO variants that combine the exploration and exploitation properties of the global and local PSO variant. Besides the basic UPSO scheme, a stochastic parameter can also be incorporated in Eq. (11) to enhance UPSO’s exploration capabilities [79]. Thus, depending on which variant UPSO is mostly based, Eq. (11) becomes Uij .t C 1/ D R3 u Gij .t C 1/ C .1  u/ Lij .t C 1/;

(13)

which is mostly based on lbest, or, Uij .t C 1/ D u Gij .t C 1/ C R3 .1  u/ Lij .t C 1/;

(14)

which is mostly based on gbest. The random variable R3  N .;  2 / is normally distributed, resembling the mutation operator in EAs. Yet, it is biased towards directions that are consistent with the PSO dynamic, instead of the pure random mutation used in EAs. Following the assumptions of Matyas [71], a proof of convergence in probability was derived for the UPSO variants of Eqs. (13) and (14) [79]. Although PSO and UPSO have been primarily designed for real-valued optimization problems, there are various applications also in problems with integer, mixed-integer, and binary variables (e.g., see Eqs. [62, 69, 80, 81, 85, 86]). The most common and easy way to achieve this is by rounding the corresponding variables to their nearest integers when evaluating the particles. Of course, problem-dependent techniques may be additionally needed to enhance performance.

8

Ant Colony Optimization (ACO)

Ant colony optimization (ACO) is a probabilistic algorithm, primarily for solving combinatorial optimization problems. The general algorithmic concept of ACO was introduced by M. Dorigo in 1992 [30], aiming at detecting optimal paths in graphs by imitating the foraging behavior of real ants. The first approaches proved to be very promising, leading to further developments and enhancements [7, 31, 33]. Today, there is a variety of ant-based algorithms. Perhaps the most popular variants are Ant System (AS) [34], Max–Min Ant System (MMAS) [95], and Ant Colony System (ACS) [32]. Setting aside some differences, most ant-based approaches follow the same procedure flow, which can be summarized in the following actions: Procedure ACO WHILE (termination condition is false) Construct_Solutions() Optional_Further_Actions() Update_Pheromones() END WHILE

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Algorithm 1: Pseudocode of the standard PSO algorithm Input : Objective function, f W X  Rn ! R; swarm size: N ; parameters: , c1 , c2 . Output: Best detected solution: x  , f .x  /. // Initialization 1 t 0. 2 for i 1 to N do 3 Initialize xi .t / and vi .t /, randomly. xi .t /. // Initialize best position 4 pi .t / 5 Evaluate f .xi .t //. // Evaluate particle 6 end // Main Iteration 7 while (termination criterion not met) do // Position and Velocity Update 8 9 10 11 12 13 14

for i 1 to N do Determine pgi .t / from the i –th particle’s neighborhood. for j 1 to n do  vij .t C 1/  vij .t / C c1 R1 .pij .t /  xij .t // C c2 R2 .pgi ;j .t /  xij .t // : xij .t / C vij .t C 1/: xij .t C 1/ end end // Best Positions Update

15 16 17 18 19 20 end

for i 1 to N do Evaluate f .xi .t C 1//. // Evaluate new position  xi .t C 1/; if f .xi .t C 1// < f .pi .t //, pi .t C 1/ pi .t /; otherwise. end t t C 1.

In general, the algorithm assumes a number, K, of search agents, called the ants. Each ant constructs a solution component by component. The construction procedure is based on the probabilistic selection of each component’s value from a discrete and finite set. For this purpose, a table of pheromones is retained and continuously updated. The pheromones play the role of quality weights, used for determining the corresponding selection probability of each possible value of a solution component. Putting it formally, let x D .x1 ; x2 ; : : : ; xn /> 2 D; be a candidate solution of the problem, with each component xi , i D 1; 2; : : : ; n, taking values from a discrete and finite set Di . The construction of a solution begins from an initially empty partial solution, x p D ;. At each step, the next component of x p is assigned one of the possible values from its corresponding discrete set. Naturally, the values assigned in previous components may affect the feasible set of

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candidate values for the current one, prohibiting the use of some of them to retain feasibility. Assume that x p is already built up to the .i  1/-th component, and let Di D fdi1 ; di 2 ; : : : ; diM g be the set of candidate values for the i -th component. Each possible value dij is associated with a pheromone level ij . The selection of a specific value for the i -th component is based on the probabilistic selection among the different candidate values, using the selection probabilities: ija .dij /b P .dij jx p / D P a ; i l .di l /b

j D 1; 2; : : : ; M;

(15)

di l 2Di

where .:/ is a heuristic function that offers a measure of the “desirability” of each component value in the solution (e.g., in TSP problems, it can be associated with the traveled distance). The parameters a and b are fixed, and they offer flexibility in tuning the trade off between pheromone and heuristic information. Upon constructing a complete solution, it is evaluated with the objective value, and the pheromones are updated so that component values that appear in better solutions increase their pheromone levels and, consequently, their selection probabilities for subsequent iterations of the algorithm. Also, all pheromones undergo a standard reduction in their values, a procedure also known as evaporation. Thus, the pheromones are updated as follows:  ij D

.1  / ij C  ; if dij appears in a good candidate solution; otherwise; .1  / ij ;

(16)

where  2 .0; 1 is the evaporation rate and is an increment that can be either fixed or proportional to the quality of the corresponding candidate solution. Different ant-based approaches can differ from the basic scheme described above. For example, in AS with K ants, the pheromone update takes place after all ants have constructed their solutions and they all contribute to the update: ij D .1  / ij C

K X

.k/

ij ;

kD1 .k/

where ij is the contributed pheromone from the k-th ant and it is related to its quality. On the other hand, ACS uses a different rule for selecting component values, where the value of the i -th component is determined as n o dig D arg max ija .dij /b ; dij 2Di

if q 6 q0 ;

where q 2 Œ0; 1 is a uniformly distributed random number and q0 2 Œ0; 1 is a user-defined parameter; otherwise, the scheme of Eq. (15) is used. Also, only the best ant contributes to the pheromones, instead of the whole swarm. Finally, MMAS employs pheromone bounds Œ min ; max . Each pheromone is initialized to

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its maximum value and updated only from the best ant, either of the current iteration or overall. There is a rich literature of applications of ant-based approaches in combinatorial optimization problems [7, 31, 33]. Recently, such approaches were used also for the solution of autocorrelation problems [69].

9

Combinatorial Optimization

Combinatorial optimization methods seem like a natural candidate of methods that should be employed to tackle extremely hard optimization problems such as the search for binary and ternary sequences that satisfy autocorrelation constraints. This is because combinatorial optimization methods routinely deal with problems featuring thousands of discrete variables. The missing link that allows one to formalize autocorrelation problems in a manner suitable for combinatorial optimization methods was provided in [66] and [61]. This formalism made it in fact possible to solve in [66] the entire B¨omer-Antweiler diagram, which was proposed 20 years before, in [6]. This formalism is based on the fact that autocorrelation can be expressed in terms of certain symmetric matrices, which correspond to the matrices associated to autocorrelation viewed as quadratic form. We shall illustrate the formalism for D-optimal matrices, following the exposition in [61]. Let n be an odd positive integer and set m D n1 . D-optimal matrices correspond 2 to two binary sequences of length n each, with constant periodic autocorrelation 2; see [61]. D-optimal matrices are 2n  2n f1; C1g-matrices that attain Ehlich’s determinant bound. See [29] for a state-of-the-art survey of existence questions for D-optimal matrices with n < 200. We reproduce the following definition and lemma from [66]. Definition 8 Let a D Œa1 ; a2 ; : : : ; an T be a column n  1 vector, where a1 ; a2 ; : : : ; an 2 f1; C1g and consider the elements of the periodic autocorrelation function vector PA .1/; : : : ; PA .m/. Define the following m symmetric matrices (which are independent of the sequence a), for i D 1; : : : ; m  Mi D .mj k /; s.t.

mj k D mkj D 12 ; when aj ak 2 PA .i /; j; k 2 f1; : : : ; ng 0; otherwise

Lemma The matrices Mi can be used to write equations involving autocorrelation in a matrix form: • For n odd: aT Mi a D PA .i /; i D 1; : : : ; m: • For n even: aT Mi a D PA .i /; i D 1; : : : ; m  1 and aT Mm a D

1 PA .m/: 2

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Now one can formulate the D-optimal matrices problem as a binary feasibility problem. Binary Feasibility version of the D-optimal matrices problem Find two sequences a D Œa1 ; : : : ; an , b D Œb1 ; : : : ; bn , (viewed as n  1 column vectors) such that aT Mi a C b T Mi b D 2; i D 1; : : : ; m; where ai ; bi 2 f1; C1g; i D 1; : : : ; n. It is now clear that one can use combinatorial optimization algorithms to search for D-optimal matrices, as well as several other combinatorial matrices and sequences referenced in Table 1. Definition 8 can easily be adapted for aperiodic autocorrelation as well. Note that the D-optimal matrices problem also features certain Diophantine constraints, as proved in [61] and subsequently generalized in [29]. It is not clear what is the right way to incorporate this kind of Diophantine constraints into the combinatorial optimization formalism described here, and this is certainly an issue that deserves to be investigated further. The papers [73–76] demonstrate how to use various optimization techniques to find new designs.

10

Applications

There are several applications of combinatorial matrices and sequences described in Table 1, in such areas as telecommunications, coding theory, and cryptography. The reader can consult [45] for applications in signal design for communications, radar, and cryptography applications. The book [53] describes applications in signal processing, coding theory, and cryptography. The book [104] describes applications in communications, signal processing, and image processing. The paper [99] discusses how partial Hadamard matrices (see [20]) are used in the area of compressed sensing. The paper [93] describes in detail several applications of Hadamard matrices on code division multiple access (CDMA) communication systems as well as in telecommunications. The 55-page paper [49] describes applications of Hadamard matrices in binary codes, information processing, maximum determinant problems, spectrometry, pattern recognition, and other areas. The papers [91] and [92] and the thesis [9] discuss cryptography applications of Hadamard matrices. Chapter 10 of the book [51] is devoted to several combinatorial problems that can be tackled with stochastic search methods.

11

Conclusion

Combinatorial matrices and the associated binary and ternary sequences provide a wide array of extremely challenging optimization problems that can be tackled with a variety of metaheuristic and combinatorial methods. We tried to summarize

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some of these methods in the present chapter. Although there are no polynomial complexity algorithms to solve the problem of searching for such matrices and sequences, the combination of new insights, new algorithms, and new implementations will continue to produce new results. The second edition of the Handbook of Combinatorial Designs [16], edited by Charles J. Colbourn and Jeffrey H. Dinitz, is a very valuable resource regarding the state of the art on open cases for several types of such matrices and sequences. The on-line updated webpage maintained by the authors is also quite useful, as many open cases have been resolved since the publication of the handbook in 2007. We firmly believe that cross-fertilization of disciplines is a very important process, from which all involved disciplines benefit eventually. It seems reasonable to predict that the area of combinatorial matrices and their associated sequences will continue to experience strong interactions with other research areas, for many years to come.

Cross-References Algorithms for the Satisfiability Problem Binary Unconstrained Quadratic Optimization Problem

Recommended Reading ´ 1. V. Alvarez, J.A. Armario, M.D. Frau, P. Real, A genetic algorithm for cocyclic Hadamard matrices, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 3857 (Springer, Berlin, 2006), pp. 144–153 2. D. Ashlock, Finding designs with genetic algorithms, in Computational and Constructive Design Theory. Mathematics and Its Applications, vol. 368 (Kluwer, Dordrecht, 1996), pp. 49–65 3. T. B¨ack, D. Fogel, Z. Michalewicz, Handbook of Evolutionary Computation (IOP Publishing/Oxford University Press, New York, 1997) 4. A. Banks, J. Vincent, C. Anyakoha, A review of particle swarm optimization. Part i: background and development. Nat. Comput. 6(4), 467–484 (2007) 5. A. Banks, J. Vincent, C. Anyakoha, A review of particle swarm optimization. Part ii: hybridisation, combinatorial, multicriteria and constrained optimization, and indicative applications. Nat. Comput. 7(1), 109–124 (2008) 6. L. B¨omer, M. Antweiler, Periodic complementary binary sequences. IEEE Trans. Inf. Theory 36(6), 1487–1494 (1990) 7. E. Bonabeau, M. Dorigo, G. Th´eraulaz, Swarm Intelligence: From Natural to Artificial Systems (Oxford University Press, New York, 1999) 8. P.B. Borwein, R.A. Ferguson, A complete description of Golay pairs for lengths up to 100. Math. Comput. 73(246), 967–985 (2004) 9. A. Braeken, Cryptographic Properties of Functions and S-Boxes. PhD thesis, Katholieke Universiteit Leuven, Belgium, 2006 10. R.A. Brualdi, Combinatorial Matrix Classes. Encyclopedia of Mathematics and Its Applications, vol. 108 (Cambridge University Press, Cambridge, 2006) 11. R.A. Brualdi, H.J. Ryser, Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications, vol. 39 (Cambridge University Press, Cambridge, 1991)

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Algorithms for the Satisfiability Problem John Franco and Sean Weaver

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Representations and Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 .0 ˙ 1/ Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Binary Decision Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Implication Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Propositional Connection Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Variable-Clause Matching Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Formula Digraph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Satisfiability Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 And/Inverter Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Consistency Analysis in Scenario Projects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Testing of VLSI Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Diagnosis of Circuit Faults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Functional Verification of Hardware Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Bounded Model Checking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Combinational Equivalence Checking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Transformations to Satisfiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Boolean Data Mining. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 General Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Efficient Transformation to CNF Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Extended Resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Davis-Putnam Resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Davis-Putnam Loveland Logemann Resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conflict-Driven Clause Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Satisfiability Modulo Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Stochastic Local Search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.9 Binary Decision Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Decompositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Branch and Bound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Algebraic Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Algorithms for Easy Classes of CNF Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 2-SAT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Horn Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Renamable Horn Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Linear Programming Relaxations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 q-Horn Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Matched Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Generalized Matched Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Nested and Extended Nested Satisfiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Linear Autark Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Minimally Unsatisfiable Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Bounded Resolvent Length Resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Comparison of Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Probabilistic Comparison of Incomparable Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The k-SAT Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Other Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

375 388 395 397 404 405 407 408 409 412 414 415 415 421 425 432 434 436 439 443 443 443 447 447

Abstract

This chapter discusses advances in SAT algorithm design, including the use of SAT algorithms as theory drivers, classic implementations of SAT solvers, and some theoretical aspects of SAT. Some applications to which SAT solvers have been successfully applied are also presented. The intention is to assist someone interested in applying SAT technology in solving some stubborn class of combinatorial problems.

1

Introduction

The satisfiability problem (SAT) has gained considerable attention over the past decade for two reasons. First, the performance of competitive SAT solvers has improved enormously due to the implementation of new algorithmic concepts. Second, so many real-world problems that are not naturally expressed as instances of SAT can be transformed to SAT instances and solved relatively efficiently using one of the ever-improving SAT solvers or solvers based on SAT. This chapter attempts to clarify these successes by presenting the important advances in SAT algorithm design as well as the classic implementations of SAT solvers. Some applications to which SAT solvers have been successfully applied are also presented. Finally, for the sake of developing some intuition on the subject, some classic theoretical results are presented. The next section of the chapter presents some logic background and notation that will be necessary to understand and express the algorithms presented. The third

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section presents several representations for SAT instances in preparation for discussing the wide variety of SAT solver implementations that have been tried. The fourth section presents some applications of SAT. The next two sections present algorithms for SAT, including search and heuristic algorithms plus systems that use SAT to manage the logic of complex computations. The seventh section presents some theoretical results regarding k-SAT instances and relates these with the performance of practical SAT solvers. The last section summarizes the results of the chapter.

2

Logic

An instance of satisfiability is a propositional logic expression in conjunctive normal form (CNF), the meaning of which is described in the next few paragraphs. The most elementary object comprising a CNF expression is the Boolean variable, shortened to variable in the context of this chapter. A variable takes one of two values from the set f0; 1g. A variable v may be negated in which case it is denoted :v. The value of :v is opposite that of v. A literal is either a variable or a negated variable. The term positive literal is used to refer to a variable and the term negative literal is used to refer to a negated variable. The polarity of a literal is positive or negative accordingly. The building blocks of propositional expressions are binary Boolean operators. A binary Boolean operator is a function Ob W f0; 1g  f0; 1g  ! f0; 1g. Often, such functions are presented in tabular form, called truth tables, as illustrated in Fig. 10. There are 16 possible binary operators and the most common (and useful) are _ (or), ^ (and), ! (implies), $ (equivalent), and ˚ (xor, alternatively exclusive-or). Mappings defining the common operators are shown in Fig. 1. The only interesting unary Boolean operator, denoted : (negation), is a mapping from 0 to 1 and 1 to 0. If :l is a literal, then ::l is the same as l. A formula is a propositional expression consisting of literals, parentheses, and operators which has some semantic content and whose syntax is described recursively as follows: 1. Any single variable is a formula. 2. If is a formula, then so is : . 3. If 1 and 2 are both formulas and O is a Boolean binary operator, then . 1 O 2 / is a formula, 1 is called the left operand of O, and 2 is called the right operand of O. Formulas can be simplified by removing some or all parentheses. Parentheses around nestings involving the same associative operators such as _, ^, and $ may be removed. For example, . 1 _ . 2 _ 3 //, .. 1 _ 2 / _ 3 /, and . 1 _ 2 _ 3 / are considered to be the same formula. In the case of non-associative operators such as !, parentheses may be removed, but right associativity is then assumed. The following is an example of a simplified formula: .:v0 _ v1 _ :v7 / ^ .:v2 _ v3 / ^ .v0 _ :v6 _ :v7 / ^ .:v4 _ v5 _ v9 /;

314 Fig. 1 A .0 ˙ 1/ matrix representation and associated CNF formula where clauses are labeled c0 , c1 ,. . . ,c7, from left to right, top to bottom

J. Franco and S. Weaver

⎛ c0 c1 c2 c3 c4 c5 c6 c7

v0 v1 v2 v3 v4 v5 v6 v7 v8 v9

⎜ −1 1 0 0 0 0 0 −1 ⎜ ⎜ 0 −1 1 0 0 0 0 0 ⎜ ⎜ 0 0 −1 1 0 0 0 0 ⎜ ⎜ −1 0 0 −1 0 0 0 0 ⎜ ⎜ 1 0 0 0 0 −1 −1 0 ⎜ ⎜ 0 0 0 0 0 −1 1 0 ⎜ ⎜ 0 0 0 0 −1 1 0 0 ⎜ ⎝ 1 0 0 1 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

((¬v0 ∨ v1 ∨ ¬v7 ) ∧ (¬v1 ∨ v2 ) ∧ (¬v2 ∨ v3 ) ∧ (¬v0 ∨ ¬v3 ∨ v8 ) ∧ (v0 ∨ ¬v6 ∨ ¬v7 ) ∧ (¬v5 ∨ v6 ) ∧ (¬v4 ∨ v5 ∨ v9 ) ∧ (v0 ∨ v4 ))

Table 1 The most common binary Boolean operators and their mappings

Operator Or And Implies Equivalent XOR

Symbol _ ^ ! $ ˚

Mapping f00  ! 0; 10; 01; 11  ! 1g f00; 01; 10  ! 0; 11  ! 1g f10  ! 0; 00; 01; 11  ! 1g f01; 10  ! 0; 00; 11  ! 1g f00; 11  ! 0; 01; 10  ! 1g

A useful parameter associated with a formula is depth. The depth of a formula is determined as follows: 1. The depth of a formula consisting of a single variable is 0. 2. The depth of a formula : is the depth of plus 1. 3. The depth of a formula . 1 O 2 / is the maximum of the depth of 1 and the depth of 2 plus 1. A truth assignment or assignment is a set of variables all of whom have value 1. If M is an assignment and V is a set of variables, then, if v 2 V and v … M , v has value 0. Any assignment of values to the variables of a formula induces a value on the formula. A formula is evaluated from innermost : or parentheses out using mappings associated with the Boolean operators that are found in Table 1. Many algorithms that will be considered later iteratively build assignments, and it will be necessary to distinguish variables that have been assigned a value from those that have not been. In such cases, a variable will be allowed to hold a third value, denoted ?, which means the variable is unassigned. This requires that the evaluation of operations be augmented to account for ? as shown in Table 2. If M is an assignment of values to a set of variables where at least one variable has value ?, then M is said to be a partial assignment. Formulas that are dealt with in this chapter often have many components of the same type which are called clauses. Two common special types of clauses are disjunctive and conjunctive clauses. A disjunctive clause is a formula consisting only of literals and the operator _. If all the literals of a clause are negative (positive), then the clause is called a negative clause (respectively, positive clause).

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Table 2 Boolean operator mappings with ?. Symbol‹ means 1 or 0 Operator

Symbol

Mapping

Or And Implies Equivalent XOR Negate

_ ^ ! $ ˚ :

f00  ! 0; 10; 01; 11; 1?; ?1  ! 1; 0?; ?0; ??  ! ?g f00; 01; 10  ! 0; 11  ! 1; ‹?; ?‹; ??  ! ?g f10  ! 0; 00; 01; 11; 0?  ! 1; 1?; ?‹  ! ?g f01; 10  ! 0; 00; 11  ! 1; ‹?; ?‹; ??  ! ?g f00; 11  ! 0; 01; 10  ! 1; ‹?; ?‹; ??  ! ?g f0  ! 1; 1  ! 0, ?  ! ?g

In this chapter disjunctive clauses will usually be represented as sets of literals. When it is understood that an object is a disjunctive clause, it will be referred to simply as a clause. The following two lines show the same formula of four clauses expressed, above, in conventional logic notation and, below, as a set of sets of literals: .:v0 _v1 _ :v7 / ^ .:v2 _ v3 / ^ .v0 _ :v6 _ :v7 / ^ .:v4 _ v5 _ v9 / ff:v0 ; v1 ; :v7 g; f:v2 ; v3 g; fv0 ; :v6 ; :v7 g; f:v4 ; v5 ; v9 gg: A conjunctive clause is a formula consisting only of literals and the operator ^. A conjunctive clause will also usually be represented as a set of literals and called a clause when it is unambiguous to do so. The number of literals in any clause is referred to as the width of the clause. Often, formulas are expressed in some normal form. Four of the most frequently arising forms are defined as follows: A CNF formula is a formula consisting of a conjunction of two or more disjunctive clauses. Given CNF formula and L , the set of all literals in , a literal l is said to be a pure literal in if l 2 L but :l … L . A clause c 2 is said to be a unit clause if c has exactly one literal. A k-CNF formula, k fixed, is a CNF formula restricted so that the width of each clause is exactly k. A Horn formula is a CNF formula with the restriction that all clauses contain at most one positive literal. Observe that a clause .:a_:b_:c_g/ is functionally the same as .a^b^c ! g/, so Horn formulas are closely related to logic programming. In fact, logic programming was originally the study of Horn formulas. A DNF formula is a formula consisting of a disjunction of two or more conjunctive clauses. When discussing a CNF or DNF formula , V is used to denote its variable set and C is used to denote its clause set. The subscripts are dropped when the context is clear. As mentioned earlier, evaluation of a formula is from innermost parentheses out using the truth tables for each operator encountered and a given truth assignment. If the formula evaluates to 1, then the assignment is called a satisfying assignment, model, or a solution.

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There are 2n ways to assign values to n Boolean variables. Any subset of those assignments is a Boolean function on n variables. Thus, the number of such n functions is 22 . Any Boolean function f on n variables can be expressed as a CNF formula where the set of assignments for which f has value 1 is identical to the set of assignments satisfying [129]. However, k-CNF formulas express only a proper subset of Boolean functions: for example, since a width k clause eliminates the fraction 2k of potential models, any Boolean function comprising more than 2n .1  2k / assignments cannot be represented by a k-CNF formula. Similarly, all Boolean functions can be expressed by DNF formulas but not by k-DNF formulas [129]. The partial evaluation of a given formula is possible when a subset of its variables are assigned values. A partial evaluation usually results in the replacement of by a new formula which expresses exactly those assignments satisfying under the given partial assignment. Write jvD1 to denote the formula resulting from the partial evaluation of due to assigning value 1 to variable v. An obvious similar statement is used to express the partial evaluation of when v is assigned value 0 or when some subset of variables is assigned values. For example, .v1 _ :v2 / ^ .:v1 _ v3 / ^ .:v2 _ :v3 / jv1 D1 D .v3 / ^ .:v2 _ :v3 / since .v3 /^.:v2 _:v3 / expresses all solutions to .v1 _:v2 /^.:v1 _v3 /^.:v2 _:v3 / given v1 has value 1. If an assignment M is such that all the literals of a disjunctive (conjunctive) clause have value 0 (respectively, 1) under M , then the clause is said to be falsified (respectively, satisfied) by M . If M is such that at least one literal of a disjunctive (conjunctive) clause has value 1 (respectively, 0) under M , then the clause is said to be satisfied (respectively, falsified) by M . If a clause evaluates to ?, then it is neither satisfied nor falsified. A formula is satisfiable if there exists at least one assignment under which has value 1. In particular, a CNF formula is satisfiable if there exists a truth assignment to its variables which satisfies all its clauses. Otherwise, the formula is unsatisfiable. Every nonempty DNF formula is satisfiable, but a DNF formula that is satisfied by every truth assignment to its variables is called a tautology. The negation of a DNF tautology is an unsatisfiable CNF formula. Several assignments may satisfy a given formula. Any satisfying assignment containing the smallest number of variables of value 1 among all satisfying assignments is called a minimal model with respect to 1. Thus, consistent with our definition of model as a set of variables of value 1, a minimal model is a set of variables of least cardinality. The usual semantics for Horn formula logic programming is the minimal model semantics: the only model considered is the (unique) minimal one.1

1

Each satisfiable set of Horn clauses has a unique minimal model with respect to 1, which can be computed in linear time by a well-known algorithm [53, 78] which is discussed in Sect. 6.2. Implications of the minimal model semantics may be found in Sect. 6.5, among others.

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If a CNF formula is unsatisfiable but removal of any clause makes it satisfiable, then the formula is said to be minimally unsatisfiable. Minimally unsatisfiable formulas play an important role in understanding the difference between easy and hard formulas. This section ends with a discussion of formula equivalence. Three types of equivalence are defined along with symbols that are used to represent them as follows: 1. Equality of formulas ( 1 D 2 ): two formulas are equal if they are the same string of symbols. Also “=” is used for equality of Boolean values, for example, v D 1. 2. Logical equivalence ( 1 , 2 ): two formulas 1 and 2 are said to be logically equivalent if, for every assignment M to the variables of 1 and 2 , M satisfies 1 if and only if M satisfies 2 . For example, in the following expression, the two leftmost clauses on each side of “,” force v1 and v3 to have the same value so .v2 _ v3 / may be substituted for .v1 _ v2 /. Therefore, the expression on the left of “,” is logically equivalent to the expression on the right. .:v1 _ v3 / ^ .v1 _ :v3 / ^ .v1 ^ v2 / , .:v1 _ v3 / ^ .v1 _ :v3 / ^ .v2 ^ v3 /: Another example is .v1 _ :v2 / ^ .:v1 _ v3 / ^ .:v2 _ :v3 / jv1 D1 , .v3 / ^ .:v2 _ :v3 /: In the second example, assigning v1 D 1 has the effect of eliminating the leftmost clause and the literal :v1 . After doing so, equivalence is clearly established. It is important to point out the difference between 1 , 2 and 1 $ 2 . The former is an assertion that 1 and 2 are logically equivalent. The latter is just a formula of formal logic upon which one can ask whether there exists a satisfying assignment. The symbol “,” may not be included in a formula, and it makes no sense to ask whether a given assignment satisfies 1 , 2 . It is easy to show that 1 , 2 if and only if 1 $ 2 is a tautology (i.e., it is satisfied by every assignment to the variables of 1 and 2 ). Similarly, define 1 logically implies 2 ( 1 ) 2 ): for every assignment M to the variables in 1 and 2 , if M satisfies 1 , then M also satisfies 2 . Thus, 1 , 2 if and only if 1 ) 2 and 2 ) 1 . Also, 1 ) 2 if and only if 1 ! 2 is a tautology. 3. Functional equivalence ( 1 V 2 ): two formulas 1 and 2 , with variable sets V 1 and V 2 , respectively, are said to be functionally equivalent with respect to variable base set V  V 1 \V 2 if, for every assignment MV to just the variables of V , either: a. There is an assignment M1 to V 1 n V and an assignment M2 to V 2 n V such that MV [ M1 satisfies 1 and MV [ M2 satisfies 2 or b. There is no assignment to V 1 [ V 2 which contains MV as a subset and satisfies either 1 or 2 . The assignments MV [ M1 and MV [ M2 are called extensions to the assignment MV .

«

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«

The notation 1 V 2 is used to represent the functional equivalence of formulas 1 and 2 with respect to base set V . The notation 1 2 is used if V D V 1 \ V 2 . In this case logical equivalence and functional equivalence are the same. For example, .:a/ ^ .a _ b/ fag .:a/ ^ .a _ c/ because 9b W .:a/ ^ .a _ b/ if and only if 9c W .:a/ ^ .a _ c/. But .:a/ ^ .a _ b/ 6D .:a/ ^ .a _ c/ since a D 0, b D 1, c D 0 satisfies .:a/ ^ .a _ b/ but falsifies .:a/ ^ .a _ c/. Functional equivalence is useful when transforming a formula to a more useful formula. For example, consider the Tseitin transformation [139] (Sect.5.3) which extends resolution: for CNF formula containing variables from set V , any pair of variables x; y 2 V , and variable z … V ,

«

«

«

V

^ .z _ x/ ^ .z _ y/ ^ .:z _ :x _ :y/:

The term functional equivalence is somewhat of a misnomer. For any set V of variables, V is an equivalence relation but is not transitive: for example, a_c a _ b and a _ b a _ :c but a _ c 6 a _ :c. The foundational problem considered in this chapter is the satisfiability problem, commonly called SAT. It is stated as follows:

«

«

«

« «

Satisfiability (SAT) Given: A Boolean formula . Question: Determine whether is satisfied by some truth assignment to the variables of . For the class of CNF, even 3-CNF, formulas, SAT is N P-hard. However, for the class of CNF formulas of maximum width 2, SAT can be solved in linear time using Algorithm 19 of Sect. 6.1. For the class of Horn formulas, SAT can be solved in linear time using Algorithm 20 of Sect. 6.2. If is a CNF formula such that reversing the polarity of some subset of its variables results in a Horn formula, then is renamable Horn. Satisfiability of a renamable Horn formula can be determined in linear time by Algorithm 21 of Sect. 6.4. The problem of determining the satisfiability of k-CNF formulas is referred to as k-SAT. In that context, input formulas are said to be instances of k-SAT or k-SAT formulas. A second problem of interest is a generalization of the common problem of determining a minimal model for a given formula. The question of minimal models is considered in Sect. 6.2. Variable-Weighted Satisfiability Given: A CNF formula and a function w W V  ! Z C where w.v/ is the weight of variable v. Question: Determine whether is satisfied by some truth assignment and, if so, determine the assignment M such that X v2M

w.v/ is minimized:

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A third problem of interest is the fundamental optimization version of SAT. This problem is probably more important in terms of practical use than even SAT. Maximum Satisfiability (MAX-SAT) Given: A CNF formula . Question: Determine the assignment M that satisfies the maximum number of clauses in . The following is an important variant of MAX-SAT: Weighted Maximum Satisfiability (Weighted MAX-SAT) Given: A CNF formula and a function w W C  ! Z C where w.c/ is the weight of clause c 2 C . Question: Determine the assignment M such that X

w.c/  sM .c/ is maximized;

c2

where sM .c/ is 1 if clause c is satisfied by M and is 0 otherwise.

3

Representations and Structures

Algorithmic concepts are usually easier to specify and understand if a formula or a particular part of a formula is suitably represented. Many representations are possible for Boolean formulas, particularly when expressed as CNF or DNF formulas. As mentioned earlier, CNF and DNF formulas will often be represented as sets of clauses and clauses as sets of literals. In this section additional representations are presented; these will later be used to express algorithms and explain algorithmic behavior. Some of these representations involve only a part of a given formula.

3.1

.0 ˙ 1/ Matrix

A CNF formula of m clauses and n variables may be represented as an mn .0˙1/matrix M where the rows are indexed on the clauses, the columns are indexed on the variables, and a cell M.i; j / has the value C1 if clause i contains variable j as a positive literal, the value 1 if clause i contains variable j as a negative literal, and the value 0 if clause i does not contain variable j as a positive or negative literal. Figure 1 shows an example of a CNF formula and its .0 ˙ 1/ matrix representation. It is well known that the question of satisfiability of a given CNF formula can be cast as an integer program as follows: M α C b  Z; ˛i 2 f0; 1g; for all 0  i < n;

(1)

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where M is the .0 ˙ 1/ matrix representation of , b is an integer vector with bi equal to the number of 1 entries in row i of M , and Z is a vector of 1s. A solution to this system of inequalities certifies is satisfiable. In this case a model can be obtained directly from α as ˛i is the value of variable vi . If there is no solution to the system, then is unsatisfiable. In addition to the well-known matrix operations, two matrix operations that are relevant to SAT algorithms on .0 ˙ 1/ matrix representations are applied in this chapter. The first is column scaling: a column may be multiplied or scaled by 1 which has the effect of reversing the polarity of a single variable. Every solution before scaling corresponds to a solution after scaling; the difference is that the values of variables associated with scaled columns are reversed. The second is row and column reordering: rows and columns may be permuted with the effect on a solution being only a possible relabeling of variables taking value 1.

3.2

Binary Decision Diagrams

Binary decision diagrams [6, 103] are a general, graphical representation for arbitrary Boolean functions. Various forms have been put into use, especially for solving VLSI design and verification problems. A canonical form [27, 28] has been shown to be quite useful for representing some particular, commonly occurring, Boolean functions. An important advantage of BDDs is that the complexity of binary and unary operations such as existential quantification, logical or, and logical and, among others, is efficient with respect to the size of the BDD operands. Typically, a given formula is represented as a large collection of BDDs and operations such as those stated above are applied repeatedly to create a single BDD which expresses the models, if any, of . Intermediate BDDs are created in the process. Unfortunately, the size of intermediate BDDs may become extraordinarily and impractically large even if the final BDD is small. So in some applications BDDs are useful and in some they are not. A binary decision diagram (BDD) is a rooted, directed acyclic graph. A BDD is used to compactly represent the truth table, and therefore complete functional description, of a Boolean function. Vertices of a BDD are called terminal if they have no outgoing edges and are called internal otherwise. There is one internal vertex, called the root, which has no incoming edge. There is at least one terminal vertex, labeled 1, and at most two terminal vertices, labeled 0 and 1. Internal vertices are labeled to represent the variables of the corresponding Boolean function. An internal vertex has exactly two outgoing edges, labeled 1 and 0. The vertices incident to edges outgoing from vertex v are called then.v/ and else.v/, respectively. Associated with any internal vertex v is an attribute called index.v/ which satisfies the properties index.v/ < minfindex.then.v//; index.else.v//g and index.v/ D index.w/ if and only if vertices v and w have the same labeling (i.e., correspond to the same variable). Thus, the index attribute imposes a linear ordering on the variables of a BDD. An example of a formula and one of its BDD representations is given in Fig.2.

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v1 1

0

v2

v3

0

1

0

1

v3 1 1

0 0

Fig. 2 The formula .v1 _ :v3 / ^ .:v1 _ v2 / ^ .:v1 _ :v2 _ v3 / represented as a BDD. The topmost vertex is the root. The two bottom vertices are terminal vertices. Edges are directed from upper vertices to lower vertices. Vertex labels (variable names) are shown inside the vertices. The 0 branch out of a vertex labeled v means v takes the value 0. The 1 branch out of a vertex labeled v means v takes the value 1. The index of a vertex is, in this case, the subscript of the variable labeling that vertex

Clearly, there is no unique BDD for a given formula. In fact, for the same formula, one BDD might be extraordinarily large and another might be rather compact. It is usually advantageous to use the smallest BDD possible. At least one canonical form of BDD, called reduced ordered BDD, does this [27, 28]. The idea is to order the variables of a formula and construct a BDD such that (1) variables contained in a path from the root to any leaf respect that ordering and (2) each vertex is the root of a BDD that represents a Boolean function that is unique with respect to all other vertices. Two Boolean functions are equivalent if their reduced ordered BDDs are isomorphic. A more detailed explanation is given in Sect. 5.9.

3.3

Implication Graph

E .V; E/ E where V An implication graph of a CNF formula is a directed graph G consists of one special vertex T which corresponds to the value 1, other vertices which correspond to the literals of , and the edge set EE such that there is an edge hvi ; vj i 2 EE if and only if there is a clause .:vi _ vj / in and an edge hT; vi i 2 EE E if and only if there is a unit clause .vi / (respectively, .:vi /) in . (hT; :vi i 2 E) Figure 3 shows an example of an implication graph for a particular 2-CNF formula. Implication graphs are most useful for, but not restricted to, 2-CNF formulas. In this role the meaning of an edge hvi ; vj i is as follows: if variable vi is assigned the value 1 then variable vj is inferred to have value 1, or else the clause .:vi _ vj / will be falsified.

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Fig. 3 An implication graph and associated 2-CNF formula

T

v0

¬ v0

v1

¬ v1

v2

¬ v2

v3

¬ v3

((v0 ) ∧ (¬v0 ∨¬ v2 ) ∧ (v1 ∨ v2 ) ∧ (¬v1 ∨¬ v3 ) ∧ (¬v1 ∨ v3 ) ∧ (v1 ∨ v3 ) ∧ (v0 ∨¬ v3 ))

3.4

Propositional Connection Graph

A propositional connection graph for a CNF formula is an undirected graph G .V; E/ whose vertex set corresponds to the clauses of and whose edge set is such that there is an edge fci ; cj g 2 E if and only if the clause in represented by vertex ci has a literal that appears negated in the clause represented by vertex cj , and there is no other literal in ci ’s clause that appears negated in cj ’s clause. An example of a connection graph for a particular CNF formula is given in Fig. 4. Propositional connection graphs are a specialization of the first-order connection graphs developed by Kowalski [102].

3.5

Variable-Clause Matching Graph

A variable-clause matching graph for a CNF formula is an undirected bipartite graph G D .V1 ; V2 ; E/ where V1 vertices correspond to clauses and V2 vertices correspond to variables and whose edge set contains an edge fvi ; vj g if and only if vi 2 V1 corresponds to a variable that exists, either as positive or negative literal, in clause vj 2 V2 . An example of a variable-clause matching graph is shown in Fig. 5.

3.6

Formula Digraph

Formulas are defined recursively on Page 313. Any Boolean formula can be adapted to fit this definition with the suitable addition of parentheses, and its structure

Algorithms for the Satisfiability Problem Fig. 4 A propositional connection graph and associated CNF formula

323

{¬v1 ; v2 }

{¬v0 ; v1 ; ¬v7 }

{v0 ; ¬v6 ; ¬v7 }

{¬v5 ; v6 }

{¬v2 ; v3 }

{¬v0 ; ¬v3 ; v8 }

{v0 ; v4 }

{¬v4 ; v5 ; v9 }

((¬v0 ∨ v1 ∨ ¬v7 ) ∧ (¬v1 ∨ v2 ) ∧ (¬v2 ∨ v3 ) ∧ (¬v0 ∨ ¬v3 ∨ v8 ) ∧ (v0 ∨ ¬v6 ∨ ¬v7 ) ∧ (¬v5 ∨ v6 ) ∧ (¬v4 ∨ v5 ∨ v9 ) ∧ (v0 ∨ v4 ))

v0

Fig. 5 A variable-clause matching graph and associated CNF formula

v1

{v0 ; ¬v2 ;v3 }

v2

v3

{v1 ; ¬v2 ; ¬v4 }

v4

{¬v0 ; ¬v2 ;v4 }

↔ ∧ ↔

∧ →

∨

→ v0

¬

v1

v2

¬

v3

¬

v4

Fig. 6 A formula digraph and associated formula

(as opposed to its functionality) can be represented by a binary rooted acyclic digraph. In such a digraph, each internal vertex corresponds to a binary operator or a unary operator : that occurs in the formula. A vertex corresponding to a binary operator has two outward-oriented edges: the left edge corresponds to the left subformula and the right edge to the right subformula operated on. A vertex corresponding to : has one outward directed edge. The root represents the operator applied at top level. The leaves are variables. Call such a representation a wff digraph. An example is given in Fig. 6. An efficient algorithm for constructing a wff digraph is given in Sect. 5.1.

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Satisfiability Index

Let be a CNF formula and let M be its .0 ˙ 1/ matrix representation. Let α D .˛0 ; ˛1 ; : : : ; ˛n1 / be an n-dimensional vector of real variables. Let z be a real variable and let Z = .z; z; : : : ; z/ be an m dimensional vector where every component is the variable z. Finally, let b D .b0 ; b1 ; : : : ; bm1 / be an m dimensional vector such that for all 0  i < m, bi is the number of negative literals in clause ci . Form the system of inequalities M α C b  Z;

(2)

0  ˛i  1 for all 0  i < n: The satisfiability index of is the minimum z for which no constraints of the system are violated. For example, the CNF formula ..v1 _ :v2 / ^ .v2 _ :v3 _ v5 / ^ .v3 _ :v4 _ :v5 / ^ .v4 _ :v1 // has satisfiability index of 5/4.

3.8

And/Inverter Graphs

An AIG is a directed acyclic graph where all gate vertices have in-degree 2, all input vertices have in-degree 0, all output vertices have in-degree 1, and edges are labeled as either negated or not negated. Any combinational circuit can be equivalently implemented as a circuit involving only 2-input and gates and not gates. Such a circuit has an AIG representation: gate vertices correspond directly to the and gates, negated edges correspond directly to the not gates, and inputs and outputs represent themselves directly. Negated edges are typically labeled by overlaying a white circle on the edge: this distinguishes them from non-negated edges which are unlabeled. Examples are shown in Fig. 7.

4

Applications

This section presents a sample of real-world problems that may be viewed as, or transformed to, instances of SAT. Of primary interest is to explore the thinking process required for setting up logic problems and gauging the complexity of the resulting systems. Since it is infeasible to meet this objective and thoroughly discuss a large number of known applications, a small but varied and interesting collection of problems are discussed.

Algorithms for the Satisfiability Problem

a

325

b

d

A

c

u X

B w

C

Y

v

Fig. 7 Examples of AIGs. Edges with white circles are negated. Edges without a white circle are not negated. (a) and gate. (b) or gate. (c) xor gate. (d) 1-bit add circuit of Fig. 9

4.1

Consistency Analysis in Scenario Projects

This application, taken from the area of scenario management [8, 54, 63], is contributed by Feldmann and Sensen [57] of Burkhard Monien’s old PC2 group at Universit¨at Paderborn, Germany. A scenario consists of (i) a progression of events from a known base situation to a possible terminal future situation and (ii) a means to evaluate its likelihood. Scenarios are used by managers and politicians to strategically plan the use of resources needed for solutions to environmental, social, economic, and other such problems. A systematic approach to scenario management due to Gausemeier, Fink, and Schlake [62] involves the realization of scenario projects with the following properties: 1. There is a set S of key factors. Let the number of key factors be n. 2. For each key factor si 2 S , there is a set Di D fdi:1 ; di:2 ; : : : di:mi g of mi possible future developments. In the language of databases, key factors are attributes and future developments are attribute values. 3. For all 1  i  n, 1  k  mi , denote by .si ; di:k / a feasible projection of development di:k 2 Di from key factor si . For each pair of projections .si ; di:k /, .sj ; dj:l /, a consistency value, usually an integer ranging from 0 to 4, is defined. A consistency value of 0 typically means two projections are completely inconsistent, a value of 4 typically means the two projections support each other strongly, and the other values account for intermediate levels of support.

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P P Consistency values may be organized in a niD1 mi  niD1 mi matrix with rows and columns indexed on projections. 4. Projections for all key factors may be bundled into a vector x D .xs1 ; : : : ; xsn / where xsi is a future development of key factor si , i D 1; 2; : : : ; n. In the language of databases, a bundle is a tuple which describes an assignment of values to each attribute. 5. The consistency of bundle x is the sum of the consistency values of all pairs .si ; xsi /; .sj ; xsj / of projections represented by x if no pair has consistency value of 0, and is 0 otherwise. Bundles with greatest (positive) consistency are determined and clustered. Each cluster is a scenario. To illustrate, consider a simplified example from [57] which is intended to develop likely scenarios for the German school system over the next 20 years. It was felt by experts that 20 key factors are needed for such forecasts; to keep the example small, only the first 10 are shown. The first five key factors and their associated future developments are as follows: [s1 ] continuing education ([d1:1 ] lifelong learning), [s2 ] importance of education ([d2:1 ] important, [d2:2 ] unimportant), [s3 ] methods of learning ([d3:1 ] distance learning, [d3:2 ] classroom learning), [s4 ] organization and policies of universities ([d4:1 ] enrollment selectivity, [d4:2 ] semester schedules), and [s5 ] adequacy of trained people ([d5:1 ] sufficiently many, [d5:2 ] not enough). The table in Fig. 8, called a consistency matrix, shows the consistency values for all possible pairs of future developments. The si ; sj cell of the matrix shows the consistency values of pairs f.si ; di:x /; .sj ; dj:y / W 1  x  mi ; 1  y  mj g, with all pairs which include .si ; di:x / on the xth row of the cell. For example, jD5 j D 2 and jD2 j D 2, so there are 4 numbers in cell s5 ; s2 and the consistency value of .s5 ; d5:2 /; .s2 ; d2:2 / is the bottom right number of that cell. That number is 0 because experts have decided the combination of there being too few trained people (d5:2 ) at the same time education is considered unimportant (d2:2 ) is unlikely. It is relatively easy to compute the consistency of a given bundle from this table. One possible bundle of future developments is f.si ; di:1 / W 1  i  10g. Since the consistency value of f.s5 ; d5:1 /; .s2 ; d2:1 /g is 0, this bundle is inconsistent. Another possible bundle is f.s1 ; d1:1 /; .s2 ; d2:1 /; .s3 ; d3:1 /; .s4 ; d4:1 /; .s5 ; d5:2 /; .s6 ; d6:3 /; .s7 ; d7:1 /; .s8 ; d8:2 /; .s9 ; d9:1 /; .s10 ; d10:2 /g: Its consistency value is 120 (the sum of the 45 relevant consistency values from the table). Finding good scenarios from a given consistency matrix requires efficient answers to the following questions: 1. Does a consistent bundle exist? 2. How many consistent bundles exist? 3. What is the bundle having the greatest consistency?

Algorithms for the Satisfiability Problem

s2 s3 s4 s5 s6 s7 s8 s9 s10

4 1 2 3 3 3 2 3 2 2 3 3 1 2 2 2 3 3 3 2 2

s1

23 23 23 13 04 40 23 23 23 40 04 23 23 23 23 23 30 23 23

s2

23 23 04 40 32 23 23 30 03 42 31 02 23 23 04 40 20

s3

23 23 23 32 23 23 23 23 23 23 23 23 23 23 23

s4

327

13 21 23 04 40 03 04 32 13 14 40 04 23

s5

3 1 2 2 2 2 2 2 2 2

03 32 34 32 31 32 32 32 32 32

s6

40 23 04 40 41 23 32 23

s7

4 3 1 2 3

2 2 2 3 2

s8

1 1 3 2 3

12 23 23

s9

Fig. 8 A consistency matrix for a scenario project Table 3 Formula to determine existence of a consistent bundle Clause of S;D .:vi;j _ :vi;k / .vi;1 _ : : : _ vi;mi / .:vi;k _ :vj;l /

Subscript range 1  i  n; 1  j < k  mi 1i n i; j; k; l W Ci:k;j:l D 0

Meaning 1 development/key factor 1 development/key factor Consistent developments only

Finding answers to these questions is N P-hard [57]. But transforming to CNF formulas, in some cases with weights on proposition letters, and solving a variant of SAT are sometimes reasonable ways to tackle such problems. This context provides the opportunity to use our vast knowledge of SAT structures and analysis to apply an algorithm that has a reasonable chance of solving the consistency problems efficiently. It is next shown how to construct representative formulas so that, often, a large subset of clauses is polynomial-time solvable and the whole formula is relatively easy to solve. Consider, first, the question whether a consistent bundle exists for a given consistency matrix of n key factors with mi projections for factor i , 1  i  n. Let Ci:k;j:l denote the consistency of the pair .si ; di:k /.sj ; dj:l / and let D D [niD1 Di . For each future development di:j , define variable vi;j which is intended to take the value 1 if and only if di;j is a future development for key attribute si . The CNF formula S;D with the clauses described in Table 3 then says that there is a consistent bundle. That is, the formula is satisfiable if and only if there is a consistent bundle.

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The second question to be considered is how many consistent bundles exist for a given consistency matrix? This is the same as asking how many satisfying truth assignments there are for S;D . A simple inclusion-exclusion algorithm exhibits very good performance for some actual problems of this sort [57]. The third question is which bundle has the greatest consistency value? This question can be transformed to an instance of the variable-weighted satisfiability problem (defined on Page 318). The transformed formula consists of S;D plus some additional clauses as follows. For each pair .si ; di:k /.sj ; dj:l /, i 6D j , of projections such that Ci;k;j;l > 0, create a new Boolean variable pi;k;j;l of weight Ci;k;j;l and add the following subexpression to S;D : .:vi;k _ :vj;l _ pi;k;j;l / ^ .vi;k _ :pi;k;j;l / ^ .vj;l _ :pi;k;j;l /: Observe that a satisfying assignment requires pi;j;k;l to have value 1 if and only if vi;k and vj;l both have value 1, that is, if and only if the consistency value of f.si ; di:k /; .sj ; dj;l /g is included in the calculation of the consistency value of the bundle. If weight 0 is assigned to all variables other than the pi;k;j;l ’s, a maximum weight solution specifies a maximum consistency bundle. It should be pointed out that, although the number of clauses added to S;D might be significant compared to the number of clauses originally in S;D , the total number of clauses will be linearly related to the size of the consistency matrix. Moreover, the set of additional clauses is a Horn subformula.2 A maximum weight solution can be found by means of a branch-and-bound algorithm such as that discussed in Sect. 5.11.

4.2

Testing of VLSI Circuits

A classic application is the design of test vectors for VLSI circuits. At the specification level, a combinational VLSI circuit is regarded to be a function mapping n 0-1 inputs to m 0-1 outputs.3 At the design level, the interconnection of numerous 0-1 logic gates are required to implement the function. Each connection entails an actual interconnect point that can fail during or soon after manufacture. Failure of an interconnect point usually causes the point to become stuck-at value 0 or 1. Traditionally, VLSI circuit testing includes testing all interconnect points for stuck-at faults. This task is difficult because interconnect points are encased in plastic and are therefore inaccessible directly. The solution is to apply an input pattern to the circuit which excites the point under test and sensitizes a path through the circuit from the test point to some output so that the correct value at the test point can be determined at the output.

2 3

Horn formulas are solved efficiently (see Sect. 6.2). Actual circuit voltage levels are abstracted to the values 0 and 1.

Algorithms for the Satisfiability Problem

A

329

u X

gate 1

gate 4

B w gate 3

Y

v

C

gate 5

gate 2

Fig. 9 A 1-bit full adder circuit

a

c

b

a

c

b

a

c

b

a

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1

Fig. 10 Truth tables for logic elements of a 1-bit full adder

The following example illustrates how such an input pattern and output is found for one internal point of a 1-bit full adder: an elementary but ubiquitous functional hardware block that is depicted in Fig. 9. For the sake of discussion, assume a given circuit will have at most one stuck-at failure. The 1-bit full adder uses logic gates that behave according to the truth tables in Fig. 10 where a and b are gate inputs and c is a gate output. Suppose none of the interconnect points enclosed by the dashed line of Fig. 9 are directly accessible, and suppose it is desired to develop a test pattern to determine whether point w is stuck at 0. Then inputs must be set to give point w the value 1. This is accomplished by means of the Boolean expression 1 D .A ^ B/. The value of point w can only be observed at output Y . But this requires point v be set to 0. This can be accomplished if either the value of C is 0 or u is 0, and u is 0 if and only if A and B have the same value. Therefore, the Boolean expression representing sensitization of a path from w to Y is 2 D .:C _ .A ^ B/ _ .:A ^ :B//. The conjunction 1 ^ 2 is the CNF expression .A ^ B/. This expression is satisfied if and only if A and B are set to 1. Such an

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input will cause output Y to have value 1 if w is not stuck at 0 and value 0 if w is stuck at 0, assuming no other interconnect point is stuck at some value. Test patterns must be generated for all internal interconnect points of a VLSI circuit. There could be millions of these and the corresponding expressions could be considerably more complex than that of the example above. Moreover, testing is complicated by the fact that most circuits are not combinational: that is, they contain feedback loops. This last case is mitigated by adding circuitry for testing purposes only. The magnitude of the testing problem, although seemingly daunting, is not great enough to cause major concern at this time because it seems that SAT problems arising in this domain are usually easy. Thus, at the moment, the VLSI testing problem is considered to be under control. However, this may change in the near future since the number of internal points in a dense circuit is expected to continue to increase dramatically.

4.3

Diagnosis of Circuit Faults

A natural extension of the test design discussed in Sect. 4.2 is finding a way to automate the diagnosis of, say, bad chips, starting with descriptions of their bad outputs. How can one reason backwards to identify likely causes of the malfunction? Also, given knowledge that some components are more likely to fail than others, how can the diagnosis system be tailored to suggest the most likely causes first? The first of these questions is discussed in this section. The first step is to write the Boolean expression representing both the normal and abnormal behavior of the analyzed circuit. This is illustrated using the circuit of Fig. 9. The expression is assembled in stages, one for each gate, starting with gate 3 of Fig. 9, which is an and gate. The behavior of a correctly functioning and gate is specified in the right-hand truth table in Fig. 10. The following formula expresses the truth table: .a ^ b ^ c/ _ .:a ^ :b ^ c/ _ .:a ^ b ^ :c/ _ .:a ^ :b ^ :c/: If gate 3 is possibly stuck at 0, its functionality can be described by adding variable Ab3 (for gate 3 is abnormal) and substituting A,B, and w for a, b, and c, to get the following abnormality expression: .A ^ B ^ w ^ :Ab3 / _ .:A ^ :B ^ w ^ :Ab3 / _ .:A ^ B ^ :w ^ :Ab3 / _ .:A ^ :B ^ :w ^ :Ab3 / _ .:w ^ Ab3 / which has value 1 if and only if gate 3 is functioning normally for inputs A and B or it is functioning abnormally and w is stuck at 0. The extra variable may be regarded as a switch which allows toggling between abnormal and normal states for gate 3. Similarly, switches Ab1 ; Ab2 ; Ab4 ,and Ab5 may be added for all the other gates in

Algorithms for the Satisfiability Problem

331

Table 4 Possible stuck-at-0 failures of 1-bit adder gates (see Fig. 9) assuming given inputs are A D 0 and B; C D 1 and observed outputs are X; Y D 0. This is the list of assignments to  which satisfy the abnormality predicates for the 1-bit adder. A “1” in the column for Abi means gate i is stuck at 0 IDs

Ab1

Ab2

Ab3

Ab4

Ab5

1–16 17–24 25–26

* * 1

* 1 0

* * *

* * 1

1 0 0

The symbol “*” means either “0” or “1”

the circuit, and corresponding expressions may be constructed using those switches. Then the set  D fAb1 ; Ab2 ; Ab3 ; Ab4 ; Ab5 g represents all possible explanations for stuck-at-0 malfunctions. The list of assignments to the variables of  which satisfy the collection of abnormality expressions, given particular inputs and observations, determines all possible combinations of stuck-at-0 failures in the circuit. The next task is to choose the most likely failure combination from the list. This requires some assumptions which are motivated by the following examples. Suppose that, during some test, inputs are set to A D 0 and B; C D 1 and the observed output values are Y D 0 and X D 1 whereas Y D 1 and X D 0 are the correct outputs. One can reason backwards to try to determine which gates are stuck at 0. Gate 4 cannot be stuck at 0 since its output is 1. Suppose gate 4 is working correctly. Since the only gate that gate 4 depends on is gate 1, that gate must be stuck. One cannot tell whether gates 2, 3, and 5 are functioning; normally it would be assumed that they are functioning correctly until evidence to the contrary is obtained. Thus, the natural diagnosis is gate 1 is defective (only Ab1 has value 1). Alternatively, suppose under the same inputs it is observed that X; Y D 0. Possibly, gates 5 and 4 are malfunctioning. If so, all other combinations of gate outputs will lead to the same observable outputs. If gate 5 is defective but gate 4 is good, then u D 1 so gate 1 is good, and any possible combinations of w and v lead to the same observable outputs. If gate 5 is good and gate 4 is defective, the bad Y value may be caused by a defective gate 2. In that case gates 1 and 3 conditions do not affect the observed outputs. But if gate 2 is not defective, the culprit must be gate 1. If gates 4 and 5 are good, then u D 1 so gate 2 is defective. Nothing can be determined about the condition of gate 3 through this test. The results of this paragraph lead to 26 abnormal  values that witness the observed outputs: these are summarized in Table 4, grouped into three cases. In the first two of these cases, the minimum number of gates stuck at 0 is 1. As before, it is assumed that the set of malfunctioning gates is as small as possible, so only those two diagnoses are considered: that is, either (i) gate 5 or (ii) gate 2 is defective. In general, it is argued, common sense leads us to consider

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only minimal sets of abnormalities: sets, like gate 2 above, where no proper subset is consistent with the observations. This is Reiter’s principle of parsimony [117]: A diagnosis is a conjecture that some minimal set of components are faulty.

This sort of inference is called non-monotonic because it is inferred, above, that gate 3 was functioning correctly, since there is no evidence it was not. Later evidence may cause that inference to be withdrawn. Yet a further feature of non-monotonic logic may be figured into such systems; the following illustrates the idea. Suppose it is known that one component, say, gate 5, is the least likely to fail. Then, if there are any diagnoses in which gate 5 does not fail, it will report only such diagnoses. If gate 5 fails in all diagnoses, then it will report all the diagnoses. Essentially, this reflects a kind of preference relationship among diagnoses. There is now software which automates this diagnosis process (e.g., [66]). Although worst-case performance of such systems is provably bad, such a system can be useful in many circumstances. Unfortunately, implementations of non-monotonic inference are new enough that there is not yet a sufficiently large body of standard benchmark examples.

4.4

Functional Verification of Hardware Design

Proving correctness of the design of a given block of hardware has become a major concern due to the complexity of present day hardware systems and the economics of product delivery time. Prototyping is no longer feasible since it takes too much time and fabrication costs are high. Breadboarding no longer gives reliable results because of the electrical differences between integrated circuits and discrete components. Simulation-based methodologies are generally fast but do not completely validate a design since there are many cases left unconsidered. Formal verification methods can give better results, where applicable, since they will catch design errors that may go undetected by a simulation. Formal verification methods are used to check correctness by detecting errors in translation between abstract levels of the design hierarchy. Design hierarchies are used because it is impractical to design a VLSI circuit involving millions of components at the substrate, or lowest, level of abstraction. Instead, it is more reasonable to design at the specification or highest level of abstraction and use software tools to translate the design, through some intermediate stages such as the logic-gate level, to the substrate level. The functionality between a pair of levels may be compared. In this case, the more abstract level of the pair is said to be the specification and the other level is the implementation level of the pair. If functionality is equivalent between all adjacent pairs of levels, the design is said to be verified. Determining functional equivalence (defined on Page 317) between levels amounts to proving a theorem of the form implementation I realizes specification S in a particular, suitable formal proof system. For illustration purposes only, consider the 1-bit full adder of Fig. 9. Inputs A and B represent a particular bit position of

Algorithms for the Satisfiability Problem

333

two different binary addends. Input C is the carry due to the addition at the next lower valued bit position. Output X is the value of the same bit position of the sum and output Y is the carry to the next higher valued bit position. The output X must have value 1 if and only if all inputs have value 1 or exactly one input has value 1. The output Y has value 1 if and only if at least two out of three inputs have value 1. Therefore, the following simple Boolean expression offers a reasonable specification of any 1-bit full adder: .X , .A ^ :B ^ :C / _ .:A ^ B^:C / _ .:A ^ :B ^ C / _ .A ^ B ^ C // ^ .Y , .A ^ B/ _ .A ^ C / _ .B ^ C //: A proposed implementation of this specification is given in the dotted region of Fig. 9. Its behavior may be described by a Boolean expression that equates each gate output to the corresponding logical function applied to its inputs. The following is such an expression: .u , .A ^ :B/ _ .:A ^ B// ^ .v , u ^ C / ^ .w , A ^ B/ ^ .X , .u ^ :C / _ .:u ^ C // ^ .Y , w _ v/: Designate these formulas S .A; B; C; X; Y / and I .A; B; C; X; Y; u; v; w/, respectively. The adder correctly implements the specification if, for all possible inputs A; B; C 2 f0; 1g, the output of the adder matches the specified output. For any individual inputs, A; B; C , that entails checking whether I .A; B; C; X; Y; u; v; w/ has value 1 for the appropriate u; v; w, S .A; B; C; X; Y /

, 9u; v; w

I .A; B; C; X; Y; u; v; w/;

where the quantification 9u; v; w is over Boolean values u; v; w. For specific A; B; C , the problem is a satisfiability problem: can values for u; v; w which make the formula have value 1 be found? (Of course, it is an easy satisfiability problem in this case.) Thus, the question of whether the adder correctly implements the specification for all 32 possible input sequences is answered using the formula: 8A; B; C; X; Y .

S .A; B; C; X; Y /

, 9u; v; w

I .A; B; C; X; Y; u; v; w//;

which has a second level of Boolean quantification. Such formulas are called quantified Boolean formulas. The example of the 1-bit full adder shows how combinational circuits can be verified. A characteristic of combinational circuits is that output behavior is strictly a function of the current values of inputs and does not depend on past history.

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Table 5 The operators of temporal logic Op. name

.S; si / ˆ

If and only if

Not And Or Henceforth Eventually Next

p (p a variable) : 1 1^ 2 1_ 2  1 ˘ 1 ı 1

si .p/ D 1 .S; si / 6ˆ 1 .S; si / ˆ 1 and .S; si / ˆ 2 .S; si / ˆ 1 or .S; si / ˆ 2 .S; sj / ˆ 1 for all states sj ; j  i .S; sj / ˆ 1 for some state sj ; j  i .S; siC1 / ˆ 1 For some j  i; .S; si /; .S; siC1 /; : : : ; .S; sj 1 / ˆ and .S; sj / ˆ 2

Until

1

U

2

1;

However, circuits frequently contain components, such as registers, which exhibit some form of time dependency. Such effects may be modeled by some form of propositional temporal logic. Systems of temporal logic have been applied successfully to the verification of some sequential circuits including microprocessors. One may think of a sequential circuit as possessing one of a finite number of valid states at any one time. The current state of such a circuit embodies the complete electrical signal history of the circuit beginning with some distinguished initial state. A change in the electrical properties of a sequential circuit at a particular moment in time is represented as a fully deterministic movement from one state to another based on the current state and a change in some subset of input values only. Such a change in state is accompanied by a change in output values. A description of several temporal logics can be found in [149]. For illustrative purposes, one of these is discussed below, namely, the linear time temporal logic (LTTL). LTTL formulas take value 1 with respect to an infinite sequence of states S D fs0 ; s1 ; s2 ; : : :g. States of S obey the following: state s0 is a legal initial state of the system, state si is a legal state of the system at time step i , and every pair si ; si C1 must be a legal pair of states. Legal pairs of states are forced by some of the components of the formula itself (the latch example at the end of this section illustrates this). Each state is just an interpretation of an assignment of values to a set of Boolean variables. LTTL is an extension of the propositional calculus that adds one binary and three unary temporal operators which are described below and whose semantics are outlined in Table 5 along with the standard propositional operators :, ^, and _ (the definition of .S; si / ˆ is given below). The syntax of LTTL formulas is the same as for propositional logic except for the additional operators. Let be a LTTL expression that includes a component representing satisfaction of some desired property in a given circuit. An example of a property for a potential JK flip-flop might be that it is eventually possible to have a value of 1 for output Q while input J has value 1 which has a corresponding formula ˘.J ^ Q/. The event that has value 1 for state si in S is denoted by

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a

s r

Set-Reset Latch

q

b

c

s

...

...

r

...

...

q

...

...

d

e

f

...

q

...

Fig. 11 A set-reset latch and complete description of signal behavior. Inputs are s and r, output is q. The horizontal axis represents time. The vertical axis represents signal value for both inputs and output. Values of all signals are assumed to be either 0 (low) or 1 (high) at any particular time. Two rows for q values are used to differentiate between the cases where the initial value of q is 0 or 1

.S; si / ˆ : Also, say Sˆ Finally, two LTTL formulas

if and only if .S; s0 / ˆ : 1

Sˆ

and 1

2

are equivalent if, for all sequences S ,

if and only if S ˆ

2:

Our example concerns a hardware device of two inputs and one output called a set-reset latch. The electrical behavior of such a device is depicted in Fig. 11 as a set of six waveforms which show the value of the output q in terms of the history of the values of inputs r and s. For example, consider waveform (a) in the Fig. 11. This shows the value of the output q, as a function of time, if initially q, r, and s have value 0 and then s is pulsed or raised to value 1 then some time later dropped to value 0. The waveform shows the value of q rises to 1 some time after s does and stays at value 1 after the value of s drops. The waveform (a) also shows that if q has value 1 and r and s have value 0 and then r is pulsed, the value of q drops to 0. Observe the two cases where (i) r and s have value 1 at the same moment and (ii) q changes value after s or r pulses are not allowed. The six waveforms are enough to specify the behavior of the latch because the device is simple enough that only recent history matters. The specification of this behavior is given by the LTTL formula of Table 6. Observe that the first three expressions of Table 6 represent assumptions needed for the latch to work correctly and do not necessarily reflect requirements that can

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Table 6 Temporal logic formula for a set-reset latch Expressions

Comments

:.s ^ r/ ..s ^ :q/ ! ..s U q/ _ s// ..r ^ q/ ! ..r U :q/ _ r// .s ! ˘q/ .r ! ˘:q/ ..:q ! ..:q U s/ _ :q/// ..q ! ..q U r/ _ q///

No two inputs have value 1 simultaneously Input s cannot change if s is 1 and q is 0 Input r cannot change if r is 1 and q is 1 If s is 1, q will eventually be 1 If r is 1, q will eventually be 0 Output q rises to 1 only if s becomes 1 Output q drops to 0 only if r becomes 1

be realized within the circuitry of the latch itself. Care must be taken to insure that the circuitry in which a latch is placed meets those requirements. Latch states are triples representing values of s, r, and q, respectively. Some examples, corresponding to state sequences depicted by waveforms (a)–(f) in Fig. 11, that satisfy the formula of Table 6 are as follows: Sa W .h000i; Sb W .h000i; Sc W .h000i; Sd W .h001i; Se W .h001i; Sf W .h001i;

h100i; h100i; h010i; h101i; h101i; h011i;

h101i; h101i; h000i; h001i; h001i; h010i;

h001i; h001i; h010i; h011i; h101i; h000i;

h011i; h010i; h000i; : : :/ h101i; h001i; : : :/ h000i; : : :/ h010i; h000i; : : :/ h001i; : : :/ h010i; h000i; : : :/

Clearly, infinitely many sequences satisfy the formula of Table 6, so the problem of verifying functionality for sequential circuits appears daunting. However, by means of careful algorithm design, it is sometimes possible to produce such verifications and successes have been reported. In addition, other successful temporal logic systems such as computation tree logic and interval temporal logic have been introduced, along with algorithms for proving theorems in these logics. The reader is referred to [149], Combinatorial Optimization Techniques for Network-Based Data Mining for details and citations.

4.5

Bounded Model Checking

Section 4.4 considered solving verification problems of the form S ˆ 1  S ˆ 2 . If it is desired instead to determine whether there exists an S such that S ˆ temporal operators can be traded for Boolean variables, the sentence can be expressed as a propositional formula, and a SAT solver applied. The propositional formula must have the following parts: 1. Components which force the property or properties of the time-dependent expression to hold 2. Components which establish the starting state

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3. Components which force legal state transitions to occur In order for the Boolean expression to remain of reasonable size, it is generally necessary to bound the number of time steps in which the time-dependent expression is to be verified, thus the name bounded model checking. As an example, consider a simple 2-bit counter whose outputs are represented by variables v1 (LSB) and v2 (MSB). Introduce variables vi1 and vi2 whose values are intended to be the same as those of variables v1 and v2 , respectively, on the i th time step. Suppose the starting state is the case where both v01 and v02 have value 0. The transition relation is Current output 00 01 10 11

: : : :

Next output 01 10 11 00

the i th line of which can be expressed as the following Boolean function: .vi1C1 $ :vi1 / ^ .vi2C1 $ vi1 ˚ vi2 /: Suppose the time-dependent expression to be proved is as follows: Can the two-bit counter reach a count of 11 in exactly three time steps?

Assemble the propositional formula having value 1 if and only if the above query holds as the conjunction of the following three parts: 1. Force the property to hold: .:.v01 ^ v02 / ^ :.v11 ^ v12 / ^ :.v21 ^ v22 / ^ .v31 ^ v32 //: 2. Express the starting state: .:v01 ^ :v02 /: 3. Force legal transitions (repetitions of the transition relation): .v11 $ :v01 / ^ .v12 $ v01 ˚ v02 / ^ .v21 $ :v11 / ^ .v22 $ v11 ˚ v12 / ^ .v31 $ :v21 / ^ .v32 $ v21 ˚ v22 /: Since .a $ b/ , .a _ :b/ ^ .:a _ b/, the last expression can be directly turned into a CNF expression. Therefore, the entire formula can be turned into a CNF expression and solved with an off-the-shelf SAT solver. The reader may check that the following assignment satisfies the above expressions: v01 D 0; v02 D 0; v11 D 1; v12 D 0; v21 D 0; v22 D 1; v31 D 1; v32 D 1:

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It may also be verified that no other assignment of values to vi1 and vi2 , 0  i  3, satisfies the above expressions. Information on the use and success of bounded model checking may be found in [18, 36].

4.6

Combinational Equivalence Checking

The power of bounded model checking is not needed to solve the combinational equivalence checking (CEC) problem which is to verify that two given combinational circuit implementations are functionally equivalent. CEC problems are easier, in general, because they carry no time dependency and there are no feedback loops in combinational circuits. It has recently been discovered that CEC can be solved very efficiently, in general [92, 94, 95], by incrementally building a singleoutput and/inverter graph (AIG), representing the miter [26] of both input circuits, and checking whether the output has value 0 for all combinations of input values. The AIG is built one vertex at a time, working from input to output. Vertices are merged if they are found to be functionally equivalent. Vertices can be found functionally equivalent in two ways: (1) candidates are determined by random simulation [26, 93] and then checked by a SAT solver for functional equivalence, and (2) candidates are hashed to the same location in the data structure representing vertices of the AIG (for the address of a vertex to represent function, it must depend solely on the opposite endpoints of the vertex’s incident edges, and therefore, an address change typically takes place on a merge). AIG construction continues until all vertices have been placed into the AIG and no merging is possible. The technique exploits the fact that checking the equivalence of two topologically similar circuits is relatively easy [65] and avoids testing all possible input-output combinations, which is CoN P-hard. In CEC the AIG is incrementally developed from gate level representations. For example, Fig. 7a shows the AIG for an and gate, Fig. 7b shows the AIG for an or gate, Fig.7c shows the AIG for exclusive-or, and Fig.7d shows the AIG for the adder circuit of Fig. 9. Vertices may take 0-1 values. Values are assigned independently to input vertices. The value of a gate vertex (a dependent vertex) is the product x1 x2 where xi is either the value of the non-arrow side endpoint of incoming edge i , 1  i  2, if the edge is not negated or 1 minus the value of the endpoint if it is negated. CEC begins with the construction of the AIG for one of the circuits as exemplified in Fig. 12a which shows an AIG and a circuit it is to be compared against. Working from input to output, a node of the circuit is determined to be functionally equivalent to an AIG vertex and is merged with the vertex. The output lines from the merged node become AIG edges outgoing from the vertex involved in the merge. Figure 12b shows the first two nodes of the circuit being merged with the AIG and Fig. 12c shows the completed AIG.

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a

b a

b

b

3

c

1

5

4 2 6 Fig. 12 Creating the AIG. (a) The beginning of the equivalency check – one circuit has been transformed to an AIG. (b) Node a of the circuit (Fig. 12a) has been merged with vertex b of the AIG. (c) The completed AIG

CEC proceeds with the application of many random input vectors to the input vertices of the AIG for the purpose of partitioning the vertices into potential equivalence classes (two vertices are in the same equivalence class if they take the same value under all random input vectors). In the case of Fig. 12c, suppose the potential equivalence classes are f1; 2g; f3; 4g, and f5; 6g. The next step is to use a SAT solver to verify that the equivalences actually hold. Those that do are merged, resulting in a smaller AIG. The cycle repeats until merging is no longer possible. If the output vertices hash to the same address, the circuits are equivalent. Alternatively, the circuits are equivalent if the AIG is augmented by adding a vertex corresponding to an xor gate with incident edges connecting to the two output vertices of the AIG, and the resulting graph represents an unsatisfiable formula, determined by using a SAT solver. Above, CEC has been shown to check the equivalence of two combinational circuits, but it can also be used for checking a specification against a circuit implementation or for reverse engineering a circuit.

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J. Franco and S. Weaver

Transformations to Satisfiability

It is routine in the operations research community to transform a given optimization problem into another, solve the new problem, and use the solution to construct a solution or an approximately optimal solution for the given problem. Usual targets of transformations are linear programming and network flows. In some cases, where the given problem is N P-complete, it may be more efficient to obtain a solution or approximate solution by transformation to a satisfiability problem than by solving directly. However, care must be taken to choose a transformation that keeps running time down and supports low error rates. In this section a successful transformation from network Steiner tree problems to weighted MAX-SAT problems is considered (taken from [87]). The network Steiner tree problem originates from the following important network cost problem (see, e.g., [4]). Suppose a potential provider of communications services wants to offer private intercity service for all of its customers. Each customer specifies a collection of cities it needs to have connected in its own private network. Exploiting the extraordinary bandwidth of fiber-optic cables, the provider intends to save cable costs by piggy-backing the traffic of several customers on single cables when possible. Assume there is no practical limit on the number of customers piggybacked to a single cable. The provider wants an answer to the following question: Through what cities should the cables be laid to meet all customer connectivity requirements and minimize total cabling cost? This problem can be formalized as follows. Let G.V; E/ be a graph whose vertex set V D fc1 ; c2 ; c3 ; : : : g represents all cities and whose edge set E represents all possible connections between pairs of cities. Let w W E  ! Z C be such that w.fci ; cj g/ is the cost of laying fiber-optic cable between cities ci and cj . Let R be a given set of vertex-pairs fci ; cj g representing pairs of cities that must be able to communicate with each other due to at least one customer’s requirement. The problem is to find a minimum total weight subgraph of G such that there is a path between every vertex-pair of R. Consider the special case of this problem in which the set T of all vertices occurring in at least one vertex-pair of R is a proper subset of all vertices of G (i.e., the provider has the freedom to use as connection points cities not containing customers’ offices) and R requires that all vertices of T be connected. This is known as the network Steiner tree problem. An example and its optimal solution are given in Fig. 13. This problem is one of the first shown to be N P-complete [85]. The problem appears in many applications and has been extensively studied. Many enumeration algorithms, heuristics, and approximation algorithms are known (see [4] for a list of examples). A feasible solution to an instance .G; T; w/ of the network Steiner tree problem is a tree spanning all vertices of a subgraph of G which includes T . Such a subgraph is called a Steiner tree. A transformation from .G; T; w/ to an instance of satisfiability is a uniform method of encoding of G, T , and w by a CNF formula with nonnegative weights on its clauses. A necessary (feasibility) property of any transformation is that a truth assignment to the variables of which maximizes

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Seattle

New York 112 511 231

58

254

160

San Francisco 25

476

45

975

23

200

Los Angeles

153

Washington

98

64 572

216

184

36 823

963

26

56

184 112

Miami

Fig. 13 An example of a network Steiner tree problem (top) and its optimal solution (bottom). The white nodes are terminals and the numbers represent hypothetical costs of laying fiber-optic cables between pairs of cities

the total weight of all satisfied clauses specifies a feasible solution for .G; T; w/. A desired (optimality) property is, in addition, that feasible solution have minimum total weight. Unfortunately, known transformations that satisfy both properties produce formulas of size that is superlinearly related to the number of edges and vertices in G. Such encodings are useless for large graphs. More practical linear transformations satisfying the feasibility property are possible but these do not satisfy the optimality property. However, it has been demonstrated that, using a carefully defined linear transformation, one can achieve close to optimal results. As an example, consider the linear transformation introduced in [87]. This transformation has a positive integer parameter k which controls the quality of the

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approximation. Without losing any interesting cases, assume that G is connected. It is also assumed that no two paths in G between the same nodes of T have the same weight; this can be made true, if necessary, by making very small adjustments to the weights of the edges. Preprocessing 1. Define an auxiliary weighted graph ..T; E 0 /; w0 / as follows: Graph G 0 D .T; E 0 / is the complete graph on all vertices in T . For edge e 0 D fci ; cj g of G 0 , let w0 .e 0 / be the total cost of the minimum cost path between ci and cj in G. (This can be found, e.g., by Dijkstra’s algorithm). 2. Let W be a minimum-cost spanning tree of .G 0 ; w0 /. It is intended to choose, for each edge fci ; cj g of W , (all the edges on) one entire path in .G; w/ between ci and cj to be included in the Steiner tree. This will produce a Steiner tree for T; G, although perhaps not a minimum-cost tree, as long as no cycles in G are included. For some prespecified, fixed k: Find the k minimum-cost paths in G between ci and cj (again, e.g., by a variation of Dijkstra’s algorithm); call them Pi;j;1 ; : : : ; Pi;j;k . Thus, the algorithm will choose one of these k paths between each pair of elements of W . The Formula The output of the preprocessing step is a tree W spanning all vertices of T . Each edge fci ; cj g 2 W represents one of the paths Pi;j;1 ; : : : ; Pi;j;k in G. The tree W , the list of edges in G, the lists of edges comprising each of the k shortest paths between pairs of vertices of W , and the number k are input to the transformation step. The path weights are not needed by the transformation step and are therefore not provided as an input. The transformation is then carried out as shown in Table 7. In the table, E is the set of edges of G and I is any number greater than the sum of the weights of all edges of G. A maximum weight solution to a formula of Table 7 must satisfy all non-unit clauses because the weights assigned to those clauses are so high. Satisfying those clauses corresponds to choosing all the edges of at least one path between pairs of vertices in the list. Therefore, since the list represents a spanning tree of G 0 , a maximum weight solution specifies a connected subgraph of G which includes all vertices of T . On the other hand, the subgraph must be a tree. If there is a cycle in the subgraph, then there is more than one path from a T vertex u to a non-T vertex v, so there is a shorter path that may be substituted for one of the paths going from u through v to some T vertex. Choosing a shorter path is always possible because it will still be one of the k shortest. Doing so removes at least one edge and cycle. This process may be repeated until all cycles are broken and the number of edges remaining is minimal for all T vertices to be connected. Due to the unit clauses, all edge variables whose values are not set to 1 by an assignment add their edge weights to that of the formula. Therefore, the maximum weight solution contains only edges forced by p variables and must specify a Steiner tree.

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Table 7 Transformation from an instance of the network Steiner tree problem to a CNF formula. The instance has already been preprocessed to spanning tree W plus lists of k shortest paths Pi;j;x , 1  x  k, corresponding to edges fci ; cj g of W . The edges of the original graph are given as set E. Boolean variables created expressly for the transformation are defined at the top and clauses are constructed according to the middle chart. Weights are assigned to clauses as shown at the bottom. The number I is chosen to be greater than the sum of all weights of edges in E Variable

Subscript range

Meaning

ei;j pi;j;l Clause .:ei;j / .pi;j;1 _ : : : _ pi;j;k /

Edge fci ; cj g 2 E fci ; cj g 2 W , 1  l  k Subscript range fci ; cj g 2 E fci ; cj g 2 W

.:pi;j;l _ em;n / Clause .:ei;j / .pi;j;1 _ : : : _ pi;j;k /

fcm ; cn g 2 Pi;j;l Weight w.fi; j g/ I

.:pi;j;l _ em;n /

I

fci ; cj g 2 the Steiner tree Pi;j;l  the Steiner tree Meaning ei;j 62 Steiner tree Include at least one of k shortest paths between ci , cj Include all edges of that path Significance Maximize weights of edges not included Force connected subgraph of G containing all vertices of T Force connected subgraph of G containing all vertices of T

A maximum weight solution specifies an optimal Steiner tree only if k is sufficiently large to admit all the shortest paths in an optimal solution. Generally this is not practical since too large a k will cause the transformation to be too large. However, good results can be obtained even with k values up to 30. Once the transformation to satisfiability is made, an incomplete algorithm such as Walksat (see Sect. 5.8.1) or even a branch-and-bound variant (see Sect. 5.11) can be applied to obtain a solution. The reader is referred to [87] for empirical results showing speed and approximation quality.

4.8

Boolean Data Mining

The field of data mining is concerned with discovering hidden structural information in databases; it is a form of (or variant of) machine learning. In particular, it is concerned with finding hidden correlations among apparently weakly related data. For example, a credit card company might look for patterns of charging purchases that are frequently correlated with using stolen credit card numbers. Using this data, the company can look more closely at suspicious patterns in an attempt to discover thefts before they are reported by the card owners. Many of the methods for data mining involve essentially numerical calculations. However, others work on Boolean data. Typically, the relevant part of the database consists of a set B of m Boolean n-tuples (plus keys to distinguish tuples, which presumably are unimportant here). Each of these n attributes might be the results of some monitoring equipment or experts’ answers to questions.

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Some of the tuples in B are known to have some property; the others are known not to have the property. Thus, what is known is a partially defined Boolean function fB of the n variables. An important problem is to predict whether tuples to be added later will have the same property. This amounts to finding a completely defined Boolean function f which agrees with fB at every value for which it is defined. Frequently, the goal is also to find such an f with, in some sense, a simple definition: such a definition, it is hoped, will reveal some interesting structural property or explanation of the data points. For example, binary vectors of dimension two describing the condition of an automobile might have the following interpretation: Vector

Value

Overheating

Coolant level

00 01 10 11

0 0 0 1

No No Yes Yes

Low Normal Low Normal

where the value associated with each vector is 1 if and only if the automobile’s thermostat is defective. One basic method is to search for such a function f with a relatively short disjunctive normal form (DNF) definition. An example formula in DNF is .v1 ^ v2 ^ :v3 / _ .:v1 ^ v4 / _ .v3 ^ v4 ^ v5 ^ v6 / _ .:v3 /: DNF formulas have the same form as that of CNF formulas except that the positions of the ^’s and the _’s are reversed. In this case the formula is in 4-DNF since each disjunct contains at most 4 conjuncts. Each term, for example, .v1 ^ v2 ^ :v3 / is an implicant: whenever it is true, the property holds. There is extensive research upon the circumstances under which, when some random sample points fB from an actual Boolean function f are supplied, a program can, with high probability, learn a good approximation to f within reasonable time (e.g., [140]). Of course, if certain additional properties of f are known or assumed, more such functions f can be determined. Applications are driving interest in the field. The problem has become more interesting due to the use of binarization which allows nonbinary data sets to be transformed to binary data sets [25]. Such transformations allow the application of special binary tools for cause-effect analysis that would otherwise be unavailable and may even sharpen explainability. For example, for a set of 290 data points containing 9 attributes related to Chinese labor productivity, it was observed in [24] that the short clause Not in the northwest region and time is later than 1987,

where time ranges over the years 1985–1994, explains all the data and the clause SOE is at least 71.44% and time is earlier than 1988,

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where SOE means state-owned enterprises, explains 94 % of the data. The technique of logical analysis of data has been successfully applied to inherently nonbinary problems of oil exploration, psychometric analysis, and economics, among others.

5

General Algorithms

In this section some algorithms for solving various SAT problems are presented. Most of these algorithms operate on CNF formulas. Therefore, the following efficient transformation to CNF formulas is needed to demonstrate the general applicability of these algorithms.

5.1

Efficient Transformation to CNF Formulas

An algorithm due to Tseitin [139] efficiently transforms an arbitrary Boolean formula  to a CNF formula such that has a model if and only if  has a model and if a model for exists, it is a model for . The transformation is important because it supports the general use of many of the algorithms which are described in following sections and require CNF formulas as input. The transformation can best be visualized graphically. As discussed in Sect. 3.6, any Boolean formula  can be represented as a binary rooted acyclic digraph W where each internal vertex represents some operation on one or two operands (an example is given in Fig. 6). Associate with each internal vertex x of W a new variable vx not occurring in . If x represents a binary operator Ox , let vl and vr be the variables associated with the left and right endpoints, respectively, of the two outward-oriented edges of x (vl and vr may be variables labeling internal or leaf vertices). If x represents the operator :, then call the endpoint of the outwardoriented edge vg . For each internal vertex x of W , write vx , .vl Ox vr / vx , :vg

if Ox is binary or if vertex x represents :.

For each equivalence there is a short, functionally equivalent CNF expression. The table of Fig. 14 shows the equivalent CNF expression for all 16 possible binary operators where each bit pattern in the left column expresses the functionality of an operator for each of four assignments to vl and vr , in increasing order, from 00 to 11. The equivalent CNF expression for : is .vg _ vx / ^ .:vg _ :vx /. The target of the transformation is a CNF formula consisting of all clauses in every CNF expression from Fig. 14 that corresponds to an equivalence expressed at a non-leaf vertex of W plus a unit clause that forces the root expression to evaluate to 1. For example, the expression of Fig. 6, namely, v0 $ ..:v0 $ .v1 _ :v2 // ^ .v1 _ :v2 / ^ .:v2 ! :v3 ! v4 //;

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Ox

Equivalent CNF Expression

0000 1111 0011 1100 0101 1010 0001 1110 0010 1101 0100 1011 1000 0111

(¬vx ) (vx ) (vl ∨ ¬vx ) ∧ (¬vl ∨ vx ) (vl ∨ vx ) ∧ (¬vl ∨ ¬vx ) (vr ∨ ¬vx ) ∧ (¬vr ∨ vx ) (vr ∨ vx ) ∧ (¬vr ∨ ¬vx ) (vl ∨ ¬vx ) ∧ (vr ∨ ¬vx ) ∧ (¬vl ∨ ¬vr ∨ vx ) (vl ∨ vx ) ∧ (vr ∨ vx ) ∧ (¬vl ∨ ¬vr ∨ ¬vx ) (¬vl ∨ ¬vx ) ∧ (vr ∨ ¬vx ) ∧ (vl ∨ ¬vr ∨ vx ) (vl ∨ vx ) ∧ (¬vr ∨ vx ) ∧ (¬vl ∨ vr ∨ ¬vx ) (vl ∨ ¬vx ) ∧ (¬vr ∨ ¬vx ) ∧ (¬vl ∨ vr ∨ vx ) (¬vl ∨ vx ) ∧ (vr ∨ vx ) ∧ (vl ∨ ¬vr ∨ ¬vx ) (¬vl ∨ ¬vx ) ∧ (¬vr ∨ ¬vx ) ∧ (vl ∨ vr ∨ vx ) (¬vl ∨ vx ) ∧ (¬vr ∨ vx ) ∧ (vl ∨ vr ∨ ¬vx ) (vl ∨¬vr ∨¬vx )∧(¬vl ∨vr ∨¬vx )∧ (¬vl ∨ ¬vr ∨ vx ) ∧ (vl ∨ vr ∨ vx ) (¬vl ∨ vr ∨ vx ) ∧ (vl ∨ ¬vr ∨ vx )∧ (vl ∨vr ∨¬vx )∧(¬vl ∨¬vr ∨¬vx )

1001 0110

Comment vx ⇔ 0 vx ⇔ 1 vx ⇔ ¬vl vx ⇔ (vl ∧ vr ) vx ⇔ (¬vl ∨ ¬vr ) vx ⇔ (vl → vr ) vx ⇔ (vl ← vr ) vx ⇔ (¬vl ∧ ¬vr ) vx ⇔ (vl ∨ vr ) vx ⇔ (vl ↔ vr ) vx ⇔ (vl ⊕ vr )

Fig. 14 CNF expressions equivalent to vx , .vl Ox vr / for Ox as shown

transforms to .v0 _ vx1 / ^ .:v0 _ :vx1 / ^ .v2 _ vx2 / ^ .:v2 _ :vx2 / ^ .v3 _ vx3 / ^ .:v3 _ :vx3 / ^ .:v1 _ vx4 / ^ .:vx2 _ vx4 / ^ .v1 _ vx2 _ :vx4 / ^ .vx3 _ vx5 / ^ .:v4 _ vx5 / ^ .:vx3 _ v4 _ :vx5 / ^ .vx1 _ :vx4 _ :vx6 / ^ .:vx1 _ vx4 _ :vx6 / ^ .:vx1 _ :vx4 _ vx6 /^ .vx1 _ vx4 _ vx6 / ^ .vx2 _ vx7 / ^ .:vx5 _ vx7 / ^ .:vx2 _ vx5 _ :vx7 / ^ .vx4 _ :vx8 / ^ .vx7 _ :vx8 / ^ .:vx4 _ :vx7 _ vx8 / ^ .vx6 _ :vx9 / ^ .vx8 _ :vx9 / ^ .:vx6 _ :vx8 _ vx9 / ^ .v0 _ :vx9 _ :vx10 / ^ .:v0 _ vx9 _ :vx10 / ^ .:v0 _ :vx9 _ vx10 /^ .v0 _ vx9 _ vx10 / ^ .vx10 /: where each line except the last corresponds to an internal vertex xi of W of Fig. 6, i increasing from 1 in left-to-right and bottom-to-top order, and the new variables labeling those vertices are vx1 ; : : : ; vx10 correspondingly. The last line forces the root expression to evaluate to 1. The algorithm of Fig. 15 expresses these ideas formally.

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Fig. 15 Algorithm for transforming a formula to a functionally equivalent CNF formula

For simplicity, it is assumed that the input is a formula with syntax consistent with that described on Page 313. The next lemma and theorems show that the algorithm correctly transforms a given expression to a CNF formula and the size of the target formula is linearly related to the size of the original expression.

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Lemma 1 Let O be any binary Boolean operator except the two trivial ones that evaluate to 0 or 1 regardless of the value of their operands. Given formulas 1 and 2 , variables v1 2 V 1 n V 2 and v2 2 V 2 n V 1 , and some base set V such that 1 jv1 D1 V 1 jv1 D0 and 2 jv2 D1 V 2 jv2 D0 ,

«

.v1 O v2 / ^

1

^

2

«

V

«

:.v1 O v2 / ^

1

^

2.

Proof It must be shown that for every truth assignment MV to V (1) if there is an extension that satisfies 1 , 2 , and .v1 O v2 /, then there is an extension that satisfies 1 , 2 , and :.v1 O v2 / and (2) if there is an extension that satisfies 1 , 2 , and :.v1 O v2 /, then there is an extension that satisfies 1 , 2 , and .v1 O v2 /. It is only necessary to take care of (1) since a similar argument applies for (2). Assume there is an extension M1 that satisfies 1 , 2 , and .v1 O v2 /. From 1 jv1 D1 V 1 jv1 D0 and 2 jv2 D1 V 2 jv2 D0 and the fact that M1 satisfies both and , all four extensions identical to M1 except in v1 and v2 will satisfy 1 and 1 2 . One of those will also satisfy :.v 2 1 O v2 / since O is nontrivial. Hence, if there is an extension to MV that satisfies 1 , 2 , and .v1 O v2 /, there is also an extension that satisfies 1 , 2 , and :.v1 O v2 /. 

«

«

Theorem 1 Let  be a formula and V D fv0 ; v1 ; : : : ; vn1 g the set of n variables contained in it. The output of Algorithm 1 on input  represents a CNF formula , written .v0 / if  has no operators and otherwise written .vi / ^  , n  i , where clause .vi / is due to the line “Set ffPop S gg” and  is such that  jvi D1 4 j . In addition, . That is, any truth assignment MV V is a V  vi D0 V model for  if and only if there is a truth assignment M1 fvn ; vnC1 ; : : : ; vi g such that MV [ M1 is a model for .

«

«

Proof The output is a set of sets of variables and therefore represents a CNF formula. The line “Set ffPop S gg” is reached after all input symbols are read. This happens only after either the “Evaluate ‘:s’ ” block or the “Push S s” line. In the former case, vi is left at the top of the stack. In the latter case, s must be v0 , in the case of no operators, or vi from execution of the “Evaluate ‘.vOw/’ ” block immediately preceding execution of the “Push S s” line (otherwise, the input is not a formula). Therefore, either v0 or vi is the top symbol of S when “Set ffPop S gg” is executed so fv0 g or fvi g is a clause of . Formulas 0 and may be shown to have the stated properties by induction on the depth of the input formula . For improved clarity, is used instead of V  0 below. The base case is depth 0. In this case  is the single variable v0 and n D 1. The line “Push S s” is executed in the first Repeat block, then the line “Set ffPop S gg” is executed so D ffv0 gg. Since L D ;, execution terminates. Obviously, D  so both hypotheses are satisfied.

«

4

The meaning of

« is given on Page 318.

«

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For the induction step, suppose  is a formula of positive depth k C 1 and the hypotheses hold for all formulas of depth k or less. It is next shown that they also hold for . Consider first the case  D : 0 ( 0 has depth k). Algorithm 1 stacks : in the line “Push S s” and then, due to the very next line, proceeds as though it were operating on  0 . The algorithm reaches the same point it would have if  0 were input. However, in this case, : and a variable are on the stack. This requires one pass through the “Evaluate ‘:s’ ” block which adds “vi , :s” to L and leaves vi as the only symbol in S . Thus, upon termination, D .vi / ^ and vi is not in  jvi D1

0 .

D .vi / ^ .vi 1 _ vi / ^ .:vi 1 _ :vi / ^



Hence,

D ..vi 1 _ vi / ^ .:vi 1 _ :vi // jvi D1 ^ D .:vi 1 / ^

« .v

i 1 /

^

 jvi D0

0

0

(by the induction hypothesis and Lemma 1, Page 348)

0

D ..vi 1 _ vi / ^ .:vi 1 _ :vi // jvi D0 ^ D

0

0

:

«

 for this case. The expression .vi / ^ Next, it is shown that to 1 only when the value of vi is 1. Therefore, D .vi / ^

«



D .:vi 1 / ^

« .v « « :

i 1 /

0

^

(by



can evaluate

 jvi D1

0 0

(from above)

«  induction hypothesis)

Finally, consider the case that  D .l O r / (l and r have depth at most k). The algorithm stacks a “(” then, by the inductive hypothesis and the recursive definition of a formula, completes operations on a which results in l in L. The line “Set ffPop S gg” is avoided because there are still unread input symbols. Thus, there are two symbols on the stack at this point: a “(” which was put there initially and a variable. The symbol O is read next and pushed on S by the line “Push S s.” Then r is put in L, but again the line “Set ffPop S gg” is avoided because there is still a “)” to be read. Thus, the stack now contains “(;” a variable, say, vl ; an operator symbol O; and another variable, say, vr . The final “)” is read and the “Evaluate ‘.vOw/’ ” section causes the stack to be popped, v , .vl Ovr / to be added to L, and variable vi to be put on S (in the final iteration of the first loop). Therefore, upon termination,

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D .vi / ^



D .vi / ^ .vi , .vl O vr // ^

l

^

r ;

where .vl / ^ l is what the algorithm output represents on input l and .vr / ^ r is represented by the output on input r (some clauses may be duplicated to exist both in l and r ). Then, D .vi , .vl O vr // jvi D1 ^

 jvi D1

D .vl O vr / ^

« : .v O v / ^ l

r

l

^ l

l

 jvi D0

r

r

^

(induction hypothesis and Lemma 1)

r

D .vi , .vl O vr // jvi D0 ^ D

^

l

^

r

:

«

It remains to show  for this case. By the induction hypothesis, l jvl D1 l jvl D0 . Therefore, for a given truth assignment MV , if there is an extension such that vl has value 1 (0) and l has value 0, then there is always an extension such that vl has value 0 (1), respectively, and l has value 1. Since, by the induction hypothesis, l .vl / ^ l , there is always an extension such that l has value 1 and vl has the same value as l . The same holds for vr and r . Therefore,

«

«

D .vi / ^



«

 jvi D1

D ..vi / ^ .vi , .vl O vr /// jvi D1 ^

« .v O v / ^ ^ « . O  / « : l

l

r

r

l

l

^

r

r

t u

Theorem 2 Algorithm 1 produces a representation using a number of symbols that is no greater than a constant times the number of symbols the input formula has if there are no double negations in the input formula. Proof Each binary operator in the input string accounts for a pair of parentheses in the input string plus itself plus a literal. Hence, if there are m binary operators, the string has at least 4m symbols. If there are m binary operators in the input string, m new variables associated with those operators will exist in due to the Append of the “Evaluate ‘.vOw/’ ” block. For each, at most 12 symbols involving literals, 12 involving operators (both ^ and _ represented as “,”), and 8 involving parentheses (represented as “f” “g”) are added to (see Fig. 14), for a total of at most 32m symbols. At most m C n new variables associated with negations are added to (both original variables and new variables can be negated) due to the Append of the “Evaluate ‘:s’ ” block. For each, at most 4 literals, 4 operators, and 4 parentheses are added to (from fvi ; vj g; f:vi ; :vj g), for a total of 12.mCn/ symbols. Therefore, the formula output by Algorithm 1 has at most 44m C 12n symbols.

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Since n  1  m, 44m C 12n  56m C 12. Therefore, the ratio of symbols of to  is not greater than .56m C 12/=4m D 14 C 3=m < 17. t u The next result shows that the transformation is computed efficiently. Theorem 3 Algorithm 1 has O.m/ worst-case complexity if input formulas do not have double negations, where m is the number of operators, and one symbol is used per parenthesis, operator, and variable. Proof Consider the first Repeat block. Every right parenthesis is read once from the input, is never stacked, and results in an extra loop of the Repeat block due to the line “Set s vi ” in the first Otherwise block. Since all other symbols are read once during an iteration of the Repeat block, the number of iterations of this block is the number of input symbols plus the number of right parentheses. Since the number of operators (m) must be at least one fifth of the number of input symbols, the number of iterations of the Repeat block is no greater than 10m. Since all lines of the Repeat block require unit time, the complexity of the block is O.m/. The second Repeat block checks items in L, in order, and for each makes one of a fixed number of substitutions in which the number of variables is bounded by a constant. An item is appended to L every time a : or right parenthesis is encountered, and there is one right parenthesis for every binary operator. Thus, there are m items in L and the complexity of the second Repeat block is O.m/. Therefore, the complexity of the algorithm is O.m/. t u If  contains duplicate subformulas, the output will consist of more variables and CNF blocks than are needed to represent a CNF formula equivalent to . This is easily remedied by implementing L as a balanced binary search tree keyed on vl , vr , and O and changing the Append to a tree insertion operation. If, before insertion in the “Evaluate ‘.vOw/’ ” block, it is discovered that “vj , .vOw/” already exists in L for some vj , then s can be set to vj , i need not be incremented, and the insertion can be avoided. A similar change can be made for the “Evaluate ‘:s’ ” block. With L implemented as a balanced binary search tree, each query and each insertion would take at most log.m/ time since L will contain no more than m items. Hence, the complexity of the algorithm with the improved data structure would be O.m log.m//. As a final remark, there is no comparable, efficient transformation from a formula to a DNF formula.

5.2

Resolution

Resolution is a general procedure that is primarily used to certify that a given CNF formula is unsatisfiable, but can also be used to find a model, if one exists. The idea originated as consensus in [20] and [129] (ca. 1937) and was applied to DNF formulas in a form exemplified by the following: .x ^ y/ _ .:x ^ z/ , .x ^ y/ _ .:x ^ z/ _ .y ^ z/:

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Consensus in DNF was rediscovered in [120] and [116] (ca. 1954) where it was given its name. A few years later its dual, as resolution in propositional logic, was applied to CNF (M. Davis and H. Putnam, 1958, Computational methods in the propositional calculus, unpublished report), for example, .x _ y/ ^ .:x _ z/ , .x _ y/ ^ .:x _ z/ ^ .y _ z/: The famous contribution of Robinson [118] in 1965 was to lift resolution to firstorder logic. Let c1 and c2 be disjunctive clauses such that there is exactly one variable v that occurs negated in one clause and unnegated in the other. Then, the resolvent of c1 and c2 , denoted by Rcc12 , is a disjunctive clause which contains all the literals of c1 and all the literals of c2 except for v or its complement. The variable v is called a pivot variable. If c1 and c2 are treated as sets, their resolvent is fl W l 2 c1 [ c2 n fv; :vgg. The usefulness of resolvents derives from the following Lemma. Lemma 2 Let a pair c1 ; c2 2

be a CNF formula represented as a set of sets. Suppose there exists of clauses such that Rcc12 … exists. Then , [ fRcc12 g.

Proof Clearly, any assignment that does not satisfy CNF formula cannot satisfy a CNF formula which includes a subset of clauses equal to . Therefore, no satisfying assignment for implies none for [ fRcc12 g. Now suppose M is a model for . Let v be the pivot variable for c1 and c2 . Suppose v 2 M . One of c1 or c2 contains :v. That clause must also contain a literal that has value 1 under M or else it is falsified by M . But that literal exists in the resolvent Rcc12 , by definition. Therefore, Rcc12 is satisfied by M and so is [ fRcc12 g. A similar argument applies if v … M . t u The resolution method makes use of Lemma 2 and the fact that a clause containing no literals cannot be satisfied by any truth assignment. A resolution algorithm for CNF formulas is presented in Fig. 16. It uses the notion of pure literal (Page 315) to help find a model, if one exists. Recall, a literal is pure in formula if it occurs in , but its complement does not occur in . If the algorithm outputs unsatisfiable, then the set of all resolvents generated by the algorithm is a resolution refutation for the input formula. Theorem 4 Let be a CNF formula represented as a set of sets. The output of Algorithm 2 on input is unsatisfiable, if and only if is unsatisfiable. If the output is a set of variables, then it is a model for . Proof If the algorithm returns unsatisfiable, one resolvent is the empty clause. From Lemma 2 and the fact that an empty clause cannot be satisfied, is unsatisfiable. If the algorithm does not return “unsatisfiable” the Otherwise block is entered because new resolvents cannot be generated from . Next, a Repeat block to

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Fig. 16 Resolution algorithm for CNF formulas

remove clauses containing pure literals is executed, followed by a final Repeat block which adds some variables to M . Consider the result of the Repeat block on pure literals. All the clauses removed in this block are satisfied by M because all variables associated with negative pure literals are absent from M and all variables associated with positive pure literals are in M . Call the set of clauses remaining after this Repeat block 0 . Now, consider the final Repeat block. If 0 D ;, then all clauses are satisfied by M from the previous Repeat block, and M1Wi will never falsify a clause for any i . In that case M is returned and is a model for . So suppose 0 6D ;. The only way a clause can be falsified by M1W1 is if it is fv1 g. In that case, the line “Set M M [ fvi g” changes M to satisfy that clause. The clause f:v1 g … 0 because otherwise it would have resolved with fv1 g to give ; 2 which violates the hypothesis that the pure literal Repeat block was entered. Therefore, no clauses are falsified by M1W1 at the beginning of the second iteration of the Repeat block when M1W2 is considered. Assume the general case that no clause is falsified for M1Wi 1 at the beginning of the i th iteration of the final Repeat block. A clause c1 will be falsified by M1Wi but not by M1Wi 1 if it contains literal vi , which cannot be a pure literal that was processed in the previous Repeat block. Then, the line “Set M M [ fvi g” changes M to satisfy c1 without affecting any previously satisfied clauses. However, it is possible that a clause that was not previously satisfied becomes falsified by the change. If there were such a clause c2 , it must contain literal :vi and no other literal in c2 would have a complement in c1 ; otherwise, it would already have been satisfied by M1Wi 1 . That means , before the pure literal Repeat block, would contain Rcc12 . Moreover, Rcc12 could not be satisfied by M1Wi 1 because such an assignment

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would have satisfied c1 and c2 . But then Rcc12 must be falsified by M1Wi 1 because it contains literals associated only with variables v1 ; : : : ; vi 1 . But this is impossible by the inductive hypothesis. It follows that the line “Set M M [ fvi g” does not cause any clauses to be falsified and that no clause is falsified for M1Wi . Since no clause is falsified by M1Wn , all clauses must be satisfied by M which is then a model. The statement of the theorem follows. t u Resolution is a powerful tool for mechanizing the solution to variants of SAT. However, in the case of an unsatisfiable input formula, since the resolution algorithm offers complete freedom to choose which pair of clauses to resolve next, it can be difficult to control the total number of clauses resolved and therefore the running time of the algorithm. The situation in the case of satisfiable input formulas is far worse since the total number of resolvents generated can be quite large even when a certificate of satisfiability can be generated quite quickly using other methods. It is easy to find examples of formulas on which the resolution algorithm may perform poorly or well, depending on the order of resolvents, and one infinite family is presented here. Let there be n normal variables fv0 ; v1 ; : : : ; vn1 g and n  1 selector variables fz1 ; z2 ; : : : ; zn1 g. Let each clause contain one normal variable and one or two selector variables. For each 0  i  n  1, let there be one clause with vi and one with :vi . The selector variables allow the clauses to be chained together by resolution to form any n-literal clause consisting of normal variables. Therefore, the number of resolvents generated could be as high as 2O.n/ although the input length is O.n/. The family of input formulas is specified as follows: ffv0 ; :z1 g; fz1 ; v1 ; :z2 g; fz2 ; v2 ; :z3 g; : : : ; fzn2 ; vn2 ; :zn1 g; fzn1 ; vn1 g; f:v0 ; :z1 g; fz1 ; :v1 ; :z2 g;

:::

; fzn2 ; :vn2 ; :zn1 g; fzn1 ; :vn1 gg:

The number of resolvents generated by the resolution algorithm greatly depends on the order in which clauses are chosen to be resolved. Resolving vertically, the 0th column adds resolvent f:z1 g. This resolves with the clauses of column 1 to add fv1 ; :z2 g and f:v1 ; :z2 g, and these two add resolvent f:z2 g. Continuing, resolvents f:z3 g,. . . ,f:zn1 g are added. Finally, f:zn1 g resolves with the two clauses of column n  1 to generate resolvent ; showing that the formula is unsatisfiable. The total number of resolutions executed is O.n/ in this case. On the other hand, resolving horizontally (resolve one clause of column 0 with a clause of column 1, then resolve the resolvent with a clause from column 2, and so on), all 2n n-literal clauses of normal variables can be generated before ; is. The number of resolution steps executed on a satisfiable formula can be outrageous. Remove column n  1 from the above formulas. They are then satisfiable. But that is not known until 2O.n/ resolutions fail to generate ;. This example may suggest that something is wrong with resolution and that it should be abandoned in favor of faster algorithms. While this is reasonable for satisfiable formulas, it may not be for unsatisfiable formulas. As illustrated by Theorem 5, in the case of providing a certificate of unsatisfiability, the resolution algorithm can always simulate the operation of many other algorithms in time

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bounded by a polynomial on input length. So the crucial problem for the resolution algorithm, when applied to unsatisfiable formulas, is determining the order in which clauses should be resolved to keep the number of resolvents generated at a minimum. A significant portion of this chapter deals with this question. However, first consider an extension to resolution that has shown improved ability to admit short refutations.

5.3

Extended Resolution

In the previous section it was shown that the size of a resolution refutation can vary enormously depending on the order in which resolvents are formed. That is, for at least one family of formulas, there are both exponentially long and linearly long refutations. But for some families of formulas, nothing shorter than exponential size resolution refutations is possible. However, the simple idea of extended resolution can sometimes yield short refutations in such cases. Extended resolution is based on a result of Tseitin [139] who showed that, for any pair of variables v, w in a given CNF formula , the following expression may be appended to : .z _ v/ ^ .z _ w/ ^ .:z _ :v _ :w/; where z is a variable that is not in

(3)

. From Fig. 14 this is equivalent to

z $ .:v _ :w/; which means either v and w both have value 1 (then z D 0) or at least one of v or w has value 0 (then z D 1). Observe that as long as z is never used again, any of the expressions of Fig. 14 can be used in place of or in addition to (3). More generally, z $ f .v1 ; v2 ; : : : ; vk / can be used as well where f is any Boolean function of arbitrary arity. Judicious use of such extensions can result in polynomial size refutations for problems that have no polynomial size refutations without extension, a notable example being the pigeonhole formulas. By adding variables not in , one obtains, in linear time, a satisfiabilitypreserving translation from any propositional expression to CNF with at most a constant factor blowup in expression size as shown in Sect. 5.1.

5.4

Davis-Putnam Resolution

The Davis-Putnam procedure (DPP) [51] is presented here mainly for historical reasons. In DPP, a variable v is chosen and then all resolvents with v as pivot are

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generated. When no more such resolvents can be generated, all clauses containing v or :v are removed and the cycle of choosing a variable, generating resolvents, and eliminating clauses is repeated to exhaustion. The fact that it works is due to the following result. Lemma 3 Let

be a CNF formula. Perform the following operations:

. 1 Choose any variable v from 1 . Repeat the following until no new resolvents with v as pivot can be added to 1 : If there is a pair of clauses c1 ; c2 2 1 such that Rcc12 with v as pivot exists and Rcc12 … 1 , c1 1 1 [ fRc2 g. 2 1. Repeat the following while there is a clause c 2 2 such that v 2 c or :v 2 c: 2 2 n fcg. Then

is satisfiable if and only if

2

is satisfiable.

Proof By Lemma 2 1 is functionally equivalent to . Since removing clauses cannot make a satisfiable formula unsatisfiable, if and therefore 1 is satisfiable, then so is 2 . Now, suppose is unsatisfiable. Consider any pair of assignments M (without v) and M [ fvg. Either both assignments falsify some clause not containing v or :v or else all clauses not containing v or :v are satisfied by both M and M [ fvg. In the former case, some clause common to and 2 is falsified by M , so both formulas are falsified by M . In the latter case, M must falsify a clause c1 2 containing v, and M [ fvg must falsify a clause containing :v. Then the resolvent Rcc12 2 2 is falsified by M . Therefore, since every assignment falsifies , 2 is unsatisfiable. t u Despite the apparent improvement in the management of clauses over the resolution algorithm, DPP is not considered practical. However, it has spawned some other commonly used variants, especially the next algorithm to be discussed.

5.5

Davis-Putnam Loveland Logemann Resolution

Here a reasonable implementation of the key algorithmic idea in DPP is presented. The idea is to repeat the following: choose a variable, take care of all resolutions due to that variable, and then erase clauses containing it. The algorithm, called DPLL [52], is shown in Fig. 17. It was developed when Loveland and Logemann attempted to implement DPP but found that it used too much RAM. So they changed the way variables are eliminated by employing the splitting rule: assignments are

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Fig. 17 DPLL algorithm for CNF formulas

recursively extended by one variable in both possible ways, looking for a satisfying assignment. Thus, DPLL is of the divide-and-conquer family of algorithms. The DPLL algorithm of Fig. 17 is written iteratively instead of recursively for better comparison with other algorithms to be discussed later. Omitted is a formal proof of the fact that the output of DPLL on input is unsatisfiable if and only if is unsatisfiable, and if the output is a set of variables, then it is a model for . The reader may consult [52] for details. A common visualization of the flow of control of DPLL and similar algorithms involves a graph structure known as a search tree. Since search trees will be used several times in this chapter to assist in making some difficult points clear, this concept is explained here. A search tree is a rooted acyclic digraph where each

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v0

¬v0

v0

v1

v2

v1

v2

v4 ¬v4 {v0 ; v4 }

¬v2

v1

v3

v4

¬v1

v1

{v0 ; ¬v4 }

v3

{v1 ; ¬v2 } v3

¬v3 {¬v2 ; v3 }

¬v3 {v2 ; v3 }

v3 {v2 ; ¬v3 }

{¬v0 ; ¬v2 ; ¬v3 }

{{v0 ; v4 }; {v0 ; ¬v4 }; {v0 ; v1 }; {v1 ; ¬v2 }; {¬v2 ; v3 }; {¬v0 ; ¬v1 ; ¬v2 }; {¬v0 ; ¬v2 ; ¬v3 }; {v2 ; ¬v3 }; {v2 ; v3 }} Fig. 18 A DPLL refutation tree for the above CNF formula

vertex has out-degree at most 2 and in-degree 1 except for the root. Each internal vertex represents some Boolean variable and each leaf may represent a clause or may have no affiliation. If an internal vertex represents variable v and it has two outward-oriented edges, then one is labeled v and the other is labeled :v; if it has one outward-oriented edge, that edge is labeled either v or :v. All vertices encountered on a path from root to a leaf represent distinct variables. The labels of edges on such a path represent a truth assignment to those variables: :v means set the value of v to 0 and v means set the value of v to 1. The remaining details are specific to a particular CNF formula which is input to the algorithm modeled by the search tree. A leaf is such that the partial truth assignment represented by the path from the root either minimally satisfies all clauses of or minimally falsifies at least one clause of . In the latter case, one of the clauses falsified becomes the label for the leaf. If is unsatisfiable, all leaves are labeled and the search tree is a refutation tree. A fully labeled refutation tree modeling one possible run of DPLL on a particular unsatisfiable CNF formula is shown in Fig. 18. With reference to Fig. 17, a path from the root to any vertex represents the state of VP at some point in the run. For a vertex with out-degree 2, the left edge label is a tagged literal. For vertices with out-degree 1, the single edge label is a pure literal. The DPLL algorithm is a performance compromise. Its strong point is that sets of resolvents are constructed incrementally, allowing some sets of clauses and resolvents to be entirely removed from consideration long before the algorithm terminates. But this is offset by the weakness that some freedom in choosing the order of forming resolvents is lost, so more resolvents may have to be generated. Applied to a satisfiable formula, DPLL can be a huge winner over the resolution

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algorithm: if the right choices for variable elimination are made, models may be found in O.n/ time. However, for an unsatisfiable formula it is not completely clear which algorithm performs best generally. Theorem 5 below shows that if there is a short DPLL refutation for a given unsatisfiable formula, then there must be a short resolution refutation for the same formula.5 But, in [7] a family for formulas for which a shortest resolution proof is exponentially smaller than the smallest DPLL refutation is presented. These facts seem to give the edge to resolution. On the other hand, it may actually be relatively easy to find a reasonably short DPLL refutation but very hard to produce an equally short or shorter resolution refutation. An important determining factor is the order in which variables or resolutions are chosen, and the best order is often sensitive to structural properties of the given formula. For this reason, quite a number of choice heuristics have been studied and some results on these are presented later in this chapter. Many modern SAT solvers augment DPLL with other features such as restarts and conflict-driven clause learning. The refutation complexity of a solver so modified is then theoretically similar to that of resolution. More will be said about this in subsequent sections. This section concludes with the following classic result. Theorem 5 Suppose DPLL is applied to a given unsatisfiable CNF formula and suppose the two lines “Set d C1 fc  f:lg W c 2 d ; l … cg” are together executed p times. Then there is a resolution refutation in which no more than p=2 resolvents are generated. Proof Run DPLL on and, in parallel, generate resolvents from clauses of and other resolvents. At most one resolvent will be generated every time the test “If VP D ;” succeeds or the line “Pop l VP ” is executed and l is neither tagged nor a pure literal. But this is at most half the number of times d C1 is set. Hence, when DPLL terminates on an unsatisfiable formula, at most p=2 resolvents will be generated. Next, it is shown how to generate resolvents and show that ; is a resolvent of two of them. The idea can be visualized as a series of destructive operations on a DPLL refutation tree. Assume clauses in label leaves as discussed on Page 358. Repeat the following: if there are two leaf siblings, replace the subtree of the two siblings and their parent either with the resolvent of the leaf clauses, if they resolve, or with the leaf clause that is falsified by the assignment represented by the path from root to the parent (there must be one if is unsatisfiable); otherwise, there is a pair of vertices consisting of one leaf and a parent, so replace this pair with the leaf. Replacement entails setting the edge originally pointing to the parent to point to the replacement vertex. Eventually, the tree consists of a root only and its label is the empty clause.

5

See [39] for other results along these lines.

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This visualization is implemented as follows. Define a stack S and set S ;. The stack will hold clauses of and resolvents. Run DPLL on . In parallel, manipulate S as follows: 1. The test “If ; 2 d ”: Whenever this test succeeds, there is a clause c 2 whose literals can only be complements of those in VP so push S c. 2. The line “Pop l VP ”: Immediately after execution of this line, if l is not tagged and is not a pure S and pop c2 S . If c1 and c2 can literal, then do the following. Pop c1 resolve, push S Rcc12 ; otherwise, at least one of c1 or c2 , say, c1 , does not contain l or :l, so push S c1 . 3. The line “If VP D ;, Output ‘unsatisfiable.’ ”: S , pop c2 S , and form the resolvent The algorithm terminates so pop c1 ct D Rcc12 . We claim that, when the algorithm terminates, ct D ;. To prove this claim, it is shown that when c1 and c2 are popped in Step 2, c2 contains literals that are only complementary to those of a subset of VP [ flg and c1 contains literals that are only complementary to those of a subset of VP [ f:lg, and if l is a pure literal in the same step, then c1 contains literals complementary to those of a subset of VP . Since VP D ; when c1 and c2 are popped to form the final resolvent, ct , it follows that ct D ;. By induction on the maximum depth of stacking of c2 and c1 . The base case has c2 and c1 as clauses of . This can happen only if c2 had been stacked in Step 1 then Step 2 was executed and then c1 was stacked soon after in Step 1. It is straightforward to check that the hypothesis is satisfied in this case. Suppose the hypothesis holds for maximum depth up to k and consider a situation where the maximum depth of stacking is k C 1. There are two cases. First, suppose l is a pure literal. Then neither l nor :l can be in c1 so, by the induction hypothesis, all of the literals of c1 must be complementary to literals of VP . Second, suppose l is not tagged and not a pure literal. Then c1 and c2 are popped. By the induction hypothesis, c1 contains only literals complementary to some in VP [ f:lg and c2 contains only literals complementary to some in VP [flg. Therefore, if they resolve, the pivot must be l and the resolvent contains only literals complementary to some in VP . If they do not resolve, by the induction hypothesis, one must not contain l or :l. That one has literals only complementary to those of VP , so the hypothesis holds in this case. t u

5.6

Conflict-Driven Clause Learning

Early SAT solvers used refinements and enhancements to DPLL to solve SAT instances. Recently, these refinements and enhancements have been standardized and developed into a parameterized algorithm. This algorithm is far enough removed from DPLL that it is commonly referred to as conflict-driven clause learning (CDCL) [119, 130, 131].

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The reasons for the increased solving power and efficiency of modern SAT solvers lie in specialized CDCL techniques tailored to support core CDCL operations such as binary constraint propagation (BCP) and heuristic computation. Each technique on its own is not very impressive but, when all of these techniques are fitted together in the DPLL framework, orders of magnitude improvement can be seen. For example, conflict clause learning helps to create a dynamic search space that is DAG-like instead of treelike as was the case for early DPLL-based algorithms [16]. This technique, when used in conjunction with dynamic restarts, is as strong as general resolution and exponentially more powerful than DPLL [16]. The combination of such techniques has allowed SAT solvers to be used practically in a wide range of fields such as bioinformatics [107], cryptography [110, 132], and planning [46, 151] as well as on many hard combinatorial problems such as van der Waerden numbers [100, 101]. Also, the recent integration of SAT-based techniques with theory solvers (this technology is named SMT [126] – details are given in Sect. 5.7) is enabling push-button solutions to industrial strength problems that previously required expert human interaction to solve (e.g., see [77]). Two chapters in the Handbook of Satisfiability [19] cover CDCL extensively. The reader is referred to [50] and [131] for details. The remainder of this section highlights the main points.

5.6.1 Conflict Analysis DPLL explores both sides of a choicepoint before backtracking over it. This is called chronological backtracking and is not generally conducive to good SAT solving because it is often the case that the same conflict exists on both sides of a choicepoint. Analyzing the conflict on one side may prevent re-computation of the same conflict on the other side, in which case the choicepoint can be backjumped over. Performing conflict analysis in the framework of DPLL enables nonchronological backtracking, which, in effect, works to counteract bad decisions made by the search heuristic. Conflict analysis is often depicted in terms of an implication graph, though, in practice, the implication graph is not actually built. In modern SAT solvers, new clauses are generated by resolution steps ordered by the current search tree path. Specifically, a conflict is reached when two asserting clauses infer opposing literals during BCP. These two clauses can be resolved together to create a new clause called a conflict clause. This clause will either be the empty clause (in which case the formula is unsatisfiable) or it will, if evaluated under the current partial assignment, be falsified. Consider the most recently assigned literal in the conflict clause. If the literal was assigned by an asserting clause, the conflict clause and the asserting clause are resolved, creating a new conflict clause, and the process is repeated. If the literal was assigned by a choicepoint, the current search tree up to the literal is undone and the value assigned by the choicepoint is flipped. All new conflict clauses may be safely added to the original CNF formula (i.e., learned) because they are logical consequences of the original formula [130]. Learning conflict clauses helps to prune the search space by preventing re-exploration of shared search structure, especially when used in conjunction with restarts. Also, prior to adding

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newly learned clauses, it may be beneficial to attempt to minimize the clauses by performing a restricted version of resolution with other asserting clauses [15, 143]. There is at least one special case to consider when learning conflict clauses. If a conflict clause contains only one literal at the current decision level, called a unique implication point (UIP), the search state can backjump to the second most recently assigned literal in the clause. This will cause the UIP literal to be naturally inferred via BCP [112] and has the effect of promoting those parts of the search tree relevant to the current conflict. Conflict analysis is also used to generate resolution proofs of unsatisfiable instances. In the same way that an assignment to a satisfiable instance acts as a polynomial-time checkable certificate that an instance is satisfiable, a resolution proof acts as a certificate of unsatisfiability, checkable in linear time and space on the number of learned clauses (though the number of learned clauses can be exponential on the number of variables in the best case). More information on this topic can be found in [15, 150], and [142].

5.6.2 Conflict Clause Memory Management Not all derived conflict clauses can be stored. Too many conflict clause increases the overhead of finding an applicable one, and too few conflict clauses reduce the chance that a good one can be found to prune the search space. Therefore, the size of the conflict clause database, what goes into it, and how long entries should stay there have been well studied. The number of conceivable schemes is quite large. A good one was recently discovered and reported in [11]. In that paper the learned clause measure LBD (for literals blocks distance) is defined to be the number of different levels at which literals of a clause were assigned values as choicepoints up to the current assignment. A small LBD suggests a learned clause that was useful in generating a long sequence of unit resolutions. It was found in experiments on all SAT-Race ’06 benchmarks that 40 % of unit resolutions occur in clauses of LBD = 2 versus 20 % on 2-literal clauses, and generally, the lower the LBD, the more significant the contribution to BCP. Due to this, the following cleaning strategy was proposed: remove half of the learned clauses, not including a class of easy discovered clauses of least LBD, every 20000 C 500 x conflicts, where x is the number of times this operation has been performed before on the current input. Adding this strategy to existing solvers resulted in a significant improvement in performance. Many other ideas for determining the life of conflict clauses have been proposed and tried. Several of these are mentioned in [131] near the end of the chapter. 5.6.3 Lazy Structures An important aspect of CDCL associated with performance has to do with maintaining data structures that store solver state. The degree to which these structures are managed lazily has an impact on the optimal choice of other solver mechanisms. For example, lazy structures result in faster backtracking, so “the authors of Chaff proposed to always stop clause learning at the first UIP” and “always take the backtrack step” [131]. In addition, the implementation of BCP and conflict analysis

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is influenced by the implementation of lazy structures. Using lazy structures, the fact that a clause becomes satisfied by a state change may not be reported to solver mechanisms immediately or even at all. This presents potential savings by preventing some unnecessary computation and does not affect the correctness of the solver. Next, in this section the classic notion of watched literals [112] is used to illustrate this point. Suppose a clause is stored as a sequence of literals and suppose the number of literals is at least 2. For each clause, assign two pointers each pointing to one of the literals but both pointing to different literals of the clause. The literals pointed to are called watched literals and are distinguished in name as the first and second watched literal. The following properties are maintained always: 1. A clause is satisfied if and only if its first watched literal has value 1. 2. A clause is falsified if and only if its first and second watched literals have value 0. 3. A clause is unit if and only if one watched literal is unassigned and the other has value 0. 4. A clause is unassigned and not unit if and only if both watched literals are unassigned. Initially, a clause is unassigned and the pointers are pointing to arbitrary literals. If a clause is unassigned, not unit, and a literal other than the second watched literal takes value 1, then that literal becomes the first watched literal. If the literal taking value 1 is the second watched literal, the first and second watched literals are swapped. If both watched literals are unassigned and one of them takes value 0, a check must be made to determine whether the clause has another unassigned literal. If it does not, the clause becomes a unit clause and the single unassigned literal takes value 1 and becomes the first watched literal swapping title, if necessary, with the watched literal of value 0. Otherwise, the pointer referencing the recently assigned watched literal gets set to point to the unassigned literal that was found during the check. The search for an unassigned literal can proceed in any direction among the literals of the clause. Observe that the watched literal pointers do not have to be reset on backtrack. Further savings are obtained by associating a level and value field with every literal: when a clause is satisfied or falsified, it is the level and value of the first watched literal that is set. Since watched literals do not change during backtrack, this information represents clause status as well. Earlier schemes for managing clause status during backtrack associate level and value fields with clauses and require searching the literals of a clause to determine status. Thus, implementing the watched literal concept results in much less backtracking overhead and significantly increases the number of backtracks handled per second. As can be observed from the above discussion, watched literals are useful in reacting to single-variable inferences. But more advanced techniques look at multivariable inferences, for example, two-variable equivalence. Watched literals cannot help in this case. Additional clause references, of course, may be used to support these advances [141] at added expense. Another idea [106] is to maintain an order of assigned literals in a clause by nondecreasing level, while the unassigned literals remain watched as above. As a variable is assigned a value, it is sifted into

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its proper place in the ordered list. The problem with this idea is that all literals in the ordered list may need to be visited on a backtrack.

5.6.4 CDCL Heuristics Lazy data structures can drastically increase the speed of BCP, but since the current state of the search is not readily available, traditional heuristics such as MOMs (see [60] for a complete discussion), Jeroslow-Wang [80], and Johnson [81] cannot be used. The first CDCL heuristic is the variable state independent decaying sum (VSIDS) [112] heuristic. VSIDS maintains one counter per literal in the SAT instance, initialized to the number of clauses each literal occurs in. VSIDS branches on the literal with the highest count, though this requires sorting the counters, which is expensive, and so is only done periodically. When a new clause is learned, the counts for each literal in the clause are incremented. Also periodically, the counters are halved. This heuristic and many similar variants efficiently glean information from learned clauses to guide the solver towards unsatisfiable cores, enabling modern solvers to solve large SAT instances (millions of variables and clauses) that have small resolution proofs. 5.6.5 Restarts Conflict-directed heuristics can sometimes cause a solver to focus too heavily on one particular region of the search space, leading to heavy-tailed runtime distribution (where progress seems to exponentially decrease the longer a solver runs). One way to positively modify this behavior [68] is to restart. A dynamic restart [76, 88] clears a solver’s current partial assignment; heuristic information and learned clauses are kept, and a short preprocessing phase where the learned clause database is garbage collected may typically follow a restart. As reported empirically in the literature, “it is advantageous to restart often” [50].

5.7

Satisfiability Modulo Theories

Despite highly significant and broad advances to SAT solver design, there are still many problems for which DPLL and CDCL variants are a challenge. For example, it is well known that verifying correctness of arithmetic circuits is generally difficult for SAT solvers. The problem is that datapath operations are over a fixed bit-width and, beyond modular arithmetic, some operations such as multiplication are not linear and particularly vexing for SAT solvers. An accepted method, called bitblasting, for dealing with small bit-widths is to describe operations as propositional formulas over vectors of Boolean variables, convert to CNF using the results of Sect. 5.1, and solve with a SAT solver. But it is doubtful that bit-blasting is going to be universally practical for 64-bit or even 32-bit logic. As another, more universally occurring example, SAT solvers have a difficult time handling cardinality constraints and, more generally, constraints comparing numeric expressions. To mitigate these and other problems, a significant body of research has studied the use of a SAT solver to manage a collection of special purpose solvers for first order theories,

Algorithms for the Satisfiability Problem Fig. 19 An SMT instance with uninterpreted functions and quantifiers

Fig. 20 An SMT instance with bit-vector expressions

365 (set-evidence! true) (define f::(-> int int)) (define g::(-> int int)) (define a::int) (assert (forall (x::int) (= (f x) x))) (assert (forall (x::int) (= (g (g x)) x))) (assert (/= (g (f (g a))) a)) (check)

(set-evidence! true) (define b1::(bitvector (define b2::(bitvector (define v1::(bitvector (assert (= v1 (mk-bv 8 (define v2::(bitvector

8)) 8)) 8)) 39))) 8))

(assert (= (bv-mul b1 (mk-bv 8 17)) b2)) (assert (= (bv-add (bv-shift-right0 b2 2) v2) v1)) (check)

including bit-vector arithmetic. This extension of SAT has found applications in hardware verification, software model checking and testing, and vulnerability detection, among other applications, and is called Satisfiability Modulo Theories or SMT. A formal specification of SMT is not given here; the reader is invited to check [12] for the syntax and semantics of SMT-LIB, the official documented language of SMT solvers. Most importantly, with respect to SAT, an instance of SMT is a formula in first-order logic where uninterpreted functions are admitted and Boolean variables are used to tie predicates from disparate theories and allow a SAT solver to decide the consistency of the constraints of the instance. The scope of SMT is made apparent through the SMT examples shown in Figs. 19 and 20. The examples are expressed in a lisp-like language that is native to YICES [55], one of the leading available SMT solvers, and may actually be input to YICES. In Fig. 19, f and g are uninterpreted functions: note that only their signatures, namely, a single int as input and an int as output, are defined in the second and third lines. However, the fifth and sixth lines force f(x)= x, g(y)= x, and g(x)= y for all integers. This is inconsistent with the seventh line which requires, for any integer a, that g(f(g(a))), which is g(g(a)) and therefore a from the above, is not equal to a. On this input YICES returns unsat. If /= (not equal) is changed to =, YICES returns unknown with the following interpretation: a = 2, g(2) = 1, f(1)= 1, g(1)= a2. The first and eighth lines are YICES-specific directives. Figure 20 shows constraints expressed in terms of bit-vector operations. This instance is satisfied by 8-bit numbers b1 and v2 such that multiplying b1 by 17, shifting right by 2 bits, and adding v2 result in 39.

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YICES output on this example is as follows: sat (= v1 (= b1 (= b2 (= v2

0b00100111) 0b01110110) 0b11010110) 0b11110010)

That is, b1 is 118 and v2 is 242, which is one of many possible solutions. Clearly, admitting bit-vector operations is important as it makes circuit design and verification problems easier to express and solve: trying to express the constraints of such problems as a CNF formula is unnatural and time-and-space consuming. A special solver for the theory of bit-vector arithmetic and logic may be developed to handle bit-vector predicates such as those in Fig. 20. Other theory solvers may be designed to handle other problems that are difficult for CNF solvers. Examples of such theories, including the theory of bit-vectors, are as follows: 1. Linear arithmetic (LA): constraints are equations or inequalities involving addition and subtraction over rational variables, constants, or constants times variables. 2. Uninterpreted functions (UF): all uninterpreted function, constant, and predicate symbols together with Leibniz’s law that allows substitution of equals for equals. 3. Difference arithmetic: this is linear arithmetic restricted to equations or inequalities of the form x  y  c where x and y are variables and c is a constant. 4. Nonlinear arithmetic: constraints are like linear arithmetic except that variables may be multiplied by other variables. 5. Arrays: array constraints ensure that the value of a memory location does not change unless a specific write operation on that location with a different value is executed and that a value written to a location may be accessed by a read operation. Symbols of this theory might be select and store with semantics given by the following: select.store.v; i; e/; j / D if i D j then e else select.v; j / store.v; i; select.v; i // D v store.store.v; i; e/; i; f / D store.v; i; f / i 6D j ) store.store.v; i; e/; j; f / D store.store.v; j; f /; i; e/, where select.v; j / means read the j th value from array v and store.v; i; e/ means store the value e into the i th element of array v. 6. Lists: symbols of this theory might be car, cdr, cons, and atom with semantics given by the following: car.cons.x; y// D x cdr.cons.x; y// D y :atom.x/ ) cons.car.x/; cdr.x// D x :atom.cons.x; y//

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7. Fixed-width bit-vectors: the main reason for including a bit-vector theory is to remove the burden of solving bit-level formulas from the SAT solver. As seen in Fig. 20, constraints are expressed at word level and may involve any machinelevel operations on words. 8. Recursive data types: this is an abstraction for common data structures that may be defined recursively such as linked lists and for common data types that may be defined recursively such as a tree or the natural numbers. It is not within the scope of this chapter to discuss the construction of decision procedures for theories. The interested reader may begin a search for such information by consulting [13]. The remainder of this section will show how a SAT solver can be made to interact with constraints from several theories. Some definitions are given now to make the procedures to be discussed later easier to express and analyze. A first-order formula is an expression consisting of a set V of variables and symbols .; /; :; _; ^; !; 9; 8; a set P of predicate symbols; a set F of function symbols, each with associated positive arity; and a set C F of constant symbols which are just 0-arity function symbols, for example, x or f . A signature, denoted †, is a subset of F [ P. A first-order structure A specifies a semantics for the above: that is, a domain A of elements, mappings of every nary function symbol f 2 ˙ to a total function f A W An 7! A and mappings of every n-ary predicate symbol p 2 ˙ to a relation p A  An . The symbols 9 and 8 are quantifiers and the scope of one of these is the entire formula. A sentence is a formula all of whose variables are bound by a quantifier (i.e., the formula has no free variables) and therefore evaluates either to true or false, depending on the structure it is evaluated over. A formula is a ground formula if it has no variables. Observe that x C 1  y has no variables because x and y are constants in the theory of linear arithmetic and is therefore a ground formula. Denote .A; ˛/ ˆ  to mean the sentence  evaluates to true in structure A under variable assignment ˛ W V 7! AjV j . Sentence  is said to be satisfiable in A if .A; ˛/ ˆ  for some ˛. Sentence  is said to be valid in A if .A; ˛/ ˆ  for every ˛. If A is finite, the question .A; ˛/ ˆ  for some ˛ and given  is decidable. However, for infinite structures, the question may be undecidable. A theory T is a set of first-order sentences expressed in the language of some signature †. Equality is implicit in the theories mentioned earlier so their signatures do not contain the symbol D. A model of T is a structure A in which every sentence of T is valid. Denote by T ˆ  the condition where A is a model for T and .A; ˛/ ˆ  for all ˛. Then  is said to be T -valid. Let ? denote a formula that is not satisfied by any structure. Then  is said to be T -satisfiable if T [ fg j6D?, also written T;  j6D?. T -validity and T -satisfiability are seen as dual concepts since T; : ˆ? if and only if T ˆ . One is generally interested in combining theories: that is, proving that statements with terms from more than one theory are valid over those combined theories. Suppose, for two theories T1 of ˙1 and T2 of ˙2 , there exist procedures P1 and P2 that determine whether a given formula 1 expressed in the language of ˙1 is T1 valid and whether a given formula 2 expressed in the language of ˙2 is T2 -valid,

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respectively. It is generally not possible to the solve the problem of determining whether f1 g [ f2 g in the language of ˙1 [ ˙2 is .T1 [ T2 /-valid with P1 and P2 . However, it can be under some modest restrictions that usually do not get in the way in practice. These restrictions will be assumed for the rest of this section and are as follows: theories T1 and T2 are decidable, ˙1 \ ˙2 D ;, and T1 and T2 are stably infinite – let C be an infinite set of constants not in signature † and define a theory T of † to be stably infinite if every T -satisfiable ground formula of signature ˙ [C is satisfiable in a model of T with an infinite domain of elements. By the duality noted above, the validity problem can be treated as a satisfiability problem where it is more conveniently solved. Suppose  is a formula that is logically equivalent to a DNF formula with m conjuncts i;1 ^i;2 , i 2 f1; 2; : : : ; mg, where i;1 and i;2 are expressed in the language of ˙1 [C and ˙2 [C , respectively, and C is a set of constants that are shared by i;1 and i;2 . Terms i;1 ^ i;2 are said to be in pure form. Clearly,  is .T1 [ T2 /-unsatisfiable if and only if for all i 2 f1; 2; : : : ; mg, i;1 ^ i;2 is .T1 [ T2 /-unsatisfiable. It is desirable to use P1 and P2 individually to determine this. By Craig’s interpolation lemma [45], a pure-form term i;1 ^ i;2 is .T1 [ T2 /-unsatisfiable if and only if there is a formula , called an interpolant, whose free variables are a subset of the free variables contained in both i;1 and i;2 such that T1 ; i;1 ˆ and T2 ; i;2 ; ˆ?. Finding an interpolant is a primary goal of an SMT solver and the subject of much of the remainder of this section, the exposition of which is strongly influenced by Klaedtke [90]. Although combining only two theories is considered here, the process generalizes to more than two theories. The first step in combining theories is to produce a pure-form formula from . This is accomplished by introducing variables to C which remove the dependence on objects from an alien theory through substitution. More specifically, suppose function f 2 ˙1 and one of its arguments t is a term, possibly a constant, expressed in ˙2 . Then create a new w, set C D C [ fwg, add a new constraint w D t, and replace t by w in the argument list of f . Similarly, for predicates p 2 ˙1 , p 2 ˙2 and function f 2 ˙2 . If there is a constraint s D t where s 2 ˙1 and t 2 ˙2 , then create a new w, and replace the constraint with w D s and w D t. If there is a constraint s 6D t, then add a new w1 and new w2 and replace the constraint with w1 D s, w2 D t, and w1 6D w2 . For example, consider  D fx  y; y  x C car.cons.0; x//; p.h.x/  h.y//; :p.0/g; which mixes uninterpreted functions, linear arithmetic, and lists. For this , C D fx; yg. The constraint on the left need not be touched because it only contains symbols from linear arithmetic. The subterm h.x/  h.y/ is mixed, so create new constants g1 D h.x/ and g2 D h.y/ and substitute g1  g2 for h.x/  h.y/. The term p.g1  g2 / is mixed, so create constant g3 D g1  g2 and substitute. The remaining term p.g3 / has no alien subterms. The symbol 0 is alien to the theory of uninterpreted functions. Hence, create g4 D 0 and substitute making the rightmost constraint :p.g4 /, and changing the list theory subterm to car.cons.g4 ; x//, both

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of which may be left alone. However, the second constraint from the left is still mixed, so create g5 D car.cons.g4 ; x// and substitute. This leaves C D fx; y; g1 ; g2 ; g3 ; g4 ; g5 g, (Shared constants) 1 D fx  y; y  x C g5 ; g3 D g1  g2 ; g4 D 0g, (LA) 2 D fg1 D h.x/; g2 D h.y/; p.g4 / D false; p.g3 / D trueg, (UF) 3 D fg5 D car.cons.g4 ; x//g, (Lists) so  has become a pure formula whose constraints are partitioned according to the theories of linear arithmetic, uninterpreted functions, and lists and are tied by newly created constants and equality constraints. The next step is to propagate inferred equalities between the partitioned sections. First, this is illustrated using the above example, then a general procedure for propagation is presented. By the first axiom of the theory of lists, the constraint g5 D car.cons.g4 ; x// infers g4 D g5 . This propagates to section LA where g5 D 0 is inferred. Then y  x is inferred and x D y is inferred from x  y and y  x. The inferred constraint x D y propagates to section UF where g1 D g2 is inferred. Propagating back to LA infers g3 D 0 and finally g3 D g4 . But this contradicts UF constraints which say p.g3 / D true and p.g4 / D false. Hence, the conclusion is that  is unsatisfiable in the combined theories of linear arithmetic, lists, and uninterpreted functions. Observe that 1 , 2 , and 3 are all satisfiable in their respective theories. Figure 21 shows an algorithm for determining satisfiability of a formula 1 ^ 2 that is in pure form with 1 of signature ˙1 [ C , 2 of signature ˙2 [ C , C a set of shared constants, and ˙1 \ ˙2 D ; and assumes there exist decision procedures P1 and P2 for theories T1 and T2 over ˙1 and ˙2 , respectively. It is straightforward to generalize this to any number of theories; it is also possible to use the generalized algorithm to decide any mixed theory formula by converting to DNF in pure form and applying the algorithm to the conjuncts separately. Step 1 applies decision procedures P1 and P2 to 1 and 2 . Both P1 and P2 not only decide whether their respective theories are consistent but also infer the maximum number of equalities from 1 and 2 within their respective theories. If no inconsistencies are found, these inferences are propagated out of their theories and the step is repeated; otherwise, unsatisfiable is returned. Step 2 does a case split in case a disjunction of equalities is inferred by one of the partitioned sets of constraints. Step 3 returns “satisfiable” if nothing else can be inferred and no inconsistencies between current sets 1 and 2 have been discovered. The algorithm always terminates since the number of shared constants is finite and the number of equalities added is directly related to that number. The algorithm closely follows the algorithm of the original Nelson-Oppen paper [113]. Completeness of Algorithm 4 is not difficult to see. Soundness is proved by observing that the algorithm produces an interpolant for 1 and 2 : namely, the residue of 1 ^ 2 . The residue of a formula , denoted Res./, is the strongest collection of equalities between constants that are inferred by the formula. The important property of residues is that if 1 and 2 are formulas whose signatures

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Fig. 21 Combined theory solver for two theories

overlap only in constants, then Res.1 ^ 2 / is logically equivalent to Res.1 / ^ Res.2 /. Thus, the residue of the conjunction of all formulas derived within each theory is the conjunction of the residues of formulas for each theory. The algorithm returns “satisfiable” only if all theories are consistent and no additional equalities of constants can be inferred. In this case the residues of all theories are satisfiable. It may be observed that the conjunction of the residues is therefore satisfiable. Hence, the residue of the conjunction of theories is satisfiable. This means if the algorithm returns satisfiable, then 1 ^ 2 is satisfiable. The purpose of this section is to acquaint the reader with procedures that comprise SMT solver implementations. As such, much has been left out. This chapter has not considered elimination of the purification step by shifting the burden of handling alien terms to the theory solvers themselves, application of theory specific rewrites, Shostak’s theories and method, and how to lift the requirement of stably infinite theories for completeness by propagating cardinality constraints in addition to equalities. Nelson-Oppen only works for quantifier free formulas, but there are some techniques for allowing quantifiers which are not discussed here. For more information about these topics, the reader may start by consulting [13].

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371

Stochastic Local Search

In DPLL and CDCL variants, a search for a satisfying assignment proceeds by extending a partial assignment as long as no constraint is falsified. In stochastic local search (SLS), there is a “current” complete assignment at every point in the search: if some clauses are not satisfied by the current assignment, one of a set of functions, each designed to create a new assignment from the given formula and current assignment, is applied according to some probability schedule to determine a new current assignment. This process continues until either all clauses are satisfied or until some function determines it is time to give up. Thus, SLS algorithms are generally incomplete and are not typically designed to provide a certificate for an unsatisfiable formula. A main strength of SLS algorithms is that they admit “long jumps” through the search space and thereby avoid the condition of becoming mired in an unnecessarily long exploration of a search subspace, particularly that for which all search trajectories lead to a local minimum number of satisfied clauses: deterministic solvers must take special measures to avoid that condition. On the other hand, it is difficult for SLS solvers to use the systematic tabulation and learning techniques that have made the deterministic variants successful.

5.8.1 Walksat A successful early implementation of SLS for SAT is Walksat [127]. The Walksat algorithm is shown in Fig.22. It is safe to assume that all clauses in the input formula have at least two literals since elementary and prudent preprocessing of would have assigned a value to any variable in a unit clause to satisfy that clause and would have propagated the effect of that change to the remainder of . This algorithm has a few interesting features worth pointing out. One is the probability p which is used to choose a variable whose value should be reversed (from now on the word flipped is used to mean just this): whether a random literal or the one whose flipped value results in the minimum number of satisfied clauses that become falsified. Experiments suggest the optimal value for p depends on and is sensitive to the particular class of problem the algorithm is applied to [89]. For example, it has been found that p D 0:57 is best for random 3-SAT formulas6 and this point sits at the minimum of an upward facing, cup-shaped curve with steeply increasing slopes on either side of the minimum. A second interesting feature is that a flip is chosen randomly only if there is not a variable in c which does not break any satisfied clauses. Thus, emphasizing positive movement to the goal of satisfying the most clauses as early as possible, Walksat has weakened some of the protection against loop stagnation that random flips provide. A third feature is the parameter mf which limits exploration of a local search space: if significant progress is not being made for a while, then exploration of this local space terminates and a long jump to a different local search space is made and search resumed. This restart

6

Random 3-SAT formulas are defined in Sect. 7, Page 440.

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Fig. 22 Walksat

mechanism compensates for the possible problem nonrandom flips may get into as mentioned above. Walksat may be applied to MAX-SAT problems. A version for weighted MAX-SAT problems is available [86].

5.8.2 Novelty Family Novelty [109] is a variant of Walksat that uses a more sophisticated variable selection algorithm. Once a falsified clause c is chosen, a score is computed for each variable v represented in c, based on the total number of satisfied clauses that would result if v were flipped. If the variable with highest score is not the variable that was most recently flipped, it is flipped now. Otherwise, it becomes the flipped variable with probability 1  p or the second highest scoring variable becomes the flipped variable with probability p. Although Novelty is an improvement over Walksat in many empirical cases, it has the problem that its inner loop may not terminate for a given M [73]. Novelty+ is a slight modification that eliminates this problem by introducing a second probability wp: a variable is randomly selected from c and flipped with probability wp and otherwise the Novelty flip heuristic is applied. It is reported that wp D 0:01 is sufficient to eliminate the non-termination problem [75] (i.e., a solution will be found with arbitrarily high probability, if one exists, even if mt D 1, as long as mf is set arbitrarily high).

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Adaptive Novelty+ provides one more tweak to the Walksat/Novelty+ approach: the noise parameter p is allowed to vary during search. It has been mentioned above that algorithm performance is sensitive to p: thus, the optimal value for p depends significantly on the particular class of input formulas. Adaptive Novelty+ [74] attempts to mitigate this problem. If p is increased, the probability of inner loop stagnation is reduced. If p is decreased, the ability of the algorithm to follow a gradient to a solution is improved. These two counteracting conditions are thought to explain the sensitivity of p to problem domain. Adaptive Novelty+ begins at p D 0 and witnesses a rapid improvement in the number of clauses satisfied. When stagnation is detected, p is increased. Such increases continue periodically until stagnation is overcome and then p is reduced. The cycle is continued until termination. Stagnation is detected when it is noticed that an objective function has not changed over a certain number of steps, called the Stagnation Detection Interval. The usual objective function is the number of satisfied clauses. One specific adaptation strategy is the following: Stagnation Detection Interval: Incremental increase in p: Incremental decrease in p:

m=6, m = number of clauses p D p C .1  p/ 0:2 p D p  0:4 p

The asymmetry between increases and decreases in p is motivated by the fact that detecting search stagnation is computationally more expensive than detecting search progress and that it is advantageous to approximate optimal noise levels from above rather than from below [74]. After p has been reset, the current objective function value is stored and becomes the basis for detecting stagnation. Observe that the factors 1/6, 0.4, and 0.2 are parameters that could be adjusted, but according to [74], performance appears to be less sensitive to these parameters than it is to p.

5.8.3 Discrete Lagrangian Methods Discrete Lagrangian methods are another class of SLS algorithms which control the ability of a search to escape a subspace that contains a local minimum [128]. Let D fc1 ; c2 ; : : : ; cm g be a CNF formula with clauses c1 ; : : : ; cm ; let V D fv W 9 c 2 s:t: v 2 c or :v 2 cg; and attach labels to variables to write V D fv1 ; v2 ; : : : ; vn g. Define for all ci 2 and vj 2 V as follows: 8 < 1  vj Qi;j .vj / D v : j 1

if vj 2 ci if :vj 2 ci otherwise.

Let x be an n-dimensional 0-1 vector where xj holds the value of variable vj , and Q let Ci .x/ D nj D1 Qi;j .vj /. Define N.x/ D

m X i D1

Ci .x/:

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Then N.x/ is the number of unsatisfied clauses in Consider the following optimization problem: minimize over x 2 f0; 1gn N.x/ D

m X

under the assignment x.

Ci .x/

i D1

subject to

Ci .x/ D 0 8i 2 f1; 2; : : : ; mg:

The objective function (the first line of the above) is enough to specify the SAT problem. The extra constraints are added to guide the search. In particular, the extra constraints help bring the search out of a local minimum. The above overconstrained optimization problem may be converted to an unconstrained optimization problem by applying a Lagrange multiplier i to each constraint Ci .x/ and adding the result to the objective function. Writing  to represent the vector h1 ; : : : ; m i of Lagrange multipliers and C.x/ to represent the vector hC1 .x/; : : : ; Cm .x/i, define L.x; / D N.x/ C T C.x/; where x 2 f0; 1gn and i can be real valued. A saddle point (x  ;  ) of L.x; / is defined to satisfy the following condition: L.x  ; /  L.x  ;  /  L.x;  / for all  sufficiently close to  and for all x whose Hamming distance between x  and x is 1. The discrete Lagrangian method (DLM) for solving CNF SAT can be defined as a set of equations which update x and : x kC1 D x k  x L.x k ; k / kC1 D k C C.x k /; respect to x such that where x L.x; / is the discrete gradient operator with P x L.x; / D hı1 ; ı2 ; : : : ; ın i, 81i n ıi 2 f1; 0; 1g, 1i n jıi j D 1, and .x  x L.x; // 2 f0; 1gn (i.e., x L.x; / identifies a variable to be flipped and x  x L.x; / is a neighbor of x – i.e., at Hamming distance 1 from x). There are two ways to calculate x L.x; /. One way, termed greedy, replaces the current assignment with a neighboring assignment leading to the maximum decrease in the Lagrangian function value. Another way, termed hill climbing, replaces the current assignment with a neighboring assignment that leads to a decrease in the value of the Lagrangian function. Both involve some local search in the neighborhood of the current assignment, but for every greedy update, the complexity of this search is O.n/, whereas the hill-climbing update generally is less expensive.

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Fig. 23 Discrete Lagrangian method

An algorithmic sketch of DLM is shown in Fig. 23. The parameter  controls the magnitude of the changes to i . A value of  D 1 is reported to be sufficient for many benchmarks, but a smaller  has resulted in improved performance for some difficult ones. Experiments have shown that  should not be updated every time the current assignment is replaced. This is because the changes in L.x; / are usually very small and relatively larger changes in  can overcompensate causing the search to become lost. Therefore, the DLM algorithm contains a line stating that a condition must be satisfied before  can be updated. One possibility is that  is not updated until x L.x; / D 0. When that happens, the local minimum is reached and the change in  is needed to jump to a different section of the search space. This strategy may cause a problem if a search-space plateau is entered. In that case, it will be prudent to change  sooner. For improved variants of DLM, see [147, 148].

5.9

Binary Decision Diagrams

Binary decision diagrams (BDDs) as graphs were discussed in Sect. 3.2. In this section the associated BDD data structure and efficient operations on that data structure are discussed. Attention is restricted to reduced ordered binary decision diagrams (ROBDDs) due to its compact, efficient, canonical properties. The following is a short review of Sect. 3.2. A ROBDD is a BDD such that (1) there is no vertex v such that the n.v/ D else.v/ and (2) the subgraphs of two distinct vertices v and w are not isomorphic. A ROBDD represents a Boolean function uniquely in the following way (symbol v will represent both a vertex of a ROBDD and a variable labeling a vertex). Define f .v/ recursively as follows: 1. If v is the terminal vertex labeled 0, then f .v/ D 0. 2. If v is the terminal vertex labeled 1, then f .v/ D 1. 3. Otherwise, f .v/ D .v ^ f .the n.v/// _ .:v ^ f .else.v///.

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Fig. 24 Procedure for finding a node or creating and inserting it

Then f .root.v// is the function represented by the ROBDD. A Boolean function has different ROBDD representations, depending on the variable order imposed by index, but there is only one ROBDD for each ordering. Thus, ROBDDs are known as a canonical representation of Boolean functions. From now on BDD will be used to refer to a ROBDD. A data structure representing a BDD consists of an array of Node objects (or nodes), each corresponding to a BDD vertex and each containing three elements: a variable label v, then.v/, and else.v/, where the latter two elements are BDD array indices. The first two nodes in the BDD array correspond to the 0 and 1 terminal vertices of a BDD. For both, then.::/ and else.::/ are empty. Denote by terminal.1/ and terminal.0/ the BDD array locations of these nodes. All other nodes fill up the remainder of the BDD array in the order they are created. A node that is not in the BDD array can be created and added to the BDD array, and its array index returned in constant time. A hashtable, commonly referred to as a unique table, is maintained for the purpose of finding a node’s array location given v; then.v/; else.v/. If there is no corresponding entry in the hashtable, a new node is created and recorded in the hashtable with key v; then.v/; else.v/. A single procedure called findOrCreateNode takes v; then.v/; else.v/ as arguments and returns the BDD array index of a node, either a newly created one or an existing one. This procedure is shown in Fig. 24. The main BDD construction operation is to find and attach two descendant nodes (the n.v/ and else.v/) to a parent node (v). The procedure findOrCreateNode is used to ensure that no two nodes in the final BDD data structure represent the same function. The procedure for building a BDD data structure is buildBDD, shown in Fig. 25. It is assumed that variable indices match the value of i ndex applied to that variable (thus, i D i ndex.vi /). The complexity of buildBDD is proportional to the number of nodes that must be created. In all interesting applications, many BDDs are constructed. But they may all share the BDD data structure above. Thus, a node may belong to many BDDs.

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Fig. 25 Algorithm for building a BDD: invoked using buildBDD(f ,1)

Fig. 26 Operations reduce1 and reduce0

The operations reduce1 and reduce0 , shown in Fig. 26, are used to describe several important BDD operations in subsequent sections. Assuming v is the root of the BDD representing f , the operation reduce1 .v; f / returns f constrained by the assignment of 1 to variable v and reduce0 .v; f / returns f constrained by the assignment of 0 to the variable v. Details on performing the common binary operations of ^ and _ on BDDs will be ignored here. The reader may refer to [9] for detailed descriptions. Here, it is only mentioned that, using a dynamic programming algorithm, the complexity of these operations is proportional to the product of the sizes of the operands and the size of the result of the operation can be that great as well. Therefore, using ^ alone, for example (as so many problems would require), could lead to intermediate structures that are too large to be of value. This problem is mitigated somewhat by operations of the kind discussed in the next four subsections, particularly existential quantification. The operations considered next are included not only because they assist BDDoriented solutions but mainly because they can assist search-oriented solutions when used properly. For example, if inputs are expressed as a collection of BDDs, then they may be preprocessed to reveal information that may be exploited later,

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Fig. 27 Algorithm for existentially quantifying variable v away from BDD f . The _ denotes the or of BDDs

during search. In particular, inferences7 may be determined and used to reduce input complexity. The discussion of the next four sections emphasizes this role.

5.9.1 Existential Quantification A Boolean function which can be written E _ .:v ^ h2 .x// E f .v; x/ E D .v ^ h1 .x// can be replaced by f .x/ E D h1 .x/ E _ h2 .x/; E where xE is a list of one or more variables. There is a solution to f .x/ E if and only if there is a solution to f .v; x/, E so it is sufficient to solve f .x/ E to get a solution to f .v; x/. E Obtaining f .x/ E from f .v; x/ E is known as existentially quantifying v away from f .v; x/. E This operation is efficiently handled if f .v; x/ E is represented by a BDD. However, since most interesting BDD problems are formulated as a conjunction of functions and therefore as conjunctions of BDDs, existentially quantifying away a variable v succeeds easily only when just one of the input BDDs contains v. Thus, this operation is typically used together with other BDD operations for maximum effectiveness. The algorithm for existential quantification is shown in Fig. 27. If inferences can be revealed in preprocessing, they can be applied immediately to reduce input size and therefore reduce search complexity. Although existential quantification can, by itself, uncover inferences (see, e.g., Fig. 28), those same inferences are revealed during BDD construction if inference lists for each node are built and maintained. Therefore, a more effective use of existential quantification is

7

Finding inferences is referred to in the BDD literature as finding essential values to variables, and a set of inferences (a conjunction of literals) is referred to as a cube.

Algorithms for the Satisfiability Problem Fig. 28 Two examples of existentially quantifying a variable away from a function. Functions are represented as BDDs on the left. Variable v3 is existentially quantified away from the top BDD leaving 1, meaning that regardless of assignments given to variables v1 and v2 , there is always an assignment to v3 which satisfies the function. Variable v2 is existentially quantified away from the bottom BDD leaving the inference v1 D 1

379

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in support of other operations, such as strengthening (see Sect. 5.9.5), to uncover those inferences that cannot be found during BDD construction or in tandem with ^ to retard the growth of intermediate BDDs. Existential quantification, if applied as a preprocessing step prior to search, can increase the number of choicepoints expanded per second but can increase the size of the search space. The increase in choicepoint speed is because existentially quantifying a variable away from the function has the same effect as branching from a choicepoint in both directions. Then overhead is reduced by avoiding heuristic computations. However, search-space size may increase since the elimination of a variable can cause subfunctions that had been linked only by that variable to become merged with the result that the distinction between the subfunctions becomes blurred. This is illustrated in Fig. 29. The speedup can overcome the lost intelligence but it is sometimes better to turn it off.

5.9.2 Reductions and Inferences Consider the truth tables corresponding to two BDDs f and c over the union of variable sets of both f and c. Build a new BDD g with variable set no larger than the union of the variable sets of f and c and with a truth table such that on rows which c maps to 1, g maps to the same value that f maps to and on other rows g maps to any value, independent of f . It should be clear that f ^ c and g ^ c are identical so g can replace f in a collection of BDDs without changing its solution space. There are at least three reasons why this might be done. The superficial reason is that g can be made smaller than f . A more important reason is that inferences can

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Fig. 29 Existential quantification can cause blurring of functional relationships. The top function is seen to separate variables v6 , v7 , and v8 from v3 , v4 , and v5 if v2 is chosen during search first. Existentially quantifying v2 away from the top function before search results in the bottom function in which no such separation is immediately evident. Without existential quantification, the assignment v1 D 0, v2 D 1, v3 D 1 reveals the inference v4 D 1. With existential quantification the assignment must be augmented with v7 D 0 and v8 D 0 (but v2 is no longer necessary) to get the same inference

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be discovered. The third reason is that BDDs can be removed from the collection without loss. Consider, for example, BDDs representing functions f D .v1 _ :v2 _ v3 / ^ .:v1 _ v2 / and c D .v1 _ :v2 /: Let a truth table row be represented by a 0-1 vector which reflects assignments of variables indexed in increasing order from left to right. Let g have the same truth table as f except for row h011i which c maps to 0 and g maps to 1. Then g D .v1 $ v2 / and f ^ c is the same as g ^ c but g is smaller than f . As an example of discovering inferences, consider

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f = (v1 → v2 ) ∧ (¬v1 → (¬v3 ∧ v4 )) v1

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g = (v1 ) ∧ (v2 )

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Fig. 30 A call to restrict.f; c/ returns the BDD g shown on the right. In this case inferences v1 D 1 and v2 D 1 are revealed. The symbol ⇓ denotes the operation

f D .v1 ! v2 / ^ .:v1 ! .:v3 ^ v4 // and c D .v1 _ v3 /: Let g have the same truth table as f except g maps rows h0001i and h0101i to 0, as does c. Then g D .v1 / ^ .v2 / which reveals two inferences. The BDDs for f , c, and g of this example are shown in Fig. 30. The example showing BDD elimination is deferred to Theorem 6, Sect. 5.9.4. Clearly, there are numerous strategies for creating g from f and c and replacing f with g. An obvious one is to have g map to 0 all rows that c maps to 0. This strategy, which will be called zero-restrict in this chapter, turns out to have weaknesses. Its obvious dual, which has g map to 1 all rows that c maps to 0, is no better. For example, applying zero-restrict to f and c of Fig. 32 produces g D :v3 ^ .v1 _ .:v1 ^ :v2 // instead of the inference g D :v3 which is obtained from a more intelligent replacement strategy. An alternative approach, one of many possible ones, judiciously chooses some rows of g to map to 1 and others to map to 0 so that g’s truth table reflects a logic pattern that generates inferences. The truth table of c has many 0 rows and this is exploited. Specifically, c maps rows h010i, h011i, h101i, and h111i to 0. The more intelligent strategy lets g map rows h011i and h111i to 0 and rows h010i and h101i to 1. Then g D :v3 . Improved replacement strategies might target particular truth table patterns, for example, equivalences, or they might aim for inference discovery. Since there is more freedom to manipulate g if the truth table of c has many zeros, it is important to choose c as carefully as the replacement strategy. This is illustrated by the examples of Figs. 31 and 32 where, in the first case, no inference is generated but after f and c are swapped, an inference is generated. The next two subsections show two replacement strategies that are among the more commonly used.

5.9.3 Restrict The original version of restrict is what is called zero-restrict above. That is, the original version of restrict is intended to remove paths to termi nal.1/ from f

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f = (v1 ∨ ¬v2 ) ∧ (¬v1 ∨ ¬v3 ) 1 v3 0

v1

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Fig. 31 A call to restrict.f; c/ results in no change

f = (v2 ∨ ¬v3 )

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Fig. 32 Reversing the roles of f and c in Fig. 31, a call to restrict.f; c/ results in the inference g D :v3 as shown on the right. In this case, the large number of 0 truth table rows for c was exploited to advantage

that are made irrelevant by c. The idea was introduced in [43]. In this chapter an alternative version which is implemented as Algorithm 11 of Fig. 33 is considered. Use the symbol ⇓ to denote the restrict operator. Then g D f ⇓ c is the result of zero-restrict after all variables in c that are not in f are existentially quantified away from c. Figures 30–32 show examples that were referenced in the previous subsection. Procedure restrict is similar to a procedure called generalized cofactor (gcf) or constrain (see the next subsection for a description). Both restrict.f; c/ and gcf.f; c/ agree with f on interpretations where c is satisfied but are generally somehow simpler than f . Procedure restrict can be useful in preprocessing because the BDDs produced from it can never contain more variables than the BDDs they replace. On the negative side, it can, in odd cases, cause a garbling of local information. Moreover, although restrict may reveal some of the inferences that strengthening would (see below), it can still cause the number of search choicepoints to increase. Both these issues are related: restrict can spread an inference that is evident in one BDD over multiple BDDs (see Fig. 34 for an example).

5.9.4 Generalized Cofactor The generalized cofactor operation, also known as constrain, is denoted here by j and implemented as gcf (Algorithm 12) in Fig. 35. It takes BDDs f and c as input

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Fig. 33 Algorithm for restricting a BDD f by a BDD c

f = (v1 ∨ ¬v2 ) ∧ (¬v1 ∨ v3 ) v1

1 v3 1 1

0

c = (¬v1 ∨ v3 ) v1

0 0

g = (v1 ∨ ¬v2 )

v2 1 0

⇓

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Fig. 34 A call to restrict.f; c/ spreads an inference that is evident in one BDD over multiple BDDs. If v3 is assigned 0 in f , then v1 D 0 and v2 D 0 are inferred. After replacing f with g D restrict.f; c/, to get the inference v2 D 0 from the choice v3 D 0, visit c to get v1 D 0 and then g to get v2 D 0. Thus, restrict can increase work if not used properly. In this case, restricting in the reverse direction leads to a better result

and produces g D f jc by sibling substitution. BDD g may be larger or smaller than f , but, more importantly, systematic use of this operation can result in the elimination of BDDs from a collection. Unfortunately, by definition, the result of this operation depends on the underlying BDD variable ordering, so it cannot be regarded as a logical operation. It was introduced in [42]. BDD g is a generalized cofactor of f and c if for any truth assignment t, g.t/ has the same value as f .t 0 / where t 0 is the nearest truth assignment to t that maps c to 1. The notion of nearest truth assignment depends on a permutation  of the numbers

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Fig. 35 Algorithm for finding a greatest common cofactor of a BDD

1; 2; : : : ; n which states the variable ordering of the input BDDs. Represent a truth assignment to n variables as a vector in f0; 1gn and, for truth assignment t, let ti denote the i th bit of the vector representing t. Then the distance between two truth P assignments t 0 and t 00 is defined as niD1 2ni .t0 i ˚ t00i /. One pair of assignments is nearer to each other than another pair if the distance between that pair is less. It should be evident that distances between pairs are unique for each pair. For example, Fig. 36 shows BDDs f and c under the variable ordering given by  D h1; 2; 3; 4i. For assignment vectors h 01i, h 10i, and h 11i (where

is a wildcard meaning 0 or 1), gcf.f; c/, shown as the BDD at the bottom of Fig. 36, agrees with f since those assignments cause c to evaluate to 1. The closest assignment to h0000i, h0100i, h1000i, and h1100i causing c to evaluate to 1 is h0001i, h0101i, h1001i, and h1101i, respectively. On all these inputs gcf.f; c/ has value 1, which the reader can check in Fig. 36. The following expresses the main property of j that makes it useful. Theorem 6 Given BDDs f1 ; : : : ; fk , for any 1  i  k, f1 ^ f2 ^ : : : ^ fk is satisfiable if and only if .f1 jfi / ^ : : : ^ .fi 1 jfi / ^ .fi C1 jfi / ^ : : : ^ .fk jfi / is satisfiable. Moreover, any assignment satisfying the latter can be mapped to an assignment that satisfies f1 ^ : : : ^ fk . Proof If it can be shown that .f1 jfi / ^ : : : ^ .fi 1 jfi / ^ .fi C1 jfi / ^ : : : ^ .fk jfi /

(4)

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is satisfiable if and only if .f1 jfi / ^    ^ .fi 1 jfi / ^ .fi C1 jfi / ^ : : : ^ .fk jfi / ^ fi

(5)

is satisfiable, then, since (5) is equivalent to f1 ^: : :^fk , the first part of the theorem will be proved. Suppose (5) is satisfied by truth assignment t. That t represents a truth table row that fi maps to 1. Clearly that assignment also satisfies ( 4). Suppose no assignment satisfies (5). Then all assignments for which fi maps to 1 do not satisfy (4) since otherwise (5) would be satisfied by any that do. Only truth assignments t which fi maps to 0 need to be considered. Each .fj jfi / in (4) and (5) maps to the same value that fj maps the nearest truth assignment, say, r, to t that satisfies fi . But r cannot satisfy (5) because it cannot satisfy (4) by the argument above. Hence, there is no truth assignment falsifying fi but satisfying (4) so the first part is proved. For the second part, observe that any truth assignment that satisfies (4) and (5) also satisfies fi ^: : :^fk , so it is only necessary to consider assignments t that satisfy (4) but not (5). In that case, by construction of .fj jfi /, the assignment that is nearest to t and satisfies fi also satisfies .fj jfi /. That assignment satisfies f1 ^ : : : ^ fk . u t This means that, for the purposes of a solver, generalized cofactoring can be used to eliminate one of the BDDs among a given conjoined set of BDDs: the solver finds an assignment satisfying gcf.f1 ; fi /^: : : ^gcf.fk ; fi / and then extends the assignment to satisfy fi ; otherwise, the solver reports that the instance has no solution. However, unlike restrict, generalized co-factoring cannot by itself reduce the number of variables in a given collection of BDDs. Other properties of the gcf operation, all of which are easy to show, are: 1. f D c^gcf.f; c/ _ :c^gcf.f; :c/. 2. gcf.gcf.f; g/; c/ D gcf.f; g ^ c/. 3. gcf.f ^ g; c/ D gcf.f; c/^ gcf.g; c/. 4. gcf.f ^ c; c/ D gcf.f; c/. 5. gcf.f ^ g; c/ D gcf.f; c/^ gcf.g; c/. 6. gcf.f _ g; c/ D gcf.f; c/_ gcf.g; c/. 7. gcf.f _ :c; c/ D gcf.f; c/. 8. gcf.:f; c/ D : gcf.f; c/. 9. If c and f have no variables in common and c is satisfiable, then gcf.f; c/ D f . Care must be taken when cofactoring in both directions (exchanging f for c). For example, f ^ g ^ h cannot be replaced by .gjf / ^ .f jg/ ^ h since the former may be unsatisfiable when the latter is satisfiable. Examples of the application of gcf are shown in Figs. 36 and 37. Figure 36 illustrates the possibility of increasing BDD size. Figure 37 presents the same example after swapping v1 and v3 under the same variable ordering and shows that the result produced by gcf is sensitive to variable ordering. Observe that the functions produced by gcf in both figures have different values under the assignment v1 D 1, v2 D 1, and v3 D 0. Thus, the function returned by gcf depends on the variable ordering as well.

386 Fig. 36 Generalized cofactor operation on f and c as shown. In this case the result is more complicated than f . The variable ordering is v1 < v2 < v3

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f = (v1 → ¬v2 ) ∨ (¬v1 → v3 )

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5.9.5 Strengthen This binary operation on BDDs helps reveal inferences that are missed by restrict due to its sensitivity to variable ordering. Given two BDDs, b1 and b2 , strengthening conjoins b1 with the projection of b2 onto the variables of b1 : that is, b1 ^ 9Ev b2 , where Ev is the set of variables appearing in b2 but not in b1 . Strengthening each bi against all other bj s sometimes reveals additional inferences or equivalences. Algorithm strengthen is shown in Fig. 38. Figure 39 shows an example. Strengthening provides a way to pass important information from one BDD to another without causing a size explosion. No size explosion can occur because, before b1 is conjoined with b2 , all variables in b2 that do not occur in b1 are existentially quantified away. If an inference (of the form v D 1, v D 0, v D w, or v D :w) exists due to just two BDDs, then strengthening those BDDs against each other (pairwise) can move those inferences, even if originally spread across both BDDs, to one of the BDDs. Because strengthen shares information between BDDs, it can be thought of as sharing intelligence and strengthening the relationships between functions; the added intelligence in these strengthened functions can be exploited by a smart search heuristic. It has been found empirically that strengthen usually decreases the number of choicepoints when a particular search heuristic is employed, but sometimes it causes more choicepoints. It may be conjectured this is due to the delicate nature of some problems where duplicating information in the BDDs leads the heuristic astray.

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f = (v3 → ¬v2 ) ∨ (¬v3 → v1 ) 1 v2 1 v3

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Fig. 37 Generalized cofactor operation on the same f and c of Fig. 36 and with the same variable ordering but with v1 and v3 swapped. In this case the result is less complicated than f and the assignment fv1 ; v2 g causes the output of gcf in this figure to have value 1, whereas the output of gcf in Fig. 36 has value 0 under the same assignment

Fig. 38 Algorithm for strengthening a BDD by another

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b2 : 1 v3 0

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Strengthening example: Existentially quantify v1 away from b2 ... ∃v1 b2 :

b1 :

v2

v2

1 0 1

0

v3

v3

⇒

0 1

1 0

1

0

v3

1

0 1

1 0

0

Fig. 39 : : : then conjoin the two BDDs. Inference v3 D 0 is revealed

Procedure strengthen may be applied to CNF formulas and in this case it is the same as applying Davis-Putnam resolution selectively on some of the clauses. When used on more complex functions, it is clearer how to use it effectively as the clauses being resolved are grouped with some meaning. Evidence for this comes from bounded model checking examples. This section concludes by mentioning that for some classes of problems, resolution has polynomial complexity, while strictly BDD manipulations require exponential time, and for other classes of problems, resolution has exponential complexity, while BDD manipulations require polynomial time [67].

5.10

Decompositions

The variable elimination methods of Sects. 5.5 and 5.11 recursively decompose a given formula into overlapping subformulas 1 and 2 such that the solution to can be inferred from the solutions to 1 and 2 . The decompositions are based on occurrences of a selected variable v in clauses of and each subformula has at least as many clauses as those of which contain neither literal v nor literal :v. Intuitively, the speed of the methods usually depends on the magnitude of the size reduction from to 1 and 2 . However, it is often the case that the number of occurrences of most variables in a formula or subformula is small which usually means small size reductions for most variables. For example, the average number of occurrences of a variable in a random k-SAT formula8 of m clauses developed

8

Random k-SAT formulas are defined in Sect. 7, Page 440.

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from n variables is km=n: so, if k D 3 and m=n is, say, 4, then the average number of occurrences of a randomly chosen variable is only 12. Hence, the methods often suffer computational inadequacies which can make them unusable in some cases. On the positive side, such methods can be applied to any CNF formula. But there are other decomposition methods that sacrifice some generality for the sake of producing subformulas of relatively small size. Truemper’s book [138] presents quite a few of these, all of which are capable of computationally efficient solutions to some problems that would be considered difficult for the more general variable elimination methods. This section presents one of these, called monotone decomposition, for illustration and because it is related to material that is elsewhere in this chapter.

5.10.1 Monotone Decomposition Let CNF formula of m clauses and n variables be represented as a m  n .0 ˙ 1/-matrix M (defined in Sect. 3.1). A monotone decomposition of M is a permutation of rows and columns of M and the multiplication by 1 of some or all of its columns, referred to below as a column scaling, resulting in a partition into four submatrices as follows: 0 @

A1 E

1 A;

(6)

D A

2

where the submatrix A1 has at most one C1 entry per row; the submatrix D contains only 1 or 0 entries; the submatrix A2 has no restrictions other than the three values of 1, C1, and 0 for each entry; and the submatrix E has only 0 entries. The submatrix A1 represents a Horn formula. In Sect. 6.2 Horn formulas are shown to have the following two important properties: they are solved efficiently, for example, by Algorithm 20 of Fig. 46, and, by Theorem 15, there is always a unique minimum model for a satisfiable Horn formula. The second property means there is always a satisfying assignment M such that, for any other satisfying assignment M 0 , the variables that have value 1 according to M 0 are a superset of the variables set to 1 according to M (more succinctly, M M 0 ). This property, plus the nature of submatrices D and E, effectively allows a split of the problem of determining the satisfiability of into two independent problems: namely, determine satisfiability for the Horn formula represented by A1 and determine satisfiability for the subformula represented by A2 . The algorithm of Fig. 40 shows this in more detail. The following theorem proves correctness of this algorithm: Theorem 7 Let CNF formula be represented as a monotone decomposition .0 ˙ 1/-matrix. On input , Algorithm 14 outputs “unsatisfiable” if and only if is unsatisfiable, and if is satisfiable, then the output set M1 [ M2 is a model for .

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Fig. 40 Algorithm for determining satisfiability of a monotone decomposition

Proof Clearly, if Horn formula A1 is unsatisfiable, then so is . So suppose there is a model M1 for A1 and consider the rows of A2 remaining after rows common to those of D which are satisfied by M1 are removed. Since M1 is a unique minimum model for A1 , no entries of D are +1, and since variables of A1 are distinct from variables of A2 , no remaining row of A2 can be satisfied by any model for A1 . Therefore, if these rows of A2 are unsatisfiable, then so is . On the other hand, if these rows are satisfied by model M2 , then clearly, M1 [ M2 is a model for .  A .0 ˙ 1/ matrix M representing CNF formula may have more than one monotone decomposition. Of particular interest is the maximum monotone decomposition of M , that is, the monotone decomposition of such that A1 has the greatest number of rows and columns. A monotone decomposition is said to be maximal with respect to the dimensions of A1 . The following theorem says a unique maximal monotone decomposition is always possible. Theorem 8 Any .0˙1/ matrix M has a unique maximal monotone decomposition. Proof Suppose M has two distinct maximal monotone decompositions, say, M1 and M2 . Let A1i , A2i , and Di , i 2 f1; 2g, be the partition of M, after column scaling, corresponding to Mi (see the partition (6) on page 389). Construct a new partition M0 of M into A01 , A02 , and D0 such that A01 includes all rows and columns of A11 and A12 . For those columns of M0 that are also columns of A11 , use a column scaling that is exactly the same as the one used in M1 . For all other columns, use the same scaling as in M2 . The submatrix of A01 that includes rows and columns of A11 is the same as A11 because the scaling of those columns is the same as for M1 . The submatrix of A01 including rows of A11 and columns not in A11 must be a 0 submatrix by the monotone decomposition M1 . The submatrix of A01 including columns of A11 and no rows of A11 must contain only 0 or 1 entries due to the M1 scaling and the submatrix including neither columns or rows of A11 must be Horn due to M2 column scaling. It follows that submatrix A01 is Horn (at most one +1 in each row).

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It is similarly easy to check that the submatrix of M0 consisting of rows of A01 and columns other than those of A01 is 0 and that the submatrix of M0 consisting of columns of A01 and rows other than those of A01 contains no +1 entries. It follows that M0 is a monotone decomposition. Since A11  A01 and A12  A01 , neither M1 nor M2 is a maximal monotone decomposition in violation of the hypothesis. The theorem follows. t u From Theorem 8 there is always a maximum monotone decomposition for M . A maximum monotone decomposition is useful because (1) A1 , representing a Horn formula, is as large as possible so A2 is as small as possible; (2) Horn formulas may be efficiently solved by Algorithm 20 and (3) a maximum monotone decomposition can be found efficiently, as will now be shown. A maximum monotone decomposition can be found using Algorithm 15 of Fig. 41. The algorithm completes one or more stages where each stage produces a proper monotone decomposition of some matrix. All submatrices change dimensions during the algorithm, so primes are used as in E 0 to refer to the current incantation of corresponding submatrices. Initially, that matrix is M . At the end of a stage, if the algorithm needs another stage to produce a bigger decomposition, A02 of the current stage becomes the entire input of the next stage and the next stage proceeds independently of previous stages. This can be done since the operation to be mentioned next does not multiply by 1 any of the rows and columns of the A01 and D0 matrices of previous stages. The important operation is to move a nonpositive column that intersects A02 to just right of the border of the current stage’s A01 matrix, move the border of A01 and D0 to the right by one column, tag and move the rows containing 1 on the right boundary of the changed D0 up to just below the border of A01 , and finally lower the border of A01 and E 0 down to include the tagged rows. Doing so keeps A01 Horn and D0 nonpositive and enlarges A01 . If no nonpositive column exists through A02 , no column can be made nonpositive through A02 by a 1 multiplication, and the initial moved column is not multiplied by 1, then the initial moved column of the stage is multiplied by 1 and the stage is restarted. Because of the following theorem, backtracking is limited to just one per stage and is used only to try to decompose with the initial moved column of the stage multiplied by 1. Theorem 9 Refer to Algorithm 15 for specific variable names and terms: 1. If z is not a nonpositive column in E 0 and z multiplied by 1 is not nonpositive in E 0 , then there is no monotone decomposition at the current stage with the initial moved column v of the stage left as is. 2. If multiplying v by 1 also fails because a z cannot be made nonpositive in E 0 , then not only does z block a monotone decomposition but multiplying any of the other columns in A01 except v by 1 blocks a monotone decomposition as well. Proof 1. There is no way to extend the right boundary of A01 and stay Horn while making E 0 0 because column z prevents it.

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Fig. 41 Algorithm for finding the maximum monotone decomposition of a .0 ˙ 1/ matrix

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2. Consider columns in A01 first. The proof is by induction on the number of columns processed in A01 . The base case has no such column: that is, A01 only contains the column v and is trivially satisfied. For the inductive step, change the column scaling to 1 for all columns and run the algorithm in the same column order it had been when it could not continue. Assume the hypothesis holds to k columns and consider processing at the k C 1st column, call it column x. At this point A01 has one 1 in each row, D0 is nonpositive, and since x is multiplied by 1, it is nonzero and nonpositive through E 0 . If there is a monotone decomposition where x is multiplied by 1, then x goes through A1 of that decomposition. The multiplication by 1 changes the nonzero nonpositive elements of x through E 0 to nonzero nonnegative elements. Therefore, at least one of these elements, say, in row r, is +1. But A1 of the decomposition must have a +1 in each row, so it must be that row r has this +1 in, say, column c, a column of A01 that is multiplied by 1. But c cannot be the same as v since v multiplied by 1 blocks a monotone decomposition by hypothesis. On the other hand, if c is not v, then by the inductive hypothesis c cannot be multiplied by 1 in a monotone decomposition. Therefore, by contradiction, there is no monotone decomposition and the hypothesis holds to k C 1 columns. Now consider column z. No scaling of column z can make z nonpositive in E 0 . Then that part of z that goes through E 0 has 1 and 1 entries. The hypothesis follows from the same induction argument as above. t u Any column blocking a monotone decomposition need never be checked again. The algorithm keeps track of blocking columns with set N , the long-term record of blocking columns, and set L, the temporary per stage record. If column indicator w is placed in N , it means the unmultiplied column w blocks, and if :w is placed in N , it means column w multiplied by 1 blocks. The algorithm has quadratic complexity. Complexity can be made linear by running the two possible starting points of each stage, namely, using column v as is and multiplying column v by 1, concurrently, and breaking off computation when one of the two succeeds. A formula which has a maximum monotone decomposition where A2 is a member of an efficiently solved subclass of satisfiability obviously may be solved efficiently by Algorithm 14 if A2 can efficiently be recognized as belonging to such a subclass. Section 6 discusses several efficiently solved subclasses of satisfiability problems which may be suitable for testing. If A2 represents a 2-SAT formula (see Sect. 6.1), then is said to be q-Horn. The class of q-Horn formulas was discovered and efficiently solved in [21, 22], and it was the results of that work that led to the development of maximum monotone decompositions [137].

5.10.2 Autarkies and Safe Assignments Definition 1 An assignment to a set of variables is said to be autark or an autarky if all clauses that contain at least one of those variables are satisfied by the assignment. We will call a set of variables that is associated with an autarky an autark set.

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If all clauses satisfied by an autarky are removed from a CNF formula , then the resulting formula is equivalent in satisfiability to . A simple example of an autark set is a collection of one or more pure literals. A simple decomposition is to remove all clauses that contain pure literals. One can do the same for any autarky. However, discovering autarkies can be expensive. In some cases, though, an autarky can be found in polynomial time. This is treated in Sect. 6.9. A similar decomposition exists for BDDs. Let f be a Boolean function and let f jv (f j:v ) denote the function obtained by setting variable v to 1 (respectively, 0). Lemma 4 ([146]) Given a conjunction of BDDs f D f1 ^ : : : ^ fm and variable v occurring in one or more BDDs of f , let f 0 be the conjunction of all BDDs in f 0 which contain v. Let fv0 D :.f 0 jv / ^ .f 0 j:v /. Let f:v D .f 0 jv / ^ :.f 0 j:v /. 0 1. If fv has value 0, then f jv is satisfiable if and only if f is satisfiable. 0 2. If f:v has value 0, then f j:v is satisfiable if and only if f is satisfiable. Lemma 4 states that if any of the BDDs in f are falsified or if all of the BDDs in f are satisfied by setting v to 1 (respectively, 0), then it is safe to make that assignment because the satisfiability of f does not change by doing so. We emphasize that the safe value for v is not necessarily inferred. This lemma provides 0 a way to test whether a safe value exists for a variable, that is, if fv0 .f:v / has value 0, then it is safe to set v to 1 (0) in f . However, to use this lemma directly requires conjoining all BDDs containing v, and from Sect. 5.9, this could be expensive. In pursuit of an efficient method for finding safe assignments, Lemma 4 may be used to derive the following weaker result: Theorem 10 ([146]) Given a conjunction of BDDs f D f1 ^: : : ^fm and variable v occurring in one or more BDDs of f , let f 0 be the conjunction of all BDDs in f which contain v. Without loss of generality, suppose f 0 D f1 ^ : : : ^ fn 0 where n  m. Let fv0 D .:.f1 jv / ^ f1 j:v / _ : : : _ .:.fn jv / ^ fn j:v /. Let f:v D .f1 jv ^ :.f1 j:v // _ : : : _ .fn jv ^ :.fn j:v //. 1. If fv0 has value 0, then f jv is satisfiable if and only if f is satisfiable. 0 2. If f:v has value 0, then f j:v is satisfiable if and only if f is satisfiable. According to Theorem 10, a safe assignment may be found without having to conjoin BDDs containing v. This is practical when BDDs are fairly small and is more practical than conjoining BDDs if v appears in many of them. However, it is possible that a safe assignment discovered using Lemma 4 may be undiscoverable using Theorem 10. The following is the corresponding theorem for safe assignments involving more than one variable: Theorem 11 ([146]) Given a conjunction of BDDs f D f1 ^ : : : ^ fm and a set of variables V D fv1 ; : : : ; vk g each occurring in one or more BDDs of f , let f 0 be the conjunction of all BDDs in f which contain at least one of the variables in V . Let M D fM1 ; : : : ; M2k g be the set of all possible truth assignments to the variables

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in V . Without loss of generality, suppose f 0 D f1 ^ : : : ^ fn where n  m. 81i 2k if .:.f 0 jMi / ^ .f 0 jM1 _ : : : _ f 0 jM2k // has value 0, then f jMi is satisfiable if and only if f is satisfiable. It is straightforward to turn Theorems 10 and 11 into an algorithm, but this is not done here because numerous variants to speed up the search are possible and it is impractical to list all of them.

5.11

Branch and Bound

The DPLL algorithm of Fig. 17 is sequential in nature. At any point in the search, only one node, representing a partial assignment M1W , of the search tree is active: that is, open to exploration. This means exploration of a promising branch of the search space may be delayed for a potentially considerable period until search finally reaches that branch. Branch and bound aims to correct this to some extent. In branch and bound, quite a few nodes of the search space may be active at any point in the search. Each of the active nodes has a number l.M1W / which is an aggregate estimate of how close the assignment represented at a node is to a solution or confirms that assignment cannot be extended to a best solution. Details concerning how l.M1W / is computed will follow. A variable v is chosen for assignment from the subformula of the active node of lowest l value and that node is expanded. The expansion eliminates one active node and may create up to two others, one for each value to v. To help control the growth of active nodes, branch and bound maintains a monotonically decreasing number u for preventing nodes known to be unfruitful from becoming active. If the l value of any potential active node is greater than u, it is thrown out and not made active. Eventually, there are no active nodes left and the algorithm completes. Branch and bound is intended to solve more problems than SAT, one of the most important being MAX-SAT (Page 319). It requires a function g .M1W / which maps a given formula and partial or total truth assignment M1W to a nonnegative number. The objective of branch and bound is to return an assignment M such that g .M / is minimum over all truth assignments, partial or complete. That is, M W 8X ; g .X /  g .M /: For example, if g .M1W / is just the number of clauses in M1W , then branch and bound seeks to find M which satisfies the greatest number of clauses, maybe all. Branch and bound discovers and discards search paths that are known to be fruitless, before they are explored, by means of a heuristic function h. M1W / where by removing clauses satisfied by and literals falsified by M1W is obtained from M1W . The heuristic function returns a nonnegative number which, when added to g .M1W /, is a lower bound on g. X / over all possible extensions X to M1W . That sum is the l.M1W / that was referred to above. That is,

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Fig. 42 Classic branch-and-bound procedure adapted to satisfiability

l.M1W / D h.

M1W /

C g .M1W /  minfg .X / W X is an extension of M1W g:

The algorithm maintains a number u that records the lowest g value that has been seen so far during search. If partial assignment M1Wi is extended by one variable to M1Wi C1 and g .M1Wi C1 / < u then u, is updated to that value and M1Wi C1 , the assignment that produced that value, is saved as M . In that case, l.M1Wi C1 / must also be less than u because it is less than g .M1Wi C1 /. But if l.M1Wi C1 / > u, then there is no chance, by definition of l.M1W /, that any extension to M1Wi C1 will yield the minimum g . Hence, if that test succeeds, the node that would correspond to M1Wi C1 is thrown out, eliminating exploration of that branch. The algorithm in its general form for satisfiability is shown in Fig. 42 as Algorithm 16. A priority queue P is used to hold all active nodes as pairs where each pair contains a reduced formula and its corresponding partial assignment. Pairs are stored in P in increasing order of l.M1W /. It is easy to see, by definition of l.M1W /, that no optimal assignment gets thrown out. It is also not difficult to see that, given two heuristic functions h1 and h2 , if h1 . M1W /  h2 . M1W / for all M , then the search

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explored using h1 will be no larger than the search space explored using h2 . Thus, to keep the size of the search space down, as tight a heuristic function as possible is desired. However, since overall performance is most important and since tighter heuristic functions typically mean more overhead, it is sometimes more desirable to use a weaker heuristic function which generates a larger search space in less time. Section 5.12.2 shows how linear programming relaxations of integer programming representations of search nodes can be used as heuristic functions. This section concludes with an alternative to illustrate what else is possible. Recall the problem of variable-weighted satisfiability which was defined in Sect. 2: given CNF formula and positive weights on variables, find a satisfying assignment for , if one exists, such that the sum of weights of variables of value 1 is minimized. Let M1W be defined as above and let Q M1W be a subset of positive clauses of M1W such that no variable appears twice in Q M1W . For example, Q M1W might look like this: .v1 _ v3 _ v7 / ^ .v2 _ v6 / ^ .v4 _ v5 _ v8 /: A strictly lower bound on the minimum weight solution over all extensions to M1W is g .M1W / C h. M1W /, the sum of the weights of the minimum weight variable in each of the clauses of Q M1W . Clearly, this is not a very tight bound. But it is computationally fast to acquire this bound, and the trade-off of accuracy for speed often favors this approach [61], particularly when weight calculations are made incrementally. Additionally, there are some tricks that help to find a good Q M1W . For example, a greedy approach may be used as follows: choose a positive clause c with variables independent of clauses already in Q M1W and such that the ratio of the weight of the minimum weight variable in c to the number of variables in c is maximum [108]. The interested reader can consult [108] and [41] for additional ideas.

5.12

Algebraic Methods

A collection of Boolean constraints may be expressed as a system of algebraic equations or inequalities which has a solution if and only if the constraints are satisfiable. The attraction of these methods is that, in some cases, a single algebraic operation can simulate a large number of resolution operations. The problem is that it is not always obvious how to choose the optimal sequence of operations to take advantage of this, and often performance is disappointing due to nonoptimal choices.

5.12.1 Grobner ¨ Bases Applied to SAT It is interesting that at the same time resolution was being developed and understood as a search tool in the 1960s, an algebraic tool for computing a basis for highly nonlinear systems of equations was introduced: the basis it found was given the name Gr¨obner basis and the tool was called the Gr¨obner basis algorithm [29].

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But it was not until the mid-1990s that Gr¨obner bases crossed paths with satisfiability when it was shown that Boolean expressions can be written as systems of multilinear equations which a simplified Gr¨obner basis algorithm can solve using a number of derivations that is guaranteed to be within a polynomial of the minimum number possible [37]. Also in that paper, it is shown that the minimum number of derivations cannot be much greater than, and may sometimes be far less than, the minimum number needed by resolution. Such powerful results led the authors to say “these results suggest the Gr¨obner basis algorithm might replace resolution as a basis for heuristics for NP-complete problems.” This has not happened, perhaps partly because of the advances in the development of CNF SAT solvers in the 1990s and partly because, as is the case for resolution, it is generally difficult to find a minimum sequence of derivations leading to the desired conclusion. However, the complementary nature of algebraic and logic methods makes them an important alternative to resolution. Generally, in the algebraic world, problems that can be solved essentially using Gaussian elimination with a small or modest increase in the degree of polynomials are easy. A classic example where this is true is the systems of equations involving only the exclusiveor operator. By contrast, just expressing the exclusive-or of n variables in CNF requires 2n1 clauses. In the algebraic proof system outlined here, facts are represented as multilinear equations and new facts are derived a database of existing facts using rules described below. Let hc0 ; c1 ; : : : ; c2n 1 i be a 0-1 vector of 2n coefficients. For 0  j < n, let bi;j be the j th bit in the binary representation of the number i . An input to the proof system is a set of equations of the following form: n 1 2X

i D0

b

b

b

ci v1i;0 v2i;1 : : : vni;n1 D 0;

(7)

where all variables vi can take value 0 or 1, and addition is taken modulo 2. An b b b equation of the form (7) is said to be multilinear. A product ti D v1i;0 v2i;1 : : : vni;n1 , n for any 0  i  2  1, will be referred P to as a multilinear term or simply a term. The degree of ti , denoted deg.ti /, is 0j j ; among terms of the same degree, the order of index is decided lexicographically. Call the equations and create set B, initially empty. Repeat the following until is empty. Pick an equation e of that has the highest index, nonzero term. As long as there is an equation g in B whose highest nonzero term has the same index as the highest index nonzero term of e, replace e with e C g. If 0 D 0 is not produced, add e to B. This ensures B remains linearly independent. Create as many as n new equations by multiplying e by every variable and add those equations to that have never been in . This sets up the addition of e with all other equations in B. When is empty, all original equations have been replaced by equations with the same solution space and are such that no two of them have the same highest index nonzero term. t u

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Next some examples of inputs are shown and then two short derivations. The CNF clause .v1 _ v2 _ v3 / is represented by the equation v1 .1 C v2 /.1 C v3 / C v2 .1 C v3 / C v3 C 1 D 0; which may be rewritten v1 v2 v3 C v1 v2 C v1 v3 C v2 v3 C v1 C v2 C v3 C 1 D 0: The reader can verify this from the truth table for the clause. Negative literals in a clause are handled by replacing variable symbol v with .1 C v/. For example, the clause .:v1 _ v2 _ v3 / is represented by .1 C v1 /.1 C v2 /.1 C v3 / C v2 .1 C v3 / C v3 C 1 D 0; which reduces to v1 v2 v3 C v1 v2 C v1 v3 C v1 D 0:

(8)

As can be seen, just the expression of a clause introduces nonlinearities. However, this is not the case for some Boolean functions. For example, the exclusive-or formula v1 ˚ v2 ˚ v3 ˚ v4 is represented by v1 C v2 C v3 C v4 C 1 D 0: An equation representing a BDD (Sect. 5.9) can be written directly from the BDD as a sum of algebraic expressions constructed from paths to 1 because each path represents one or more rows of a truth table and the intersection of rows represented by any two paths is empty. Each expression is constructed incrementally while tracing a path as follows: when a 1 branch is encountered for variable v, multiply by v, and when a 0 branch is encountered for variable v, multiply by .1 C v/. Observe that for any truth assignment, at most one of the expressions has value 1. The equation corresponding to the BDD at the upper left in Fig. 28 is .1 C v1 /.1 C v2 /.1 C v3 / C .1 C v1 /v2 v3 C v1 .1 C v2 /v3 C v1 v2 C 1 D 0;

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which reduces to v1 C v2 C v3 C v1 v2 v3 D 0: Since there is a BDD for every Boolean function, this example illustrates the fact that a single equation can represent any complex function. It should be equally clear that a single equation addition may have the same effect as many resolution steps. Addition of equations and the Gaussian-elimination nature of algebraic proofs is illustrated by showing steps that solve the following simple formula: .v1 _ :v2 / ^ .v2 _ :v3 / ^ .v3 _ :v1 /:

(9)

The equations corresponding to (9) are expressed below as (1), (2), and (3). All equations following those equations are derived as stated on the right. Cv2

v1 v2

Cv3

=0 =0 =0

(1) (2) (3)

v1 v2 v3 +v2 v3 v1 v2 v3 Cv3 v1 v2 v3 Cv1 v3 v1 v2 v3 Cv1 v1 v2 v3 Cv1 v2 v1 v2 v3 Cv2 v1 Cv2 v1 Cv3

=0 =0 =0 =0 =0 =0 =0 =0

.4/ ( v3  .1/ .5/ ( .4/ C .2/ .6/ ( v1  .2/ .7/ ( .6/ C .3/ .8/ ( v2  .3/ .9/ ( .8/ C .1/ .10/ ( .9/ C .7/ .11/ ( .5/ C .7/

v2 v3 v1 v3

Cv1

The solution is given by the bottom two equations which state that v1 D v2 D v3 . If, say, the following two clauses are added to (9) .:v1 _ :v2 / ^ .v3 _ v1 /; the equation v1 C v2 C 1 D 0 could be derived. Adding this to (10) would give 1 D 0 which proves that no solution exists. Ensuring a derivation of reasonable length is difficult. One possibility is to limit derivations to equations of bounded degree where the degree of a term t, deg.t/, is defined in the proof of Theorem 12 and the degree of an equation is degree.e/ D maxfdeg.t/ W t is a nonzero term in eg. An example is Algorithm 17 of Fig. 43 which is adapted from [37]. In the algorithm terms are re-indexed as in Theorem 12. Then first non-zero(ei ) is used to determine the highest index of a nonzero term of ei . The function red uce.e/ is an explicit statement that says reduction rules 1 and 2 are applied as needed to produce a multilinear equation.

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Fig. 43 Simple algebraic algorithm for SAT

This section concludes by comparing equations and the algebraic method with BDDs and BDD operations. Consider an example taken from Sect. 5.9. Equations corresponding to f and c in Fig. 31 are f W v1 v3 C v2 C v1 v2 D 0 c W v2 v3 C v3 D 0: Multiply f by v2 v3 to get v2 v3 D 0 which adds with c to get v3 D 0, the inference that is missed by restrict.f; c/ in Fig. 31. The inference can be derived from BDDs by reversing the role of f and c as shown in Fig. 32. Consider what multiplying f by v2 v3 and adding to c means in the BDD world. The BDD representing v2 v3 , call it d , consists of two internal nodes v2 and v3 , a path to 0 following only 1 branches, and all other paths terminating at 1. Every path that terminates at 1 in f also terminates at 1 in d . Therefore, d ^ c can safely be added as a BDD as long as f remains. But it is easy to check that d ^ c is simply v3 D 0. The process used in the above example can be applied more generally. All that is needed is some way to create a best factor d from f and c. This is something a generalized cofactor, which is discussed in Sect. 5.9.4, can sometimes do. However, the result of finding a generalized cofactor depends on BDD variable ordering. For the ordering v1 < v2 < v3 , the generalized cofactor g D gcf.f; c/ turns out to be .v1 _ :v2 _ v3 / ^ .:v2 _ :v3 / which is different from d in the leading clause but

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is sufficient to derive the inference when conjoined with c. By the definition of gcf, since f ^ c D g ^ c, g may replace f – this is not the case for d above. Existentially quantifying v away from a BDD has a simple counterpart in algebra: just multiply two polynomials, one with restriction v D 1 and the other with restriction v D 0. For example, the BDD of Fig. 28 may be expressed as v1 v2 v3 C v1 v3 C v1 C 1 D 0: The equations under restrictions v2 D 1 and v2 D 0, respectively, are v1 C 1 D 0 and v1 v3 C v1 C 1 D 0: The result of existential quantification is .v1 C 1/.v1 v3 C v1 C 1/ D v1 C 1 D 0; which reveals the same inference. As with BDDs, this can be done only if the quantified variable is in no other equation. The counterpart to strengthening is just as straightforward. The BDDs of Fig. 39 have equation representations b2 W v1 v3 C v2 C v1 v2 D 0 b1 W v3 C v2 v3 D 0: Existentially quantify v1 away from b2 to get v2 v3 D 0 and add this to b1 to get v3 D 0.

5.12.2 Integer Programming An integer program models the problem of maximizing or minimizing a linear function subject to a system of linear constraints, where all n variables are integral: maximize or minimize c α

(10)

subject to Mα  b lα u ˛i integral; 1  i  n; where M is a constraint matrix, c is a linear objection function, b is a constant vector, and α is a variable vector. The integer programming problem and its relaxation to linear programming are very well studied, and a large body of techniques have been developed to assist in establishing an efficient solution to (10). They are divided into the categories of preprocessing and solving. However, an important third aspect concerns the matrix M that is used to model a given instance.

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Modeling is important because the effective solution of integer programs often entails the use of linear programming relaxations. A solution to such a relaxation generally provides a bound on the actual solution, and the relaxation of one formulation of the input may provide a tighter bound than another. Generally, the tighter the bound, the better. For example, consider two formulations of the pigeonhole problem. The pigeonhole problem is as follows: can n C 1 pigeons be placed in n holes so that no two pigeons are in the same hole? Define Boolean variables vi;j , 1  i  n, 1  j  n C 1 with the interpretation that vi;j will take the value 1 if and only if pigeon j is in hole i and otherwise will take value 0. The following equations, one per pigeon, express the requirement that every pigeon is to be assigned to a single hole: n X

vi;j D 1; 1  j  n C 1;

(11)

i D1

and the following inequalities express the requirement that two pigeons cannot occupy the same hole: vi;j C vi;k  1; 1  i  n; 1  j < k  n C 1:

(12)

There is no solution to this system of equations and inequalities. Relaxing integrality constraints, there is a solution at vi;j D 1=n for all i; j . If running the algorithm to be shown later, practically complete enumeration is necessary before finally it is determined that no solution exists [82]. However, the requirement that at most one pigeon is in a hole may alternatively be represented by nC1 X

vi;j  1; 1  i  n:

(13)

j D1

which may be used instead of (12). The new constraints are much tighter than (12) and the system (11) and (13) is easily solved [82]. The purpose of preprocessing is to reformulate a given integer program and tighten its linear programming relaxation. In the process it may eliminate redundant constraints and may even be able to discover unsatisfiability.

6

Algorithms for Easy Classes of CNF Formulas

For certain classes of CNF formulas, the satisfiability problem is known to be solved efficiently by specially designed algorithms. Some classes, such as the 2-SAT and Horn classes are quite important because they show up in real applications and others are quite interesting because results on these add to our understanding of

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Fig. 44 Unit resolution for CNF formulas

the structural properties that make formulas hard or easy. Such an understanding can help develop a search heuristic that will obtain a solution more efficiently. In particular, knowledge that a large subset of clauses of a given formula belongs to some easy class of formulas can help reduce the size of the search space needed to determine whether some partial truth assignment can be extended to a solution. Because CNF formulas are so rich in structure, much space in this section is devoted to a few special cases. Hopefully this will help to fully appreciate the possibilities. In particular, results on minimally unsatisfiable and nested formulas are greatly detailed. The reader may have the impression that the number of polynomial-time solvable classes is quite small due to the famous dichotomy theorem of Schaefer [121]. But this is not the case. Schaefer proposed a scheme for defining classes of propositional formulas with a generalized notion of clause. He proved that every class definable within his scheme was either N P -complete or polynomial-time solvable, and he gave criteria to determine which. But not all classes can be defined within his scheme. The class of Horn formulas can be but several others, which will be described in this section, including q-Horn, extended Horn, CC-balanced, and SLUR cannot be so defined. The reason is that Schaefer’s scheme is limited to classes that can be recognized in log space. Below, some of the more notable easy classes and algorithms for solving them are presented. A crucial component of many of these algorithms is unit resolution. An implementation is given in Fig. 44.

6.1

2-SAT

Every clause of a 2-SAT formula contains at most two literals. A given 2-SAT E formula may be solved efficiently by constructing the implication graph G

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Fig. 45 Algorithm for determining satisfiability of 2-CNF formulas

of (see Sect. 3.3) and traversing its vertices, ending either at a cycle containing complementary literals or with no additional vertices to explore. The algorithm of Fig. 45 implicitly does this. E . Unit resolution drives the exploration of strongly connected components of G The initial application of unit resolution, if necessary, is analogous to traversing all vertices reachable from the special vertex F . Choosing a literal l arbitrarily and temporarily assigning it the value 1 after unit resolution completes is analogous to starting a traversal of some strongly connected component with another round of unit resolution. If that round completes with an empty clause, a contradiction exists so l is set to 0,  and M are reset to what they were just before l was set to 1, and exploration resumes. If an empty clause is encountered before the entire component is visited (i.e., while there still exist unit clauses), then the formula is unsatisfiable. Otherwise, the value of l is made permanent and so are values that were given to other variables during traversal of the component. This process repeats until the formula is found to be unsatisfiable or all components have been explored. The variable named s keeps track of whether variable l has been given one value or two,

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the variable M 0 holds the temporary assignments to variables during traversal of a component, and the variables  0 and l 0 save the point to return to if a contradiction is found. This algorithm is adapted from [56]. The next two theorems are stated without proof. Theorem 13 On input CNF formula , Algorithm 2-SAT Solver outputs “unsatisfiable” if and only if is unsatisfiable, and if it outputs a set M , then M is a model for . t u Theorem 14 On input CNF formula containing m clauses and n variables, Algorithm 2-SAT Solver has O.m C n/ worst-case complexity. t u

6.2

Horn Formulas

A CNF formula is Horn if every clause in it has at most one positive literal. This class is widely studied, in part because of its close association with logic programming. To illustrate, a Horn clause .:v1 _ :v2 _ : : : _ :vi _ v/ is equivalent to the rule v1 ^ v2 ^ : : : ^ vi ! v or the implication v1 ! v2 ! : : : ! vi ! v. However, the notion of causality is generally lost when translating from rules to Horn formulas. The following states an important property of Horn formulas. Theorem 15 Every Horn formula has a unique minimum model. Proof Let be a Horn formula. Let M be a minimal model for , that is, a smallest subset of variables of value 1 that satisfies . Choose any v 2 M . Since M n fvg is not a model for , there must be a clause c 2 such that positive literal v 2 c. Since all other literals of c are negative and M is minimal, all assignments not containing v cannot satisfy c and therefore . It follows that all models other than M must have cardinality greater than jM j. Hence, M is a unique minimum model for . t u The satisfiability of Horn formulas can be determined in linear time using unit resolution [53, 78, 125]. One of several possible variants is shown in Fig. 46. Theorem 16 Given Horn formula as input, Algorithm Horn Solver outputs “unsatisfiable” if and only if is unsatisfiable, and if it outputs a set of variables M , then M is a unique minimum model for .

Proof When Algorithm Horn Solver completes with output set M , all remaining clauses have at least one negative literal. Since none of the remaining clauses are null and since v added to M serves to falsify negative literals, at least one of the remaining negative literals in a remaining clause has not been added to M and is therefore satisfied by M . Therefore, all remaining clauses are satisfied by M .

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Fig. 46 Algorithm for determining satisfiability of Horn formulas

A clause is removed when adding v to M only because it contains literal v. Therefore, all removed clauses are satisfied by M . Hence, M satisfies . Suppose M is not the unique minimum model for . Then, for some variable v, the assignment M n fvg satisfies . Variable v was added to M because some clause c 2 containing positive literal v became a unit clause after all variables associated with the negative literals of c were placed in M . But then M n fvg cannot satisfy c. Therefore, M is the unique minimum model. Now suppose the algorithm outputs unsatisfiable but there is a model for . Let M 0 be the unique minimum model. Run the algorithm until reaching the point at which an empty clause is generated. Let this happens on the i th iteration of the Repeat block and let 0 be the set of all clauses removed up to the test for an empty clause on the i first iteration. Let v be the last variable added to M and c be the unit clause from which it was obtained. Clearly, 0 is Horn and M n fvg is its unique minimum model. M 0 must contain all variables of M n fvg since it is the unique minimum model for . Therefore, it cannot contain fvg. But then c is not satisfied by M 0 , a contradiction. t u Algorithm Horn Solver is only useful if the input formula is known to be Horn. It is easy to see that this can be checked in linear time.

6.3

Renamable Horn Formulas

Given CNF formula and variable subset V 0 V , define swi tch. ; V 0 / to be the formula obtained from by reversing the polarity of all occurrences of v and :v in for all v 2 V 0 . If there exists a V 0 V such that swi tch. ; V 0 / is Horn, then is said to be renamable Horn or hidden Horn.

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Renamable Horn formulas can be recognized and solved in O.j j/ time [10, 104]. Algorithm SLUR on Page 410 solves renamable Horn formulas in linear time.

6.4

Linear Programming Relaxations

Let M be a .0 ˙ 1/ matrix representing a CNF formula . If M has a particular structure, it is possible to solve the inequalities (1) with non-integer constraints 0  ˛i  1 to obtain a solution to either directly or by rounding. Notable classes based on particular matrix structures are the extended Horn formulas and what in this chapter are called the CC-balanced formulas. The class of extended Horn formulas was introduced by Chandru and Hooker [31] who were looking for conditions under which a linear programming relaxation could be used to find solutions to propositional formulas. It is based on a theorem of Chandrasekaran [30] which characterizes sets of linear inequalities for which 0-1 solutions can always be found (if one exists) by rounding a real solution obtained using an LP relaxation. Extended Horn formulas can be expressed as linear inequalities that belong to this family of 0-1 problems. For the sake of clarity, an equivalent graph theoretic definition is presented here. A formula is in the class of extended Horn formulas if one can construct a rooted directed tree T , called an extended Horn tree, indexed on the variables of such that, for every clause c 2 : 1. All the positive literals of c are consecutive on a single path of T . 2. There is a partition of the negative literals of c into sets N1 ; N2 : : : ; Nnc , where nc is at least 1, but no greater than the number of negative literals of c, such that for all 1  i  nc , all the variables of Ni are consecutive on a single path of T . 3. For at most one i , the path in T associated with Ni begins at the vertex in T from which the path associated with positive literals begins. 4. For all remaining i , the path in T associated with Ni begins at the root of T . Disallowing negative paths that do not originate at the root (point 3 above) gives a subclass of extended Horn called simple extended Horn [133]. Extended Horn formulas can be solved in polynomial time by Algorithm SLUR on Page 410 [122]. Chandru and Hooker showed that unit resolution alone can determine whether or not a given extended Horn formula is satisfiable. This is due to the following two properties of an extended Horn formula: 1. If is extended Horn and has no unit clauses, then is satisfiable. 2. If is extended Horn and v is a variable in , then 1 D fc  fvg W c 2 ; :v … cg and 2 D fc  f:vg W c 2 ; v … cg are both extended Horn. Chandru and Hooker proposed an algorithm that finds a model for a satisfiable extended Horn formula. First, apply unit resolution, setting values of unassigned variables to 1/2 when no unit clauses remain. Then round the result by a matrix multiplication. Their algorithm cannot, however, be reliably applied unless it is

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Fig. 47 Algorithm for determining satisfiability of SLUR formulas

known that a given formula is extended Horn. Unfortunately, the problem of recognizing extended Horn formulas is not known to be solved in polynomial time. As will be shown later in this section, this problem has become moot since Algorithm SLUR solves extended Horn formulas in linear time without the need for recognition. The class of CC-balanced formulas has been studied by several researchers (see [38] for a detailed account of balanced matrices and a description of CCbalanced formulas). The motivation for this class is the question, for SAT, when do linear programming relaxations have integer solutions? A formula with .0 ˙ 1/ matrix representation M is CC-balanced if in every submatrix of M with exactly two nonzero entries per row and per column, the sum of the entries is a multiple of 4 (this definition is taken from [138]). Recognizing that a formula is CC-balanced takes linear time. However, the recognition problem is moot because Algorithm SLUR solves CC-balanced formulas in linear time without the need for recognition. As alluded to above, both extended Horn and CC-balanced formulas are subsets of a larger efficiently solved class of formulas solved by single lookahead unit resolution [122] (SLUR). The SLUR class is peculiar in that it is defined based on an algorithm rather than on properties of formulas. Algorithm SLUR of Fig. 47 selects variables sequentially and arbitrarily and considers a one-level lookahead,

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under unit resolution, of both possible values that the selected variable can take. If unit resolution does not result in an empty clause in one direction, the assignment corresponding to that value choice is made permanent and variable selection continues. If all clauses are satisfied after a value is assigned to a variable (and unit resolution is applied), the algorithm returns a satisfying assignment. If unit resolution, applied to the given formula or to both subformulas created from assigning values to the selected variable on the first iteration, results in a clause that is falsified, the algorithm reports that the formula is unsatisfiable. If unit resolution results in falsified clauses as a consequence of both assignments of values to the selected variable on any iteration except the first, the algorithm reports that it has given up. A formula is in the class SLUR if, for all possible sequences of selected variables, Algorithm SLUR does not give up on that formula. Observe that due to the definition of this class, the question of class recognition is avoided. The worst-case complexity of Algorithm SLUR, as written, is quadratic in the length of the input formula. The complexity is dominated by the execution of fc  fvg W c 2 ; :v … cg, fc  f:vg W c 2 ; v … cg, and the number of unit clauses eliminated by unit resolution. The total number of times a clause is checked and a literal removed due to the first two expressions is at most the number of literals existing in the given formula if the clauses are maintained in a linked list indexed on the literals. However, the same unit clause may be removed by unit resolution on successive iterations of the Repeat block of Algorithm SLUR since once branch of execution is always cut. This causes quadratic worst-case complexity. A simple modification to Algorithm SLUR brings the complexity down to linear time: run both calls of unit resolution simultaneously, alternating execution of their Repeat blocks. When one terminates without an empty clause in its output formula, abandon the other call. Theorem 17 Algorithm SLUR has O.j j/ worst-case complexity if both calls to unit resolution are applied simultaneously and one call is immediately abandoned if the other finishes first without falsifying a clause. Proof For reasons mentioned above, only the number of steps used by unit resolution is considered. The number of times a literal from a unit clause is chosen and satisfied clauses and falsified literals removed in non-abandoned calls of unit resolution is O.j j/ since no literal is chosen twice. Since the Repeat blocks of abandoned and non-abandoned calls alternate, the time used by abandoned calls is no greater than that used by non-abandoned ones. Thus, the worst-case complexity of Algorithm SLUR, with the interleave modification, is O.j j/. t u All Horn, renamable Horn, extended Horn, and CC-balanced formulas are in the class SLUR. Thus, an important outcome of the results on SLUR is the observation that no special preprocessing or testing is needed for some of the special polynomial-time solvable classes of SAT when using a reasonable variant of the DPLL algorithm.

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A limitation of all the classes of this section is they do not represent many interesting unsatisfiable formulas. There are several possible extensions to the SLUR class which improve the situation. One is to add a 2-SAT solver to the unit resolution steps of Algorithm SLUR. This extension is at least able to handle all 2-SAT formulas which is something Algorithm SLUR cannot do. It can be elegantly incorporated due to the following observation: whenever Algorithm SLUR completes a sequence of unit resolutions and if at that time the remaining clauses are nothing but a subset of the original clauses (which they would have to be if all clauses have at most two literals), then effectively the algorithm can start all over. That is, if fixing of a variable to both values leads to an empty clause, then the formula has been proved to be unsatisfiable. Thus, one need not augment Algorithm SLUR by the 2-SAT algorithm, because the 2-SAT algorithm (at least one version of it) does exactly what the extended algorithm does. Another extension of Algorithm SLUR is to allow a polynomial number of backtracks, giving up if at least one branch of the search tree does not terminate at a leaf where a clause is falsified. This enables unsatisfiable formulas with short search trees to be solved efficiently by Algorithm SLUR.

6.5

q-Horn Formulas

This class of propositional formulas was developed in [22] and [23]. The class of q-Horn formulas may be characterized as a special case of maximum monotone decomposition of matrices [137, 138]. Express a CNF formula of m clauses and n variables as an m  n .0 ˙ 1/-matrix M. In the monotone decomposition of M, columns are scaled by 1 and the rows and columns are partitioned into submatrices as follows: ! A1 E ; D A2 where the submatrix A1 has at most one C1 entry per row; the submatrix D contains only 1 or 0 entries; the submatrix A2 has no restrictions other than the three values of 1, C1, and 0 for each entry; and the submatrix E has only 0 entries. If A1 is the largest possible over columns, then the decomposition is a maximum monotone decomposition. If the maximum monotone decomposition of M is such that A2 has no more than two nonzero entries per row, then the formula represented by M is q-Horn. Truemper [138] shows that a maximum monotone decomposition for a matrix associated with a q-Horn formula can be found in linear time (this is discussed in Sect. 5.10.1). Once a q-Horn formula is in its decomposed form, it can be solved in linear time by Algorithm q-Horn Solver of Fig. 48. Theorem 18 Given q-Horn formula , Algorithm q-Horn Solver outputs “unsatisfiable” if and only if is unsatisfiable, and if is satisfiable, then the output set M1 [ M2 is a model for .

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Fig. 48 Algorithm for determining satisfiability of q-Horn formulas

Proof Clearly, if Horn formula A1 is unsatisfiable, then so is . Suppose A2 is unsatisfiable after rows whose columns in D are removed because they are satisfied by M1 . Since M1 is a unique minimum model for A1 and no entries of D are +1, no remaining row of A2 can be satisfied by any model for A1 . Therefore, is unsatisfiable in this case. The set M1 [ M2 is a model for if it is output since M1 satisfies rows in A1 and M2 satisfies rows in A2 . t u An equivalent definition of q-Horn formulas comes from the following. Theorem 19 A CNF formula is q-Horn if and only if its satisfiability index is no greater than 1. Proof Let be a q-Horn formula with variable set V and suppose jV j D n. Let M be a monotone decomposition for . Let na be such that for 0  i < na , column i of M coincides with submatrices A1 and D and for na  i < n, column i coincides with submatrix A2 . Form the inequalities (2) from M . Assign value 1 to all ˛i , 0  i < na , and value 1/2 to all ˛i , na  i < n. This satisfies 0  ˛i  1 of system (2). Since rows of A1 have nonzero entries in columns 0 to na 1 only and at most one of those is +1, the maximum sum of the elements of the corresponding row of inequality (2) is 1. Since all rows of D and A2 have at most two nonzero entries in columns na to n  1 and no +1 entries in columns 0 to na  1, the sum of elements of a corresponding row of inequality (2) has maximum value 1 too. Thus, inequality (2) is satisfied with z D 1. Now, suppose has satisfiability index I which is no greater than 1. Choose values for all ˛i terms such that inequality (2) is satisfied for z D I . Let  be a permutation of the columns of M so that ˛i  ˛j if and only if i < j . Form M0 from M by permuting columns according to . Let nb be such that ˛i < 1=2 for 0  i < nb and 1=2  ˛i for nb  i < n. Scale columns 0 through nb  1 of M0 by 1. Then inequality (2), using M0 for M , is satisfied with z D I and

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1=2  ˛i for all 0  i < n. It follows that each row of M0 can have at most two +1 entries or else the elements of the corresponding row of inequality (2) sums to greater than 1. Consider all rows of M0 with two +1 entries and mark all columns that contain at least one of those entries. Let  be a permutation of the columns of M0 so that all marked columns have higher index than all unmarked columns. Form M00 from M0 by permuting columns according to  and permuting rows so that all rows with only 0 entries in marked columns are indexed lower than rows with at least one nonzero entry in a marked column. Let nc be such that columns nc to n1 in M00 are exactly the marked columns. The value of ˛i , nc  i < n, must be exactly 1/2 or else the elements of some row of inequality (2) must sum to greater than 1. It follows that the number of nonzero entries in columns nc to n  1 in any row of M00 must be at most two or else the elements of some row of the inequality sum to greater than 1. By construction of M00 , every row can have at most one +1 entry in columns 0 to nc  1. Hence, M00 is a monotone decomposition for . u t The following result from [23] is also interesting in light of Theorem 19. It is stated without proof. Theorem 20 The class of all formulas with a satisfiability index greater than 1 C 1=n , for any fixed  < 1, is NP-complete. t u There is an almost obvious polynomial-time solvable class larger than that of the q-Horn formulas: namely, the class of formulas which have a satisfiability index no greater than 1 C a ln.n/=n, where a is any positive constant. The M matrix for any formula in this class can be scaled by 1 and partitioned as follows: 0 @

A1 E B 1 D A B 2

1 A;

2

where submatrices A , A , E, and D have the properties required for q-Horn formulas and the number of columns in B 1 and B 2 is no greater than O.ln.n//. Satisfiability for such formulas can be determined in polynomial time by solving the q-Horn system obtained after substitution of each of 2O.ln.n// partial truth assignments to the variables of B 1 and B 2 . 1

6.6

2

Matched Formulas

The matched formulas have been considered in the literature (see [136]) but not extensively studied, probably because this seems to be a rather useless and small class of formulas. Our interest in matched formulas is to provide a basis of comparison with other, well-known, well-studied classes. Let G .V1 ; V2 ; E/ be the variableclause matching graph (see Sect. 3.5) for CNF formula , where V1 is the set of clause vertices and V2 is the set of variable vertices. A total matching with respect

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to V1 is a subset of edges E 0 E such that no two edges in E 0 share an endpoint but every vertex v 2 V1 is an endpoint for some edge in E 0 . Formula is a matched formula if its variable-clause matching graph has a total matching with respect to V1 . A matched formula is trivially satisfied: for each edge e 2 E 0 , assign the variable represented by the variable vertex in e a value that satisfies the clause represented by the clause vertex in e. The comparison with other classes is discussed in Sect. 6.12.

6.7

Generalized Matched Formulas

The class of matched formulas has been generalized by Szeider [135]. Let G .V1 ; V2 ; E/ be the variable-clause matching graph of as above. Let V10 V1 and V20 V2 and suppose the subgraph G 0 .V10 ; V20 ; E 0 / induced by V10 and V20 is complete: that is, for every pair of vertices v1 2 V10 and v2 2 V20 , there is an edge e 2 E 0 . Call such a subgraph a biclique. . Let Theorem 21 Let G .V1 ; V2 ; E/ be the variable-clause graph for fX1 ; X2 ; : : : ; Xr g be a collection of bicliques in G .V1 ; V2 ; E/ and suppose (1) the number of clause vertices in X1 is less than 2 raised to the number of variable vertices in Xi , 1  i  r; (2) every v1 2 V1 (representing a clause) is in some Xi , 1  i  r; and (3) every v2 2 V2 (representing a variable) is in at most one Xi , 1  i  r. Then is satisfiable. Proof Let V Xi be the variables represented by vertices of Xi and let C Xi be the clauses represented by the remaining vertices of Xi . Remove from all c 2 C Xi all literals not associated with variables of V Xi . Since Xi is complete, the number of literals in c is jV Xi j and the number of assignments to the variables of V Xi which are falsified by c is 1. By condition (1), the total number of falsifying assignments for C Xi is less than all possible assignments to V Xi ; hence, some assignment satisfies C Xi . By condition (3), V Xi \ V Xj D ;, i 6D j , so satisfying assignments for both C Xi and C Xj cannot conflict. By condition (2), every clause in is in some biclique and is therefore satisfied by some assignment. t u Clause width is typically fixed. In that case, a model for a formula satisfying the conditions of Theorem 21 can be found in time linear in the number of clauses of . If the bicliques of G are all single edges, then is a matched formula. Unfortunately, the problem of recognizing a generalized matched formula is N P -complete, even for 3-CNF formulas.

6.8

Nested and Extended Nested Satisfiability

The complexity of nested satisfiability, inspired by Lichtenstein’s theorem of planar satisfiability [105], has been studied in [91]. Index all variables in a CNF formula consecutively from 1 to n and let positive and negative literals take the index of

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their respective variables. A clause ci is said to straddle another clause cj if the index of a literal of cj is strictly between two indices of literals of ci . Two clauses are said to overlap if they straddle each other. A formula is said to be nested if no two clauses overlap. For example, the following formula is nested: .v6 _ :v7 _ v8 / ^ .v2 _ v4 / ^ .:v6 _ :v9 / ^ .v1 _ :v5 _ v10 /: The class of nested formulas is quite limited in size. A variable cannot show up in more than one clause of a nested formula where its index is strictly between the greatest and least of the clause: otherwise, two clauses overlap. Therefore, a nested formula of m clauses and n variables has at most 2m C n literals. Thus, no CNF formula consisting of k-literal clauses is a nested formula unless m=n < 1=.k  2/. This particular restriction will be understood better from the probabilistic perspective taken in Sect. 6.13. Despite this, the class of nested formulas is not contained in either of the SLUR or q-Horn classes as shown in Sect. 6.12. This adds credibility to the potential usefulness of the algorithm presented here. However, our main interest in nested formulas is due to an enlightening analysis and efficient dynamic programming solution which appears in [91] and is presented here. Strong dependencies between variables of nested formulas may be exploited for fast solutions. In a nested formula, if clause ci straddles clause cj , then cj does not straddle ci , and if clause ci straddles clause cj and clause cj straddles ck , then ci straddles ck . Thus, the straddling relation induces a partial order on the clauses of a nested formula. It follows that the clauses can be placed in a total ordering using a standard linear time algorithm for topologically sorting a partial order: in the total ordering, a given clause does not straddle any clauses following it. In the example above, if clauses are numbered c0 to c3 from left to right, c2 straddles c0 , c3 straddles c1 and c2 , and no other clause straddles any other, so these clauses are in the desired order already. Once clauses are topologically sorted into a total order, the satisfiability question may be solved in linear time by a dynamic programming approach where clauses are processed one at a time, in order. The idea is to maintain a partition of variables as a list of intervals such that all variables in any clause seen so far are in one interval. Associated with each interval are four D values that express the satisfiability of corresponding processed clauses under all possible assignments of values to the endpoints of the interval. As more clauses are considered, intervals join and D values are assigned. By introducing two extra variables, v0 and vnC1 and an extra clause .v0 _ vnC1 / which straddles all others and is the last one processed, there is one interval Œv0 ; vnC1 remaining at the end. A D value associated with that interval determines satisfiability for the given formula. What remains is to determine how to develop the interval list and associated D values incrementally. This task is made easy by the fact that a variable which appears in a clause c with index strictly between the highest and lowest indices of variables in c never appears in a following clause. An efficient algorithm for determining the satisfiability of nested formulas is shown in Fig. 49. The following lemma is needed to prove correctness. The actual dependence of h values on i is not shown to prevent the notation from going out of control. However, from the context, this dependence should be clear.

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Fig. 49 Algorithm for determining satisfiability of nested formulas

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Lemma 5 Assume, at the start of any iteration 0  i  m of the outer Repeat loop of Algorithm Nested Solver, that D D fDh0 ;h1 . ; /; Dh1 ;h2 . : /; : : : ; Dhk ;hkC1 . ; /g; where h0 D 0 and hkC1 D n C 1. Let i p;pCj Dfc

W c 2 fc0 ; : : : ; ci 1 g; hp  mi nI ndex.c/ < maxI ndex.c/  hpCj g;

for any 0  j  k  p C 1. Then the following hold: 1. At the start of iteration i of the outer Repeat loop, each variable in ci is the same as one of fvh0 ; vh1 ; : : : ; vhkC1 g, and for every clause c 2 fc0 ; : : : ; ci 1 g, there exists an 0  r  k such that all the variables of c have index between hr and hrC1 . Moreover, for every hr and hhC1 , there is at least one clause whose minimum indexed variable has index hr and whose maximum indexed variable has index hrC1 . 2. At the start of iteration i of the outer Repeat loop, Dhj ;hj C1 .s; t/, 0  j  k, i has value 1 if and only if j;j C1 is satisfiable with variable vhj set to value s and variable vhj C1 set to value t. 3. At the start of iteration 0  j < q  p of the main inner Repeat loop, i Ehp ;hpCj .s; t/ has value 1 if and only if p;pCj [ fflx W lx 2 ci ; x  hpCj gg is satisfiable with variable vhp set to value s and variable vhpCj set to value t. 4. At the start of iteration 0  j < q  p of the main inner Repeat loop, i Ghp ;hpCj .s; t/ has value 1 if and only if p;pCj [ fflx W lx 2 ci ; x  hpCj gg i is not satisfiable, but p;pCj is satisfiable with variable hp set to value s and variable vhpCj set to value t. Proof Consider point 1. At the start of iteration 0 of the outer Repeat loop, point 1 holds because all variables are in the set fh0 ; : : : ; hnC1 g. Suppose point 1 holds at the beginning of iteration i . As a result of the topological sort, ci cannot be straddled by any clause in fc0 : : : ci 1 g. If one variable of ci is indexed strictly between hr and hrC1 , by point 1, there is a clause in fc0 ; : : : ; ci 1 g whose variable indices are as high as hrC1 and as low as hr . But such a clause would straddle ci . Hence, for every variable v 2 ci , v 2 fvhp ; vhpC1 ; : : : ; vhq g. Since the last line of the outer Repeat loop replaces all Dhp ;hpC1 : : : Dhq1 ;q with Dhp ;hq , point 1 holds at the beginning of iteration i C 1 of the outer loop. Consider point 2. At the start of iteration i D 0, all defined variables are i Dj;j C1 . ; /, 0  j  n, and these have value 1. From the definition of p;j , 0 D ;. Thus, point 2 holds before iteration i D 0. Suppose point 2 holds j;j C1 at the start of iteration i . Since ci contains no variables indexed less than hp or greater than hq and all Dhr ;hrC1 . ; / are unchanged by the algorithm for r < p and r  q, then these D values are correct for iteration i C 1 (but the subscripts on h values change because there are fewer D variables on the next iteration). So, due to the short inner Repeat loop following the main inner Repeat loop, point 2 holds

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i at the start of iteration i C 1 if Ehp ;hq .s; t/ has value 1 if and only if p;q [ fci g is satisfiable with variable vhp set to value s and variable vhq set to value t. This is shown below, thus taking care of point 2. Assume, during iteration i of the outer loop and before the start of the main inner loop, that points 1 and 2 hold. Consider the iteration j D 0 of the main inner 0 loop. As above, ; D ;. Moreover, by point 1, flx W lx 2 c0 ; x  hp g D flhp g 0 so p;p [ fflhp gg D fflhp gg. There is no satisfying assignment for this set if the value of the highest and lowest indexed variables of c0 are opposite each other since the highest and lowest indexed variables are the same. Accordingly, Ep;p .s; t/ and Gp;p .s; t/ are set to 0 in the algorithm when s and t are of opposite value. 0 However, if s D t D 1 and if lhp D vhp , then p;p [ ffvhp gg is satisfiable. Accordingly, Ep;p .1; 1/ is set to 1 and Gp;p .1; 1/ is set to 0 in the algorithm. 0 Otherwise, if s D t D 1 and lhp D :vhp , then p;p [ ff:vhp gg is unsatisfiable. Accordingly, Ep;p .1; 1/ is set to 0 and Gp;p .1; 1/ is set to 1 in the algorithm. Similar reasoning applies to the case s D t D 0. Thus, points 3 and 4 hold for the case j D 0. Now consider iteration i of the outer loop and iteration j > 0 of the main inner loop. Assume, at the start of iteration j  1, that points 3 and 4 hold. Points 1 and 2 hold from before since no changes to these occur in the main inner loop. Let j 1 ci D flx W lx 2 ci ; x  p C j  1g be the subset of literals of ci that have index no greater than hpCj 1 . From Point 1, for every clause c 2 fc0 ; : : : ; ci 1 g, there exists a positive integer 0  r  k such that all variables of c have index between hr and hrC1 . It follows that

i p;pCj

j 1

[ fci

gD.

i p;pCj 1

j 1

[ fci

g/ [

i pCj 1;pCj ;

j 1

i i g/ \ pCj and . p;pCj 1 [ fci 1;pCj D ;. For the sake of visualization, define i i D and D 1 2 p;pCj 1 pCj 1;pCj . At most one variable, namely, vhpCj 1 , is common to both 1 and 2 . Hence, when vhpCj 1 is set to some value, the following j 1 j 1 i holds: both 1 [fci g and 2 are satisfiable if and only if p;pCj [fci g is satisfi-

able. Now consider

i p;pCj

j

[ fci g. Since every variable of ci has an index matching j

j 1

one of fhp ; hpC1 ; : : : ; hq g, ci n ci , either is the empty set or fvhpCj g or f:vhpCj g. j i Therefore, given vhpCj 1 , p;pCj 1 [ fci g is satisfiable if and only if either both j 1

j 1

[ fci g and 2 are satisfiable or 1 and 2 are satisfiable, 1 [ fci g is j 1 unsatisfiable, but 1 [ fci [ flg is satisfiable where l 2 fvhpCj ; :vhpCj g or flg D j 1 ;. But since l does not occur in 1 , 1 [fci g can be unsatisfiable with 1 and 1 [ j 1 [ flg satisfiable only for assignments which satisfy l. Then, by hypothesis, fci i i the association of the E and G variables with p;pCj 1 and pCj 1;pCj , and the assignments to E and G in the main inner loop of Algorithm 23, the values of E and G match points 3 and 4 for the j th iteration of the main inner loop. 1

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From point 3, upon completion of the main inner loop, Ehp ;hq .s; t/ has value 1 if and only if hi p ;hq [ fci g is satisfiable. This matches the hypothesis of point 2, thereby completing the proof that point 2 holds. t u Corollary 1 Algorithm Nested Solver correctly determines satisfiability for a given nested formula. Proof Algorithm Nested Solver determines is satisfiable if and only if D0;nC1 .1; 1/ has value 1. But, from Lemma 5, D0;nC1 .1; 1/ has value 1 if and only if [ fv0 ; vnC1 g is satisfiable and v0 and vnC1 are set to value 1. The corollary follows. t u The operation of Nested Solver is demonstrated by means of an example taken from [91]. Suppose the algorithm is applied to a nested formula containing the following clauses which are shown in proper order after the topological sort: .v1 _v2 /^.v2 _v3 /^.:v2 _:v3 /^.v3 _v4 /^.v3 _:v4 /^.:v3 _v4 /^.:v1 _v2 _v4 /: The following table shows the D values that have been computed prior to the start of iteration 6 of the outer loop of the algorithm. The columns show s; t values and the rows show variable intervals. D; .s; t /

0, 0

0, 1

1, 0

1, 1

1; 2 2; 3 3; 4

0 0 0

1 1 0

1 1 0

1 0 1

Such a table could result from the following set of processed clauses at the head of the topological order (and before c): processing clause c, using the initial values and recurrence relations for E and G variables given above, produces values as shown in the following tables. Ep;pCj .s; t /

0, 0

0, 1

1, 0

1, 1

1; 1 1; 2 1; 3 1; 4

1 0 1 0

0 1 0 0

0 0 1 0

0 1 0 1

Gp;pCj .s; t / 1; 1 1; 2 1; 3

0, 0 0 0 0

0, 1 0 0 0

1, 0 0 1 0

1, 1 1 0 1

The last line of the E table holds the new D values for the interval Œv1 ; v4 as shown by the following table.

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D; .s; t /

0, 0

0, 1

1, 0

1, 1

1; 4

0

0

0

1

Linear time is achieved by careful data structure design and from the fact that no more than 2m C n literals exist in a nested formula. Theorem 22 Algorithm Nested Solver has worst-case time complexity that is linear in the size of . u t The question of whether the variable indices of a given formula can, in linear time, be permuted to make the formula nested appears to be open. An extension to nested satisfiability has been proposed in [72]. The details are skipped. This extension can be recognized and solved in linear time. For details, the reader is referred to [72].

6.9

Linear Autark Formulas

This class is based on the notion of an autark assignment which was introduced in [111]. Repeating the definition from Page 393, an assignment to a set of variables is an autark assignment if all clauses that contain at least one of those variables are satisfied by the assignment. An autark assignment provides a means to partition the clauses into two groups: one that is satisfied by the autark assignment and one that is completely untouched by that assignment. Therefore, autark assignments provide a way to reduce a formula to one that is equivalent in satisfiability. The following shows how to find an autark assignment in polynomial time. Let CNF formula of m clauses and n variables be represented as a .0 ˙ 1/ matrix M . Let α be an n-dimensional real vector with components ˛1 ; ˛2 ; : : : ; ˛n . Consider the following system of inequalities: M α  0;

(14)

α 6D 0:

Theorem 23 ([145]) A solution to (14) implies an autark assignment for

.

Proof Create the autark assignment as follows: if ˛i < 0 then assign vi D 0, and if ˛i > 0 then assign vi D 1, if ˛i D 0 then keep vi unassigned. It is shown that either a clause is satisfied by this assignment or it contains only unassigned variables. For every clause, by (1), it is necessary to satisfy the following: a1 v1 C a2 v2 C : : : C an vn  1  b;

(15)

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where ai factors are 0, 1, or +1 and the number of negative ai terms is b. Suppose α is a solution to (14). Write an inequality of (14) as follows:

a1 ˛1 C a2 ˛2 C : : : C an ˛n  0:

(16)

Suppose at least one term, say, ai ˛i , is positive. If ˛i is positive, then ai D 1 so, with vi D 1 and since there are at most b terms of value 1, the left side of (15) must be at least 1  b. On the other hand, if ˛i is negative, then ai D 1, vi D 0, and the product ai vi in (15) is 0. Since there are b  1 negative factors left in (15), the left side must be at least 1  b. Therefore, in either case, inequalities of the form (16) with at least one positive term represent clauses that are satisfied by the assignment corresponding to α. Now consider the case where no terms in (16) are positive. Since the left side of (16) is at least 0, there can be no negative terms either. Therefore, all the variables for the represented clause are unassigned.  A formula for which the only solution to ( 14) is α = 0 is said to be linear autarky-free. System (14) is an instance of linear programming and can therefore be solved in polynomial time. Hence, an autark assignment, if one exists, can be found in polynomial time. Algorithm LinAut of Fig. 50 is the central topic of study in this section and it will be used to define a polynomial-time solvable class of formulas called linear autarky formulas. The algorithm repeatedly applies an autark assignment as long as one can be found by solving (14). In a departure from [145], the algorithm begins with a call to Unit Resolution to eliminate all unit clauses. This is done to remove some awkwardness in the description of linear autarky formulas that will become clear shortly. The following states an important property of Algorithm LinAut. Lemma 6 Let be a CNF formula that is input to Algorithm LinAut and let 0 be the formula that is output. If 0 D ; then is satisfiable and αS transforms to a satisfying assignment for . Proof An autark assignment t to 0 induces a partition of clauses of 0 into those that are satisfied by t and those that are untouched by t. Due to the independence of the latter group, a satisfying assignment for that group can be combined with the autark assignment for the former group to satisfy 0 . If there is no satisfying assignment for either group, then 0 cannot be satisfiable. Therefore, since each iteration finds and applies an autark assignment (via α), if 0 D ; then the composition of the autark assignments of each iteration is a satisfying assignment for . t u The algorithm has polynomial-time complexity. In some cases it solves its input formula.

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Fig. 50 Repeated decomposition of a formula using autark assignments. VP is the set of variables whose values are set in partial assignment P . Output real vector αS can be transformed to a truth assignment as described in the proof of Theorem 23

Theorem 24 Let be Horn or renamable Horn. If is satisfiable, then LinAut returns an αS that transforms to a solution of as described in the proof of Theorem 23, and the formula returned by LinAut is ;. If is unsatisfiable, LinAut returns “unsatisfiable.”

Proof Horn or renamable Horn formula is unsatisfiable only if there is at least one positive unit clause in . In this case Unit Resolution. / will output a formula containing ; and LinAut will output unsatisfiable. Otherwise, Unit Resolution. / outputs a Horn or renamable Horn formula 0 with no unit clauses and a partial assignment P which is recorded in αS . In the next paragraph, it will be shown that any such Horn or renamable Horn formula is not linear autarky-free, so there exists an autark assignment for it. By definition, the clauses that remain after the autark assignment is applied must be Horn or renamable Horn. Therefore, the Repeat loop of LinAut must continue until there are no clauses left. Then, by Lemma 6, αS transforms to a satisfying assignment for . Now it is shown that there is always an α 6D 0 that solves (14). Observe that any Horn formula without unit clauses has at least one negative literal in every clause. Therefore, some all-negative vector α of equal components solves (14). In the case of renamable Horn, the polarity of α components in the switch set is reversed to get the same result. Therefore, there is always at least one autark assignment for a Horn or renamable Horn formula. Note that Algorithm LinAut may find an α that does not zero out all rows, so the Repeat loop may have more than 1 iteration, but since

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the reduced formula is Horn or renamable Horn, there is always an α 6D 0 that solves (14) for the clauses that are left. t u The following Horn formula would violate Theorem 24 if the call to Unit Resolution had not been added to LinAut: .v1 / ^ .v2 / ^ .v3 / ^ .v1 _ :v2 _ :v3 / ^ .:v1 _ v2 _ :v3 / ^ .:v1 _ :v2 _ v3 /: This formula is satisfied with variables set to 1 but is linear autarky-free. Theorem 25 If is a 2-SAT formula that is satisfiable, then LinAut outputs an αS that represents a solution of and the formula output by LinAut is ;. Proof In this case Unit Resolution. / outputs and causes no change to αS . The only nonzero autark solutions to M α  0 have the following clausal interpretation: choose a literal in a clause and assign its variable a value that satisfies the clause; for all non-satisfied clauses that contain the negation of that literal, assign a value to the variable of the clause’s other literal that satisfies that clause; continue the process until some clause is falsified or no variable is forced to be assigned a value to satisfy a clause. This is one iteration of Algorithm 2-SAT Solver, so since is satisfiable, the process never ends in a falsified clause. Since what is left is a satisfiable 2-SAT formula, this step is repeated until ; is output. As in the proof of Theorem 24, αS transforms to a satisfying assignment for . t u If is an unsatisfiable 2-SAT formula, then LinAut. / will output an unsatisfiable 2-SAT formula. The class of linear autarky formulas is the set of CNF formulas on which the application of LinAut either outputs unsatisfiable or a formula that is ; or a linear autarky-free 2-SAT formula. In the middle case, the input formula is satisfiable; in the other two cases, it is unsatisfiable. Observe that in the last case, it is unnecessary to solve the remaining 2-SAT formula. Theorem 26 All q-Horn formulas are linear autark formulas. Proof Apply LinAut to q-Horn formula . Then a linear autarky-free q-Horn formula is output. Assume that all clauses in the output formula have at least 2 literals – this can be done because all clauses of 0 entering the Repeat loop for the first time have at least 2 literals and any subsequent autark assignment would not introduce a clause that is not already in 0 . The rows and columns of M 0 representing the output q-Horn formula 0 may be permuted, and columns may be scaled by 1 to get a form as shown on Page 389 where A1 represents a Horn formula, D is nonpositive, and A2 represents a 2-SAT formula. Construct vector α as follows. Assign equal negative values to all ˛i where i is the index of a column through A1 (reverse the value if the column had been scaled by 1) and 0 to all other ˛i . Then, since there are at least as many positive as negative entries in every

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row through A1 , the product of any of those rows, unscaled, and α is greater than 0. Since there are only negative entries in rows through D and all the entries through columns of A2 multiply by ˛i values that are 0, the product of any of the rows through D, unscaled, and α is also positive. Therefore, α shows that 0 is not linear autarky-free, a contradiction. It follows that A1 D D D ; and the output formula is either 2-SAT or ;. If it is 2-SAT, it must be unsatisfiable as argued in Theorem 25. t u The relationship between linear autarky-free formulas and the satisfiability index provides a polynomial-time test for the satisfiability of a given CNF formula. Theorem 27 If the shortest clause of a CNF formula has width k and if the satisfiability index of is less than k=2, then is satisfiable. Proof This is seen more clearly by transforming variables as follows. Define real n-dimensional vector β with components ˇi D 2˛i  1 for all 0  i < n: The satisfiability index ( 2) for the transformed variables may be expressed, by simple substitution, as follows: M β  2Z  l ;

(17)

where l is an n-dimensional vector expressing the number of literals in all clauses and Z D hz; z; : : : ; zi. The minimum z that satisfies (17) is the satisfiability index of . Apply Algorithm LinAut to which, by hypothesis, has shortest clause of width k. The algorithm removes rows, so z can only decrease and k can only increase. Therefore, if z < k=2 for , it also holds for the 0 that is output by LinAut. Suppose 0 6D ;. Formula 0 is linear autarky-free so the only solution to 0  M 0 β is β = 0 which implies 0  2Z  l . Since the shortest clause of 0 is at least k, it follows that k=2  z. This contradicts the hypothesis that z < k=2. Therefore, 0 D ; and, by Lemma 6, the input formula is satisfiable. t u The effectiveness of this test on random k-CNF formulas9 will be discussed in Sect. 6.13.

6.10

Minimally Unsatisfiable Formulas

An unsatisfiable CNF formula is minimally unsatisfiable if removing any clause results in a satisfiable formula. The class of minimally unsatisfiable formulas is

9

Random k-SAT formulas are defined in Sect. 7, Page 440

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easily solved if the number of clauses exceeds the number of variables by a fixed positive constant k. This difference is called the formula’s deficiency. This section begins with a discussion of the special case where deficiency is 1, then considers the case of any fixed deficiency greater than 1. The discussion includes some useful properties which lead to an efficient solver for this case and helps explain the difficulty of resolution methods for many CNF formulas. The following result was proved in one form or another by several people (e.g., [5, 99]). A simple argument due to Truemper is presented. Theorem 28 If is a minimally unsatisfiable CNF formula with jV j D n variables, then the number of clauses in must be at least n C 1. Proof Let be a minimally unsatisfiable CNF formula with n C 1 clauses. Suppose jV j  n C 1. Let G .V1 ; V2 ; E/ be the variable-clause matching graph for (see Sect. 3.5) where V1 is the set of clause vertices and V2 is the set of variable vertices. There is no total matching with respect to V1 since this would imply is satisfiable, a contradiction. Therefore, by Hall’s Theorem [70] (stated as Theorem 36 in Sect. 6.13), there is a subset V 0 V1 with neighborhood smaller than jV 0 j. Let V 0 be such a subset of maximum cardinality. Define 1 to be the CNF formula consisting of the clauses of corresponding to the neighborhood of V 0 . By the minimality property of , 1 must be satisfiable. Delete from the clauses of 1 and from the remaining clauses all variables occurring in 1 . Call the resulting CNF formula 2 . There must be a matching of the clauses of 2 into the variables of 2 since otherwise V 0 and therefore 1 were not maximal in size. Hence, 2 is satisfiable. But if 1 and 2 are satisfiable, then so is , a contraction. t u Most of the remaining ideas of this section have been inspired by Oliver Kullmann [96]. A saturated minimally unsatisfiable formula is a minimally unsatisfiable formula such that adding any literal l, existing in a clause of , to any clause of not already containing l or :l results in a satisfiable formula. The importance of saturated minimally unsatisfiable formulas is grounded in the following lemma. Lemma 7 Let be a saturated minimally unsatisfiable formula and let l be a literal from . Then 0 D fc n f:lg W c 2 ; l … cg is minimally unsatisfiable. In other words, if satisfied clauses and falsified literals due to l taking value 1 are removed from , what is left is minimally unsatisfiable. Proof Formula 0 is unsatisfiable if is; otherwise, there is an assignment which sets l to 1 and satisfies . Suppose there is a clause c 2 0 such that 0 n fcg is unsatisfiable. Clause c must contain a literal l 0 that is neither l nor :l. Construct 00 by adding l 0 to all clauses of containing :l. Since is saturated, there is a truth assignment M satisfying 00 and therefore 0 with literals l 0 added as

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above. Assignment M must set l 0 to 1; otherwise, 0 n fcg is satisfiable. Then, adjusting M so the value of l is 0 gives an assignment which also satisfies 0 , a contradiction. t u The following is a simple observation that is needed to prove Theorem 29 below. Lemma 8 Every variable of a minimally unsatisfiable formula occurs positively and negatively in the formula. Proof Let be any minimally unsatisfiable formula. Suppose there is a positive literal l such that 9c 2 W l 2 c and 8c 2 ; :l … c. Let 1 D fc W c 2 ; l … cg. Let 2 D n 1 denote the clauses of which contain literal l. Clearly, j 2 j > 0. Then, by definition of minimal unsatisfiability, 1 is satisfiable by some truth assignment M . Hence, M [ flg satisfies 1 [ 2 D . This contradicts the hypothesis that is unsatisfiable. A similar argument applies if literal l is negative, proving the lemma. t u Theorem 29 Let be a minimally unsatisfiable formula with n > 0 variables, and n C 1 clauses. Then there exists a variable v 2 V such that the literal v occurs exactly one time in and the literal :v occurs exactly one time in . Proof If is not saturated, there is some set of literals already in that may be added to clauses in to make it saturated. Doing so does not change the number of variables and clauses of . So suppose from now on that is a saturated minimally unsatisfiable formula with n variables and n C 1 clauses. Choose a variable v such that the sum of the number of occurrences of literal v and literal :v in is minimum. By Lemma 8, the number of occurrences of literal v in is at least one and the number of occurrences of literal :v in is at least one. By Lemma 7, 1

D fc n f:vg W c 2 ; v … cg and

2

D fc n fvg W c 2 ; :v … cg

are minimally unsatisfiable. Clearly, jV 1 j D jV 2 j D n  1 or else the minimality of variable v is violated. By Theorem 28, the fact that 1 and 2 are minimally unsatisfiable and the fact that j i j < j j, i 2 f1; 2g, it follows that j 1 j D j 2 j D n. Therefore, the number of occurrences of literal v in is one and the number of occurrences of literal :v in is one. t u Lemma 9 Let be a minimally unsatisfiable CNF formula with n > 0 variables and n C 1 clauses. Let v be the variable of Theorem 29 and define clauses cv and c:v such that literal v 2 cv and literal :v 2 c:v . Then there is no variable which appears as a positive literal in one of cv or c:v and as a negative literal in the other. Proof Suppose there is a variable w such that literal w 2 cv and literal :w 2 cvN . By the minimality of and the fact that variable v appears only in cv and c:v , there exists an assignment M which excludes setting a value for variable v and

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Fig. 51 Algorithm for determining whether a CNF formula with one more clause than variable is minimally unsatisfiable

satisfies n fcv ; c:v g. But M must also satisfy either cv , if w has value 1, or c:v if w has value 0. In the former case M [ f:vg satisfies and in the latter case M [ fvg satisfies , a contradiction. The argument can be generalized to prove the lemma. t u The next series of results shows the structure of minimally unsatisfiable formulas and how to exploit that structure for fast solutions. Theorem 30 Let be a minimally unsatisfiable formula with n > 1 variables and n C 1 clauses. Let v be the variable of Theorem 29, let cv be the clause of containing the literal v, and let c:v be the clause of containing the literal :v. Then 0 D n fcv ; c:v g [ fRccv:v g is a minimally unsatisfiable formula with n  1 variables and n clauses. Proof By Lemma 9 the resolvent Rccv:v of cv and c:v exists. Moreover, Rccv:v ¤ ; if n > 1 or else is not minimally unsatisfiable. Therefore, 0 is unsatisfiable, contains n  1 variables, and has n nonempty clauses. Remove a clause c from n fcv ; c:v g. By the minimality of , there is an assignment M which satisfies n fcg. By Lemma 2, M also satisfies n fc; cv ; c:v g [ fRccv:v g. Hence, M satisfies 0 n fcg. Now remove cv and c:v from . Again, by the minimality of , there is an assignment M which satisfies n fcv ; c:v g and hence 0 n fRccv:v g. Thus, removing any clause from 0 results in a satisfiable formula and the theorem is proved. t u Theorem 30 implies the correctness of the polynomial-time algorithm of Fig. 51 for determining whether a CNF formula with one more clause than variable is

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minimally unsatisfiable. If the output is “munsat(1)” the input formula is minimally unsatisfiable, with clause-variable difference of 1; otherwise, it is not. The following results provide an interesting and useful characterization of minimally unsatisfiable formulas of the type discussed here. Theorem 31 Let be a minimally unsatisfiable formula with n > 1 variables and n C 1 clauses. Then there exists a variable v, occurring once as a positive literal and once as a negative literal, and a partition of the clauses of into two disjoint sets v and :v such that literal v only occurs in clauses of v , literal :v only occurs in clauses of :v , and no variable other than v that is in v is also in :v and no variable other than v that is in :v is also in v . Proof By induction on the number of variables. The hypothesis clearly holds for minimally unsatisfiable formulas of one variable, all of which have the form .v/ ^ .:v/. Suppose the hypothesis is true for all minimally unsatisfiable CNF formulas containing k > 1 or fewer variables and having deficiency 1. Let be a minimally unsatisfiable CNF formula with k C1 variables and k C2 clauses. From Theorem 29 there is a variable v in such that literal v occurs in exactly one clause, say, cv , and literal :v occurs in exactly one clause, say, c:v . By Lemma 9 the resolvent Rccv:v of cv and c:v exists and Rccv:v 6D ; or else is not minimally unsatisfiable. Let 0 D . n fcv ; c:v g/ [ fRccv:v g. That is, 0 is obtained from by resolving cv and c:v on variable v. By Theorem 30 0 is minimally unsatisfiable with k variables and k C 1 clauses. Then, by the induction hypothesis, there is a partition v00 and 0 0 and a variable v0 such that literal v0 occurs only in one clause :v0 of clauses of 0 0 0 0 of v0 , literal :v occurs only in one clause of :v 0 , and, excluding v , there is 0 0 cv 0 no variable overlap between v0 and :v0 . Suppose Rc:v 2 :v0 . Then v00 and 0 cv . :v 0 n fRc:v g/ [ fcv ; c:v g (denoted v and :v , respectively) form a nonvariable overlapping partition of clauses of (excluding v0 and :v0 ), literal v0 occurs 0 once in a clause of v , and literal :v occurs once in a clause of :v . A similar statement holds if Rccv:v 2 v00 . Therefore, the hypothesis holds for or any other minimally unsatisfiable formula of k C 1 variables and k C 2 clauses. The theorem follows. t u Theorem 32 A CNF formula with n variables and n C 1 clauses is minimally unsatisfiable if and only if there is a refutation tree for in which every clause of labels a leaf exactly one time, every variable of labels exactly two edges (once as a positive literal and once as a negative literal), and every edge label of the tree appears in at least one clause of . Proof ( ) By induction on the size of the refutation tree. Suppose there is such a refutation tree T for . If T consists of two edges, they must be labeled v and :v for some variable v. In addition, each clause labeling a leaf of T must consist of one literal that is opposite to that labeling the edge the leaf is the endpoint of. Hence, must be .v/ ^ .:v/ which is minimally unsatisfiable.

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Now suppose the theorem holds for refutation trees of k > 2 or fewer edges and suppose T has k C1 edges. Let v be the variable associated with the root of T , and let v and :v be the sets of clauses labeling leaves in the subtree joined to the root edges labeled v and :v, respectively. Define 1 D fc n f:vg W c 2 v g and 2 D fc n fvg W c 2 :v g. It is straightforward to see that the relationships between each refutation subtree rooted at a child of the root of T and 1 and 2 are as described in the statement of the theorem. Hence, by the induction hypothesis, 1 and 2 are minimally unsatisfiable. Let c be a clause in 1 . Let M1 be an assignment to variables of 1 satisfying 1 nfcg. Let M2 D M1 [fvg. Clearly, M2 satisfies v as well as all clauses of :v which contain literal v. The remaining clauses of :v are a proper subset of clauses of 2 and are satisfied by some truth assignment M3 to variables of 2 since 2 is minimally unsatisfiable. Since the variables of 1 and 2 do not overlap, M2 [M3 is an assignment satisfying vN [ v nfcg and therefore nfcg. The same result is obtained if c is removed from 2 . Thus, is minimally unsatisfiable. (!) Build the refutation tree in conjunction with running Algorithm Min Unsat Solver as follows. Before running the algorithm, construct n C 1 (leaf) nodes and distinctly label each with a clause of . While running the algorithm, add a new (non-leaf) node each time a resolvent is computed. Unlike the case for a normal refutation tree, label the new node with the resolvent. Construct two edges from the new node to the two nodes which are labeled by the two clauses being resolved (cv and c:v in the algorithm). If the pivot variable is v, label the edge incident to the node labeled by the clause containing v (alternatively :v) :v (v, respectively). Continue until a single tree containing all the original leaf nodes is formed. The graph constructed has all the properties of the refutation tree as stated in the theorem. It must include one or more trees since each clause, original or resolvent, is used one time in computing a resolvent. Since two edges are added for each new non-leaf node and n new non-leaf nodes are added, there must be 2n C 1 nodes and 2n edges in the structure. Hence, it must be a single tree. Any clause labeling a node contains all literals in clauses labeling leaves beneath that node minus all literals of pivot variables beneath and including that node. In addition, the label of the root is ;. Therefore, all literals of a clause labeling a leaf are a subset of the complements of edge labels on a path from the root to that leaf. Obviously, the complement of each edge label appears at least once in leaf clauses. t u An example of such a minimally unsatisfiable formula and corresponding refutation tree is shown in Fig. 52. Now, attention is turned to the case of minimally unsatisfiable formulas with deficiency ı > 1. Algorithms for recognizing and solving minimally unsatisfiable formulas in time nO.k/ were first presented in [98] and [58]. The fixed-parameter tractable algorithm that is presented in [134] is discussed here. Let be a CNF formula with m clauses and n variables. The deficiency of is the difference m  n. Each subformula of has its own deficiency, namely, the difference between the number of clauses of the subformula and the number of variables it contains. The maximum deficiency of is then the maximum of the deficiencies of all its subformulas. If, for every nonempty subset V 0 of variables

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¬v1 ¬v2

v1 ¬v3

v2

(v1 ∨ v2 )

(¬v2 ) ¬v6

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¬v4

v6

v4

v3 ¬v5

v5

(¬v1 ∨ ¬v4 ) (¬v1 ∨ v5 ) (¬v3 ∨ ¬v5 )

(v3 ∨ v4 ∨ v6 ) (¬v1 ∨ v4 ∨ ¬v6 )

= (v1 ∨ v2 ) ∧ (¬v2 ) ∧ (v3 ∨ v4 ∨ v6 ) ∧ (¬v1 ∨ v4 ∨ ¬v6 ) ∧ (¬v1 ∨ ¬v4 ) ∧ (¬v1 ∨ v5 ) ∧ (¬v3 ∨ ¬v5 ) Fig. 52 A minimally unsatisfiable CNF formula of six variables and seven clauses and a corresponding refutation tree. Edge labels are variable assignments (e.g., :v1 means v1 has value 0). Each leaf is labeled with a clause that is falsified by the assignment indicated by the path from the root to the leaf

of , there are at least jV 0 j C q clauses C such that some variable of V 0 is contained in C , then is said to be q-expanding. The following four lemmas are stated without proof. Lemma 10 The maximum deficiency of a formula can be determined in polynomial time. t u Lemma 11 Let be a minimally unsatisfiable CNF formula which has ı more clauses than variables. Then the maximum deficiency of is ı. t u Lemma 12 Let be a minimally unsatisfiable formula. Then for every nonempty set V 0 of variables of , there is at least one clause c 2 such that some variable of V 0 occurs in c. t u Lemma 13 Let be a CNF formula with maximum deficiency . A maximum matching of the bipartite variable-clause graph G of does not cover  clause vertices. t u Lemma 13 is important because it is used in the next lemma and because it shows that the maximum deficiency of a CNF formula can be computed with a O.n3 / matching algorithm.

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Lemma 14 Let be a 1-expanding CNF formula with maximum deficiency ı. Let 0 be a subformula of . Then the maximum deficiency of 0 is less than ı. Proof It is necessary to show that for any subset 0 , if the number of variables contained in 0 is n0 , then j 0 j  n0 < ı. Let G be the bipartite variable-clause graph of . In what follows symbols will be used to represent clause vertices in G and clauses in interchangeably. Choose a clause c 2 n 0 . Let MG be a maximum matching on G that does not cover vertex c. There is always one such matching because, by hypothesis, every subset of variable vertices has a neighborhood which is larger than the subset, and this allows c’s cover to be moved to another clause vertex, if necessary. Let C be the set of all clause vertices of G that are not covered by MG . By Lemma 13 jC j D ı so since c … 0 , jC \ 0 j < ı. Since MG matches every clause vertex in 0 n C to a variable that is in 0 , the number of clauses in 0 n C must be no bigger than n0 . Therefore, j 0 j  n0  j 0 j  j 0 n C j. But j 0 j  j 0 n C j is the number of clauses in C \ 0 which is less than ı. t u Theorem 33 ([134]) Let be a CNF formula with maximum deficiency ı. The satisfiability of can be determined in time O.2ı n3 / where n is the number of variables in . Proof Let m be the number of clauses in . By hypothesis, m  n C ı. Let G D .V1 ; V2 ; E/ be the variable-clause matching graph for . Find a maximum matching MG for G. Since nm  n.nCı/ D O.n2 /, this can be done in O.n3 / time by the well-known Hopcroft-Karp maximum cardinality matching algorithm for bipartite graphs [40]. The next step is to build a refutation tree for of depth ı. u t Theorem 34 ([134]) Let be a minimally unsatisfiable CNF formula with ı more clauses than variables. Then can be recognized as such in time O.2ı n4 / where n is the number of variables in . Proof By Lemma 11 the maximum deficiency of is ı which, by Lemma 13, can be checked in O.n3 / time. Since, by Lemma 14, the removal of a clause from reduces the maximum deficiency of , the algorithm inferred by Theorem 33 may be used to check that, for each c 2 , n fcg is satisfiable. The algorithm may also be used to check that is unsatisfiable. Since the algorithm is applied m C 1 times and m D n C ı, the complexity of this check is O.2ı n4 /. Therefore, the entire check takes O.2ı n4 / time. t u

6.11

Bounded Resolvent Length Resolution

This class is considered here because it is a counterpoint to minimally unsatisfiable formulas. A CNF formula is k-BRLR if Algorithm BRLR of Fig. 53 either generates all resolvents of or returns unsatisfiable.

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Fig. 53 Finding a k-bounded refutation

Algorithm BRLR implements a simplified form of k-closure [144]. It repeatedly applies the resolution rule to a CNF formula with the restriction that all resolvents are of size no greater than some fixed k. In other words, the algorithm finds what are sometimes called “k-bounded” refutations. For a given CNF formula with n variables and m clauses, the worst-case number of resolution steps required by the algorithm is k X i D1

2

i

! ! n k n D2 .1 C O.k=n//: i k

This essentially reflects the product of the cost of finding a resolvent and the maxi  mum number of times a resolvent is generated. The latter is 2k nk . The cost of finding a resolvent depends on the data structures used in implementing the algorithm. For every clause, maintain a linked list of literals it contains, in order by index, and a linked list of possible clauses to resolve with such that the resolvent has no  greater than k   literals. Maintain a list T of matrices of dimension n  2; n2  4; : : : ; kn  2k such that each cell has value 1 if and only if a corresponding original clause of or resolvent exists at any particular iteration of the algorithm. Also, maintain a list of clauses that may resolve with at least one other clause to generate a new resolvent: the list is threaded through the clauses. Assume a clause can be accessed in constant time and a clause can access its corresponding cell in T in constant time by setting a link just one time as the clause is scanned the first time or created as a resolvent. Initially, set to 1 all the  cells of T which correspond to a clause of and set all other cells to 0. Over m2 pairs of clauses, use 2L literal comparisons to determine whether the pair resolves. If so and their resolvent has k or fewer literals and the cell in T corresponding to the resolvent has value 0, then set the cell to 1, add a link to the clause lists of the two clauses involved, and add to the potential resolvent

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list any of the two clauses that is not already threaded through it. This accounts for complexity O.Lm2 /. During an iteration, select a clause c1 from the potential resolvent list. Scan through c1 ’s resolvent list checking whether the cell of T corresponding to the other clause, c2 , has value 1. If so, delete c2 from c1 ’s list. If c1 ’s list is scanned without finding a cell of value 0, delete c1 from the potential resolvent list and try the next clause in the potential resolvent list. When some clause is paired with another having a cell of value 0 in T , a new resolvent is generated. In this case, construct a new clause and create a resolvent list for the clause by checking for resolvents with all existing clauses.

6.12

Comparison of Classes

This section provides some intuition about the relative size and scope of the easy classes discussed above. Observe that the SLUR, q-Horn, nested, and matched classes are incomparable; the class of q-Horn formulas without unit clauses is subsumed by the class of linear autark formulas and SLUR is incomparable with the linear autark formulas. All the other classes considered above are contained in one or more of the three. For example, Horn formulas are in the intersection of the q-Horn and SLUR classes and all 2-SAT formulas are q-Horn. Any Horn formula with more clauses than distinct variables is not matched, but is both SLUR and q-Horn. The following is a matched and q-Horn formula but is not a SLUR formula: .v1 _ :v2 _ v4 / ^ .v1 _ v2 _ v5 / ^ .:v1 _ :v3 _ v6 / ^ .:v1 _ v3 _ v7 /: In particular, in Algorithm SLUR, initially choosing 0 values for v4 , v5 , v6 , and v7 leaves an unsatisfiable formula with no unit clauses. To verify q-Horn membership, set ˛1 D ˛2 D ˛3 D 1=2 and the remaining ˛’s to 0 in (2). The following formula is matched and SLUR but is not q-Horn: .:v2 _ v3 _ :v5 / ^ .:v1 _ :v3 _ v4 / ^ .v1 _ v2 _ :v4 /: In particular, the satisfiability index of this formula is 4/3. To verify SLUR membership, observe that in Algorithm SLUR no choice sequence leads to an unsatisfiable formula without unit clauses. The following formula is nested but not q-Horn (minimum Z is 5/4): .:v3 _ :v4 / ^ .v3 _ v4 _ :v5 / ^ .:v1 _ v2 _ :v3 / ^ .v1 _ v3 _ v5 /: The following formula is nested but not SLUR (choose v3 first, use the branch where v3 is 0 to enter a situation where satisfaction is impossible):

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.v1 _ v2 / ^ .v1 _ :v2 _ v3 / ^ .:v1 _ v4 / ^ .:v1 _ :v4 /: The following is a linear autark formula that is not q-Horn (α = h0:5; 0:5; 0i and the minimum Z is 3/2): .v1 _ v2 _ v3 / ^ .:v1 _ :v2 _ :v3 /: The following is SLUR but is not a linear autark formula (by symmetry, any variable choice sequence in SLUR leads to the same result which is a satisfying assignment and only α = h0; 0; 0i satisfies the inequality of (14)): .:v1 _ v2 _ v3 / ^ .v1 _ :v2 _ v3 / ^ .v1 _ v2 _ :v3 / ^ .:v1 _ :v2 _ :v3 /: The following formula is minimally unsatisfiable with one more clause than variable but is not 3-BRLR: .v0 _ v1 _ v48 / ^ .v0 _ :v1 _ v56 / ^ .:v0 _ v2 _ v60 / ^ .:v0 _ :v2 _ v62 /^ .v3 _ v4 _ :v48 / ^ .v3 _ :v5 _ v56 / ^ .:v3 _ v5 _ v60 / ^ .:v3 _ :v5 _ v62 /^ .v6 _ v7 _ v49 / ^ .v6 _ :v7 _ :v56 / ^ .:v6 _ v8 _ v60 / ^ .:v6 _ :v8 _ v62 /^ .v9 _ v10 _ :v49 / ^ .v9 _ :v10 _ :v56 / ^ .:v9 _ v11 _ v60 / ^ .:v9 _ :v11 _ v62 /^ .v12 _ v13 _ v50 / ^ .v12 _ :v13 _ v57 / ^ .:v12 _ v14 _ :v60 / ^ .:v12 _ :v14 _ v62 /^ .v15 _ v16 _ :v50 / ^ .v15 _ :v16 _ v57 / ^ .:v15 _ v17 _ :v60 / ^ .:v15 _ :v17 _ v62 /^ .v18 _ v19 _ v51 / ^ .v18 _ :v19 _ :v57 / ^ .:v18 _ v20 _ :v60 / ^ .:v18 _ :v20 _ v62 /^ .v21 _ v22 _ :v51 / ^ .v21 _ :v22 _ :v57 / ^ .:v21 _ v23 _ :v60 / ^ .:v21 _ :v23 _ v62 /^ .v24 _ v25 _ v52 / ^ .v24 _ :v25 _ v58 / ^ .:v24 _ v26 _ v61 / ^ .:v24 _ :v26 _ :v62 /^ .v27 _ v28 _ :v52 / ^ .v27 _ :v28 _ v58 / ^ .:v27 _ v29 _ v61 / ^ .:v27 _ :v29 _ :v62 /^ .v30 _ v31 _ v53 / ^ .v30 _ :v31 _ :v58 / ^ .:v30 _ v32 _ v61 / ^ .:v30 _ :v32 _ :v62 /^ .v33 _ v34 _ :v53 / ^ .v33 _ :v34 _ :v58 / ^ .:v33 _ v35 _ v61 / ^ .:v33 _ :v35 _ :v62 /^ .v36 _ v37 _ v54 / ^ .v36 _ :v37 _ v59 / ^ .:v36 _ v38 _ :v61 / ^ .:v36 _ :v38 _ :v62 /^ .v39 _ v40 _ :v54 / ^ .v39 _ :v40 _ v59 / ^ .:v39 _ v41 _ :v61 / ^ .:v39 _ :v41 _ :v62 /^ .v42 _ v43 _ v55 / ^ .v42 _ :v43 _ :v59 / ^ .:v42 _ v44 _ :v61 / ^ .:v42 _ :v44 _ :v62 /^ .v45 _ v46 _ :v55 / ^ .v45 _ :v46 _ :v59 / ^ .:v45 _ v47 _ :v61 / ^ .:v45 _ :v47 _ :v62 /:

This formula was obtained from a complete binary refutation tree, variables v0 to v47 labeling edges of the bottom two levels and v48 to v62 labeling edges of the top four levels. Along the path from the root to a leaf, the clause labeling that leaf contains all variables of the bottom two levels and one variable of the top four levels. Thus, resolving all clauses in a subtree rooted at variable v3i , 0  i  15, leaves a resolvent of four literals, all taking labels from edges in the top four levels. To any k-BRLR formula that is unsatisfiable, add another clause arbitrarily using the variables of . The result is a formula that is not minimally satisfiable. The above ideas can be extended to show that each of the classes contains a formula that is not a member of any of the others. These examples may be extended to infinite families with the same properties. The subject of comparison of classes will be revisited in Sect. 6.13 using probabilistic measures to determine relative sizes of the classes.

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Probabilistic Comparison of Incomparable Classes

Formulas that are members of certain polynomial-time solvable classes that have been considered in this section appear to be much less frequent than formulas that can usually be solved efficiently by some variant of DPLL. This statement is supported by a probabilistic analysis of these classes where a random formula is an instance of k-SAT with m clauses chosen uniformly and without replacement from all possible width k clauses taken from n variables. A few examples may help to illuminate. In what follows is used to denote a random k-SAT formula. Consider the class of Horn formulas (Sect. 6.2) first. The probability that a randomly generated clause is Horn is .k C 1/=2k , so the probability that is Horn is ..k C 1/=2k /m . This tends to 0 as m tends to 1 for any fixed k. For a hidden Horn formula (Sect. 6.3), regardless of switch set, there are only k C 1 out of 2k ways (k ways to place a positive literal and 1 way to place only negative literals) that a random clause can become Horn. Therefore, the expected number of successful switch sets is 2n ..k C 1/=2k /m . This tends to 0 for increasing m and n if m=n > 1=.k  log2 .k C 1//. Therefore, by Markov’s inequality, is not hidden Horn, with probability tending to 1, if m=n > 1=.k  log2 .k C 1//. Even when k D 3, this is m=n > 1. This bound can be improved considerably by finding complex structures that imply a formula cannot be hidden Horn. Such a structure is presented next. The following result for q-Horn formulas (Sect. 6.5) is taken from [59]. For p D bln.n/c  4, call a set of p clauses a c-cycle if all but two literals can be removed from each of p  2 clauses, all but three literals can be removed from two clauses, the variables can be renamed, and the clauses can be reordered in the following sequence: .v1 _ :v2 / ^ .v2 _ :v3 / ^ : : : ^ .vi _ :vi C1 _ vpC1 / ^

(18)

: : : ^ .vj _ :vj C1 _ :vpC1 / ^ : : : ^ .vp _ :v1 /; where vi 6D vj if i 6D j . Use the term cycle to signify the existence of cyclic paths through clauses which share a variable: that is, by jumping from one clause to another clause only if the two clauses share a variable, one may eventually return to the starting clause. Given a c-cycle C , if no two literals removed from C are the same or complementary, then C is called a q-blocked c-cycle. If has a q-blocked c-cycle, then it is not q-Horn. Let a q-blocked c-cycle in be represented as above. Develop satisfiability index inequalities (2) for . After rearranging terms in each, a subset of these inequalities is as follows ˛1  Z  1 C ˛2  : : : ::: ˛i  Z  1 C ˛i C1  ˛pC1  : : : :::

(19)

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˛j  Z  1 C ˛j C1  .1  ˛pC1 /  : : : ::: ˛p  Z  1 C ˛1  : : : :

(20)

From inequalities (19) to (20), it can be deduced that ˛1  pZ  p C ˛1  .1  ˛pC1 C ˛pC1 /  : : : or 0  pZ  p  1 C : : : where all the terms in : : : are nonpositive. Thus, all solutions to (19) through (20) require Z > .p C1/=p D 1C1=p D 1C1=bln2 nc > 1C1=nˇ for any fixed ˇ < 1. This violates the requirement (Theorem 19) that Z  1 in order for to be q-Horn. The expected number of q-blocked c-cycles can be found and the second moment method applied to give the following result. Theorem 35 A random k-SAT formula is not q-Horn, with probability tending to 1, if m=n > 4=.k 2  k/. t u For k D 3 this is m=n > 2=3. A similar analysis yields the same results for hidden Horn, SLUR, CCbalanced, or extended Horn classes. The critical substructure which causes not to be SLUR is called a crisscross loop. An example is shown in Fig. 54 as a propositional connection graph. Just one crisscross loop in prevents it from being SLUR. Comparing Fig. 54 and expression (18), it is evident that both the SLUR and q-Horn classes are vulnerable to certain types of cyclic structures. Most other polynomial-time solvable classes are similarly vulnerable to cyclic structures of various kinds. But random k-SAT formulas are constructed without consideration of such cyclic structures: at some point as m=n is increased, cycles begin to appear in in abundance, and when this happens, cycles that prevent membership in one of the above named polynomial-time solvable classes also show up. Cycles appear in abundance when m=n > 1=O.k 2 /. It follows that a random k-SAT formula is not a member of one of the above named polynomial-time solvable classes, with probability tending to 1, when m=n > 1=O.k 2 /. By contrast, simple polynomial-time procedures will solve a random k-SAT formula with high probability when m=n < 2k =O.k/. The disappointing conclusion that many polynomial-time solvable classes are relatively rare among random k-SAT formulas because they are vulnerable to cyclic structures is given added perspective by considering the class of matched formulas. A k-CNF formula is a matched formula (see Page 415) if there is a total matching in its variable-clause matching bipartite graph: a property that is not affected by cyclic structures as above. A probabilistic analysis tells us that a random k-SAT

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. . .

... ...

u2 ; ¬u1 ¬up−1 ; u p

u1 up

v0

u1 ; ¬v0 ¬up ; ¬v0

v0

v0

v0 ; ¬u3p

u3p

u3p ; ¬u3p−1

..

v0 ; u p+1

up+1 ¬u ;u p+1 p+2

..

v0

. . .

Fig. 54 A criss-cross loop of t D 3p C2 clauses represented as a propositional connection graph. Only cycle literals are shown in the nodes; padding literals, required for k  3 and different from cycle literals, are present but are not shown

generator produces many more matched formulas than SLUR or q-Horn formulas. Let Q be any subset of clauses of . Denote by V.Q/ the neighborhood of Q: that is, the set of variables that occur in Q. Then V.Q/ is the set variables corresponding to a subset of vertices in V2 of G that are adjacent to the vertices in V2 that correspond to clauses in Q. Denote the deficiency of Q by .Q/. From the definition of deficiency (Page 430), .Q/ D jQj  jV.Q/j. A subset Q is said to be deficient if .Q/ > 0. The following theorem is well known. Theorem 36 (Hall’s Theorem [70]) Given a bipartite graph with vertex sets V1 and V2 , a matching that includes every vertex of V1 exists if and only if no subset of V1 is deficient. t u Theorem 37 Random k-SAT formulas are matched formulas with probability tending to 1 if m=n < r.k/ where r.k/ is given by the following table [59].

k 3 4 5 6 7 8 9 10

r.k/ 0.64 0.84 0.92 0.96 0.98 0.990 0.995 0.997

t u Theorem 37 may be proved by finding a lower bound on the probability that a corresponding random variable-clause matching graph has a total matching. By Theorem 36 it is sufficient to prove an upper bound on the probability that there exists a deficient subset of clause vertices, and then show that the bound tends to 0 for m=n < r.k/ as given in the theorem. This bound is obtained by the first moment method, that is, by finding the expected number of deficient subsets.

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The results in this subsection up to this point are interesting for at least two reasons. First, Theorem 37 says that random k-SAT formulas are matched formulas with high probability if m=n < r.k/ which is approximately 1. But, by Theorem 35 and similar results, a random k-SAT formula is almost never a member of one of the well-studied classes mentioned earlier unless m=n < 1=O.k 2 /. As already pointed out, this is somewhat disappointing and surprising because all the other classes were proposed for rather profound reasons, usually reflecting cases when corresponding instances of integer programming present polytopes with some special properties. Despite all the theory that helped establish these classes, the matched class, mostly ignored in the literature because it is so trivial in nature, turns out to be, in some probabilistic sense, much bigger than all the others. Second, the results provide insight into the nature of larger polynomial-time solvable classes of formulas. Classes vulnerable to cyclic structures appear to be handicapped relative to classes that are not. In fact, the matched class has been generalized considerably to larger polynomial-time solvable classes such as linear autarkies [97,145] which are described in Sect. 6.9 and biclique satisfiable formulas of Sect. 6.7. But there is disappointing news concerning linear autarkies. Consider the test for satisfiability that is implied by Theorem 27: find the satisfiability index z of the given formula and compare against the width, say, k, of the formula’s shortest clause; if z < k=2, then the formula is satisfiable. This test is based on the fact that Algorithm LinAut, Page 422, will output 0 D ; on anyformula for which z < k=2. Theorem 38 The above test for satisfiability does not succeed on a random k-SAT formula with probability tending to 1 as m; n ! 1 and m=n > 1. t u

7

The k-SAT Problem

The previous sections present SAT algorithm designs and applications. Those sections are intended for people who use SAT technology, want to find out more about what is going on under the hood, and may want to experiment with new algorithm designs to suit particular problems. Those sections may also be useful to people who find SAT interesting and want to see what others have accomplished before trying out their new ideas. This section complements those sections. The success of SAT algorithms is somewhat mysterious owing to the very simple and cumbersome representation of constraints in conjunctive normal form. Is there any particular reason that SAT solvers can often perform so well, defying normal, reasonable intuition? Why can a few tweaks to SAT algorithms result in orders of magnitude performance improvements? Theoretical results have contributed partial answers to such questions and have even influenced the design of SAT solvers. Section 6 presents some of these results. This section highlights some others. In particular, the results of this section apply to the k-SAT problem. In this section a k-SAT formula is a Boolean formula expressed in conjunctive normal form where each clause has exactly k distinct literals, no two of which

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are complementary. A random k-SAT formula (restated from Sect. 6.13) is a Boolean formula in conjunctive normal form with m clauses, each of which is chosen uniformly and without replacement from all possible width k clauses taken from n variables. The clause density of a random k-SAT formula is the ratio m=n. The theoretical results on k-SAT are best understood in terms of the so-called satisfiability threshold rk which is the limiting clause density below which random k-SAT formulas are nearly always satisfiable and above which random k-SAT formulas are nearly always unsatisfiable. In [1, 2] the second moment method was applied to a variant of k-SAT to show that rk D 2k log 2O.k/. In Sect. 6.13 several algorithms that are designed to solve subclasses of CNF formulas were analyzed with respect to random k-SAT formulas. It was shown that none of these algorithms could solve a significant proportion of random k-SAT formula if density is greater than 1. Non- or limited backtracking variants of DPLL have been analyzed with respect to random k-SAT formulas with results that are significantly better than and equally insightful to those of Sect. 6.13. In [32–34] it was shown that DPLL will nearly always solve a random k-SAT formula if m=n < O.2k =k/. The analysis can be visualized as a system of clause flows between nodes that represent a collection of clauses of widths from 1 to k. Each node holds a number of clauses of a particular width and on every iteration there is some average flow out of a node as clauses are satisfied and literals are falsified. The outward flow due to falsified literals becomes an inward flow to a node holding clauses of one fewer literal. The reason for the k in the denominator of the right side becomes apparent from this analysis: the average flow of clauses into the node representing unit clauses rises above 1 if m=n D !.2k =k/, and since only 1 clause is removed from that node in one iteration, on average, the number of unit clauses rises and eventually it becomes likely that a complementary pair of unit clauses exists in the partially assigned formula. These results may be made to apply to any myopic algorithm (i.e., a straight line algorithm whose distribution of expressions corresponding to a particular coalesced state under the spectral coalescence of states) can be expressed by its spectral components alone: that is, by the number of clauses of width i , for all 1  i  k, and the number of assigned variables. But in [3] it is shown that no myopic algorithm can succeed in solving random 3-SAT formulas with high probability if m=n > 3:26. However, using a non-myopic greedy algorithm, m=n may be as high as 3.52 before finding solutions becomes unlikely [69, 83, 84]. Note this is well below the presumed threshold r3 D 4:26 and is also well inside the range of densities for which Walksat and Survey Propagation do a good job of solving random 3-SAT formulas. Analyses have been performed on the unsatisfiable side of the threshold as well. In particular, the probability that the length of a shortest resolution proof (this includes DPLL algorithms) is bounded by a polynomial in n when m=n > ck0 2k , ck0 some constant, and limm;n!1 m=n D O.1/ tends to 0 [17, 35]. But, if m is great enough, a random k-SAT formula can, with high probability, be efficiently

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proven to have no solution. For example, if m=n D ‚.nk1 /, the probability that there exist 2k clauses containing the same variables spanning all different literal complementation patterns tends to 1, and since such a pattern can be identified in polynomial time and certifies there is no solution, random expressions can be proven not to have a solution with probability tending to 1 in this case. The best resolution can do is not much different from m=n D ‚.nk1 / [14]. This lack of success has led to investigation of alternatives to resolution. A notable example is to cast SAT as a HITTING SET problem and sum the number of variables that are forced to be set to 1 and the number that are forced to be set to 0. If this sum can be shown to be greater than n in polynomial time, a short proof of unsatisfiability follows. This idea has been applied to random k-SAT and yields a polynomial-time algorithm that nearly always proves unsatisfiability when m=n D nk=2Co.1/1 or greater, a considerable improvement over resolution results [64]. This may be regarded as an example of how probabilistic analysis has driven the search for improved, alternative algorithms. Theoretical results on random k-SAT have provided a visual record of the nature of problems that are hard for some algorithms and not for others. If one were to produce a figure in which formulas appear as points at some distance above the figure floor, where distance is measured in the number of clauses that must be falsified for any assignments and where points are connected to other points which are close in terms of Hamming distance, one would see one or more inverted hills, some of which may actually touch the floor. Each hill reaches a local minimum called a local energy minimum (a contribution from physicists who have studied this). When clause density is low, there exists a single local energy minimum. Then as density is increased, there is a sudden change to several local minima. Finally, there is another jump where exponentially many local minima materialize. The first jump occurs at m=n D O.2k /, and the second jump occurs at m=n D O.2k /. Thus, it is not surprising that so many algorithms perform miserably on unsatisfiable formulas with density just above the transition point. Presumably, as density is increased further, these local minima coalesce (since the minima cannot go below 0), and verifying unsatisfiability becomes easy again. Theoretical results corroborate this [14, 64]. Theoretical results have revealed performance upper bounds on solving k-SAT. The algorithms that are inferred by these results are not considered practical, at least not on their own, and are surprisingly simple. For example, consider the probabilistic algorithm of Fig. 55. The analysis of [123] shows that this algorithm almost always terminates in time that is within a polynomial factor of 1:334n for any 3-SAT input, of 1:5n for any 4-SAT input, of 1:6n for any 5-SAT input, and so on. These complexities, although not considered to represent efficient algorithms, are far below the 2n complexity of a brute force search. Other results followed. Notably, the bound for 3-SAT was lowered to 1:324n by combining Sch¨oning’s algorithm with the probabilistic algorithm of [115] in a nontrivial way [79], and the   n.1 ln. m /CO1 ln.ln.m/ / n [48]. best probabilistic bound is currently O 2

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Fig. 55 Probabilistic algorithm of Sch¨oning

Upper bounds have also been obtained for deterministic algorithms. An early upper bound of O.˛kn j j/ was obtained in (B. Monien and E. Speckenmeyer, 1979, 3-Satisfiability is testable in O.1:62r / steps, Unpublished manuscript) [111] for a simple algorithm that seeks and eliminates autarkies (Page 393) when possible. The factor ˛kn bounds the Fibonacci-like recursion Tk .1/ D Tk .2/ D : : : D Tk .k  1/ D 1 and Tk .n/ D Tk .n  1/ C Tk .n  2/ C : : : C Tk .n  k C 1/; for n  k; where Tk .n/ expresses the worst-case complexity for solving k-SAT in this manner. For example, ˛3  1:681, ˛4  1:8393, and ˛5  1:9276. Specifically, the algorithm looks for a shortest clause c D fl1 _ : : : _ ljcj g in the current formula . If c has k  1 or fewer literals, then is split into at most k  1 subformulas according to the following subassignments: .1/

l1 D 1

.2/

l1 D 0; l2 D 1 :::

jcj/

l1 D l2 D : : : D ljcj1 D 0; ljcj D 1:

If all clauses c of  have length k, then  is split into k subformulas as described, and each subformula either contains a clause of length at most k  1 or one of the resulting subformulas only contains clauses of length k. In the latter case, the corresponding subassignment is autark (Page 393), that is, all clauses containing these variables are satisfied by this subassignment, thereby pruning a branch of the search space. Numerous results reducing the deterministic upper bounds   2 n followed. The best deterministic bound for solving k-SAT is O .2  kC1 / [49]. For CNF formulas of unrestricted clause length, the best worst-case time bound for

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a deterministic algorithm is O.2n.1 log.2m/ / / [47]. The algorithm was obtained by derandomizing a probabilistic algorithm of [124].

8

Other Topics

Several significant topics are not treated in this chapter. There are experimental algorithms that control the space of solutions with linear or quadratic cuts. The Handbook on Satisfiability [19] is a good source of information about these. Several probabilistic algorithms other than those presented here have been proposed, for example, simulated annealing and genetic algorithm variants. These were omitted in favor of more influential varieties. Message passing algorithms such as Survey Propagation have been omitted as they seem to have a special niche. More general constraint programming algorithms have been omitted as the focus here is strictly on SAT. The use of SAT in non-monotonic settings was mentioned in the applications section, but the topic is too broad to cover adequately in this chapter so stable models, well-founded semantics, and answer set programming algorithms, among others, have been omitted.

9

Conclusion

Resolution continues to be the foundation for modern SAT solvers. Resolution alone is inadequate for most inputs. However, advances known collectively as conflictdriven clause learning, plus advanced search and cache heuristics, have resulted in CNF solvers that are quite useful in a variety of roles related to problems in combinatorial optimization. One of the most useful roles of SAT to date is to support solvers of mixed logic systems, known as Satisfiability Modulo Theory solvers. Resolution-based solvers are gaining in performance and are replacing reduced ordered binary decision diagrams in some roles. Gr¨obner bases applied to SAT promise to replace resolution-based algorithms in significantly many roles, but this has not happened yet because producing an optimal order of operations has remained illusive. The same can be said for other algebraic systems such as cutting planes. For awhile it seemed stochastic local search algorithms would replace resolution-based algorithms in many roles, but this has not happened due to the advances in conflict-driven clause learning noted above and because local search techniques, as developed, are not able to prove unsatisfiability. Some classes of SAT are solvable in polynomial time. There are many such classes and many of those have been analyzed over the years. Using a probabilistic measure, the scope of such classes seems limited. Theoretical analyses illustrate the reasons why some algorithms work under certain circumstances. Theoretical results have been used to design new algorithms such as the probabilistic algorithm presented in this chapter. Theoretical results also reveal upper bounds on algorithm performance.

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Glossary Algorithm: (328, 336, 340) A specific set of instructions for carrying out a procedure or solving a problem, usually with the requirement that the algorithm terminate at some point. Boolean Function: (316) A mapping f0; 1g  f0; 1g  : : :  f0; 1g  ! f0; 1g. If the dimension of the domain n is n, the number of possible functions is 22 . Clause: (314, also see Formula, CNF) In CNF formulas a clause is a disjunction of literals such as the following: .aN _ b _ c _ d /. A clause containing only negative (positive) literals is called a negative clause (alternatively, a positive clause). In this chapter a disjunction of literals is also written as a set such as this: fa; N b; c; d g. Either way, the width of a clause is the number of literals it contains. In logic programming a clause is an implication such as the following: .a ^b ^c ! g/. In this chapter, if a formula is a conjunction or disjunction of expressions, each expression is said to be a clause. Clause Width: (see Clause) Edge: (see Graph) Endpoint: (338, 338, 345, 415, 416, 429) One of two vertices spanned by an edge. See Graph. Formula, DNF: (344) DNF stands for disjunctive normal form. Let a literal be a variable or a negated variable. Let a conjunctive clause be a single literal or a conjunction of two or more literals (see Clause). A DNF formula is an expression of the form C1 _ C2 _ : : : Cm where each Ci , 1  i  m, is a conjunctive clause: that is, a DNF formula is a disjunction of some number m of conjunctive clauses. A conjunctive clause evaluates to true under an assignment of values to variables if all its literals has value true under the assignment. Formula, CNF: (327, 329, 340, 313) CNF stands for conjunctive normal form. Let a literal be a variable or a negated variable. Let a disjunctive clause be a single literal or a disjunction of two or more literals (see Clause). A CNF formula is an expression of the form C1 ^ C2 ^ : : : Cm where each Ci , 1  i  m, is a disjunctive clause: that is, a CNF formula is a conjunction of some number m of disjunctive clauses. In this chapter a disjunctive clause, sometimes called a clause when the context is clear, is regarded to be a set of literals and a CNF formula to be a set of clauses. Thus, the following is an example of how a CNF formula is expressed in this chapter. N egg: ffa; N bg; fa; c; d g; fc; d; N A CNF formula is said to be satisfied by an assignment of values to its variables if the assignment causes all its clauses to evaluate to true. A clause evaluates to true under an assignment of values to variables if at least one of its literals has value true under the assignment.

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Formula, Horn: (315, 328) A CNF formula in which every clause has at most one positive literal. Satisfiability of Horn formulas can be determined in linear time [53, 78]. Horn formulas have the remarkable property that, if satisfiable, there exists a unique minimum satisfying assignment with respect to the value true. In other words, the set of all variables assigned value true in any satisfying assignment other than the unique minimum one includes the set of variables assigned value true in the minimum one. Formula, Minimally Unsatisfiable: (317) An unsatisfiable CNF formula such that removal of any clause makes it satisfiable. Formula, Propositional or Boolean: (327, 336, 340, 342, 315) A Boolean variable is a formula. If is a formula, then . / is a formula. If is a formula, then : is a formula. If 1 and 2 are formulas and Ob is a binary Boolean operator, then 1 Ob 2 is a formula. In some contexts other than logic programming, aN or is used instead of :a or : to denote negation of a variable or formula. Formulas evaluate to true or false depending on the operators involved. Precedence from highest to lowest is typically from parentheses, to :, to binary operators. Association, when it matters as in the case of ! (implies), is typically from right to left. Graph: (340) A mathematical object composed of points known as vertices or nodes and lines connecting some (possibly empty) subset of them, known as edges. Each edge is said to span the vertices it connects. If weights are assigned to the edges, then the graph is a weighted graph. Below is an example of a graph and a weighted graph.

1

43

5 12

123

44

2

76 56

Graph, Directed: (321) A graph in which some orientation is given to each edge. Graph, Directed Acyclic: (324, 321, 320) A directed graph such that, for every pair of vertices va and vb , there is not both a path from va to vb and a path from vb to va . Graph, Rooted Directed Acyclic: (324, 321, 320) A connected, directed graph with no cycles such that there is exactly one vertex, known as the root, whose incident edges are all oriented away from it. Literal: (313 see also Formula, CNF) A Boolean variable or the negation of a Boolean variable. In the context of a formula, a negated variable is called a negative literal and an unnegated variable is called a positive literal.

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Logic Program, Normal: (315) A formula consisting of implicational clauses. That is, clauses have the form .a^b^c ! g/ where atoms to the left of ! could be positive or negative literals. Maximum Satisfiability (MAX-SAT): () The problem of determining the maximum number of clauses of a given CNF formula that can be satisfied by some truth assignment. Model, Minimal Model: (316) For the purposes of this chapter, a model is a truth assignment satisfying a given formula. A minimal model is such that a change in value of any true variable causes the assignment not to satisfy the formula. See also Formula, Horn. N P-hard: (327) A very large class of difficult combinatorial problems. There is no known polynomial-time algorithm for solving any N P-hard problem, and it is considered unlikely that any will be found. For a more formal treatment of this topic, see M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of N P-completeness, W.H. Freeman, San Francisco, 1979. Operator, Boolean: (334, 313, 345) A mapping from binary Boolean vectors to f0; 1g. Most frequently used binary operators are Or _: f00  ! 0; 10; 01; 11  ! 1g And ^: f00; 01; 10  ! 0; 11  ! 1g Implies !: f10  ! 0; 00; 01; 11  ! 1g Equivalent $: f01; 10  ! 0; 00; 11  ! 1g XOR ˚: f00; 11  ! 0; 01; 10  ! 1g out of the 16 possible mappings. A Boolean operator O shows up in a formula like this: .vl O vr / where vl is called the left operand of O and vr is called the right operand of O. Thus, the domain of O is a binary vector whose first component is the value of vl and whose second component is the value of vr . In the text patterns of 4 bits are sometimes used to represent an operator: the first bit is the mapping from 00, the second from 01, the third from 10, and the fourth from 11. Thus, the operator 0001 applied to vl and vr has the same functionality as .vl ^ vr /, the operator 0111 has the same functionality as .vl _ vr /, and the operator 1101 has the same functionality as .vl ! vr /. The only meaningful unary operator is ::f1  ! 0; 0  ! 1g. Sometimes aN or N is written for :a or : where a is a variable and is a formula. Operator, Temporal: (334, 334) An operator used in temporal logics. Basic operators include henceforth ( 1 ), eventually (˘ 1 ), next (ı 1 ), and until ( 1 U 2 ). Satisfied Clause: see Formula, CNF Satisfiability (SAT): (327, 330, 340) The problem of deciding whether there is an assignment of values to the variables of a given Boolean formula that makes the formula true. In 1971, Cook showed that the general problem is NP-complete even if restricted to CNF formulas containing clauses of width 3 or greater or if the Boolean operators are restricted

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to any truth-functionally complete subset. However, many efficiently solved subclasses are known. State: (334, 334, 336) A particular assignment of values to the parameters of a system. Subgraph: (340, 340, 342) A graph whose vertices and edges form subsets of the vertices and edges of a given graph where an edge is contained in the subset only if both its endpoints are. Unit Clause: (342) A clause consisting of one literal. See also Clause in the glossary. Variable, Propositional or Boolean: (330, 340, 342)) An object taking one of two values f0; 1g. Vertex: see Graph Vertex Weighted Satisfiability: (328) The problem of determining an assignment of values to the variables of a given CNF formula, with weights on the variables, that satisfies the formula and maximizes the sum of the weights of true variables. The problem of finding a minimal model for a given satisfiable formula is a special case.) Weighted Maximum Satisfiability: (340, 342) The problem of determining an assignment of values to the variables of a given CNF formula, with weights on the clauses, which maximizes the sum of the weights of satisfied clauses. Acknowledgements We thank John S. Schlipf for his help in writing some of the text, particularly the sections on the diagnosis of circuit faults and Binary Decision Diagrams. We especially appreciate John’s attention to detail which was quite valuable to us.

Cross-References Hardness and Approximation of Network Vulnerability Probabilistic Verification and Non-approximability Steiner Minimal Trees: An Introduction, Parallel Computation, and Future Work

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Bin Packing Approximation Algorithms: Survey and Classification Edward G. Coffman, J´anos Csirik, G´abor Galambos, Silvano Martello and Daniele Vigo

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Definitions and Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Definitions for Classical Bin Packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 General Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Classification Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 On-Line Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Classical Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Weighting Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Any-Fit and Almost Any-Fit Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Bounded-Space Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Variations and the Best-in-Class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 APR Lower Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Semi-on-line Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Off-line Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Algorithms with Presorting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Linear Time, Randomization, and Other Approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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E.G. Coffman Department of Computer Science, Columbia University, New York, NY, USA e-mail: [email protected]; [email protected] J. Csirik Department of Computer Algorithms and Artificial Intelligence, University of Szeged, Szeged, Hungary e-mail: [email protected] G. Galambos Faculty of Education, Department of Computer Science, University of Szeged, Szeged, Hungary e-mail: [email protected] S. Martello () • D. Vigo DEI “Guglielmo Marconi”, University of Bologna, Bologna, Italy e-mail: [email protected]; [email protected] 455 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 35, © Springer Science+Business Media New York 2013

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4.3 Asymptotic Approximation Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Anomalies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Variations on Bin Sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Variable-Sized Bins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Resource Augmentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Dual Versions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Minimizing the Bin Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Maximizing the Number of Items Packed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Maximizing the Number of Full Bins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Variations on Item Packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Dynamic Bin Packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Selfish Bin Packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Bin Packing with Rejection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Item Fragmentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Fragile Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Packing Sets and Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Additional Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Item-Size Restrictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Cardinality Constrained Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Class Constrained Bin Packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 LIB Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Bin Packing with Conflicts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Partial Orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Clustered Items. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Item Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Examples of Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The survey presents an overview of approximation algorithms for the classical bin packing problem and reviews the more important results on performance guarantees. Both on-line and off-line algorithms are analyzed. The investigation is extended to variants of the problem through an extensive review of dual versions, variations on bin sizes and item packing, as well as those produced by additional constraints. The bin packing papers are classified according to a novel scheme that allows one to create a compact synthesis of the topic, the main results, and the corresponding algorithms.

1

Introduction

In the classical version of the bin packing problem, one is given an infinite supply of bins with capacity C and a list L of n items with sizes no larger than C : the problem is to pack the items into a minimum number of bins so that the sum of the sizes in each bin is no greater than C . In simpler terms, a set of numbers is to be partitioned into a minimum number of blocks subject to a sum constraint common to each block. Bin packing rather than partitioning terminology will be used, as it eases considerably the problem of describing and analyzing algorithms.

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The mathematical foundations of bin packing were first studied at Bell Laboratories by M.R. Garey and R.L. Graham in the early 1970s. They were soon joined by J.D. Ullman; then at Princeton, these three published the first [103] of many papers to appear in the conferences of the computer science theory community over the next 40 years. The second such paper appeared within months; in that paper, D.S. Johnson [125] extended the results in [103] and studied general classes of approximation algorithms. In collaboration with A. Demers at Princeton, researchers Johnson, Garey, Graham, and Ullman published the first definitive analysis of bin packing approximation algorithms [129]. In parallel with the research producing this landmark paper, Johnson completed his 1973 Ph.D. thesis at MIT which gave a more comprehensive treatment and more detailed versions of the abbreviated proofs in [129]. The pioneering work in [129] opened an extremely rich research area; it soon turned out that this simple model could be used for a wide variety of different practical problems, ranging from a large number of cutting stock applications to packing trucks with a given weight limit, assigning commercials to station breaks in television programming, or allocating memory in computers. The problem is wellknown to be N P-hard (see, e.g., Garey and Johnson [100]); hence, it is unlikely that efficient (i.e., polynomial-time) optimization algorithms can be found for its solution. Researchers have thus turned to the study of approximation algorithms, which do not guarantee an optimal solution for every instance, but attempt to find a near-optimal solution within polynomial time. Together with closely related partitioning problems, bin packing has played an important role in applications of complexity theory and in both the combinatorial and average-case analysis of approximation algorithms (see, e.g., the volume edited by Hochbaum [115]). Starting with the seminal papers mentioned above, most of the early research focused on combinatorial analysis of algorithms leading to bounds on worst-case behavior, also known as performance guarantees. In particular, letting A.L/ be the number of bins used by an algorithm A and letting OPT.L/ be the minimum number of bins needed to pack the items of L, one tries to find a least upper bound on A.L/=OPT.L/ over all lists L (for a more formal definition, see Sect. 2). Much of the research can be divided along the boundary between on-line and off-line algorithms. In the case of on-line algorithms, items are packed in the order they are encountered in a scan of L; the bin in which an item is packed is chosen without knowledge of items not yet encountered in L. These algorithms are the only ones that can be used in certain situations. For example, the items to be packed may arrive in a sequence according to some physical process and have to be assigned to a bin as soon as they arrive. Off-line algorithms have complete information about the entire list throughout the packing process. Since the early 1980s, progressively more attention has been devoted to the probabilistic analysis of packing algorithms. A book by Coffman and Lueker [46] covers the methodology in some detail (see also the book by Hofri [119, Chap. 10]). Nevertheless, combinatorial analysis remains a central research area, and from time to time, numerous new results need to be collected into survey papers. The first comprehensive surveys of bin packing algorithms were by Garey and Johnson [101]

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in 1981 and by Coffman et al. [50] in 1984. The next such survey was written some 10 years later by Galambos and Woeginger [96] who gave an overview restricted to on-line algorithms. New surveys appeared at the end of the 1990s. Csirik and Woeginger [62] concentrated on on-line algorithms, while Coffman et al. [52] extended the coverage to include off-line algorithms, and Coffman et al. [53] considered both classes of algorithms in an earlier version of the present survey. Worst-case and average-case analysis are surveyed in [62] and [52]. More recently, classical bin packing and its variants were covered in Coffman and Csirik [43] and in Coffman et al. [54, 55]. This survey gives a broad summary of results in the one-dimensional bin packing arena, concentrating on combinatorial analysis, but in contrast to other surveys, greater space is devoted to variants of the classical problem. Important new variants continue to arise in many different settings and help account for the thriving interest in bin packing research. The survey does not attempt to give a self-contained work, for example, it does not present all algorithms in detail nor does it give formal proofs, but it refers to certain proof techniques which have been used frequently to establish the most important results. Also, in the coverage of variants, the restriction to partitioning problems in one dimension is maintained. Packing in higher dimensions, for example, strip packing and two-dimensional bin packing, is itself a big subject, one deserving its own survey. The interested reader is referred to the recent survey by Epstein and van Stee [79]. Another important feature of this survey is that it adopts the novel classification scheme for bin packing papers recently introduced by Coffman and Csirik [44]. This scheme allows one to create a compact synthesis of the topic, the main results, and the corresponding algorithms. Section 2 covers the main definitions and as well as a full description of the classification scheme. For the classical bin packing problem, on-line (and semion-line) algorithms are discussed in Sect. 3 and off-line algorithms in Sect. 4. (Section 4.4 is a slight departure from this organization: it deals with anomalous behavior both for on-line and off-line algorithms.) The second part of this survey concentrates on special cases and variants of the classical problem. Variations on bin size are considered in Sect. 5, dual versions in Sect. 6, variations on item packing in Sect. 7, and additional conditions in Sect. 8. Finally, Sect. 9 gives a series of examples to familiarize the reader with the classification scheme. The classification of the papers considered in the survey is also given, when appropriate, in the Bibliography.

2

Definitions and Classification

This section introduces more formally all of the relevant notation required to define, analyze, and classify bin packing problems. Starting with the classical problem, the notation is then extended to more general versions, and finally, the new classification scheme is described. Additional definitions needed by specific variants are introduced in the appropriate sections.

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459

Basic Definitions for Classical Bin Packing

The classical bin packing problem is defined by an infinite supply of bins with capacity C and a list L D .a1 ; : : : ; an / of items (or elements). A value si  s.ai / gives the size of item ai and satisfies 0 < si  C; 1  i  n. The problem is to pack the items into a minimum number of bins under the constraint that the sum of the sizes of the items in each bin is no greater than C . In the classical problem, the bin capacity C is just a scale factor, so without loss of generality, one can adopt the normalization C D 1. Unless stated otherwise, this convention is in force throughout. One of the exceptions will be in the sections treating variants of the classical problem in which bins Bj have varying sizes. In those sections, bin sizes will be denoted by s.Bj /. In the algorithmic context, nonempty bins will be classified as either open or closed. Open bins are available for packing additional items. Closed bins are unavailable and can never be reopened. It is convenient to regard a completely full bin as open under any given algorithm until it is specifically closed, even though such bins can receive no further items. Denote the bins by B1 ; B2 ; : : : . When no confusion arises, Bj will also denote the set of items packed in the j -th bin, and jBj j will denote the number of such items. The items are always organized into a list L (or a list Lj in some indexed set of lists). The notation for list concatenation is as usual; L D L1 L2 : : : Lk means that the items of list Li are followed by the items of list Li C1 for each i D 1; 2; : : : ; k  1. When the number of items in a list is needed as part of the notation, an index in parentheses is used: L.n/ denotes a list of n items. If Bj is nonempty, its current content or level is defined as

c.Bj / D

X

si :

ai 2Bj

A bin (necessarily empty) is opened when it receives its first item. Of course, it may be closed immediately thereafter, depending on the algorithm and the item size. It is assumed, without loss of generality, that all algorithms open bins in order of increasing index. Within any collection of nonempty bins, the earliest opened, the leftmost, and the lowest indexed all refer to the same bin. In general, in the considered lists, the item sizes are taken from the interval .0; ˛, with ˛ 2 .0; 1. It is usually assumed that ˛ D 1r , for some integer r  1, and many results apply only to r D ˛ D 1. There are two principal measures of the worst-case behavior of an algorithm. Recall that A.L/ denotes the number of bins used by algorithm A to pack the elements of L; OPT denotes an optimal algorithm, one that uses a minimum number of bins. Define the set V˛ of all lists L for which the maximum size of the items is bounded from above by ˛. For every k  1, let

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 RA .k; ˛/ D sup

L2V˛

 A.L/ W OPT.L/ D k : k

Then the asymptotic worst-case ratio (or asymptotic performance ratio, APR) as a function of ˛ is given by RA1 .˛/ D limk!1 RA .k; ˛/: Clearly, RA1 .˛/  1, and this number measures the quality of the packings produced by algorithm A compared to optimal packings in the worst case. In an equivalent definition, RA1 .˛/ is a smallest number such that there exists a constant K  0 for which A.L/  RA1 .˛/OPT.L/ C K: for every list L 2 V˛ . Hereafter, if ˛ is left unspecified, the APR of algorithm A refers to RA1  RA1 .1/. The second way to measure the worst-case behavior of an algorithm A is the absolute worst-case ratio (AR)  RA .˛/ D sup

L2V˛

 A.L/ : OPT.L/

The comparison of algorithms by asymptotic bounds can be strikingly different from that by absolute bounds. Generally speaking, the number of items n must be sufficiently large (how large will depend on the algorithm) for the asymptotic bounds to be the better measure for purposes of comparison. Note that the ratios are bounded below by 1; the better algorithms have the smaller ratios. When algorithm A is an on-line algorithm, the asymptotic ratio is also called the competitive ratio. There are some further proposals to measure the quality of a packing, like differential approximation measure (see Demange et al. [66], [67]), random-order ratio (see Kenyon [136]), relative worst-order ratio (see Boyar and Favrholdt [27]), and accommodation function (see Boyar et al. [28]). It is finally worth mentioning a special sequence ti of integers which was investigated by Sylvester [170] in connection with a number theoretic problem and later generalized by Golomb [107]. (In describing the performance of various algorithms in later sections, such sequence will occasionally be considered.) For an integer r  1, define t1 .r/ D r C 1 t2 .r/ D r C 2 ti C1 .r/ D ti .r/.ti .r/  1/ C 1;

for i  2:

(The Sylvester sequence was defined for r D 1). It was conjectured by Golomb that for r D 1, this sequence gives the closest approximation to 1 from below among

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the approximations by sums of reciprocals of k integers, the basis of the conjecture being k X 1 1 C D 1: t .1/ t .1/ 1 kC1 i D1 i Furthermore, it is also easily proved that the following expression is valid for the Golomb sequences: 1 .r  1/ X 1 C C D1 t1 .r/ t .r/ tkC1 .r/  1 i D2 i k

for any r  1:

On the other hand, the following value appears in several results:

h1 .r/ D 1 C

1 X i D2

1 : ti .r/  1

The first few values of h1 .r/ are the following: h1 .1/  1:69103, h1 .2/  1:42312, h1 .3/  1:30238.

2.2

General Definitions

This survey covers results on several variants of the one-dimensional bin packing problem. In such variants, the meaning of a number of notations can be different, so more general definitions are necessary. These will be introduced in the present section, together with the basic ideas behind the adopted classification scheme. The notion of packing items into a sequence of initially empty bins helps visualize algorithms for constructing partitions. It is also helpful in classifying algorithms according to the various constraints under which they must operate in practice. The items are normally given in the form of a sequence or list L D .a1 ; : : : ; an /; although the ordering in many cases will not have any significance. To economize on notation, a harmless abuse is adopted whereby si denotes the name as well as the size of the i -th item. The generic symbol for packing is P, the number of items in P is denoted by jPj; the sum of the sizes of the items in P is denoted by c.P/, and the number of bins in P is denoted by #P. In the classical bin packing optimization problem, the objective is to find a packing P of all the items in L such that #P is minimized over all partitions of L satisfying the sum constraints. Recall that C is only a scale factor in this problem; it is usually convenient to replace the si by si =C and take C D 1. Let PA .L/ denote the packing of L produced by algorithm A. In the literature, one finds the notation A.L/ representing metrics such as #P, but since A.L/ may denote different metrics for different problems (the same algorithm A may apply to problems with different objective functions), the alternative notation will be

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necessary on occasion. The minimum of #P over all partitions P of L satisfying the sum constraints will have the notation OPT.L/ WD min.L/ #P.L/, where .L/ denotes the set of all such partitions. The notation OPT.L/ suffers from the same ambiguity as before, that is, the objective function to which it applies is determined by context. Moreover, in contrast to other algorithm notation, OPT does not denote a unique algorithm. There are two variants of classical bin packing that arise immediately from the definition. In one, a general bin capacity C and a number m of bins are part of the problem instance, and the problem is to find a maximum cardinality subset of fsi g that can be packed into m bins of capacity C . In the other, all n items must be packed into m bins, and the problem is to find a smallest bin capacity C such that there exists a packing of all the items in m bins of capacity C . The first problem suggests yet another in which the total size of the items in a subset S is the quantity of interest rather than the number jSj of items. Problems fixing the number of bins fall within scheduling theory whose origins in fact predate those of bin packing theory. In scheduling theory, which is very large in its own right, makespan scheduling is more likely to be described as scheduling a list of tasks or jobs (items) on m identical processors (bins) so as to minimize the schedule length or makespan (bin capacity). This incursion into scheduling problems will be limited to the most elementary variants and applications of bin packing problems, such as those above. Bin covering problems are also included in bin packing theory for classification purposes, and these change the sum constraints to c.Bj /  1; where again, bin capacity is normalized to 1. The classical covering problem asks for a partition, C, called a cover, which maximizes the number #C of bins satisfying the new constraint. Both the packing and covering combinatorial optimization problems are N P-hard. With problems defined on restricted item sizes or number of items per bin being the major exceptions, this will be the case for nearly all problem variants in the classification scheme. Note that there are immediate variants to bin covering, just as there were for bin packing. For example, one can take the number m of bins to be fixed and ask for a minimum cardinality subset of fai g from which a cover of m bins can be obtained. Or one can consider the problem of finding a largest capacity C such that L can be made to cover m bins of capacity C . Order-of-magnitude estimates of time complexities of fundamental algorithms and their extensions are usually easy to derive. The analysis of parallel algorithms for computing packings is an example where deriving time complexities is not always so easy. However, the research in this area, in which results take the form of complexity measures, has been very limited. Several results quantify the trade-off between the running time of algorithms and the quality of the packings. They produce polynomial-time (or fully polynomial-time) approximation schemes [100], denoted by PTAS (or FPTAS). In simplified terms, a typical form of such results is illustrated by “Algorithm A produces packings with O."/ wasted space and has a running time that is polynomial in 1=".”

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The most common approach to the analysis of approximation algorithms has been worst-case analysis by which the worst possible performance of an algorithm is compared with the performance of an optimization algorithm. (Detailed definitions will be provided shortly.) The term performance guarantee puts a more positive slant on results of this type. So also does the term competitive analysis, which usually refers to a worst-case analysis comparing an on-line approximation algorithm with an optimal off-line algorithm. Probability models also enjoy wide use and have grown in popularity, as they bring out typical, average-case behavior rather than the normally rare worst-case behavior. In probabilistic analysis, algorithms have random inputs; the items are usually assumed to be independent, identically distributed random variables. For a given algorithm A, A.Ln / is a random variable whose distribution becomes the goal of the analysis. Because of the major differences in probabilistic results and in the analysis that produces them, they are not covered in this survey, which focuses exclusively on combinatorial analysis. For a general treatment of average-case analysis of packing and related partitioning problems, see the text by Coffman and Lueker [46].

2.3

Classification Scheme

The scheme for classifying problems and solutions is aimed at giving the reader a good idea of the results in bin packing theory to be found in any given paper on the subject. Although in many, if not most cases, it is impractical to describe every result contained in a paper, an indication of the main results should be useful to the reader. The classification uses four fields, presented in the form problem j algorithm class j results j parameters For a brief preview, observe that, normally, the problem is to minimize or maximize a metric under sum constraints and refers, for example, to the number of bins of fixed capacity and the capacity of a fixed number of bins. The algorithm class refers to paradigms such as on-line, off-line, or bounded space. The results field specifies performance in terms of absolute or asymptotic worst-case ratios, problem complexity, etc. Lastly, the parameters field describes restrictions on problem parameters, such as a limit on item sizes or on the number of items per bin, and a restriction of all data to finite or countable sets. In many cases, there are no additional parameters to specify, in which case the parameters field will be omitted. The three remaining fields will normally be nonempty. In the following, a detailed description of the classification scheme is introduced. Section 9 gives a number of annotated examples to familiarize the reader with the proposed classification. The special terms or abbreviations adopted for entries will be given in bold face. In the large majority of cases, the entry will have a mnemonic quality that makes the precise meaning of an entry clear in spite of its compact form.

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2.3.1 Problem The combinatorial optimization problems of interest can easily be expressed in the notation given so far, but for readability, abbreviations of the names by which the problems are commonly known will be used. Most, but not all, of the problems below were discussed earlier. Moreover, some of them are not treated in this survey and are listed for the sake of completeness. 1. pack refers to the problem of minimizing #P.L/. The default sum constraints are c.Bj /  1. But they are c.Bj /  s.Bj /, if variable bin capacities s.Bj / are considered, and more simply c.Bj /  C if there is only a common bin capacity C . 2. maxpack problems invert the problem above, that is, #P is to be maximized. Clearly, such problems trivialize unless there is some constraint placed on opening new bins. The tacit assumption will be that packings under approximation algorithms must obey a conservative, any-fit constraint: a new bin cannot be opened for an item si unless si cannot fit into any currently open bin. Optimal packings by definition must be such that, for some permutation of the bins, no item in Bi fits into Bj for any j < i . 3. mincap refers to the problem of minimizing the common bin capacity needed to pack L into a given number m of bins. Bin-stretching analysis applies if the problem instances are restricted to those for which an optimization rule can pack L into m unit-capacity bins. Under this assumption, in the analysis of algorithm A, one asks how large must the bin capacity be (how much must it be stretched) for algorithm A to pack L into the same number of bins. 4. maxcard(subset) has the number m of bins and their common capacity C as part of the problem instance. The problem is to find a largest cardinality subset of L that can be packed into m bins of capacity C . 5. maxsize(subset) has the number m of bins and their common capacity C as part of the problem instance. The problem is to find a subset of L with maximum total item size that can be packed into m bins of capacity C . 6. cover refers to the problem of finding a partition of L into bins that maximizes #P.L/ subject to the constraints c.Bj /  1. The partition is called a cover. 7. capcover refers to the problem of finding, for a given number m of bins, the maximum capacity C such that a covering of m bins of capacity C can be obtained from L. 8. cardcover(subset) is the related problem of minimizing the cardinality of the subset of L needed to cover a given number m of bins with a given capacity C . 2.3.2 Algorithm Class 1. on-line algorithms sequentially assign items to bins, in the order encountered in L, without knowledge of items not yet packed. Thus, the bin to which ai is assigned is a function only of the sizes s1 ; : : : ; si . 2. off-line algorithms have no constraints beyond the intrinsic sum constraints; an off-line algorithm simply maps the entire list L into a packing P.L/. All items are known in advance, so the ordering of L plays no role.

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3. bounded-space algorithms decide where an item is to be packed based only on the current contents of at most a finite number k of bins, where k is a parameter of the algorithm. A more precise definition and further discussion of these algorithms appear later. 4. linear-time algorithms have O.n/ running time. More precisely, all such algorithms take constant time to pack each item. The three characterizations above are orthogonal. But the literature suggests that the following convention allows to use one term in classifying algorithms most of the time: bounded space implies linear time and linear time implies online. Exceptions will be noted explicitly: it will be seen below (under repack) how off-line algorithms can be linear time. 5. open-end refers to certain cover approximation algorithms. An open bin Bj , that is, one for which c.Bj / < s.Bj / and further packing is allowed, must be closed just as soon as c.Bj /  s.Bj /. (The item causing the “overflow” is left in the bin.) 6. conservative algorithms are those required to pack the current item into an open bin with sufficient space, whenever such a bin exists; in particular, it cannot choose to open a new bin. Scheduling algorithms satisfying a similar constraint are sometimes called work conserving. 7. repack refers to packing problems which allow the repacking (possibly limited in some way) of items, that is, moving an item, say si , from one bin to another. 8. dynamic packing introduces the time dimension; an instance L of this problem consists of a sequence of triples .si ; bi ; di / with bi and di denoting arrival and departure times, respectively. Under a packing algorithm A, A.L; t/ denotes the number of bins occupied at time t, that is, the number of bins occupied by those items ai for which bi  t < di .

2.3.3 Results Almost all results fall into the broad classes mentioned in Sect. 2.1. 1. R1 A is the asymptotic worst-case ratio, with algorithm A specified when appropriate. When only a bound is proved, the word bound is appended. 2. RA is the absolute worst-case ratio. When only a bound is proved, the word bound is appended. 3. Where possible, complexity of the problem will be given in the standard notation of problem complexity. Approximation schemes are classified as complexity results and have entries like PTAS and FPTAS as noted earlier. 4. Complexity of the algorithm may also be a result; it refers to running-time complexity and will be signaled by the entry running time. A paper classified as a worst-case analysis may also have complexity results (but not conversely, unless both types of results figure prominently in the paper, in which case both classifications will be given). 2.3.4 Parameters These typically correspond to generalizations whereby limitations can be placed on the problem instance, or further properties of the algorithm classification. In some

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cases, problems may be simplified by certain limitations, such as a specific upper limit of two to the number of items per bin. 1. fBi g means that there can be more than one bin size and an unlimited supply of each size. 2. stretch refers to certain asymptotic bounds which compare the number of bins of capacity C needed to pack L by algorithm A to the number of unit-capacity bins needed by an optimal packing. 3. mutex stands for mutual exclusion and introduces constraints, where needed, in the form of a sequence of pairs .si ; sj /; i ¤ j; meaning that si and sj cannot be put in the same bin. 4. card.B/  k gives a bound on the number of items that can be packed in a bin. 5. si  ˛ or si  ˛ denotes bounds on item sizes, the former being far more common in the literature. In some cases, ˛ is specialized to a discrete parameter 1=k, k an integer. The problems with item-size restrictions are called parametric cases in the literature. 6. restricted si refers to simplified problems where the number of different item sizes is finite. 7. discrete calls for discrete sets of item sizes, in particular, item sizes that, for a given integer r, are all multiples of 1=r with C D 1. Equivalently, the bin size could be taken as r and item sizes restricted to the set f1; : : : ; j g for some j 2 f1; : : : ; rg: While this may not be a significant practical constraint, it will affect the difficulty of the analysis and may create significant changes in the results. 8. controllable means that one has the possibility not to pack an item as is, but to first take some decision about it, such as rejecting it or splitting it into parts.

3

On-Line Algorithms

Recall that a bin packing algorithm is called on-line if it packs each element as soon as it is inspected, without any knowledge of the elements not yet encountered (either the number of them or their sizes). This review starts with results for some classical algorithms and then generalizes to the Any-Fit and the Almost Any-Fit classes of on-line algorithms. The subclass of bounded-space on-line algorithms will also be considered: an algorithm is bounded space if the number of open bins at any time in the packing process is bounded by a constant. (The practical significance of this condition is clear; e.g., one may have to load trucks at a depot where only a limited number of trucks can be at the loading dock). This section will be concluded by a detailed discussion of lower bounds on the APRs of certain classes of on-line algorithms, but before doing so, relevant variations of old algorithms and the current best-in-class algorithms will be presented. Anomalous behavior occurring in on-line algorithms is discussed in Sect. 4.4.

Bin Packing Approximation Algorithms: Survey and Classification

3.1

467

Classical Algorithms

In describing an on-line algorithm, it will occasionally be convenient, just before a decision point, to refer to the next item to be packed as the current item; right after A packs ai ; i < n, ai C1 becomes the current item. A simple approach is to pack the bins one at a time according to Next-Fit (NF): After packing the first item, NF packs each successive item in the bin containing the last item to be packed, if it fits in that bin; if it does not fit, NF closes that bin and packs the current item in an empty bin. The time complexity of NF is clearly O.n/. Note that only one bin is ever open under NF, so it is bounded space. This advantage is compensated by a relatively poor APR, however. Theorem 1 (Johnson et al. [129]) One has ( 1 .˛/ RNF

D

1 2

˛1

2

if

.1  ˛/1

if 0 < ˛
1: Since the first item of Bj C1 did not fit in Bj ; 1  j < NF.L/, the sum of the item sizes in Bj [ Bj C1

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exceeds 1, and hence the sum of the weights of the items in Bj [ Bj C1 exceeds 2. Then X X

NF.L/

bNF.L/=2c

X

X

j D1

a2B2j 1 [B2j

WNF .a/ 

j D1 a2Bj

WNF .a/

 2bNF.L/=2c  NF.L/  1; so (1) holds with K D 1. The inequality (2) with K  D 2 is immediate from the 1 definition of WNF , so RNF  2, as desired. There is no systematic way to find appropriate weighting functions, and the approach can be difficult to work out. For example, consider the proof of the following result. Theorem 3 (Johnson et al. [129]) ( 1 RFF

The proof of the function suffices

17 10

D

1 RBF

D

17 10

1C

b ˛1 c1

if

1 2

if

0 1=2 and 1  , one can combine one small element with an item in the interval .1=2; . Also recall that in this case, only a fraction of the bins belonging to interval i can be used to reserve space for a (possibly later arriving) element from the interval .1=2; : This fraction is denoted as ˛ i : (The values of ˛ i s obviously depend on the number of intervals, and they are the crucial point of any such algorithm.) On this basis, they developed four algorithms, which are applied to the intervals Œ1; 6=5/, Œ6=5; 4=3/, Œ4=3; 12=7/, and Œ12=7; 2/, respectively. An analytical solution was given only for the Tiny Modified-Fit algorithm, which was applied to interval Œ12=7; 2/. To examine the worst-case behavior of the other algorithms, they defined two weighting functions and constructed a linear program P .f /. By applying it to the cost function, they could use the method introduced by Seiden [163], thus obtaining upper bounds for the asymptotic behavior of the algorithms. The cost functions were not given explicitly, but program P was solved for many C values, and a diagram was given to show the upper bounds. Concerning lower bounds, Epstein and van Stee [80] followed the argument laid down by van Vliet [171]. In this case too, an LP was generated for many values of C , and appropriate sublists were obtained through specialized greedy algorithms. Chan et al. [37] further extended the study to dynamic bin packing. They considered the case where the items may depart at any time and where moving items among the open bins is disallowed. They first investigated Any-Fit algorithms, proving the following theorem.

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Theorem 35 (Chan et al. [37]) Consider the resource augmentation problem with bin capacity C; and let A be an Any-Fit algorithm. Then A can achieve 1competitiveness if and only if C D 2: They also gave a new lower bound, as a function of C , for on-line algorithms. Let m be the largest integer such that C  m1 < 1, and define, for any positive integer 1  i  m, ˛.1/ D mŠ.m  1/Š

ˇ.i / D

i X

˛.j /

˛.i / D

j D1

ˇ.i  1/ : .m  i C 2/.m  i C 1/

Theorem 36 (Chan et al. [37]) Let A be an on-line algorithm. Then RA1 .C /



2 ˇ.m/ ;  max mŠ.m  1/Š C

 :

They also analyzed certain classical algorithms: Theorem 37 (Chan et al. [37]) 1 RFF .C /  min 1 RBF .C / D

and



 C2 C 3 2C C 1 5  C ; ; ; 2C  1 2C  1 C.2C  1/

1 C 1



2C C 2 C 2  8C C 20 ; min 2C C.4C /



1  RWF .C / 

4 : C2

(4)

Very recently, Boyar et al. [31] gave a complete APR analysis of algorithms WF and NF for the resource augmentation case. They showed that WF is strictly better than NF in the interval 1 < C < 2 but has the same APR for all C  2: Theorem 38 (Boyar et al. [31]) Let tC D b C 11 c: Then 1 RWF .C /

 D

2C ; 3C 2 1 ; C 1

if C 2 Œ1; 2 if C 2 Œ2; 1/

and 1 .C / D RNF

2tC2 C  2tC2  4tC C 2 C 2C tC : tC2 C C 2C tC  tC2  3tC  2 C 2C

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Dual Versions

6.1

Minimizing the Bin Capacity

495

Suppose that the number of bins is fixed, and let the objective be a smallest C such that a given list L of items can be packed in m bins of capacity C . This dual of bin packing actually predates bin packing and is known as multiprocessor or parallelmachine scheduling. It is a central problem of scheduling theory and is surveyed in depth elsewhere (see, e.g., Lawler et al. [141] and Hall [114]), so only relevant results and research directions will be highlighted. The problem is referred to as P jjCmax in scheduling notation. Optimal and heuristic capacities normally differ by at most a maximum item size, so absolute worst-case ratios rather than asymptotic ones are considered. The first analysis of approximation algorithms was presented by Graham [108, 109], who studied the worst-case performance of various algorithms. For the on-line case, he investigated the Worst-Fit algorithm (better known as the List Scheduling, LS algorithm), which initially opens m empty bins and then iteratively packs the current item into the bin having the minimum current content (breaking ties in favor of the lowest bin index). Graham proved that RLS D 2  m1 , a bound that for many years was thought to be best among on-line algorithms. Galambos and Woeginger were first to prove that the bound could be improved. Let A.L; m/ denote the bin capacity needed under A for m bins, and define 

A.L; m/ RA .m/ D sup OPT.L; m/ L

 and R.m/ D inf RA .m/: A

Theorem 39 (Galambos and Woeginger [94]) There exists a sequence of nonnegative numbers "1 ; "2 ; : : :, with "m > 0; m  4 and "m ! 0 as m ! 1, such that R.m/  2  m1  "m . This result encouraged further research; important milestones on the way to the current position on the problem are as follows. Bartal et al. [19] presented an 1 algorithm with a worst-case ratio 2  70 D 1:98571 : : : for all m  70. Earlier, Faigle et al. [83] proved lower bounds for different values of m. They pointed out that for m  3, the LS algorithm is optimal among on-line algorithms, and they proved a 1 C p1 D 1:70710 : : : lower bound for m  4. Then these bounds were 2 improved, as shown below. Theorem 40 (Chen et al. [41]) For all m D 80 C 8k with k a nonnegative integer, R.m/  1:83193. Theorem 41 (Bartal et al. [20]) For all m  3;454, R.m/  1:8370.

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For some time, the following result gave the best upper bound for R.m/ and was a generalization of the methods in [20]. Theorem 42 (Karger et al. [131]) R.m/  1:945 for all m. A tighter result was found by Albers [2], who proved the next two bounds. Theorem 43 (Albers [2]) R.m/  1:923 for all m. Just as previous authors, Albers recognized that, during the packing process, a good algorithm must try to avoid packings in which the content of each of the bins is about the same, for in such cases many small items followed by a large item can create a poor worst-case ratio. Albers’ new algorithm attempts to maintain throughout the packing process b m2 c bins with small total content and d m2 e bins with large total content. The precise aim is always to have a total content in the small-content bins that is at most  times the total content in the large-content bins. With an optimal choice of  , one obtains a worst-case ratio of at most 1:923. For her new lower bound, Albers proved Theorem 44 (Albers [2]) R.m/  1:852 for all m  80. Attempts to further reduce the general gap of about :07 continue. For the special case m D 4, it is proved in Chen et al. [41] that 1:73101  R.4/  52 30 D 1:7333 : : : , but the exact value of R.4/ remains an open problem. Another enticing open problem is: does R.m/ < R.m C 1/ hold for any m? The off-line case is now considered. By preordering the elements according to nonincreasing size and applying the LS algorithm, one obtains the so-called Largest Processing Time (LPT) algorithm for P jjCmax . Graham [109] proved that 1 RLPT D 43  3m . The Multifit algorithm by Coffman et al. [47] adopts a different strategy, leading to better worst-case behavior. The algorithm determines, by binary search over an interval with crude but easily established lower and upper bounds, the smallest capacity C such that the FFD algorithm can pack all the elements into m bins. Coffman, Garey, and Johnson proved that if k binary search iterations are performed, then the algorithm, denoted by MFk , requires O.n log n C k n log m/ time and has an absolute worst-case ratio satisfying RMFk  1:22 C 2k : Friesen [87] improved the bound uniform in k to 1:2, but Yue [182] later settled the question by proving that this bound is 13 . Friesen and Langston [89] developed 11 a different version of the Multifit algorithm, proving that its worst-case ratio is bounded by 72 C 2k . 61 A different off-line approach was proposed by Graham [109]. His algorithm Zk optimally packs the minfk; ng largest elements and completes the solution by packing the remaining elements, if any, according to the LS algorithm. Graham

Bin Packing Approximation Algorithms: Survey and Classification

497

proved that RZk  1 C

1  m1  ˘ 1 C mk

and that the bound is tight when m divides k. Algorithm Zk implicitly defines an approximation scheme. By selecting k D k" D d m.1"/1 e, one obtains RZk"  " m.1"/=" /, and so the method is unlikely to 1 C ". The running time is O.n log n C m be practical. The result was improved by Sahni [161], in the sense that the functional dependence on m and " was reduced substantially. His algorithm A" has running 2 time O.n. n" /m1 / and satisfies RA"  1 C "; hence it is a fully polynomial-time approximation scheme for any fixed m. For m even moderately large, the approach of Hochbaum and Shmoys [118] offers major improvements. They developed an "-approximate algorithm that removed the running-time dependence on m and reduced the dependence on n. The dependence on 1" is exponential, which is to be expected, since a polynomial dependence would imply P D N P. Their algorithm is based on the binary search of an interval (like MFk ), but it uses the "-dual approximation algorithm of Sect. 4.3 at each iteration. Hochbaum and Shmoys show that, after k steps of the binary search, the bin capacity is at most .1 C "/.1 C 2k / times optimal. From this fact and the properties of M" given in Sect. 4.3, one obtains a linear-time approximation scheme for the capacity minimization problem. (See also Hochbaum’s [116] for a more extensive discussion of this interesting technique.) The Hochbaum and Shmoys result was generalized by Alon et al. [3], who gave conditions under which a number of scheduling problems on parallel machines admit a polynomial-time approximation scheme.

6.2

Maximizing the Number of Items Packed

This section considers another variant in which the number of bins is fixed; the objective is now to pack a maximum cardinality subset of a given list L.n/ . The problem arises in computing applications, when one wants to maximize the number of records stored in one or more levels of a memory hierarchy or to maximize the number of tasks performed on multiple processors within a given time interval. The classical bin packing notation is extended in an obvious way by defining  RO A .m; n/ D min

A.L.n/ ; m/ OPT.L.n/ ; m/

so that the APR is defined by RO A1 .m/ D lim inf RO A .m; n/ n!1

and has values less than or equal to 1.



498

E.G. Coffman et al.

First consider off-line algorithms. The problem was first studied by Coffman et al. [48], who adapted the First-Fit Increasing (FFI) heuristic. The items are preordered according to nondecreasing size, the m bins are opened, and the current item is packed into the lowest indexed bin into which it fits, stopping the process as soon as an item is encountered that does not fit into any bin. It is proved in [48] that 1 RO FFI .m/ D 34 for all m. The Iterated First-Fit Decreasing (IFFD) algorithm was investigated by Coffman and Leung [45]. The algorithm sorts the items by nonincreasing size and iteratively tries to pack all the items into the m bins following the First-Fit rule. Whenever the current item cannot be packed, the process is stopped, the largest item is removed from the list, and a new FFD iteration is performed on the shortened list. The search terminates as soon as FFD is able to pack all the items of the current list. Coffman and Leung showed that IFFD performs at least as well as FFI on every list and that 6 1  RO IFFD .m/  78 . The time complexity of IFFD is O.n log n C mn log m/, but a 7 tight APR is not known. Lower bounds on the APR are now considered. If an algorithm A packs items a1 ; : : : ; at from a list L D .a1 ; : : : ; an / so that si > 1  s.Bj / for any j and for any i > t (i.e., no unpacked item fits into any bin), then it is said to be a prefix algorithm. Note that both FFI and and IFFD are prefix algorithms. Theorem 45 (Coffman et al. [48]) Let A be a prefix algorithm and let k D min1j mfjBj jg be the least number of items packed into any bin. Then RO A1 .m/ 

mk : mk C m  1

Moreover, the bound is achievable for all m  1 and k  1. The case of divisible item sizes was investigated by Coffman et al. [51]. They proved that both FFI and IFFD produce optimal packings if .L; C / is strongly divisible. IFFD remains optimal even if .L; C / is weakly divisible, but FFI does not. The case of variable-sized bins was first analyzed by Langston [140], who proved 1 that if the bins are rearranged by nonincreasing size, then RO FFI .m/ D 12 and 23  8 1 RO IFFD .m/  11 for all m. Friesen and Kuhl [88] gave a new efficient algorithm, which is a hybrid of two algorithms defined earlier: First-Fit Decreasing (FFD) and Best-Two-Fit (B2F). The hybrid is iterative: it attempts to pack smaller and smaller suffixes of its list of items until it succeeds in packing the entire suffix. During each attempt, the hybrid partitions the current list of items and packs one part by B2F and then the other part by FFD. They proved an APR of 34 for this hybrid. Coming to the on-line case, Azar et al. [9] (see also [8]) studied the behavior of so-called fair and unfair algorithms. When there is a fixed number of bins, an on-line algorithm is fair if it rejects an item only if it does not fit in any bin, while an unfair algorithm is allowed to discard an item even if it fits. They show that an unfair algorithm may get better performance compared with the performance achieved by the fair version. In particular, they show that an unfair variant of

Bin Packing Approximation Algorithms: Survey and Classification

499

First-Fit can pack approximately 2=3 of the items for sequences for which an optimal off-line algorithm can pack all the items, while the standard (fair) First-Fit algorithm has an asymptotically tight performance arbitrarily close to 5=8. Later, Epstein and Favrholdt [74] proved that the APR of any fair, deterministic algorithm n is 12 RA  23 and that a class of algorithms including BF has an APR of 2n1 :

6.3

Maximizing the Number of Full Bins

In this variant of the problem, the objective is to pack a list of items into a maximum number of bins subject to the constraint that the content of each bin be no less than a given threshold T: Each such bin is said to be full. Potential practical applications are (i) the packing of canned goods so that each can contains at least its advertised net weight and (ii) the stimulation of economic activity during a recession by allocating tasks to a maximum number of factories, all working at or beyond the minimal feasible level. This problem is yet another dual of bin packing. A comprehensive treatment can be found in Assmann’s Ph.D. thesis [5]; Assmann’s collaboration with Johnson, Kleitman and Leung (see [6]) established the main results. They first analyzed an on-line algorithm, a dual version of NF (Dual Next-Fit, DNF): pack the elements into the current bin Bj until s.Bj /  T , then open a new empty bin Bj C1 as the current bin. If the current bin is not full when all items have been packed, then merge its contents with those of other full bins. All on-line algorithms are allowed this last 1 repacking step to ensure that all bins are full. It is proved in [6] that RQ DNF D 12 , where 

 A.L/ W OPT.L/ D m : RQ A1 D lim inf min m!1 OPT.L/ Assmann et al. [6] also studied a parametrized dual of FF, called Dual First-Fit (DFF[r]). Given a parameter r (1 < r < 2), the current item ai is packed into the first bin Bj for which c.Bj / C si  rT . If there are at least two nonempty and unfilled bins (i.e., bins Bj with 0 < c.Bj / < T ) when all items have been packed, an item is removed from the rightmost such bin and added to the leftmost, thus filling it. Finally, if a single nonempty and unfilled bin remains, then its contents 1 are merged with those of previously packed bins. It was shown that RQ DFFŒr D 12 , although a better bound might have been expected. But some years later, Csirik and Totik proved that DNF is optimal among the on-line algorithms. 1 Theorem 46 (Assmann et al. [6] and Csirik and Totik [61]) RQ DFFŒr D 1 1 Q there is no on-line dual bin packing algorithm A for which RA > 2 .

1 2

and

Turning now to off-line algorithms, first consider the observation of Assmann et al. [6] that presorting does not help the adaptations of algorithms Next-Fit Decreasing 1 1 and Next-Fit Increasing, for which RQ NFD D RQ NFI D 12 . On the other hand, the

500

E.G. Coffman et al.

off-line version DFFD[r] of DFF[r], obtained by presorting the items according to nonincreasing size, has better performance. 1 Theorem 47 (Assmann et al. [6]) RQ DFFDŒr 1 1 limr!1 RQ DFFDŒr D limr!2 RQ DFFDŒr D 12 .

D

2 3

if

4 3

< r


bi ) is the departure time, and, as usual, si is the size. Item ai remains in the packing during the time interval Œbi ; di /. Assume that bi  bj if i < j . The problem calls for the minimization

Bin Packing Approximation Algorithms: Survey and Classification

501

of the maximum number, OPTD .L/, of bins ever required in dynamically packing list L when repacking of the current set of items is allowed each time a new item arrives. More formally, if A.L; t/ denotes the number of bins used by algorithm A to pack the current set of items at time t, the objective is to minimize A.L/ D max A.L; t/: 0t bn

Note that, in this case, an open bin can later become empty, so the classical open/close terminology is no longer valid. The definition of the APR is straightforward:   A.L/ W jLj D n : RA1 D lim sup n!1 OPTD .L/ The problem models an important aspect of multiprogramming operating systems: the dynamic memory allocation for paged or other virtual memory systems (see, e.g., Coffman [42]), or data storage problems where the bins correspond to storage units (e.g., disk cylinders or tracks), and the items correspond to records which must be stored for certain specified periods of time (see Coffman et al. [49]). Research on dynamic packing is at the boundary between bin packing and dynamic storage allocation. The most significant difference between the two areas lies in the repacking assumption of dynamic bin packing; repacking is disallowed in dynamic storage allocation, so fragmentation of the occupied space can occur. Coffman et al. [49] studied two versions of the classical FF algorithm. The first is a direct generalization of the classical algorithm. The second is a variant (Modified First-Fit, MFF) in which the elements with si  12 are handled separately, in an attempt to pair each of them with smaller elements. Theorem 50 (Coffman et al. [49]) If 11 4

1  RFF .k/ 

kC1 k

C

1 k2

5 2

C 32 ln

1  RFF .k/ 

1 2:77  RMFF .1/ 

5 2

p

131 2

kC1 k

C

1 kC1

< maxfsi g  k1 , then

D 2:89674 : : : if k D 1;

1 k1

2

k ln k 2 kC1 if k  2;

C ln 43 D 2:78768 : : : .

1 For k D 2, one gets 74  RFF .2/  1:78768 : : : . Although these results are worse than their classical counterparts, relative performance is much better than one might think. This can be seen in the following lower bounds.

Theorem 51 (Coffman et al. [49]) For any on-line dynamic bin packing algorithm kC2 A, RA1  52 and RA .k/  1 C k.kC1/ if k  2; which gives RA .2/  1:666 : : : : By restricting attention to strongly divisible instances (see Sect. 8.1), then matters improve, as expected. The gap between the general lower bound and that for FF disappears. Indeed,

502

E.G. Coffman et al.

Theorem 52 (Coffman et al. [51]) If .L; C / is strongly divisible and L is such that s1 > s2 >    > sk .k  2/, then the APR of the dynamic version of FF is 1 D RFF

k1 Y

1C

i D1

si si C1  : C C

Moreover, for any on-line algorithm A that operates only on strongly divisible 1 instances, RA1  RFF . By a straightforward inductive argument, it can be shown that the continued product has the upper bound

lim 1 C

k!C1

 k2  Y

1 1 C i D 2:384 : : : : 2k2 i D1 2 1

Other algorithms were developed, which, in addition to deletion, allow at least one of the following capabilities: repacking, look ahead, and preordering. For those algorithms which may use all of the above operations, Ivkovic and Lloyd [122] (see also [120]) introduced the fully dynamic bin packing (FDBP) term and presented an algorithm, MMP, which is 54 -competitive.

7.2

Selfish Bin Packing

Consider the case in which the items are controlled by selfish agents, and the cost of a bin is split among the agents in proportion to the fraction of the occupied bin space their items require. Namely, if an agent packs its item a of size s.a/ s.a/ into bin B, then its cost is c.B/ , where c.B/ is the content of the bin. In other words, the selfish agents would like their items to be packed in a bin that is as full as possible. Hence, given a feasible solution, the agents are interested in bins s.a/ s.a/ B 0 such that s.a/ C c.B 0 /  1 and s.a/Cc.B 0 / < c.B/ : If such a bin exists, then a unilaterally migrates from B to B 0 , as in this new feasible packing its cost decreases. When no such bin exists for any agent, a stable state has been reached. The social optimum is to minimize the number of bins used, which is the aim of the standard BP problem. This problem corresponds to a noncooperative game, and the stable state is a pure Nash Equilibrium (NE). There is a Strong Nash Equilibrium (SNE) if there is no subset of agents, which can profit by jointly moving their items to different bins so that all agents in the subset obtain a strictly smaller cost. More precisely, a noncooperative strategic game is a tuple G D hN; .Si /i 2N ; .ci /i 2N i where N is a finite set of players. Each player has a finite set Si of strategies and a cost function ci , and each player has complete information about the strategy of all players. Players choose their strategies independently

Bin Packing Approximation Algorithms: Survey and Classification

503

of each other. A combination of strategies chosen by the players is denoted by s D .xj /j 2N 2 j 2N Sj and is called a strategy profile. The social cost of a game is an objective function S C.s/ W X ! R; that gives the cost of an outcome of the game for a strategy profile s 2 X , where X denotes the set of all strategy profiles. The social optimum of the game is then OPT.G/ D mins2X S C.s/: The quality of an NE is measured by the Price of Anarchy (PoA) and the Price of Stability (PoS), which are defined as follows: PoA D

SC.s/ OPT.G/ s2NE.G/ sup

PoS D

inf

s2NE.G/

SC.s/ : OPT.G/

By using the SNE, one gets the Strong Price of Anarchy (SPoA) and the Strong Price of Stability (SPoS): SP oA D

S C.s/ OPT.G/ s2SNE.G/ sup

SP oS D

inf

s2SNE.G/

S C.s/ : OPT.G/

For the bin packing problem asymptotic measures are used: P oA.BP / D lim sup

sup P oA.G/

OPT.G/!1 G2BP

P oS.BP / D lim sup

inf P oS.G/

OPT.G/!1 G2BP

and SP oA.BP /D lim sup

sup SP oA.G/ SP oS.BP /D lim sup

OPT.G/!1 G2BP

inf SP oS.G/:

OPT.G/!1 G2BP

To stress the connection with game theory the problem is called the bin packing game. The first publication which discussed this problem from a game theoretical perspective was presented by Bil´o (see [25]). He showed that starting from an arbitrary feasible packing, one can always reach an NE using an exponential number of migration steps. This result implies that, for every instance of the bin packing game, there exists an optimal packing which provides a NE, that is, P oS.BP / D 1. He also proved that computing the best NE is N P  hard . Yu and Zhang [181] constructed an O.n4 /-time Recursive First-Fit Decreasing (RFFD) algorithm for finding an NE. They also investigated the P oA.BP / and proved the following theorem: Theorem 53 (Yu and Zhang [181]) For the bin packing game, 1:6416 

1 X lD1

1 2

l.l1/ 2

p 41 C 145  1:6575:  P oA.BP /  32

Epstein and Kleiman [76] improved the upper bound to 1:64286 and conjectured that the best upper bound is equal to the lower bound.

504

E.G. Coffman et al.

Bin packing algorithms look for a good global solution; hence, in general, they do not produce a Nash equilibrium. An exception is the Subset Sum (SS) algorithm by Caprara and Pferschy [34], which produces an NE by repeatedly solving a knapsack problem to pack a bin as full as possible. Unfortunately, the knapsack problem is not polynomial unless P D N P. Indeed, Epstein and Kleiman [76] on the one hand proved equality among the SP oS.BP / and the SP oA.BP / and the asymptotic behavior of the Subset Sum algorithm and on the other hand showed that finding an SNE is N P-hard. Caprara and Pferschy proved that 1:6062  RS1S  1:6210: So, by comparing P oA.BP / and SP oA.BP /, it appears that they have similar values. Therefore, the efficiency in the bin packing game is only due to selfishness, and coalitions do not help. In addition, P oS.BP / is significantly better than SP oS.BP /, which in turn equals SP oA.BP /, implying that in the bin packing game the best equilibrium is less efficient if coalition among agents is allowed.

7.3

Bin Packing with Rejection

This variation is motivated by the following problem, arising in file management. Suppose that files are used by a local system. Either a file is downloaded in advance or the system downloads it only when it is actually requested. The first option needs space on a local server, and it has a local transmission cost, while a program uses the file. The second option has a communication cost that one incurs, while the system downloads the file from an external server. An algorithm which manages the local file system has the choice of either packing a file (item) on the local storage device or (according to the costs) downloading it when needed. The problem can be modeled as the following bin packing problem. A pair .si ; ri / is associated with each item i (i D 1; : : : ; n/, where si is the size and ri is the rejection cost of the item. An algorithm has the choice of either packing an element into the bins or rejecting it. The objective is to minimize the number of bins used, plus the sum of all rejection costs. (If all rejection costs are larger than 1 then every item will be packed, and a standard problem arises.) This model was introduced by D´osa and He [69], who considered both off-line and on-line algorithms, and investigated their absolute and asymptotic behavior. For off-line algorithms, D´osa and He gave an algorithm with absolute worstcase ratio 2. They also observed that the lower bound for the standard problem remains valid, that is, there is no algorithm with absolute worst-case ratio better than 32 , unless P D N P. The lower bound was reached by Epstein [73] (see also [70]). Based on the ratio ri =si , D´osa and He classified the elements into subclasses and used a sophisticated greedy-type algorithm, RFF4, for which they proved that 3 1 RRFF 4  2: For the on-line case, they investigated an algorithm, called RFF1, and proved that the absolute worst-case ratio of the best on-line algorithm lies in the interval .2:343; 2:618/: For the asymptotic competitive ratio, they introduced another algorithm, RFF2, and proved the following:

Bin Packing Approximation Algorithms: Survey and Classification

505

1 Theorem 54 (D´osa and He [69]) For any positive integer m  2; RRFF 2 .m/  7m3 : Hence there exists an on-line algorithm with an APR arbitrarily close to 4m2 7=4 D 1:75:

Not surprisingly, this result was beaten by a variant of the HF algorithm proposed by Epstein [73]. Her RejHk algorithm classifies the items as HF. Then thresholds are defined for the rejection choice as follows. Suppose that the next item is ai 2 Ij ; 1 where Ij D . j C1 ; j1  for some 1  j  k: Item ai will be rejected if either j D k k and ri  k1 si or j < k and ri  j1 . Using the weighting function technique, Epstein proved the following theorem.

Theorem 55 (Epstein [73]) The APR of RejHk tends to h1 .1/ D 1:6901 if k!1: No algorithm with a constant number of open bins can have a smaller APR. For the unbounded-space case, the latter result was improved by a Rejective Modified Harmonic (MHR) algorithm, which is an adaptation of the Modified Harmonic analyzed by Ramanan et al. [159]: Theorem 56 (Epstein [73]) The APR of MHR is at most

538 333 :

Starting from the classical results of Fernandez de la Vega and Lueker [84] and Hochbaum and Shmoys [118], Epstein [73] also proposed an APTAS for 4 "1

bin packing with rejection, having time complexity O.nO.." / / /. Using some results on the multiple knapsack problem (see Chekuri and Khanna [40] and 2 Kellerer [134]) a faster, O.nO." / / time, approximation scheme was constructed by Bein et al. [21].

7.4

Item Fragmentation

Menakerman and Rom [155] investigated a variant of the bin packing problem in which items may be subdivided into pieces of smaller size, called fragments. Fragmenting an item is associated with a cost, which makes the problem N P-hard. They studied two possible cost functions. In the first variant, called Bin Packing with Size-Increasing Fragmentation (BP-SIF), whenever an item is fragmented, overhead units are added to the size of every fragment. In the second variant, called Bin Packing with Size-Preserving Fragmentation (BP-SPF), each item has a size and a cost, and, whenever it is fragmented, one overhead unit is added to its cost without changing its total size. (Actually, problem BP-SIF was first introduced by Mandal et al. [150] who showed that it is N P-hard.) It is supposed that the item sizes are positive integers and the bins are all of size C . It is obvious that a good algorithm should try to perform the minimum number of fragmentations. Menakerman and Rom developed algorithms that do not fragment items unnecessarily. More precisely, an algorithm is said to prevent unnecessarily fragmentation if it follows the next two rules:

506

E.G. Coffman et al.

1. No unnecessary fragmentation: An item (or a fragment of an item) is fragmented only if it must be packed into a bin that cannot contain it. In case of fragmentation, the item (or fragment) is divided into two fragments. The first fragment must fill one of the bins, while the second is packed according to the rules of the algorithm. 2. No unnecessary bins: An item is packed into a new bin only if it cannot fit in any of the open bins. For the BP-SIF problem, it is proved that for any algorithm A that prevents unnecessary fragmentation, RA1  CC2 for every C > 2. For the obvious generalization of the Next-Fit algorithm, called Next-Fit with item fragmentation (NFf ), a CC2 bound for all C  6 was proved. For a more sophisticated algorithm, which is actually an iterated version of First-Fit Decreasing, they proved that the APR is not greater than CC1 when C  15, while it is between CC1 and CC2 when C  16. As to the BP-SPF problem, the performance of an algorithm for a given list L, OHA .L/ is measured by its overhead, that is, by the difference between the cost of the solution produced by the algorithm and the cost of an optimal solution. The worst-case performance of an algorithm is OHAm D inffh W OHA .L/  h for all L with OPT.L/ D mg: Menakerman and Rom [155] proved that for any algorithm A that prevents unnecessary fragmentation, OHAm  m  1 and that an appropriate modification of Next-Fit reaches this bound. First-Fit Decreasing performs better when the bin size is small. Naaman and Rom [158] considered two generalizations of the BP-SPF. The first one is quite simple: r overhead units are added to the size of every fragment. This C changes the performance of Next-Fit to C 2r : The other generalization is more challenging: a set BP of m bins is given, whose (integer) bin sizes may be different. Denote by C D m1 m j D1 s.Bj / the average bin size. In this case, the asymptotic C worst-case bound for Next-Fit is C 2r for every C > 4r: Shachnai et al. [168] studied slightly modified versions of BP-SIF and BPSPF. For the BP-SIF, a header of fixed size  is attached to each (whole or fragmented) item, that is, the size required for packing ai is si C . When an item is fragmented, each fragment gets the header. For the BP-SPF, the number of possible splits is bounded by a given value. They presented a dual PTAS and an APTAS for each of the problems. The dual PTASs pack all the items in OPT.L/ bins of size .1 C "/, while the APTASs use at most .1 C "/OPT.L/ C 1 bins. All of these schemes have running times that are polynomial in n and exponential in 1=". They also showed that both problems admit a dual AFPTAS, which packs the items in OPT.L/ C O.1="2/ bins of size 1 C ". Shachnai and Yehezkely [167] developed fast AFPTASs for both problems. Their schemes pack the items in .1 C "/OPT.L/ C O."1 log "1 / bins in a time that is linear in n and polynomial in 1=":

Bin Packing Approximation Algorithms: Survey and Classification

507

Epstein and van Stee [81] investigated a version where the items may be split without extra cost, but each bin may contain at most k (parts of) items, for a given constant k. They provided a polynomial-time approximation scheme and a dual approximation scheme for this problem.

7.5

Fragile Objects

Bansal et al. [17] (see also [16]) investigated two interrelated optimization problems: bin packing with fragile objects and frequency allocation in cellular networks (which is a special case of the former problem). In this problem, each item has two attributes: size and fragility. The goal is to pack the items so that the sum of the item sizes in each bin is not greater than the fragility of any item in the bin. They provided a 2-approximation algorithm for the problem of minimizing the number of bins and proved a lower bound of 32 on the asymptotical approximation ratio. They also considered the approximation with respect to fragility and gave a 2-approximation algorithm. Chan et al. [35] considered the on-line version of the problem and proved that the asymptotic competitive ratio of any on-line algorithm is at least 2. They also considered the case where the ratio between maximum and minimum fragility is bounded by a value k, showing that the APR of an Any-Fit algorithm (including First-Fit and Best-Fit) is at least k. For the case where k is bounded by a constant, they developed a class of on-line algorithms which achieves an APR of 14 C 3 2r for any r > 1:

7.6

Packing Sets and Graphs

Let the bin capacity C be a positive integer and suppose that the items are sets of elements drawn from some universal set. A collection of items can fit into a bin if and only if their union has at most C elements. As usual, the object is to pack the items in the smallest number of bins. Good approximation algorithms have yet to be analyzed for this problem; adaptations of the classical bin packing approximation algorithms do not have finite APRs. One observes immediately that while classical algorithms prioritize items based on cardinality and position in sequence, a good algorithm for set packing will have to consider the intersections among a given set of items, an orthogonal basis for prioritizing the items. It is thought that approximation algorithms with a finite APR for this problem are unlikely to exist, but a rigorous treatment of this question has yet to appear. Dror (Dror, Private communication) points out that scheduling setups in a manufacturing process is an application of set packing. As an example, he describes a machine that produces many types of circuit boards, each requiring a given set of components. The machine has a magazine (bin) that holds at most C components. At any time, only the circuit boards with their component sets (items) in the magazine

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can be in production. The problem is to plan the overall production process so as to minimize the number of setups. Khanna (Khanna, Private communication) mentions that, in connection with studies of multimedia communications, graph packing is an interesting special case. Items are edges (pairs of vertices) in a given graph G. An edge is packed in a bin if both of the vertices to which it is incident are in the bin, and so the problem is to pack the edges of G into as few bins as possible subject to the constraint that there can be at most C vertices in any bin. The approximability of this problem has yet to be studied. Katona [133] investigated a special case of graph packing. A graph is called p-polyp if it consists of p simple paths of the same length and sharing a common end vertex. Polyp Packing is the following generalization of bin packing: pack a set of paths of different lengths into a minimum number of edge disjoint polyps. He proved that the problem is N P-hard and gave bounds on the performance of a modification of First-Fit.

8

Additional Conditions

8.1

Item-Size Restrictions

Perhaps the simplest (and most practical) restriction is simply to have a finite number k of item sizes, say s1 ; : : : ; sk , and thus a finite number N of feasible item configurations in a bin. This class of problems arose in studies on approximation schemes (see [84,132]) and was considered in its own right by Blazewicz and Ecker [26]. Let the i -th configuration be denoted by pi D .pi1 ; : : : ; pi k / where pij is the P number of items of size sj in pi . By the definition of feasibility, kj D1 pij sj  1 for all i . Let yi be the number of bins in a packing of a given list L that contain configuration pi , and let nj be the number of items of size sj in L. Then a solution to the bin packing problem for L is a solution to the integer programming formulation:

min

N X

yi

i D1 N X

pij yi  nj

.j D 1; : : : ; k/

i D1

yi  0 and integer .i D 1; : : : ; N /: Note that, if 1r lower bounds the item sizes, then N D O.k r1 /, so the constraint matrix above has k rows and O.k r1 / columns. This problem can be solved in time polynomial in the number of constraints (see Lenstra [144]), which is a constant independent of the number of items. The optimal overall packing can then be constructed in linear time.

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If the exact solution is not needed, then one can use the classical approach of Gilmore and Gomory [105, 106]) and compute LP relaxations (see Sect. 4.3); this method gives solutions that are at most k off the optimal in significantly less time. Gutin et al. [113] investigated an on-line problem in which one only has two different element sizes. They gave a 43 lower bound for this problem and provided an asymptotically optimal algorithm for it. Epstein and Levin [77] focused on the parametric case, where both item sizes are bounded from above by k1 for some natural number k  1. Next consider the classical problem restricted to lists L of items whose sizes form a divisible sequence, that is, the distinct sizes s1 > s2 > : : : taken on by the items are such that si C1 divides si for all i  1. The number of items of each size is arbitrary. Given the bin capacity C , the pair .L; C / is weakly divisible if L has divisible item sizes and strongly divisible if in addition the largest item size s1 in L divides C . This variant has practical applications in computer memory allocation, where device capacities and memory block sizes are commonly restricted to powers of 2. It is important to recognize such applications when they are encountered, because approximation algorithms for many types of N P-hard bin packing problems generate significantly better packings when items satisfy divisibility constraints. In some cases, the restriction leads to algorithms that are asymptotically optimal. The problem was studied by Coffman, Garey, and Johnson, who proved the following result for classical off-line algorithms. Theorem 57 (Coffman et al. [51]) If .L; C / is strongly divisible, then NFD and FFD packings are optimal. Indeed, the residual capacity in a bin is always either zero or at least as large as the last (smallest) item packed in the bin. The last packed item is at least as large as any remaining (unpacked) item, so either the bin is totally full or it has room for the next item. Therefore, the FFD and NFD algorithms have the same behavior: they initialize a new bin only when the previous one is totally full, so the packings they produce are identical and perfect. The optimality of FFD holds in the less restrictive case of the following theorem as well. Theorem 58 (Coffman et al. [51]) If .L; C / is weakly divisible, then an FFD packing is always optimal. For strongly divisible instances, optimal performance can also be obtained without sorting the items. Theorem 59 (Coffman et al. [51]) If .L; C / is strongly divisible, then an FF packing is always optimal.

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In the Unit-Fractions Bin Packing Problem, n items are given, and the size of each item i is w1i , with wi a positive integer (1  i  n). This problem was P initially studied by Bar-Noy et al. [18]. Let H D d niD1 w1i e be a lower bound on the required number of bins. They presented an off-line algorithm which uses at most H p C 1 bins. For the on-line version, they analyzed an algorithm which uses H C O. H / bins. It is also proved that any on-line algorithm for the unit-fractions bin packing problem uses at least H C .ln H / bins. Dynamic bin packing problems with unit-fractions items were considered by Chan et al. [36]. They investigated the family of Any-Fit algorithms and proved the following results: 1 1 RBF D RWF D3 1 2:45  RFF  2:4985:

In addition they proved the following lower bound result: Theorem 60 (Chan et al. [36]) For a dynamic bin packing problem with unitfractions items, there is no on-line algorithm A with RA1 < 2:428: It would be interesting to examine how approximation algorithms behave with other special size sequences (e.g., Fibonacci numbers), as mentioned by Chandra et al. [39] in the context of memory allocation.

8.2

Cardinality Constrained Problems

The present section deals with the generalization in which the maximum number of items packed into a bin is bounded by a positive integer p. The problem has practical applications in multiprocessor scheduling with a single resource constraint, when the number p of processors is fixed, or in multitasking operating systems where the number of tasks is bounded. If the maximum number of items in a bin is bounded by p, then the problem is called the p-Cardinality Constrained Bin Packing Problem. Observe which is the difficulty when one tries to construct a good algorithm for this problem. Without the cardinality constraint, a huge number of small elements slightly affect the number of occupied bins: either they can be packed together into few bins, or they can be packed into bins which are “almost full.” In the considered case, having packed many small elements results in almost empty bins, because of the cardinality constraint. Therefore, when a large item arrives, it has to be packed into a new bin even if it would fit into an already open bin. In the context of the above applications, Krause et al. [139] modified classical algorithms so as to deal with the restricted number of items per bin. Their on-line algorithm, pFF, is FF with an additional check on the number of items in open bins. They proved that if p  3, then

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37 24 27 27 1   RpFF  :  10 10p 10 10p As p tends to 1, the bound is much worse than the corresponding APR of the unrestricted problem (2:7 versus 1:7). Whether this upper bound can be improved remains an open question. On-line algorithms were also studied by Babel et al. [11] (see also [10]), who investigated four algorithms, denoted as A1 ; : : : ; A4 . • Algorithm A1 operates with the BF strategy while packing the subsequent element. Different classes of blocked bins are defined according to the number of items in a bin. The algorithm tries to pack the element first into all blocked bins that satisfy a threshold condition, then into all formerly blocked and currently unblocked bins, and finally into the remaining bins. It is proved that for every p  3, p C kp .kp  3/ R.p/ D 2 C ; .p  kp C 1/kp

p  p C p 3  2p ; kp D p2 

where

so limp!1 A1 D 2: • In algorithm A2 , a bin is closed if c.B/  12 and it contains at least p2 items. Furthermore, a pair of bins, B1 and B2 , is also closed if c.B1 / C c.B2 /  1 and jB1j C jB2j  k. The open bins are subdivided into three classes, and the algorithm tries to pack the current item, in turn, into bins of such classes. If the item does not fit in any of these bins, a new bin is open. It is proved that for every p  3, algorithm A2 has an APR of 2. • Algorithm A3 deals with the case p D 2: The basic idea of the algorithm is that, when a small item is packed, a balance should be kept between the number of bins with only one small item and the number of bins which have two such items. It is proved that RA3 D 1 C p15 : • For the 3-cardinality problem, a Harmonic-type algorithm is defined, with four bin classes. The score intervals are as follows: I1 D .0I 13 ; I2 D . 13 ; 12 ; I3 D . 12 ; 23 ; and I4 D . 23 ; 1. The resulting Algorithm A4 is a much more complicated version of the classical HF algorithm and has an asymptotic competitive ratio of 1:8. Two lower bounds are also given: Theorem 61 (Babel et al. [11]) p There is no 2-cardinality on-line bin packing algorithm A with APR RA < 2: Theorem 62 (Babel et al. [11]) There is no 3-cardinality on-line bin packing algorithm A with APR RA < 32 : For unbounded-space algorithms, some further cases were investigated by Epstein [71]. She considered a Harmonic-type algorithm that defines five

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subintervals and packs the items into bins with a more sophisticated procedure. Her algorithm for the case p D 3 improves the APR of [11] to 74 D 1:75. For the cases p D 4, p D 5 and p D 6, she refined the basic algorithm, designing three algorithms with APRs 71 D 1:86842, 771 D 1:93719, and 284 D 1:99306. 38 398 144 She proved that an algorithm with an APR strictly better than 2 cannot be bounded space (unless p  3). For the bounded-space case, Epstein defined the Cardinality Constrained Harmonic-p (CCHp) algorithm. She considered the Sylvester sequence, defined the sum  p  X 1 1 ; ; Sp D ti  1 p i D1 and proved the following theorems: Theorem 63 (Epstein [71]) The value of Sp is a strictly increasing function of p  2 such that 32  Sp < h1 .1/ C 1 and li mp!1 Sp D h1 .1/ C 1 Ð 2:69103: Theorem 64 (Epstein [71]) For every p  2, the APR of CCHp is Sp , and no on-line algorithm which uses bounded space can have a better competitive ratio. In addition, Epstein extended her results to the variable-sized and the resourceaugmentation cases (see Sects. 5.1 and 5.2). An off-line algorithm (Largest Memory First, LMF), based on an adaptation of 1 D 2  p2 if p  2. FFD, was studied by Krause et al. [139]. They proved that RLMF Here the gap between the unrestricted and the restricted case is large (2 versus 11 ). 9 The authors’ search for an algorithm with better worst-case behavior resulted in the Iterated Worst-Fit Decreasing (IWFD) algorithm. IWFD starts by sorting the items according to nonincreasing size, and by opening q empty bins, where q WD P dmax. pn ; niD1 si /e is an obvious lower bound on OPT.L/. Then (i) IWFD tries to place the items into these q bins using the Worst-Fit rule (an item is placed into the open bin whose current number of items is less than p and whose current content is minimum, breaking ties by highest bin index); (ii) whenever the current item ai does not fit in any of the q bins, q is increased by 1, all items are removed from the bins and the processing is restarted from (i). If one implements this approach using a binary search on q, the time complexity of IWFD is O.n log2 n/. Krause, Shen, and Schwetman proved that the algorithm behaves very well in certain special cases: – If p D 2, then IWFD.L/ D OPT.L/ for any list L. – If in the final schedule no capacity constraint is operative, that is, no open bin has residual capacity smaller than the size of the smallest item, then IWFD.L/ D OPT.L/. – If in the final schedule no cardinality constraint is operative, that is, no open bin containing p items has residual capacity at least equal to the size of the smallest 1 item, then RIWFD < 43 : For the general case, where the final packing contains both bins for which the capacity constraint is operative and bins for which it is not, the constants are

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1 significantly worse but still small: 43  RIWFD  2. The authors conjecture that 4 1 RIWFD D 3 . For a long time, it was an open question whether there exists a heuristic algorithm with an APR better than 2. Kellerer and Pferschy [135] proposed a new algorithm based on a method that proved to be fruitful in scheduling theory, namely, one that performs a binary search on the possible OPT.L/ values. At each search step, they run a procedure that tries to pack all items into a number of bins equal to 32 times the current search value. The procedure basically tries to solve a multiprocessor scheduling problem using the classical LPT rule with a cardinality constraint. The time complexity of this Binary Search (BS) algorithm is O.n log2 n/, and the 1 improved bound is 43  RBS  32 . Although the gap has been reduced, the exact bound is still not known.

8.3

Class Constrained Bin Packing

In the Class Constrained Bin Packing Problem (CLCP), each item ai has two parameters: the size si and the color ci , where si 2 .0; 1 and a color is represented by a positive integer, that is, ci 2 Œ1; q: A further parameter Q 2 N is associated with the problem as a bound for the number of the different colors within a bin. Different items may have the same color, and the items with the same color are classified into color classes. The objective isPto pack the items into the minimum number of bins fB1 ; B2 ; : : : ; Bm g, such that aj 2Bi sj  1 for i D 1; : : : ; m and each such bin has items from at most Q color classes. Clearly, if q D n, then one gets the p-cardinality constrained problem (see Sect. 8.2), and if q  Q, then a classical bin packing problem arises. Shachnai and Tamir [165] proved that the problem is N P-hard even in the case of identical item sizes. For the case of on-line bounded-space algorithms, Xavier and Miyazawa [176] presented inapproximability results. More precisely, they proved that if si 2 ŒK1 ; K2 , with 0 < K1 < K2  1, for all i D 1; : : : ; n, then any bounded-space 1 on-line algorithm has an APR of . QK /: 1 Shachnai and Tamir [166] studied unbounded-space algorithms for the case of 1 identically sized items. Suppose that si 2 . rC1 ; 1r  for each ai : they denoted by SQ;r .k; h/ the set of those lists which have kQ < q  .k C 1/Q and hr < n  .h C 1/n, where n is the number of elements in the list. They proved that for any deterministic algorithm A, RA  1 C

k C 1  d kQC1 e r ; hC1

and that the lower bound can be reached by the FF algorithm. Sachnai and Tamir also introduced the notion of Color-Set algorithm (CS). This algorithm partitions the colors into dq=Qe color sets and packs the items of each color set in a greedy way. Note that only one bin is open for each color-set at a time.

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E.G. Coffman et al.

They proved that RCS < 2. The set of Any-Fit algorithms was also investigated, and it was shown that RA  min.r=Q; Q C 1/ for any such algorithm. They further proved that the absolute worst-case ratio of the Next-Fit algorithm is RNF D r=Q: The idea of CS was used to develop new algorithms. Color classes are divided on-line into sets of Q colors each, in order of appearance. Each open bin is assigned one color set. As a new item arrives, the algorithm tries to pack it into a bin having the color of the item. If the FF rule is used, then one gets the Color-Set First-Fit (CSFF) algorithm, while the NF rule gives the Color-Set Next-Fit (CSNF) algorithm. For the case of identical item sizes considered in [165], the authors proved that the competitive ratio of both CSFF and CSNF is at most 2. The case Q D 2 was exhaustively studied by Epstein et al. [82], who proved that the upper bound for identical items is tight for algorithms FF and CSFF. Furthermore, they showed that, for arbitrary item sizes, the competitive ratio is at least 9=4. A 1:5652 lower bound was also given for any on-line algorithm with arbitrary item sizes, thus showing that, even if Q D 2, the on-line algorithms for CLCP behave worse than those for the classical bin packing. For what concerns upper bounds, they proved that for arbitrary values of Q, RCSFF 1  3  Q1 : 1 Xavier and Miyazawa [176] investigated the FF algorithm and proved that RFF 2 Œ2:7; 3: They defined algorithm AQ , which subdivides the items, in a harmonic way, into three classes, and packs the next item, according to its size, into the class it belongs to. The algorithm also considers the color constraint Q. It was proved 1 that RA 2 .2:666; 2:75. Epstein et al. [82] developed an algorithm which has a Q competitive ratio of 2:65492; valid for every Q. For fixed values of q, there are approximability results for CLCP. Note that, if q is fixed, then Q is also constant since Q  q: For equal item sizes, Shachnai and Tamir [165] constructed a dual PTAS whose time complexity is polynomial in n. The scheme uses the optimal number of bins, but the bin capacity is 1 C ". Xavier and Miyazawa [176] developed an APTAS. They also gave an AFPTAS with time complexity O. "n2 /. Epstein et al. [82] proved that if q is considered as a parameter, then there is no APTAS for CLCP for any value of Q: They also gave an AFPTAS for the case of constant q, with time complexity O. "n2 /:

8.4

LIB Constraints

In certain applications, the positions of the items within a bin are restricted: a larger item cannot be put on top of a smaller one. This leads to the variant with the LIB (Larger Item on the Bottom) constraint. The problem was first discussed by Finlay and Manyem [85] and Manyem et al. [151]. They studied the Next-Fit algorithm and proved that it cannot achieve a finite APR. For the First-Fit algorithm, they 1 showed that 2  RFF  3: They also analyzed a variant of Harmonic-Fit (HARM) which partitions the items into classes following the classical rule and packs them by applying to each class of bins the FF rule (instead of NF, which performs

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1 very badly). They showed that 2  RHARM . The parametric case of HARM was considered in Epstein [72], who proved the following result:

Theorem 65 (Epstein [72]) Let r > 0 be a positive integer, and suppose that 1 1 rC1 < maxfsi g  r . Let k denote the number of bin classes. Then RHARM .r/ D ‚.k  r C 1/ for any fixed value of k < 1: This result has an interesting consequence: contrary to the standard problem, for which the performance of the Harmonic-Fit algorithm improves when the number of bin classes is increased, in this case, the competitive ratio worsens, and algorithm HARM has an unbounded ratio as k ! 1. Unfortunately, neither BF nor WF nor AWF behave better than HARM, that is, their APRs are also unbounded. In the same paper, Epstein analyzed the FF algorithm, proving that RFF .1/ D 2:5 and RFF .r/ D 2 C 1r if r  2. Very recently, her result was improved by D´osa, Tuza, and Ye (D´osa et al., 2010, Private communication), who proved that RFF .1/ D 2 C 16 1 and RFF .r/ D 2 C r.rC2/ if r  2. As to lower bounds, very little is known. Clearly, the FFD algorithm automatically fulfills the LIB constraint, so 11 9 is a lower bound for any semi-on-line algorithm which packs preordered elements. The only nontrivial lower bound was obtained by Epstein [72]: Theorem 66 (Epstein [72]) The competitive ratio of any on-line algorithm for bin packing with LIB constraints is at least 2. The result is valid for the parametric case as well, for any value of r  1:

8.5

Bin Packing with Conflicts

In this problem, a set of items L D a1 ; a2 ; : : : ; an with sizes s1 ; s2 ; : : : ; sn and a conflict graph G.L; E/ are given. The objective is to find the minimum number of independent sets such that the sum of the sizes within each set is at most one. The problem is a generalization of both the classical bin packing problem (when E D ;) and the graph coloring problem (when si D 0 for all i ). In the on-line version of the problem, when a new item arrives, one gets its size and the edges of the conflict graph which connect it to previous items. Instead of examining the APR, most investigations focused on the absolute worst-case ratio AR, since the conflict graph that proves the worst case is valid both for small and large instances. Therefore, the APR may not be better than the AR. It is known that graph coloring is hard to approximate for general graphs (see, e.g., Lov´asz et al. [149]); the problem at hand has been studied for specific graphs. The early studies on the bin packing with conflicts (BPC) dealt with perfect graphs, ¨ since the coloring problem is polynomially solvable for them. Jansen and Ohring [123] gave an algorithm with an APR of 2:7. Epstein and Levin [78] improved this result with an algorithm having the performance guarantee h1 .1/ C 1 D 2:690103:

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¨ Jansen and Ohring also proposed an algorithm that uses the so-called Precoloring Extension Problem (PEP) as a preprocessing stage to get a partition of the conflict graph. In the PEP, one is given an undirected graph G D .V; E/ and k distinct vertices v1 ; v2 ; : : : ; vk : The aim is to find a minimum coloring f of G such that f .vi / D i for i D 1; 2; : : : ; k: The minimum number of colors is denoted by

I .G/: The PEP is polynomially solvable for a number of graph classes (interval graphs, forests, K-trees, etc.), whereas it is N P-hard for bipartite graphs. Let C be the class of those graphs for which the PEP is polynomially solvable for any induced subgraph of G. Let G 2 C be a conflict graph and let BIG D faj 2 L W sj > 12 g. The ¨ algorithm by Jansen and Ohring uses BIG as the set of pre-colored vertices and then determines a feasible coloring of G using I .G/ colors: in this coloring, each pair of items in BIG has different colors. Finally, it uses a bin packing heuristic for each color class. Denote this algorithm by A.H /, where H is the bin packing heuristic. They proved the following theorem: ¨ Theorem 67 (Jansen and Ohring [123]) For the BPC problem, the A.FFD/ has a performance ratio 2:4231 D h1 .2/ C 1  RA.FFD/  2:5: Later, Epstein and Levin [78] revisited the algorithm, proving that the lower bound of RA.FFD/ is tight. Instead of using the PEP for preprocessing, they used a matching-based preprocessing (MP) algorithm, for which they proved that RMP D 2:5, and that the result remains valid for all classes of graphs for which there exists a polynomial-time algorithm to find a coloring with a minimum number of colors. They also suggested a greedy algorithm which tries to pack two or three independent items into a single bin and then applies FFD to each color class. They proved that the approximation ratio of this algorithm is 73 : As already mentioned, the PEP is N P-hard for bipartite graphs. However, Jansen ¨ and Ohring [123] proved that by defining a simple algorithm which finds a coloring of the conflict graph with two colors, and packs each color class with NF, one gets an AR of 2: They also pointed out that the AR does not change if NF is replaced by FFD. Using a more complicated pairing technique, Epstein and Levin [78] presented an algorithm with approximation ratio exactly equal to 74 . For the on-line version, the BPC is hard to approximate for many classes of graphs. Success depends on the on-line coloring phase. For interval graphs, Kierstead and Trotter [137] presented an on-line coloring algorithm, which uses 3!  2 colors in the worst case, where ! is the maximum clique size of the graph. Epstein and Levin [78] pointed out that, using NF for each color class, one obtains a 5-competitive algorithm. They also proved a combination of FFD with the algorithm in [137] produces an algorithm with competitive ratio 4:7. For interval graphs, there exists a lower bound for any on-line algorithm: Theorem 68 (Epstein and Levin [78]) The competitive ratio of any on-line algorithm for BPC on interval graphs is at least 155  4:30556: 36

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The theorem is a direct consequence of two interesting lemmas: Lemma 1 (Epstein and Levin [78]) Let c be a lower bound on the APR of any on-line algorithm for classical bin packing, which knows the value OPT in advance. Then the AR of any on-line algorithm for BPC on interval graphs is at least 3 C c. Lemma 2 (Epstein and Levin [78]) Any on-line algorithm for classical bin packing, which knows the value OPT in advance, has an AR of at least 47 36  1:30556: The on-line problem for other graph classes is still open.

8.6

Partial Orders

In this generalization, precedence constraints among the elements are given, where precedence refers to the relative ordering of bins. Let denote the partial order giving the precedence constraints. Then ai aj means that, if ai and aj are packed in Br and Bs , respectively, then r  s. Call the model strict if r < s replaces r  s in this definition. Two practical applications have been considered in the literature. The first one is the assembly line balancing problem, in which the assembly line consists of identical workstations (the bins) where the products stop for a period of time equal to the bin capacity. The item sizes are the durations of tasks to be performed, and a partial order is imposed: ai aj means that the workstation to which ai is assigned cannot be downstream of the one to which aj is assigned. The second application arises in multiprocessing scheduling; here each item corresponds to a unit-duration process having a memory (or other resource) requirement equal to the item size. The bin capacity measures the total memory availability. In the given partial order, ai aj imposes the requirement that ai must be executed before aj finishes. The objective is then to find a feasible schedule that finishes the set of processes in minimum time (number of bins). The first problem was studied by Wee and Magazine [174] and the second one by Garey et al. [104]. In both cases, the Ordered First-Fit Decreasing (OFFD) algorithm was applied. An item is called available if all its immediate predecessors have already been packed. At each stage, the set of currently available items is sorted according to nonincreasing size, and each item is packed into the lowest indexed bin where it fits and no precedence constraint is violated. Note that, if no partial order is given, this algorithm produces the same packing as FFD. In general, however, its 1 worst-case behavior is considerably worse. The APR is ROFFD D 2; except in the 27 1 strict model, where ROFFD D 10 . Liu and Sidney [148] considered the case in which there is an ordinal assumption: initially, the actual sizes of the items are unknown, but their ordering is known, that is, s1  s2      sn  0. Now assume that perfect knowledge of some of the sizes can be “purchased” by specifying the ordinal position of the desired sizes. It is shown that a worst-case performance ratio of  (where   2 is an

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integer) can be achieved if knowledge of bln .n.  1/ C 1/= ln c weights can be purchased.

8.7

Clustered Items

In this generalization, a function f .ai ; aj / is given, measuring the “distance” between items ai and aj . A distance constraint D is also given; two items ai and aj may be packed into the same bin only if f .ai ; aj /  D. The problem has several obvious practical applications in contexts where geographical location constraints are present. Chandra et al. [39] studied different cases, their main result being for the case where the items in a bin must all reside within the same unit square. They proposed a geometric algorithm A and proved that 3:75  RA1  3:8.

8.8

Item Types

In studying the packing of advertisements on Web pages, Adler et al. [1] encountered a variant of bin packing in which items are classified into types; the set of types is given, and there are no constraints on the number of items of any type. The problem is to pack the items into as few bins as possible subject to the constraint that no bin can have two items of the same type. They design optimal algorithms for restricted cases and then bound the performance of these algorithms viewed as approximations to the general problem.

9

Examples of Classification

In this section, the classification scheme proposed in Sect. 2.3 is resumed. The following examples should help familiarize the reader with it: 1. The classification of [157] shows a bin-size extension of pack. packjoff-linejFP TAS jfBi g Results: An approximation algorithm for variable-sized bin packing is presented which for any given positive " produces a scheme with approximation ratio 1 C ". This algorithm has time complexity polynomial in the number of items, the number of bins, and 1=". 2. The classification of [145] gives another such example: packjon-line; off-line; open-endjRA1 boundI FP TAS Results: For this hybrid of pack and cover, it is shown that any openend version of on-line algorithms must have an asymptotic worst-case ratio

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of at least 2. Next-Fit achieves this ratio. There is a fully polynomial-time approximation scheme for this problem. 3. The classification of [7] illustrates a capacity minimization problem with binstretching analysis: mi ncapjon-linejRA boundjst retchi ng Results: A combined algorithm achieves a worst-case bound of 1.625. The best lower bound for any on-line algorithm is 4/3. 4. Reference [48] studies a subset-cardinality objective function: maxcard.subset/joff-linejRA Results: RWFI D 12 ; RFFI D 34 . 5. Reference [61] is classified as a covering problem. coverjon-linejRA1 bound Results: Asymptotic bound: RA1  1=2 for any on-line algorithm A. There exists an asymptotically optimal on-line algorithm. 6. The classification of [90] is packjoff-linejRA1 bound Algorithm: Combined Best-Fit (CBF) which takes the better of the FirstFit Decreasing solution and Best Two-Fit (B2F) solution, where the latter algorithm is a grouping version of Best-Fit limiting the number of items per bin. 1 1 Results: RB2F D 5=4; 227  RCBF  65 195 Note that the word “variant” may be simplistic in that it occasionally hides details of relatively complicated algorithms. 7. Reference [95] shows an example for the combination of two algorithm classes: packjbounded-space; repackjRA1 bound Algorithm: REP3 , an adaptation of FFD using at most three open bins. 1 Result: RREP  1:69 : : :. 3 8. The classification of [139] illustrates a case where one is allowed to constrain the number of items per bin. packjon-line; off-linejRA1 boundjcard.B/  k Result:



  

27 37 24 27 1   RFF  ;  k 10 10k 10 10k

where FFk is the obvious adaptation of FF.

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9. For [175], the classification mentions yet another constraint: packjbounded-space; on-linejRA1

1 1 Result: RSH .k/ D RH ; where SHk is a simplified version of Hk that uses k k only O.log k/ open bins at any time. 10. Classification of [6] aggregates several results:

coverjon-line; off-line; open-endjRA1 Algorithms: Open-end variant of Next-Fit called DNF, of First-Fit Decreasing with a parameter r (FFDr ), and of an iterated version of Worst-Fit (IWFD). Results: 1 D RDNF

4 3 1 2 3 1 ; FFDr1 D for all r;  r  ; and RIWFD D : 2 3 3 2 4

11. A more recent such publication is [29, 30]: maxpackjon-line; off-linejRA1 jsi  1=k Results: a. No off-line approximation algorithm can have an asymptotic bound larger than 17/10, a bound achieved by FFD. 1 1 b. For the off-line case, RFFI D 6=5, RFFI .1=k/ D 1 C 1=k; k  2 1 1 c. For the on on-line case, RFF .1=k/ D RBF .1=k/ D 1 C 1=.k  1/; k  3: d. No deterministic algorithm A has a finite asymptotic bound RA1 for either the card(subset) min orc.subset/ min problems, except for the latter problem in the parametric case with k > 1, in which case the bound is again 1 C 1=.k  1/. 12. An analysis with similarities to bin stretching is used in [31] to compare WF and NF. packjon-linejRA1 jst retchi ng Results: A derivation of the asymptotic ratios for WF and NF under the assumption that they use bins of capacity C while OPT uses unit-capacity bins shows that WF and NF have the same asymptotic bound for all C  1; except when 1 < C < 2, in which case WF has the smaller bound. ´ Acknowledgements The second author was supported by Project “TAMOP-4.2.1/B-09/1/KONV2010-0005 - Creating the Center of Excellence at the University of Szeged,” supported by the European Union and cofinanced by the European Regional Development Fund.

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Binary Unconstrained Quadratic Optimization Problem Gary A. Kochenberger, Fred Glover and Haibo Wang

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Recasting into the Unified Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Accommodating General Linear Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Solving UQP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Illustrative Computational Experience. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Task Allocation Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Max 2-Sat Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Group Technology Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Set Packing Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Set Partitioning Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The Linear Ordering Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 The Max-Cut Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Comments on Computational Experience. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Important Alternative Model for Assignment Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Promising New Application Areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



534 534 535 540 542 542 542 543 544 545 546 547 548 549 549 550 554 555 555

This is an updated version of the previous paper given in [38].

G.A. Kochenberger () School of Business, University of Colorado, Denver, CO, USA F. Glover OptTek Systems, Inc, Boulder, CO, USA H. Wang College of Business Administration, Texas A&M International University, Laredo, TX, USA 533 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 15, © Springer Science+Business Media New York 2013

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Abstract

In recent years the unconstrained quadratic binary program (UQP) has emerged as a unified framework for modeling and solving a wide variety of combinatorial optimization problems. The unexpected versatility of the UQP model is opening doors to the solution of a diverse array of important and challenging applications. Developments in this evolving area are illustrated by describing its methodology with examples and by reporting substantial computational experience demonstrating the viability and robustness of latest methods for solving the UQP model, showing that they obtain solutions to wide-ranging instances of the model that rival or surpass the best solutions obtained by today’s best special-purpose algorithms.

1

Introduction

The unconstrained quadratic binary program (UQP) has a lengthy history as an interesting and challenging combinatorial problem. Simple in its appearance, the model is given by UQP W Opt x0 Qx where x is an n-vector of binary variables and Q is an n-by-n symmetric matrix of constants. Published accounts of this model go back at least as far as the 1960s in the work of Hammer and Rudeanu [31] and have applications in such diverse areas as spin glasses [18, 30], machine scheduling [1], the prediction of epileptic seizures [35], solving satisfiability problems [12, 13, 31, 33], and determining maximum cliques [13, 55, 56]. The application potential of UQP is much greater than might be imagined, due to the reformulation possibilities afforded by the use of quadratic infeasibility penalties as an alternative to imposing constraints in an explicit manner. In fact, any linear or quadratic discrete (deterministic) problem with linear constraints in bounded integer variables can in principle be recast in the form of UQP via the use of such penalties. As will be shown, this outcome has more than theoretical significance. A broad range of challenging problems in combinatorial optimization can not only be reexpressed as UQP problems but can be solved in a highly effective manner when expressed this way. The process of reformulating a given combinatorial problem into an instance of UQP is easy to carry out. The common modeling framework that results, coupled with recently reported advances in solution methods for UQP, serve to make the model a viable alternative to more traditional combinatorial optimization models as illustrated in the sections that follow.

1.1

Recasting into the Unified Framework

For certain types of constraints, equivalent quadratic penalty representations are known in advance making it easy to embody the constraints within the UQP

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objective function. For instance, let xi and xj be binary variables and consider the constraint xi C xj  1 (1) which precludes setting both variables to one simultaneously. A quadratic infeasibility penalty that imposes the same condition on xi and xj is Pxi xj

(2)

where P is a large positive scalar. This penalty function is positive when both variables are set to one (i.e., when (1) is violated), and otherwise the function is equal to zero. For a minimization problem then, adding the penalty function to the objective function is an alternative equivalent to imposing the constraint of (1) in the traditional manner. In the context of transformations involving UQP, a penalty function is said to be a valid infeasible penalty (VIP) if it is zero for feasible solutions and otherwise positive. Including quadratic VIPs in the objective function for each constraint in the original model yields a transformed model in the form of UQP. VIPs for several commonly encountered constraints are given below (where x and y are binary variables and P is a large positive scalar): Note that the penalty term in each case is zero if the associated constraint is satisfied, and otherwise the penalty is positive. These penalties, then, can be directly employed as an alternative to explicitly introducing the original constraints. For other more general constraints, however, VIPs are not known in advance and need to be “discovered.” A simple procedure for finding an appropriate VIP for any linear constraint is given in the next section.

1.2

Accommodating General Linear Constraints

To recast a constrained problem in the form of UQP when the VIPs are not known in advance, consider as a starting point the general constrained problem min x0 D xQx (3)

subject to Ax D b; x binary

This model accommodates both quadratic and linear objective functions since the linear case results when Q is a diagonal matrix (observing that x2j D xj when xj is a 0-1 variable). Under the assumption that A and b have integer components, problems with inequality constraints can also be put in this form by representing their bounded slack variables by a binary expansion. These constrained quadratic optimization models are converted into equivalent UQP models by adding a quadratic infeasibility penalty function to the objective function in place of explicitly imposing the constraints Ax D b.

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Specifically, for a positive scalar P, x0 D xQx C P .Ax  b/t .Ax  b/ D xQx C xDx C c

(4)

O Cc D xQx where the matrix D and the additive constant c result directly from the matrix multiplication indicated. Dropping the additive constant, the equivalent unconstrained version of the constrained problem becomes O x binary UQP W min xQx;

(5)

From a theoretical standpoint, a suitable choice of the penalty scalar P can always be chosen so that the optimal solution to UQP is the optimal solution to the original constrained problem. Remarkably, as demonstrated later, it is often easy to find such a suitable value in practice as well. The preceding general transformation that transforms (3) and (4) into (5) will be called Transformation 1. A fuller discussion of this transformation along with related material can be found in [13, 32, 34]. Transformation 1 provides the general procedure alluded to earlier that can in principle be employed to transform any problem in the form of (3) into an equivalent instance of UQP. For problems where VIPs are known in advance, as by the penalty transformations given in Table 1, it is usually preferable to use the known VIP directly rather than applying Transformation 1. One special constraint in particular xj C xk  1 where the VIP takes the simple form Pxj xk appears in many important applications. Due to the broad applicability of this constraint, it is convenient to refer to this special case as Transformation 2. This process of transforming a given problem into the unified framework of xQx is illustrated by the following three examples. Example 1 Set Packing Set packing problems have the form Max cx: Ax  e and x binary where A is a matrix of 0s and 1s and c and e are both vectors of 1s. These problems are important in the field of combinatorial optimization due to their application potential and their computational challenge. Consider the following example: max x0 D x1 C x2 C x3 C x4 st x1 C x3 C x4  1 x1 C x2  1

Binary Unconstrained Quadratic Optimization Problem Table 1 Known penalties

537

Classical constraint

Equivalent penalty (VIP)

xCy  1 xCy  1 xCy D 1 xy x1 C x2 C x3  1

P.xy/ P.1  x  y C xy/ P.1  x  y C 2xy/ P.x  xy/ P.x1 x2 C x1 x3 C x2 x3 /

The VIPs in Table 1 make it possible to immediately recast this classical model into the UQP unified framework. Representing the positive scalar penalty P by 2M, the equivalent unconstrained problem is max x0 D x1 C x2 C x3 C x4  2Mx1 x3  2Mx1 x4  2Mx3 x4  2Mx1 x2 which can be rewritten as 2

1 6M max .x1 x2 x3 x4 / 6 4M M

M 1 0 0

M 0 1 M

30 1 x1 M Bx2 C 0 7 7B C M5 @x3 A 1

x4

This model has the form max x0 Qx where Q, as shown above, is a square, symmetric matrix. The procedure illustrated here can be used with any set packing problem and has proven to be an effective approach for solving problems with thousands of variables and constraints (see [4]).

Example 2 Set Partitioning The classical set partitioning problem has the form Min dx: Ax = e, x binary where again A is a matrix of 0s and 1s, e is a vector of 1s, and (in contrast to the vector c of set packing problems) d is a vector of integers, typically nonnegative. The set partitioning problem is found in applications that range from vehicle routing to crew scheduling [36, 51]. As an illustration, consider the following small example: min x0 D 3x1 C 2x2 C x3 C x4 C 3x5 C 2x6 subject to x1 C x3 C x6 D 1 x2 C x3 C x5 C x6 D 1 x3 C x4 C x5 D 1 x1 C x2 C x4 C x6 D 1

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and x binary. Applying Transformation 1 with P = 10 gives the equivalent UQP model: O min xQx; x binary _

where the additive constant, c, is 40 and the symmetric Q matrix is 2

17 610 6 6 10 O D6 Q 6 610 6 40 20

10 18 10 10 10 20

10 10 29 10 20 20

10 10 10 19 10 10

0 10 20 10 17 10

3 20 20 7 7 7 20 7 7 10 7 7 10 5 28

Solving this UQP formulation provides an optimal solution x1 D x5 D 1 (with all other variables equal to 0) to yield x0 D 6. In the straightforward application of Transformation 1 to this example, the replacement of the original problem formulation by the UQP model does not require any new variables to be introduced. Set partitioning problems with thousands of variables have been successfully solved by reformulating them in this manner, as reported in [47]. In many applications, Transformations 1 and 2 can be used in concert to produce an equivalent UQP model, as demonstrated next. Example 3 The K-Coloring Problem Vertex coloring problems seek to assign colors to nodes of a graph in such a way that adjacent nodes receive different colors. The K-coloring problem attempts to find such a coloring using exactly K colors. A wide range of applications, ranging from frequency assignment problems to printed circuit board design problems, can be represented by the K-coloring model. These problems can be modeled as satisfiability problems using the assignment variables as follows: Let xij be 1 if node i is assigned color j, and 0 otherwise. Since each node must be colored, K X

xij D 1 i D 1; : : : ; n

(6)

jD1

where n is the number of nodes in the graph. A feasible coloring, in which adjacent nodes are assigned different colors, is assured by imposing the constraints xip C xjp  1 p D 1; : : : ; K for all adjacent nodes (i, j) in the graph.

(7)

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This problem can be recast in the form of UQP by using Transformation 1 on the assignment constraints of (6) and Transformation 2 on the adjacency constraints of (7). No new variables are required. Since the resulting model has no explicit objective function, any positive value for the penalty P will do. The following example gives a concrete illustration of the reformulation process. Find a feasible coloring of the following graph using three colors. Thus, the goal is to find a solution to the system:

1

5

2

4

3

xi1 C xi2 C xi3 D 1 xip C xjp  1

i D 1; 5

p D 1; 3

(8) (9)

(for all adjacent nodes i and j)

In this traditional form, the model has 15 variables and 26 constraints. To recast this problem in the form of UQP, it suffices to use Transformation 1 on the equations of (8) and Transformation 2 on the inequalities of (9). Arbitrarily choosing the penalty P to be 4, the equivalent problem in unified form is given by O UQP.Pen/ W min xQx O matrix is where the Q

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2

4 4 64 4 6 64 4 6 64 0 6 6 60 4 6 60 0 6 60 0 6 O D 60 0 Q 6 60 0 6 60 0 6 6 60 0 6 60 0 6 64 0 6 40 4 0 0

4 4 4 0 4 0 0 4 0 4 4 4 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 4 0

0 0 4 0 0 4 4 4 4 4 4 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4

0 0 0 0 0 0 4 0 0 4 0 0 4 4 4 4 4 4 4 0 0 4 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 4 0 0 4 0 4 4 4 0 4 0 0 4 0 4 4 4 0 4 0 0 0 0

0 0 0 0 0 0 0 0 4 0 0 4 0 0 4 0 0 4 4 4 4 4 4 4 0 0 4 0 0 4

4 0 0 4 0 0 4 0 0 4 0 0 0 0 0 0 0 0 4 0 0 4 0 0 4 4 4 4 4 4

3 0 0 7 7 4 7 7 0 7 7 7 0 7 7 4 7 7 0 7 7 0 7 7 0 7 7 0 7 7 7 0 7 7 4 7 7 4 7 7 4 5 4

O yields the feasible coloring: Solving this unconstrained model, xQx, x11 ; x22 ; x33 ; x41 ; x53 ; D 1 all other xij D 0 This approach to coloring problems has proven to be very effective for a wide variety of coloring instances from the literature as disclosed in [41].

2

Solving UQP

Employing the UQP unified framework to solve combinatorial problems requires the availability of a solution method for xQx. The recent literature reports major advances in such methods involving modern metaheuristic methodologies. The reader is referred to references [6–9, 14, 16, 24–26, 37, 45, 49, 50, 53, 54] for a description of some of the methods that have provided valuable contributions to the area. The pursuit of further advances in solution methods for xQx remains an active research arena. The computational work reported later in this chapter derives from previous studies of three methods: a basic tabu search method due to Glover, Kochenberger, and Alidaee [23–25], a tabu search method due to Lewis [47], and the more recent tabu search method of Glover, Jin-Kao, and Lu [29]. For convenience these methods will be referred to as methods 1, 2, and 3 respectively. A brief overview of each approach is given below. Complete details are provided in the aforementioned references. Method 1 Overview: This tabu search metaheuristic for UQP is centered around the use of strategic oscillation, which constitutes one of the primary strategies of tabu search. The method alternates between constructive phases that progressively

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set variables to 1 (whose steps are called “add moves”) and destructive phases that progressively set variables to 0 (whose steps are called “drop moves”). To control the underlying search process, the method uses a memory structure that is updated at critical events, identified by conditions that generate a subclass of locally optimal solutions. Solutions corresponding to critical events are called critical solutions. A parameter SPAN is used to indicate the amplitude of oscillation about a critical event. To begin span is set equal to 1 and then is gradually increased until reaching some limiting value. For each value of span, a series of alternating constructive and destructive phases is executed before progressing to the next value. At the limiting point, span is gradually decreased, allowing again for a series of alternating constructive and destructive phases. When span reaches a value of 1, a complete span cycle has been completed and the next cycle is launched. The search process is typically allowed to run for a preset number of SPAN cycles. Information stored at critical events is used to influence the search process by penalizing potentially attractive add moves (during a constructive phase) and inducing drop moves (during a destructive phase) associated with assignments of values to variables in recent critical solutions. Cumulative critical event information is used to introduce a subtle long-term bias into the search process by means of additional penalties and inducements similar to those discussed above. Other standard elements of tabu search such as short- and long-term memory structures are also included. Method 2 Overview: This method is a modification of the previous method that implements a basic multi-start, tabu search procedure with path-relinking. A local search is performed using a 1-opt mechanism. When no improvements to the current solution can be found using 1-opt local search, including tabu aspiration possibilities, a path-relinking procedure is initiated between the current solution and an elite set of diverse solutions saved during the search process. After relinking, a random portion of the current solution is perturbed and testing continues. If after a number of iterations (specified by the restart limit) no improvements are found, then a larger perturbation is invoked based on long-term memory. Method 3 Overview: The DDTS method repeatedly alternates between a simple version of tabu search (TS) and a diversification phase founded on a memory-based perturbation operator. Starting from an initial random solution, DDTS uses the TS procedure to reach a local optimum. Then, the perturbation operator is applied to displace the solution to a new region, whereupon a new round of TS is launched. To facilitate achieving effective diversification, the perturbation operator is guided by information from a special memory structure. This tabu search procedure uses a neighborhood defined by the single 1-flip moves, which consists of changing (flipping) the value of a single variable xj to its complement value 1  xj . The implementation of this neighborhood uses a fast incremental evaluation technique to calculate the cost of candidate moves. The diversification strategy utilizes a memory-based perturbation operator composed of three parts: a flip frequency memory, an elite solution memory, and an elite value frequency memory. These memory structures are used jointly by the perturbation operator to enhance the diversification of the search process. This method has proven

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to be highly effective for solving large instances of UQP. Complete details of this method are given in Glover, Lu, and Hao [29].

3

Applications

To date several important classes of combinatorial problems have been successfully modeled and solved by employing the unified framework. Results with the unified framework applied to these problems have been uniformly attractive in terms of both solution quality and computation times. While the three solution methods described above are designed for the completely general form of UQP, without any specialization to take advantage of particular types of problems reformulated in this general representation, the outcomes typically prove competitive with or even superior to those of specialized methods designed for the specific problem structure at hand. The broad base of experience with UQP as a modeling and solution framework obtained by applying the three methods above includes a substantial range of problem classes including quadratic assignment problems, capital budgeting problems, multiple knapsack problems, task allocation problems (distributed computer systems), maximum diversity problems, p-median problems, asymmetric assignment problems, symmetric assignment problems, side constrained assignment problems, quadratic knapsack problems, constraint satisfaction problems (CSPs), set partitioning problems, fixed charge warehouse location problems, maximum clique problems, maximum independent set problems, maximum cut problems, graph coloring problems, graph partitioning problems, number partitioning problems, and-linear ordering problems. Additional test problems representing a variety of other applications have also been reformulated and solved via UQP. The section below reports specific computational experience with some large-scale applications. Section 5 then suggests promising new applications areas for UQP.

4

Illustrative Computational Experience

The following summarizes results obtained on large-scale test problems of a variety of well-known problem classes from combinatorial optimization. Because methods 1, 2, and 3 were developed at different points in time, and each has been applied to classes of problems different from those treated by the other two, outcomes for these classes of problems are reported by describing the findings for the three methods in the sequence in which they have been applied.

4.1

The Task Allocation Problem

The task allocation problem, which consists of assigning tasks to processors, can be described as follows: Let P D fP1 ; P2 ; : : : ; Pm g be a set of distributed processors

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and T D fT1 ; T2 ; : : : Tn g be a set of tasks to be run on the processors. Let cij be the communications cost between tasks Ti and Tj and qtp be the execution cost of task t on processor p. Then stipulating that xtp equals 1 if task Tt is assigned to processor Pp and is otherwise equal to 0, the model becomes min

n P m P

qtp xtp C

tD1 pD1

st

m P

n P m P

cij xip .1  xjp /

i ciS ;

8;  ¤ rO ;

(11)

aj 2 SOrO H) cjFrO > cjF ;

8;  ¤ rO :

(12)

Note that the constructed classification of the features and instances FOr and SOr are not necessarily the same as classifications Fr and Sr , respectively.

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Biclustering B is referred to as a consistent biclustering if relations (11) and (12) hold for all elements of the corresponding classes, where matrices CS and CF are defined according to (9) and (10), respectively. A data set is biclustering-admitting if some consistent biclustering for it exists. Furthermore, the data set is called conditionally biclustering-admitting with respect to a given (partial) classification of some instances and/or features if there exists a consistent biclustering preserving the given (partial) classification. Busygin et al. [29] prove conic separability which also indicates convex hulls of classes do not intersect. By definition, a biclustering is consistent if Fr D FOr and Sr D SOr . However, a given data set might not have these properties. In such cases, one can remove a set of features and/or instances from the data set so that there is a consistent biclustering for the truncated data. This feature selection process may incorporate various objective functions depending on the desirable properties of the selected features. One general choice is to select the maximal possible number of features in order to lose minimal amount of information provided by the training set:

ŒCB max x

m X

xi

(13a)

i D1

Pm Pm i D1 aij fi rO xi i D1 aij fi  xi > P subject to Pm m f x i D1 i rO i i D1 fi  xi

rO ;  D 1; : : : ; k; rO ¤ ; j 2 SrO (13b)

xi 2 f0; 1g i D 1; : : : ; m:

(13c)

In this formulation, xi ; i D 1; : : : m are the decision variables. xi D 1 if i th feature is selected, and xi D 0 otherwise. fi k D 1 if feature i belongs to class k, and fi k D 0 otherwise. The objective is to maximize the number of features selected and (13b) ensures that the biclustering is consistent with respect to the selected features. The goal of the CB problem is to find the largest set of features that can be used to construct a consistent biclustering. A problem with selecting the most representative feature set is the following. Assume that there is a consistent biclustering for a given data set, and there is a feature, i , where the difference between the two largest values of ciSr is negligible. For such cases, additive and multiplicative consistent biclustering are introduced in [103] by relaxing (11)–(12) with (14)–(15) and (16)–(17), respectively: ai 2 FrO H) ciSrO > ˛iS C ciS ;

8;  ¤ rO

(14)

aj 2 SrO H) cjFrO > ˛jF C cjF ;

8;  ¤ rO

(15)

ai 2 FrO H) ciSrO > ˇiS ciS ;

8;  ¤ rO

(16)

aj 2 SrO H) cjFrO > ˇjF cjF ;

8;  ¤ r: O

(17)

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Using these relaxations, ˛-consistent and ˇ-consistent biclustering problem formulations are obtained as follows: Œ˛CB max x

m X

xi

(18a)

i D1

Pm Pm i D1 aij fi rO xi i D1 aij fi  xi subject to Pm > ˛j C P m i D1 fi rO xi i D1 fi  xi

ŒˇCB max x

8Or ;

 D 1; : : : ; k; rO ¤ ; j 2 SrO

(18b)

xi 2 f0; 1g

(18c)

m X

i D 1; : : : ; m

xi

(19a)

i D1

Pm Pm i D1 aij fi rO xi i D1 aij fi  xi > ˇj P subject to Pm m i D1 fi rO xi i D1 fi  xi  D 1; : : : ; k; rO ¤ ; j 2 SrO xi 2 f0; 1g

i 2 1; : : : ; m:

8Or ; (19b) (19c)

The information obtained from these solutions can be used to classify additional instances in the test set. These solutions are also useful for adjusting the values of vectors ˛ and ˇ to produce more characteristic features and decrease the number of misclassifications. Feature selection for consistent (13), ˛-consistent (18), and ˇ-consistent biclustering (13) is proven to be N P-hard [86].

2.3

Other Classification Approaches

A special case of biclustering introduced by Ben-Dor et al. [12] is the orderpreserving submatrix problem, which is proven to be N P-hard. The goal of this problem is to select a subset of rows and columns from the original data matrix in which there exists a permutation of columns such that in each row the values are strictly increasing. Trapp and Prokopyev [135] propose a general linear mixed 0–1 programming formulation and an iterative algorithm that makes use of a smaller linear 0–1 programming formulations. The proposed solution algorithm enhanced by a number of enhancements including valid inequalities and bounding schemes is able to solve problems with approximately 1,000 rows and 50 columns to optimality. Bertsimas and Shioda [16] introduce mixed-integer linear methods to the classical statistical problems of classification and regression and construct a software

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package called classification and regression via integer optimization (CRIO). CRIO separates data points into different polyhedral regions. In classification, each region is assigned a class, while in regression, each region has its own distinct regression coefficients. Computational experimentations show that CRIO is comparable to the leading methods in classification and regression. Logical analysis of data is a technique that is used for risk prediction in medical applications [2]. This method is based on combinatorial optimization and boolean logic. The goal is essentially classifying groups of patients at low and high mortality risk, and LAD is shown to outperform standard methods used by cardiologists. ¨ uyurt [87] consider the problem of generating the seKundakcioglu and Unl¨ quence of tests required to reach a diagnostic conclusion with minimum average cost, which is also known as a test-sequencing problem. In this setting, the training data consists of asymmetrical tests (i.e., a probabilistic outcome for a feature or test) and the decision rule is obtained using an optimal binary AND/OR decision tree. An efficient bottom-up tree construction algorithm is developed based on fundamental ideas of Huffman coding. Computational results show that the algorithm outperforms previously proposed heuristic algorithms in reasonable time.

3

Unsupervised Learning

Unsupervised learning can be defined as finding patterns in unlabeled data. The goal is finding similarities among instances that belong to same class, where no class information is available. In this group of methods, N instances .x1 ; x2 ; : : : ; xN / of a random p-vector X having joint density P r.X/ are given and a decision is to be made based on representation of instances. Unlike the supervised methods that are trying to minimize some external error criterion, in unsupervised methods, calculating the difference between the target and input data instances is not possible. This results in different methods that are not relying on any outside information. It should be noted that unsupervised methods are particularly useful since unlabeled data is usually less costly compared to labeled data. Two widely used applications of unsupervised learning are clustering and dimensionality reduction. Clustering (or cluster analysis) is defined as finding a convenient and valid organization of the data to establish rules for separating future data into categories [75]. Clustering algorithms can be divided into two major groups: partitional and hierarchical [74]. In partitional clustering, all clusters are found simultaneously as partitions of the data. In hierarchical clustering, nested clusters are found recursively. This can be done by one of the following two approaches. Agglomerative methods initially consider each data point as the head of its own cluster and try to merge the most similar pair of clusters successively to form a cluster hierarchy. On the contrary, divisive methods initially assign all data instances in one cluster and divide each cluster into smaller clusters (see [62]). Another useful application of unsupervised learning is dimensionality reduction. These techniques aim to find a lower dimensional representation of the original data, which captures the content of the original data based on some criterion. A lower

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dimensional representation of data is desired mainly because the computational advantage of handling a smaller data set outweighs the loss of information. Frequently used in the preprocessing stage, dimensionality reduction helps mitigate computational limitations of data mining algorithms for large data sets.

3.1

Latent Variable Models

In this section, probabilistic methods that utilize latent variables are introduced. Latent variable (or hidden variable) is a statistical variable that is not observed directly but can be estimated based on its relation with an observed variable. Latent variable models can be used to perform dimensionality reduction and clustering, the two cornerstones of unsupervised learning. Clustering problems that are addressed in this section are hard clustering problems and fuzzy clustering problems (FCP). In hard clustering problems, each data point needs to be assigned to one and only one cluster. On the other hand, in fuzzy clustering, each data point is a member of all clusters and it has a membership grade for each cluster showing its likelihood of belonging to that cluster.

3.1.1 k-Means k-means is a hard clustering problem that has been introduced by several researchers across different disciplines, most notably Lloyd [95], Forgy [54], Friedman and Rubin [58], Ball and Hall [6], and MacQueen [96]. Given a data set of n vectors xj 2 Rd , j D 1; : : : ; n, the goal is to find a partition S D fS1 ; S2 ; : : : ; Sk g that minimizes the expected loss (see [96]). A random point z 2 Rd with a known distribution p generates a loss proportional to the square of the error resulting from the choice of clusters, formally kz  zOk2 , where zO is the cluster centers or exemplars. The goal is to minimize the expected loss over possible choices of clusters, which is formally defined as w2 .S / D

k Z X i D1

kz  zOk2 dp.z/:

(20)

Si

Given a selection of cluster members from the data, it is easy to see that the cluster mean minimizes the squared error term. Therefore, the problem reduces to selection of cluster members, which can be formulated as the following mixedinteger programming (MIP) problem:

Œk  means min ;r

subject to

n k X X

rij kxj  i k2

(21a)

i D1 j D1 k X i D1

rij D 1

j D 1; : : : ; n (21b)

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rij 2 f0; 1g

i D 1; : : : ; k; j D 1; : : : ; n: (21c)

This problem is proven to be N P-hard [3]. One of the most widely used iterative heuristic approaches is the k-means algorithm, also known as Lloyd’s algorithm [95], which can be considered as an expectation-maximization algorithm. Initially, k initial exemplars are randomly selected. Next, by repeating the following two steps, the clusters will be iteratively refined. In the first (i.e., expectation) step, each instance will be temporarily associated with the cluster whose exemplar is the closest. In the second (i.e., maximization) step, based on the temporary cluster assignment, each exemplar is updated as the mean of instances in its cluster. The algorithm terminates when no exemplar can be further updated. The convergence is guaranteed since the objective is finite and an improvement is assured in each step. It should be noted that the term k-means is used to define the problem of minimizing the within-cluster sum of squares (i.e., (21)) as well as Lloyd’s expectation-maximization algorithm defined above. One observation is that k-means algorithm is sensitive to the initial set of exemplars especially when k is large. Thus, random initialization of exemplars is acceptable for relatively small values of k. There are alternative approaches, especially useful when number of clusters is increased, including, but not limited to, heuristic initialization schemes and parallel implementations with multiple initial exemplars. Typically, k-means problem considers spherical clusters by minimizing Euclidean distance between points and cluster centers, as presented in formulations (20) and (21). Other distances that are used in k-means are mahalanobis distance metric [99], Itakura-Saito distance [93], L1 distance [81], and Bregman distance [7]. It has been proven that the worst-case complexity of the k-means algorithm is O.k n2 2 / [4]. Although it is introduced over 50 years ago, k-means is still one of the most widely used algorithms for clustering. This algorithm is particularly useful due to its simplicity, ease of implementation, efficiency, and success in practice.

3.1.2 Message Passing In order to overcome difficulties associated with the sensitivity of k-means to the initialization process, Frey and Dueck [57] propose a message-passing method where every data instance is a potential exemplar. The problem to be solved is the N P-hard k-median problem. For computational tractability, an affinity propagation scheme is proposed, where data instances are nodes of a network, real-valued messages are transmitted along edges, and the magnitude of each message is updated based on the current affinity that one data instance has for choosing another data instance as its exemplar. Two types of messages are transmitted: First, responsibility, r.i; j /, sent from data instance i to candidate exemplar j , is a cumulative variable for how appropriate instance j is to be the exemplar

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for instance i . Second, availability, a.i; j /, sent from potential exemplar j to data instance i , reflects the accumulated evidence for how good it is for instance i to select instance j as its exemplar. Availability includes support from other instances that instance j is an exemplar. Formally, responsibility and availability are defined as r.i; j /

s.i; j /  8
0, there exists a polynomial-time approximation within a factor of 1 C " from optimal. 2

However, the running time of such an approximation is O.n1=" /. It cannot be implemented to use in the real world. Therefore, they are still making efforts to look for approximations with a smaller performance ratio and a low-degree polynomial for running time. Cheng et al. [14] designed a .1 C 1=s/-approximation for the minimum 2 Connected Dominating Set in unit disk graph. The running time is nO..s log s/ / . There is an evidence [14] to show that currently existing implemented approximations have a large space for improvement. In this section, they will introduce their main results. An algorithm A is a PTAS for a minimization problem with optimal cost OP T if the following is true: Given an instance I of the problem and a small positive error parameter ": 1. The algorithm outputs a solution with cost at most .1 C "/OP T . 2. When " is fixed, the running time is bounded by a polynomial in the size of the instance I .

If there exists a PTAS for an optimization problem, the problem’s instance can be approximated to any required degree.

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To design a PTAS for MCDS in UDG, they quote the result from [13] which is: Lemma 4 (Cheng et al. [13]) There exists a polynomial-time approximation for the MCDS in unit disk graph, with performance ratio 8. The general picture of constructing the PTAS for the MCDS in UDG [14] is as follows: 1. Divide a space and make it contain all vertices of the input unit disk graph, into a grid of small cells. For each small cell, take the points h distance away from the boundary (the central area of the cell). 2. Optimally compute a minimum union of Connected Dominating Sets in each cell for connected components of the central area of the cell. The key lemma is to prove that the union of all such minimum unions is no more than the MCDS for the whole graph. 3. For vertices not in central areas, just use the part of an 8-approximation lying in boundary areas (within distance h C 1 away from the boundary, with some overlap with the central areas) to dominate them. This part, together with the above union, forms a Connected Dominating Set for the whole input unit disk graph. 4. Using the shifting argument (i.e., shift the grid around to get partitions at different coordinates) to make sure there are at least half good partitions having good approximations overall. Corollary 1 (Cheng et al. [14]) There is a .1 C 1=s/-approximation for an 2 MCDS in connected unit disk graphs, running in time nO..s log s/ / . The proof of this corollary can be found in [14].

2.2

Node-Weighted Unit Disk Graph

In real world, different sensors do not have the same value. Therefore, when wireless sensor networks contain many types of sensors with the same transmission range but different costs, the mathematical model would be changed to the unit disk graph with node weight. For virtual backbone, they would prefer to have Connected Dominating Set with minimum total weight rather than Connected Dominating Set with minimum cardinality. If they want to use the same construction method as unit disk graph, they should construct the minimum-weight Dominating Set first. However, they could not continue using the maximal independent set method, because, there is no efficient way for constructing a minimum-weight MIS unless NP D P [42]. Actually, how to construct a constant-approximation algorithm for minimum-weight Dominating Set in UDG is still an open problem.

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In 2006, Amb¨uhl et al. [5] presented a 72-approximation algorithm for this problem by using the partition technique. Gao et al. [40] introduced a new technique, called double partition in 2008. They proposed an algorithm with performance ratio of 6 C ". A year later, Dai et al. [19] reduced the ratio to 5 C ". Zou et al. [90] further improved the performance ratio to .4 C "/ in the same year. For connecting the DS into minimum-weight Connected Dominating Set in unit disk graph, Amb¨uhl et al. [5] spent 17opt to connect their previous solution and got an 89-approximation for minimum-weight Connected Dominating Set where opt is the object function value of an optimal solution which is the minimum total weight of Connected Dominating Set. Gao et al. [40] decreased the cost for interconnection to 4opt. Zou et al. [92] further reduced the cost to 2:5  opt  3:875opt, where  is the best-known performance ratio for polynomial-time approximation of classical Steiner minimum tree problem in graphs. This result can be used to connect a Dominating Set by adding nodes of weight at most 2:5 times the weight of an optimal Connected Dominating Set, yielding the approximation ratio of 8.875 for MWCDS. Later Zou [90] and Erlebach et al. [36] both obtained 4 C "-approximation for minimum-weight Dominating Set, and this result will maintain a 7.875-approximation for MWCDS, which improves the previously best approximation ratio of 8.875. In [90], they used dynamic programming algorithm for the mi n  weight chromati c d i sk cover to propose a .4 C "/-approximation algorithm for MWDS. For a given set D of unit disks, they assign positive weight to each disk by a function c and a set P of nodes in the plane. They colored each disk D 2 D either red or blue. Assume there are m C 1  2 distinct horizontal lines y D yi for 0  i  m from the top to the bottom, where m is a nonnegative integer constant. Therefore, the m C 1 lines separate the plane into m C 2 horizontal strips. A node p 2 P is chromatically covered by a red/blue disk D 2 D if p 2 D and the center of D is above/below the strip which contains p. A chromatic cover of P is a subset D 2 D which each node in P is chromatically covered by some disk in D0 . The min-weight chromatic disk cover (MWCDC) problem is to find a min-weight chromatic cover D 2 D of P . They proved that there is a polynomial .4 C "/-approximation algorithm for the MWDS for any fixed " by proposing a -approximation algorithm for the MWCDC. In [36], Erlebach et al. proposed a .4 C "/-approximation algorithm for MWDS in unit disk graph. Their algorithm is based on several ideas of previous constantfactor approximation algorithms for the problem [5, 47]. They first partitioned the plane into K K areas where K is an arbitrary constant. They considered the following subproblem for each of these areas: Find a minimum-weight set of disks that dominate all disks that have a center in the area.

The union of feasible solutions for each subproblem produces a Dominating Set for the original problem. For the shifting technique presented in [47], the loss in the approximation factor is only .1 C O.1/=K/. Therefore, the (best) combination of the solutions is a . C O.1/=K/-approximation for the original problem if the solution for every subproblem is a -approximation by using the shifting technique.

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Thus, they can set K such that the obtained solution is a . C "/-approximation for any constant ". They presented a 4-approximation algorithm for the subproblem. By using the shifting technique and setting K appropriately, the algorithm forms a .4 C "/-approximation algorithm.

2.2.1 2 :5 Approximation Algorithm in [92] The main idea of this algorithm is based on the classical Steiner tree problem. They first constructed an edge-weighted graph G 00 with the same node and edge set as the original graph G. Then, they defined the weight of every edge in G 00 as the half of the sum of its endpoints’ weights in G. Finally, they used -approximation algorithm to obtain a Steiner tree of G 00 . From the Steiner tree of G 00 , they can get a Steiner tree of G with approximation ratio 2:5  3:875. In Corollary 2, they obtained a 8:875 C " algorithm for minimum-weight Connected Dominating Set in unit disk graph. 1. Preliminaries In this section, they introduce some useful definitions and denotations in [92]. For a node-weighted (or edge-weighted) graph G with weight function w and a subgraph H of G, they use w.H / to represent the weight sum of all nodes (or edges) in H . For any two nodes u and v of G, d i stG .u; v/Pis the weight of the shortest path between u and v, which is calculated as min vk 2P w.vk / among all the possible paths between u and v which vk is the internal node on every possible path p. Given an edge-weighted graph G and a node subset S , they use SM T .G; S / and M S T .G/ to represent the -approximation algorithm for Steiner minimum tree (SMT) and the algorithm for finding minimum spanning tree (MST). 2. 2:5 Approximation Algorithm (Zou et al. [92]) The idea of this algorithm is to convert the node-weighted Steiner tree problem to the classical Steiner tree problem. There are three steps: (a) Construct an edge-weighted graph G 00 from G as follows by initializing G 00 with the same node set and edge set as G and an edge weight function w0 . For every edge e D .u; v/ in G 00 , let the edge weight w0 .u; v/ D 12 .w.u/ C w.v//. (b) Compute a Steiner tree T of G 00 on S through the -approximation algorithm. (c) View T as the node-weighted Steiner tree of G on S and output it. The pseudo-code of this algorithm is presented in Algorithm 2 .

Algorithm 2 N W S T .G D .V; E; w; S // (Zou et al. [92]) 1: Initialize an edge-weighted graph G 0 D .V 0 ; E 0 ; w0 ; S 0 / by setting V 0 D V; S 0 D S and E0 D E 2: for each edge .vi ; vj / in graph G 0 do 3: Assign the weight of this edge w0 .vi ; vj / D .w.vi / C w.vj //=2 4: end for 5: T D SMT.G0 ; S/ 6: Output T

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Since G is a unit disk graph, for any terminal set S , there always exists a minimum Steiner tree with maximum degree no more than 5 [61]. The following lemma proves the relationship between the weight of the optimal SMT in graph G 0 and the weight of the optimal NWST they want. Lemma 5 (Zou et al. [92]) Denote TOP TSM as the optimal Steiner minimum tree for G 00 on the set S and TOP TNWS as the optimal node-weighted Steiner tree of G on the same terminal set S , respectively. Then w0 .TOP TSM /  2:5w.TOP TNWS / when G is a unit disk graph. Proof (Zou et al. [92]) Consider TOP TNWS as a Steiner tree on S of G 00 . For convenience, denote T as TOP TNWS , V as V .TOP TNWS /, E as E.TOP TNWS /, and dT .u/ as the degree of node u in tree T ; they have w0 .TOP TSM /  w0 .TOP TNWS / D

X eD.u;v/2E

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5 w.T /: 2

Hence, the lemma holds. If they consider the Steiner tree T of G 00 as a subgraph of G, they can get the following lemma: Lemma 6 (Zou et al. [92]) For any Steiner tree T of G 00 on terminal set S , if they view it as a subgraph G, then T is also a Steiner tree of G on set S and w.T /  w0 .T /. Proof (Zou et al. [92]) Since for any Steiner tree, the degree of each Steiner node is not less than 2 and all weight of nodes in terminal set is 0, the lemma holds. Theorem 3 (Zou et al. [92]) Algorithm N W S T is a 2:5-approximation for nodeweighted Steiner tree problem in unit disk graph. Proof (Zou et al. [92]) Let T be the output of N W S T . Denote TOP TSM and TOP TNWS as Lemma 5. By Lemmas 5 and 6, they have w.T /  w0 .T /  w0 .TOP TSM /  2:5w.TOP TNWS /: Since the node-weighted Steiner tree can be used in the MWCDS problem to interconnect all nodes of the CDS, they can obtain the following corollary:

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Corollary 2 (Zou et al. [92]) Using node-weighted Steiner tree to interconnect all nodes of the CDS produces .9:875 C "/-approximation for minimum-weight Connected Dominating Set in unit disk graph. Proof (Zou et al. [92]) For any node-weighted graph G and a terminal set S , denote TOP T and TOP TCDS be the optimal Steiner tree of G on S and optimal CDS of G, respectively. Since the induced graph GŒS [ TOP TCDS  is connected, this graph contains a Steiner tree of G on S . Thus, w.TOP T /  w.TOP TCDS /. By Huang et al. [47], they can get a Dominating Set C of G with w.C /  .6 C "/w.TOP TCDS /. Then, using algorithm NWST for C , they can obtain a Steiner tree T with w.T /  2:5w.TOP T /. Clearly, V .T / is a Connected Dominating Set of G and w.V .T // D w.C / C w.T /  .6 C "/w.TOP TCDS / C 2:5w.TOP T /  .9:875 C "/w.TOP TCDS /:

2.2.2 PTAS for MCDS in Node-Weighted UDG The same as the MCDS in UDG, they have the following question: Open Problem 2 Does the minimum-weight Dominating Set have a PTAS in unit disk graphs? Nieberg et al. [64] developed a technique, called the local neighborhood-based scheme technique. A PTAS for minimum Dominating Set in polynomial growth bounded graphs is constructed. Because the unit disk graph is also a polynomial growth-bounded graph, Wang et al. [80] tried to use the local neighborhood-based scheme technique to find a PTAS in node-weighted UDG. They are successful in some special case. However, it is still open in general. Because the running time for all approximations constructed with partition techniques is very high, it is hard to implement for application in the real world. Therefore, the following open problem receives more attention: Open Problem 3 Can they construct a constant approximation for the minimumweight (Connected) Dominating Set in unit disk graphs without using partition?

2.3

Unit Ball Graph

In most cases, people studied the MCDS problem in homogeneous wireless networks. They also assumed that all nodes in the networks are on a twodimensional space and used unit disk graph (UDG) to abstract the networks. However, sometimes, this assumption is very far from reality. Underwater sensor networks (USNs) is a good example. USN can be used for ocean sampling networks, undersea explorations, distributed tactical surveillance, disaster prevention, assisted navigation, ocean environmental monitoring, etc. In many applications of USNs, underwater sensor nodes are floating around water instead of staying on the bottom

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of the ocean. In this case, the height of nodes becomes considerably important. Therefore, when identical sensors are deployed into a field which is not flat, the mathematical model would be the unit ball graph. Definition 11 (Unit ball graph(UBG)) A graph is called a unit ball graph if all its nodes lie in the three-dimensional Euclidean space, and an edge .u; v/ exists if and only if the distance between u and v is at most one. In other words, if they let each node u associate a ball with center u and diameter one, called a unit ball, then edge .u; v/ exists if and only if ball u and ball v intersect each other. Since MCDS problem in UDG is NP-hard, and UDG is a special case of UBG, MCDS problem in UBG is also NP-hard. There are also two stages for constructing a Connected Dominating Set in unit ball graph. The same as in UDG, they construct a maximal independent set first. Therefore, they need to find an upper bound for the size of maximal independent set. To do this, they have to study the following sphere packing problems. Call the ball at center u with radius one the dominating ball of node u. The following is the first packing problem that they would like to study: How many independent nodes can lie in the dominating ball of a node?

This problem has a close relationship to Gregory-Newton Problem, a well-known packing problem as follows: How many unit balls can touch a unit ball such that they do not touch each other?

Gregory-Newton Problem was proposed in 1694 and conjectured as 12. Hoppe in 1874 (see [86]) proved that the answer is 12. Actually, an icosahedron has 12 vertices and the distance from the center to a vertex is smaller than the edge length. Therefore, they have Corollary 3 ([46]) The maximum degree of a minimum spanning tree T in a UBG G is at most 12. Lemma 7 ([46]) For any vertex u in a UBG G, the neighborhood NG .u/ contains at most 12 independent vertices. Hence, they can simply obtain jmi sj  11  opt C 1 in unit ball graph by the argument similar to the proof for jmi sj  4  opt C 1 in unit disk graph. The result can be improved by study the following question: Open Problem 4 How many independent nodes can lie in the union of dominating balls of two nodes with distance at most one? There are only a few work that has been done for studying the approximation algorithms for minimum Connected Dominating Set problem in unit ball graph. Hansen and Schmutz studied the expected size of a CDS in a random UBG [44]. In 2007, Butenko et al. [8] presented a distributed algorithm for MCDS in UBG

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and proved that the performance ratio of their algorithm is 22 which means there are at most 22 independent nodes that can lie in the union of dominating balls of two nodes with distance at most one. Zou et al. [91] introduced a centralized heuristic algorithm whose performance ratio is 13 C ln 10  15:302. Zhong et al. [89] proposed a distributed algorithm and claimed its approximation ratio to be 16. However, when Kim et al. [53] did the same work later, they found that Zhong et al.’s algorithm is flawed and the performance analysis is incorrect. Kim et al. [53] studied how to construct quality CDS in UBG in distributed environments and showed the performance ratio is 14.937 which is the current best result. For the second stage of connecting a maximal independent set into a Connected Dominating Set, the minimum spanning tree [8] and greedy algorithms [93] can also be used. They are all simple extensions from the Euclidean plane to the threedimensional Euclidean space.

2.3.1 Currently Best Result Kim et al. proposed the current best result in 2010. The contributions they made can be summarized as follows: 1. They first figure out how many independent nodes can be contained in the union of two adjacent unit balls. Lemma 8 (Kim et al. [53]) The number of independent nodes in .B1 [ B2 / n .B1 \ B2 / is at most 20. Lemma 9 (Kim et al. [53]) The number of independent nodes in the union of two adjacent unit balls is at most 22. By combining their new result with some existing one, they reduced the upper bound for the size of a maximal independent set in a UBG from 11OP TCDS C 1 in [8] to 10:917 OP TCDS C 1:083, where OP TCDS is the size of an optimal CDS of the UBG. Theorem 4 (Kim et al. [53]) Let M be an MIS of a UBG G. Then jM j  10:917OP TCDS C 1:083, where OP TCDS is the number of vertices in an optimal CDS of G. 2. Different from others, they first presented a centralized algorithm C-CDS-UBG. The reason they introduced C-CDS-UBG first is that it is both easier to show its performance ratio than for the distributed algorithm and simpler to understand its behavior. C-CDS-UBG has two stages: (a) Compute an MIS. (b) Connect MIS into CDS by using greedy strategy. They showed that the performance ratio of C-CDS-UBG is 14.937. Here are some notations in Algorithm 3 : • N.u/ D fvjv 2 V .G/ n fugg. • N Œx D N.x/ [ fxg.

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Algorithm 3 C  CDS  UBG.G.V; E// (Kim et al. [53]) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

Set M0 D ;, W D ;, V 0 D V Pick a root r 2 V 0 with a maximum degree Set M0 D r, W D N.r/, and V 0 D V 0 n .r [ N.r//: while V 0 ¤ ; do Pick x 2 N.W / such that jN.x/ \ V 0 j is maximized. Set M0 D M0 [ x; W D W [ .N.x/ \ V 0 /, and V 0 D V 0 n .x [ N.x//: end while Set C D r and M D M0  r. while M ¤ ; do Pick a node v 2 N.C / such that jMv;C D max jMx;C jjx 2 N.C /j Set C D C [ u [ Mv;C and M D M n Mv;C end while return C

• For a node set C , N.C / D .[x2C N.x// n C . • Mv;C is a set of MIS nodes adjacent with v and not in C . Lemma 10 (Kim et al. [53]) After Algorithm 3 is finished, jC n M0 j  4:02j OP TCDS j, where OP TCDS is an optimal CDS of G. Theorem 5 (Kim et al. [53]) The size of the node set C generated by Algorithm 3 is no more than 14:937 jOP TCDS j C 1:083, where OP TCDS is an optimal CDS of G. Proof (Kim et al. [53]) By Theorem 4, jM0 j  10:917OP TCDS C 1:083, and by Lemma 10, Algorithm 3 adds at most 4:02jOP TCDS j to connect the nodes in M0 . Therefore, jC j D

t X

X

w.x/ C jM0 j  4:02t C 10:917jOP TCDSj C 1:083

i D1 x2M0 \Si

 14:937jOP TCDS j C 1:083: 3. They introduce a distributed algorithm D-CDS-UBG which is referred from C-CDS-UBG. Therefore, the performance ratio is also 14.937.

2.3.2 PTAS in UBG Not every technique used in the Euclidean plane can be used in the threedimensional Euclidean space. For example, the technique for constructing PTAS for MCDS in UDG [14] does not work in three-dimensional Euclidean space, because Cheng et al.’s [14] proof depends on a geometrical property which holds in the plane but is not true in the space. Zhang et al. [87] discovered a new technique to overcome this difficulty and showed that in any Euclidean space, the minimum Connected Dominating Set in unit ball graph has PTAS. They introduced a new

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Algorithm 4 P TAS f or UBG(Zhang et al. [87]) Input: The geometric representation of a connected unit ball graph G and a positive real number " < 1: Output: A Connected Dominating Set D of G. 1: Let m D d300="e where  is the performance ratio of a constant approximation for MCDS in UBG. 2: Use the -approximation algorithm to compute a Connected Dominating Set D0 of G. For each a 2 0; 1; : : : ; m  1, denote by D0 .a/ the set of vertices of D0 lying in the boundary region of P .a/. Choose a with the minimum jD0 .a/j. 3: For each cell e of P .a /, denote by Ge the subgraph of G induced by the vertices in the central region Ce . Compute a minimum subset De of vertices in e, such that f or each component H of Ge ; GŒDe  has a connect ed component d omi nat i ng H: 

4: Let D D D0 .a / [ [

e2P .a /

(4)

De .

construction, which gave not only a PTAS for the minimum Connected Dominating Set in unit ball graph but also improved running time of PTAS for unit disk graph. In fact, their method can be used to compute CDS for any n-dimensional unit ball graph. In this section, they will introduce their main results. Lemma 11 (Zhang et al. [87]) The output D of the Algorithm 4 is a CDS of G 2/

Lemma 12 (Zhang et al. [87]) The Algorithm 4 runs in time nO.1="

Theorem 6 (Zhang et al. [87]) The Algorithm 4 is a .1 C "/-approximation for CDS in UBG. The proof of Algorithm 4 is in [87].

3

Connected Dominating Set Under Constraints

Wireless network is always a hot topic in research community because of its wide military and civil use. Different from wired networks, there is no physical infrastructure in wireless networks. On-demand routing and table-driven protocols cannot be used in wireless networks [75]. Hence, the idea of virtual backbone [15] in wireless network is well acknowledged and widely studied. Virtual backbone can save searching time for routing and reduce routing table size. A Connected Dominating Set (CDS) is regarded as a good option acting as a virtual backbone. CDS can also have applications in routing, broadcasting, coverage, etc.

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CDS’s construction will have a close relation to the performance of wireless networks. If the size of CDS is too large, then the cost of CDS-based broadcasting will increase since more nodes will be involved to help forward in the networks. Then more redundancy and interferences will be generated in the network too. Hence, a wireless network prefer small CDS. However, broadcasting is not the only operation in wireless network. Routing also happens very frequently in the network. If they reduce CDS size too much, routing cost will increase. In Fig. 3, nodes 4, 5, and 6 construct a minimum CDS. The hop distance between 1 and 3 is 2 and the shortest path between 1 and 3 is pshort est D f1  2  3g. However, the routing path between 1 and 3 going through the CDS becomes P S D f14563g of hop distance 4, twice of that of shortest path. Thus, when they construct a CDS, routing operation should also be taken into consideration and the size of the CDS is not the only criteria used to evaluate the performance of a CDS. In [62], Mohammed et al. proposed the concept of diameter to evaluate the performance of a CDS. Diameter is defined as the length of the longest-shortest path of the graph, where shortest path between any pair of nodes is the path having the smallest hop count among all paths between the pair of nodes. Later, Kim et al. proposed a definition of Average Backbone Path Length (ABPL) which is used to evaluate the average routing path length in a graph in [51]. If the diameter and ABPL of the subgraph induced by a CDS is small, then the CDS is regarded as an efficient CDS. However, both [62] and [51] cannot provide guaranteed routing cost. In [23], MOC-CDS was proposed, How to choose a CDS will determine backbone-based routing protocols’ performance in wireless networks. If the size of the CDS is too large, it is difficult to maintain it and searching time and routing table size cannot be reduced significantly. If the size of the CDS is too small, some characteristics in original networks may not be preserved. For instance, in routing protocols, the length of routing path is an important factor. The smaller the length is, the fewer nodes will be involved in routing process. The benefit is that delivery delay, energy cost, and interference will be reduced since fewer nodes will participate in forwarding packets. In [62], Mohammed et al. also pointed out that in wireless networks, the probability of message transmission failure often increases when a packet is sent through a longer path. Therefore, when designing a routing protocol, they should take path length into consideration, which can be viewed as routing cost. The shorter the routing path is, the less the routing cost will be. That is why shortest path is applied to many routing protocols. However, most current CDS-based routing protocols only focus on how to reduce the size of CDS while sacrificing shortest path properties in original networks. For example, in Fig. 3a, the shortest path between A and C is f1; 2; 3g with length of 2. However, through the chosen minimum CDS, the routing path between A and C will become f1; 4; 5; 6; 3g, with length twice as the original shortest path. To achieve efficient CDS-based routing and broadcasting at the same time, in [22], the author studied a CDS problem – Shortest Path Connected Dominating

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Fig. 3 Illustration of regular CDS and MOC-CDS. (a) A minimum regular CDS. (b) A minimum MOC-CDS

Set (SPCDS), which is a minimum node set including all intermediate nodes in every shortest path between any two nodes in the network. Due to this constraint, the length of path between any two nodes will not be increased even if the path is constructed through SPCDS. In [23], they study a special CDS problem – Minimum rOuting Cost Connected Dominating Set (MOC-CDS). First of all MOC-CDS is also a CDS. In addition, MOC-CDS has a constraint that between any pair of nodes, there exists at least one shortest path in the network, all of whose intermediate nodes must be included in MOC-CDS. Thus, packages can be delivered through MOC-CDS with the same routing cost as that in the original network. For example, in Fig. 3b, according to the constraints of MOC-CDS, node .2; 4; 5; 6; 8/ are selected forming a minimum MOC-CDS. Hence, the shortest path between 1 and 3 will still be f1; 2; 3g with same length of 2 as that in the original network. However, the size of CDS will increase a lot when they put shortest constraint on CDS. To solve this problem, in [24], the shortest constraint is relaxed. The problem named as ˛ Minimum rOuting Cost Connected Dominating Set (˛MOC-CDS) is studied. Besides all properties of regular CDS, ˛-MOC-CDS also has a special constraint – for any pair of nodes, there is at least one path all intermediate nodes on which belong to ˛-MOC-CDS, and the number of the intermediate nodes is smaller than ˛ times of that on the shortest path in the original network. ˛-MOC-CDS is also studied in [32]. To save the energy in the unnecessary directions, directional antennas are used in [25] and MOC-DVB is studied. In Sect. 3.1, they will recall the network models first. The details of the construction of the CDS under constraints with be introduced in Sect. 3.2.

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Network Model

The network without directional antennas is modeled as a general graph when obstacles [65] exist or transmission radius are different in one network. If no obstacles exist and transmission ranges are same in one network, the network is modeled as a unit disk graph. When the directional antennas are used, the network is modeled as a directional graph where obstacles exist and transmission ranges are same.

3.1.1 General Graph Two nodes can communicate when they satisfy two requirement. The first one is transmission range constraint. In wireless networks, nodes may be from different providers. In one network, nodes may have different transmission ranges. In general graph model, two nodes can communicate with each other when they are in each other’s transmission ranges. Two nodes cannot communicate when one is in the other one’s transmission range but the other one is not in the “one’s” transmission range. The second one is non-obstacle constraint. Communications in wireless network are based on radio wave transmissions which are easily prevented by obstacles (like buildings, mountains). If two nodes satisfy the first constraint and there is no obstacle between them, then the two nodes can communicate. Otherwise, they cannot communicate even though they are physically close to each other enough. In sum, two nodes can communicate when and only when they are close to each other and no obstacles exist between them. A general graph is denoted as G D .V; E/. V is a node set in the network. For each pair of nodes in the network, if there is a directed link between them in the network, then there exists an edge in E. 3.1.2 Unit Disk Graph In unit disk graph (UDG) networks, it is assumed that all nodes in one network have the same transmission range and no obstacles are in the network. Any two nodes can communicate when and only when they satisfy the first constraint mentioned in Sect. 3.1.1. A UDG is denoted as G D .V; E/ where V is a node set and E is an edge set. 3.1.3 Directed Graph In directed graph network, directional antennas are used. And you can choose the directions to forward messages through switching on antennas. In this kind of network, it is assumed that all nodes have the same transmission range and obstacles are there. There are two kinds of reception models – directional reception and omnidirectional reception. In their work, omnidirectional reception is used. Uniform and nonoverlapping directional antennas are used. Directional antennas will divide one node’s transmission range into uniform sectors. In Fig. 4, four uniform directional antennas are deployed on each node.

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Hence, each node’s transmission range is divided to four sectors. Sector i of node u is denoted as ui . In Fig. 4, the first sector of A is denoted as A1 . Nodes B and C are in the fourth direction of A (A4 ). If A4 is switched off, then B and C cannot receive any messages from A, even though they are in A’s transmission range. Thus, in directed graph network, if one node v wants to receive messages from node u, they must first satisfy two requirement in Sect. 3.1.1, and they also need to satisfy that u’s sector where v is switched on. Compared to the prior two models with node set and edge set, a directed graph has a direction set (sector set). Thus, one directed graph is denoted as G D .V; E; S /, where V is a node set, E is an edge set, and S is a sector set. In Fig. 4, V D fA; B; C; Dg, E D fA4 !B; A4 !C; B 4 !D; B 2 !A; C 2 ! A; C 4 ! D; D 2 ! B; D 2 ! C g, S D fA1 ; A2 ; A3 ; A4 ; B 1 ; B 2 ; B 3 ; B 4 ; C 1 ; C 2 ; C 3 ; C 4 ; D 1 ; D 2 ; D 3 ; D 4 g. The three models are used in the following algorithms in Sect. 3.2. In every model, the length of a path is equal to the hop count of the path. And the distance of two nodes is the hop distance which is equal to the hop count on the shortest path between the two nodes.

3.2

Construction of CDS Under Constraints

To construct a CDS with guaranteed routing cost, many researches are done on this topic. In [82], Wu et al. proposed an algorithm to construct a CDS. The first step in [82] is to find a CDS including all intermediate nodes of all shortest path. Actually, this CDS in the first step can solve in polynomial time, which is studied in [22].

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3.2.1 An Exact Algorithm to Construct SPCDS SPCDS is a CDS. SPCDS also has another constraint that for each pair of nodes, the intermediate nodes on all shortest path should be in the SPCDS. To solve SPCDS, an equivalent problem 2PCDS is proposed in [22]. 2PCDS is a CDS firstly. 2PCDS also requires that all intermediate nodes on shortest path of distance 2. It is proved in [22] that SPCDS is equivalent to 2PCDS. Lemma 13 SPCDS and 2PCDS is equivalent to each other. Proof The first part is to prove that an SPCDS is a 2PCDS. In SPCDS, all shortest paths of any length will be considered. Thus, SPCDS also includes the shortest path of length 2. Thus, an SPCDS is also a 2PCDS. The second part is to prove that a 2PCDS is also an SPCDS. Suppose there is a 2PCDS. For any pair of nodes u, v of distance bigger than or equal to 2, suppose that there is a shortest path between u and v denoted as pu;v D fu; w1 ; w2 ; : : : ; wk ; vg. From pu;v , they know that the distance between u and w2 should be of distance 2, then they can find a node w01 from the 2PCDS to replace w1 . The distance between w01 and w3 is 2, then they can find a node w02 to replace w2 . Finally, they can find 0 a replacement pu;v D fu; w01 ; w02 ; : : : ; w0k ; vg, where all intermediate nodes are in 2PCDS. Thus, a 2PCDS is also a SPCDS.  The algorithm proposed in [22] is based on 2hop local information. The idea of algorithm in [22] is that for one node, if there exists a pair of nodes of distance 2, then the node will be added to 2PCDS. The check time is O.ı 2 / O.n/ D O.ı 2 n/, where ı is the maximum node degree in the network and n is the number of nodes in the network. Finally, a SPCDS is also constructed optimally in polynomial time.

3.2.2 MOC-CDS Problem The size of SPCDS is relatively large, so in [23], MOC-CDS is proposed. Different from SPCDS, MOC-CDS does not require all nodes on the shortest path should be selected. MOC-CDS requires that for any pair of nodes, there exists at least one shortest path on all intermediate nodes that are in MOC-CDS. Thus, the size of MOC-CDS will be reduced greatly compared to that of SPCDS. Similar to SPCDS, another problem 2hop-CDS is proposed in [23] too. Referring to the proof of equivalence in [22], the equivalence between 2hop-CDS and MOC-CDS is proved in [23]. It is proved that MOC-CDS or 2hop-CDS is NP-hard in a general graph. The proof of NP-hard is based on the reduction of 2hop-CDS from set cover [37] problem. Definition 12 (Set cover) Given a collection M of subsets of a finite set X such that [B2M B D X , find a minimum subcollection B  M such that [B2M B D X. Decision version of 2hop-CDS: Given a positive integer k and a graph G, determine whether there is a 2hop-CDS of size at most k in G.

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Decision version of set cover: Given a collection M of subsets of a finite set X and a positive number h, determine whether there is a set cover M0 of at most h subsets in M. Theorem 7 A minimum 2hop-CDS is NP-hard in a general graph. Proof Create a node aB , 8B 2 M. Create a node bx , 8x 2 X . Two additional nodes will be created c and d . c and d are connected to all aB ’s, 8B 2 M. Meanwhile d is also connected to all bx , 8x 2 X . Edge between bx and aB represents that x is in B. It is claimed that M has a set cover B of size at most k when and only when G has a 2hop-CDS of size at most k C 1. There are two parts of the proof. The first one is to prove that a 2hop-CDS of size k C 1 can be constructed when jBj  k. If there is a set cover B of size at most k, then the corresponding nodes fd g [ faB jB 2 Bg in G construct a 2hop-CDS of size at most k C 1. c can be dominated by nodes in faB jB 2 Bg. 8z 2 fbx jx 2 X g [ fuB jB 2 g, z can be dominated by d . The node subset satisfies 2hop-CDS’s characteristic of dominating. Since d is connected to any other node in the node set, the node set is connected. For any two nodes e; f 2 G of distance 2, fe; d; f g must be a shortest path. For any two nodes c and e 2 of distance 2, it is true that e 2 fbx jx 2 X g [ fd g. There should be a path fe; aB ; cg 2 G – a shortest path between e and c. Therefore, the construction of the node set should be a 2hop-CDS of size at most k C 1. The second part is to prove that if there is a 2hop-CDS of size at most k C 1, set cover has a solution of size at most k. The distance between c and any node e in fbx jx 2 X g is 2. Any shortest path between c and e must pass a node a. Thus, the set cover is B D fBjaB 2 2hopCDS and B 2 Mg. For any two different element U; V 2 M of distance 2 in G, each shortest path must pass one node corresponding to element in M. Thus, jMj  k. In summary, a minimum 2hop-CDS is NP-hard.  Based on the above proof, it is trivial to conclude that a minimum MOC-CDS is NP-hard as well. Set cover does not have a solution of performance ratio of ln , where  is the maximum cardinality of element in M and  is a nonnegative number smaller than 1, unless NP  DT IME.nO.log log n/ /. Thus, there cannot be a solution to MOC-CDS with performance ratio of lnı, where ı is the maximum node degree. To construct a MOC-CDS, the distributed algorithm named FlagContest is proposed. For a given graph G D .V; E/, each node v is assigned a unique id, denoted as id.v/. id aims to solve the situation that more than one node can be selected at one time. The solution is to select the one having highest id . At the very beginning, each node will collect the pairs of neighbor nodes of distance 2. Each node having nonempty pair set will compute the times the node is acting as an intermediate node by counting the number of pairs in its pair set. Send the calculated

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times to its neighbors. When one node receives times, it will choose one with the biggest times and send a flag to the selected one. If one node collect flags from all its neighbors, turn to black and send out its pair set 2 hops away. Nodes receiving the pair set will remove those pairs in the received pair sets from their own pair sets. Update the pair sets and recalculate the times acting as an intermediate node. The algorithm will stop when all nodes’ pair sets are empty. The black nodes colored during the process will construct a 2hop-CDS. The algorithm proposed in [23] is proved correct. Meanwhile, the performance ratio is proved – FlagContest produces approximation solution with performance ratio H..ı2 //, where H is the harmonic function and ı is the maximum node degree of input graph.

3.2.3 CDS Under Relaxed Constraints The above CDS under shortest path constraint does achieve efficient routing; however, from their simulation results, the size of CDS will increase a lot compared to other CDS size. To reduce CDS size, [24] relaxed the shortest path constraint and proposed ˛ Minimum rOuting Cost Connected Dominating Set (˛-MOC-CDS). In ˛-MOC-CDS, the shortest path constraint is relaxed to that for any two nodes, there exists at least one path, and the number of intermediate nodes on this path is smaller than ˛ times that of on the shortest path. Meanwhile, all intermediate nodes on this path must be in ˛-MOC-CDS. An equivalent problem ˛-2hop-CDS is proposed in [24] and the equivalence is proved too. Actually, MOC-CDS is a special case of ˛-MOC-CDS, when ˛ D 1. In [24], Ling et al. proposed a distributed localized algorithm to construct ˛-MOC-CDS which is introduced as follows. The algorithm is based on the ˛ C 1 local topology information. The topology of the local information on each node v is denoted as G.v/ D .Vv ; Ev /. A weighted graph Gw .v/ D .Vw .v/; Ew .v/; Ww .v/ can be induced by G.v/, where Vw .v/ D V .v/, Ew .v/ D E.v/, and Ww .v/ record the weight between any two nodes. For any two nodes, if there is an edge between them in E.v/, then the weight is 0. Otherwise, it is 1. For each node v, it will record a pair set which stores the pairs where distance of the two nodes in each pair is 2 in G.v/ and Gw .v/, there is two edges between v and the two nodes, and the total weight of the two edges should be smaller than ˛. Send out the pair set to its neighbors in G.v/. If one node receives all pair sets from its neighbors, it will order them based on their id and cardinality of the pair set. One node has higher rank when its cardinality is bigger. When two nodes have the same cardinality, the node that has higher id will have higher rank. After ordering all neighbors, one node will send flags to those nodes whose pair set has no intersection with any pair set of any node having higher rank. If one node receives all flags from its neighbors, it will turn black, send out its own pair set two hops away, and update its weighted graph. If node v turns black, then for any pair of nodes of distance 2 in Gw .v/ and there is a path in Gw .v/ of weight w, then edge between the two nodes in the pair should be added to Gw .v/ of weight 1 C w. When one node receives the pair set, it will remove the pairs in the received pair set from its own pair set. The algorithm will stop when all pair sets are empty.

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From the simulation results in [24], the size of CDS will reduce greatly, while the routing cost will not increase a lot when ˛ D 3. Thus, there is a tradeoff between CDS size and routing cost.

3.2.4 Du’s Relaxation In [32], Du et al. proposed a CDS with guaranteed routing cost named as GOC-MCDS. GOC-MCDS is also a ˛-MOC-CDS. In [24], Ling et al. propose a distributed algorithm; however, no performance ratio is give. In [32], one centralized and one distributed algorithms are proposed with performance ratio. In the following two parts, the two algorithm will be recalled in details. 1. GOC-MCDS-C The centralized algorithm follows the steps of regular MCDS algorithms. The algorithm has two stages. In the first stage, a maximum independent set (MIS) is constructed. For any pair of MIS nodes of distance smaller than 4, find a shortest path and add all intermediate nodes on the path to the CDS. Finally, the selected MIS combined with the added nodes will construct a GOCMCDS. The constructed GOC-MCDS will guarantee that for any pair of nodes, the length of the routing path through GOC-MCDS will be smaller five times that of the shortest path between the two nodes. And the size of the GOC-MCDS will be smaller than jC j C jI j  121 jI j  443 32 optGOC M CDS C 201 32 , where jC j is the number of added nodes in the second stage, jI j is the size of the MIS in the first stage, and optGOC M CDS is the size of the optimal solution to GOC-CDS. 2. GOC-MCDS-D The distributed algorithm also has two stages. At the very beginning, the color of all nodes is white. During the algorithm, some node will keep white, some will change to black, and some will change to gray. All the black ones will construct a GOC-MCDS. Meanwhile, each node will be assigned a unique id . The algorithm will be introduced as follows. The first stage is to find an MIS too. Each white node will send its own id to its neighbors. When one node receives all id ’s from its neighbors and its own id is smaller than all id ’s it receives, then the node will change to black and send the message “black” to its neighbors. Once a white node receives the “black,” turn to gray. When the algorithm stops, there are only black or gray nodes in the graph. In the second stage, more nodes are added combining with black nodes in the first stage to form a GOC-MCDS. Every black node s will send its own id.s/ to its neighbors. When one node t receives id.s/, t will put its own id.t/ together with id.s/. t will send the pair (id.s/,id.t/) to all its neighbors. For each node w with id.w/, when w receives a pair (id.s/,id.t/) having id.s/ < id.t/, w will send the tuple (id.t/,id.w/,id.s/) to the neighbor with id.s/. When w receives (id.s/,id.t/) and (id.s  /,id.t  /) and id.s/ < id.s  /, w will send a tuple (id.s  /,id.t  /,id.w/,id.s/,id.t/) to the neighbor with id.t/. When w receives (id.s/,id.t/) and id.s  / and id.s/ < id.s  /, w will send a tuple (id.s  /,id.w/,id.t/,id.s/) to the neighbor with id.t/. Otherwise, w will send tuple (id.s/, id.t/, id.w/,id.s /) to the neighbor with id.s  /. when the node t receives a tuple (id.s  /,id.t  /,id.w/,id.s/,id.t/) or

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(id.s  /,id.w/,id.t/,id.s/), t will send the tuple to its neighbors with id.s/. Each black node of id.s/ will have a tuple set M storing tuples (id.w1 /,id.t/,id.s/), (id.w2 /,id.w1 /,id.t/,id.s/), and (id.w3 /, id.w2 /,id.w1 /,id.t/,id.s/). Perform the policy of “computation on M” on each node. When a node id.x/ receives a tuple (. . . ,id.x  1/,id.x/,. . . ), x will turn to black and pass the tuple to the node id.x  1/. Computation on M: Pick up .id.h/; : : : ; id.2/; id.1// 2 M and send the tuple to the node of id.2/. Delete all tuple from M starting with id.h/. This distributed algorithm has the same performance ratio as the centralized one.

3.2.5 MOC-DVB In wireless networks, saving energy is always the critical topic. In previous literature, directional antennas are used to help save energy in many operations, like broadcasting [21]. Thus, Ling et al. apply directional antennas to the wireless networks. The wireless networks with directional antennas are modeled as directed graphs. In such a wireless, Minimum rOuting Cost Directional VB (MOC-DVB) is studied in [25]. Fewer sectors selection will help save energy and reduce interference. MOC-DVB requires that if the source node initiates a message in omni-direction, then the message should be delivered by a shortest path on all intermediate directions that are in MOC-DVB. A MOC-DVB itself may be disconnected. For example, in Fig. 5, the gray sectors construct a minimum MOCDVB. However, the MOC-DVB cannot induce a connected graph. Even though a connected subgraph of MOC-DVB, any messages can be successfully routed or broadcasted. If A has a message with destination F , the directional path should be fA > B 1  > D 1  > F g which means that A initiate a message in omnidirection, B use omni-reception and forward it to D through the first direction, D receive the message from B then forward it through the first sector of D, and lastly, D forward the message to F through the first direction of D. Meanwhile, it can go back from F to A through path fF  > E 3  > C 3  > Ag. Originally, the directed graph is strongly connected. In [25], Ling et al. prove that MOC-DVB is NP-hard. A distributed and greedy algorithm is proposed in [25]. For each direction ui , a direction id is assigned, denoted as d.ui /. For each direction wi , it will compute a pair set which stores that the pair .u; v/ of distance 2, u’s message can be delivered to v through direction wi . Each direction will send out its times acting as an intermediate direction to its neighbors. When one node receives the times from all its neighbors, it will select the direction with the maximum times and send out a flag to the direction. When one direction collects flags from all the directions where it is, the direction will be added to MOC-DVB. The selected direction’s pair set will be sent out 2 hops away. When one direction receives the pair set, it will remove the pair in the received pair set from its own pair set. The algorithm stops when all pair sets are empty.

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Fig. 5 Comparison between CDS and MOC-DVB

E F

C A D B Directions in MOC-DVB

3.2.6 PTAS for MCDS with Routing Cost Constraint [31] Du et al. in [31] showed that ˛MOC  CDS has a PTAS for ˛  5. However, the problem is still open for 1  ˛ < 5. Actually, the main difficulty for 1  ˛ < 5 is how to connect a Dominating Set into a feasible solution for ˛MOC  CDS . So far, there is no a good method to do that without increasing too much number of nodes. They will show some of their main results. First, they use a square Œ0; q Œ0; q to cover the whole graph G D .V; E/, then construct the other square called P .0/. P .0/ is divided into p 2 cells. Each cell is an a a square called e. The side length for each cell a is equal to 2.˛ C 2/k, where p D 1 C dq=ae and k is a positive integer. Note that all cells are disjoint, so they should ensure each cell only has left and lower boundary. The union of all cells covers the square Œ0; q Œ0; q. For each cell e, construct two squares with the same center as e. The side length are .a C 4/ .a C 4/ and .a C 2˛ C 4/ .a C 2˛ C 4/. They use e c to represent the central area which is the closed area bounded by the first square and e b to represent the boundary area which is the area between the second square (exclude boundary) and cell e (include boundary). e cb is the open area bounded by the second square which is e c [ e b . Now, for each cell e, they study the following problem: [Du et al. in [31]] LOCAL(e): Find the minimum subset D of nodes in V \ e cb such that 1. D dominates all nodes in V \ e c . 2. mD .u; v/  ˛ for any two nodes u; v 2 V \ e c with m.u; v/ D 1 and fu; vg \ e 6D ;.

Let De denote the minimum solution for the LOCAL.e/ problem. Define D.0/ D [e 2 P .0/ De where e 2 P .0/ means that e is over all cells in partition P .0/. Lemma 14 (Du et al. in [31]) D0 is a feasible solution of ˛MOC  CDS and D0 4 can be computed in time nO.˛ / where n D jV j. Lemma 15 (Du et al. in [31]) jD.0/ C jD.1/j C    C jD.k  1/j  .k C 8/jD  j.

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Algorithm 5 Algorithm PTAS(Du et al. in [31]) 1: Compute D.0/, D.1/, . . . , D.k  1/; 2: Choose i  , 0  i   k  1 such that 3: jD.i  /j D min.jD.0/j; jD.1/j; : : : ; jD.k  1/j/; 4: Output D.i  /.

Theorem 8 (Du et al. in [31]) Algorithm PTAS (Algorithm 5 ) produces an approximation solution for ˛MOC-CDS with size jD.i  /j  .1 C "/jD  j 4

and runs in time nO.1=" / .

4

Strongly Connected Domination Set

In many scenarios, the communications in multi-hop wireless networks are asymmetric in real world. For example, nodes may have different transmission due to several reasons such as energy concern and cost. Therefore, it is possible to consider that node u can communicate with node v, but v cannot communicate with u. In such case, they need to model this kind of network to a directed graph G D .V; E/. In directed graph, they usually consider a strongly Connected Dominating Set (SCDS) as a virtual backbone. The problem minimum SCDS seeks an SCDS of the smallest cardinality in a specified digraph. Given a directed graph G D .V; E/, S is strongly connected if for every pair u and v 2 S , there exists a directed path from u to v in the directed graph induced by S , i.e., G.S /. Formally, the SCDS problem can be defined as follows: Definition 13 (Strongly Connected Domination Set) Given a directed disk graph G D .V; E/, find a subset S  V with minimum size, such that the subgraph induced by S , called G.S ), is strongly connected and S forms a Dominating Set in G. The SCDS problem is easy to be proved as an NP-hard problem because the MCDS problem in UDG is NP-hard and UDG is a special case of DG. Du et al. [28] consider the SCDS in directed graphs. Differ from undirected graph, an MIS is not necessary a DS in a directed graph. Therefore, they cannot find an MIS and connect it into a SCDS. Instead, they have to find a DS in the first stage, then connect it to a SCDS. Based on this approach, they present two constant-approximation algorithms for computing a minimum SCDS in DG if the transmission range is bounded which are Connected Dominating Set using the Breath First Search tree

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(CDS-BFS) and Connected Dominating Set using the minimum nodes Steiner tree (CDS-MNS) algorithms. The main differences of these two algorithm are the second stage for connecting the obtained Dominating Set to an SCDS. They assume that G is a strongly connected graph in order to guarantee that the graph G has a solution for the SCDS problem. In 2009, Li et al. [55] proposed another approximation algorithm for SCDS. In this section, they will introduce some of their main results. They used G D .V; E/ to represent a digraph. A node in G is said to be a i nternal node if its out-degree is not zero and an si nk node otherwise. I.G/ is the set of internal nodes in G. If a subgraph’s vertex set is exactly V , they said that the subgraph is spanning. They call a subgraph of G an arboresce nce rooted at a node s 2 V if the in-degree of s is zero, and the in-degree of any other node is exactly one in this subgraph. An s-arborescence is also an arborescence rooted at s. For each node v 2 V; ı C .v//ı  .v/) is the out/in-degree of v in G, and N C .v//N  .v/) is the set of out/in-neighbors of v in G. For any subset U of V , they denote by GŒU  the subgraph of G induced by U . G R is the reverse of G which is a digraph obtained from G by reversing the direction of every arc. In other words, G R D .V; E R / with E R D f.u; v/j.v; u/ 2 Eg: Lemma 16 (Li et al. [55]) Let G D .V; E/ be a strongly connected digraph and s be an arbitrary node in V . Suppose that T1 is a spanning s-arborescence in G, and T2 is a spanning s-arborescence in G R . Then I.T1 / [ I.T2 / is an SCDS of G. Theorem 9 (Li et al. [55]) Suppose that A is a -approximation for SAFIN. Then Algorithm S CDS.A/ produces an SCDS of size at most 2  opt  2 C 1, where opt is the size of a minimum SCDS. By Lemma 16, they introduce the Spanning Arborescence with Fewest Internal Nodes (SAFIN) problem: [Spanning Arborescence with Fewest Internal Nodes (SAFIN) problem] Given a digraph G D .V; E/ and a source node s 2 V , compute a spanning s-arborescence T with minimum jI.T / n fsgj.

It is easy to argue that SAFIN is at least as hard as minimum CDS. Therefore, they use SAFIN to find the minimum SCDS. Let A be an arbitrary polynomialtime approximation for SAFIN. The Algorithm 6 described a polynomial-time approximation algorithm S CDS.A/ for minimum SCDS with the parametric A as a subroutine. To reduce the running time, they selected a small subset S of candidates for the source node in the beginning. Specifically, let u be the node u which minimize satisfying that min.ı  .v/; ı C .v// over all nodes v 2 V . They said that S consists of u and all its out-neighbors if ı  .u/ > ı C .u/. Otherwise, S consists of u and all

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Algorithm 6 S CDS.A/ (Li et al. [55]) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

u arg minv2V min.ı  .v/; ı C .v//I  If ı .u/  ı C .u/ then S u [ N  .u/I else S u [ N C .u/I U VI for each s 2 S, spanning s-arborescence in G output by A; T1 T2 spanning s-arborescence in G R output by A; if jI.T1 / [ I.T2 /j < jU j then U I.T1 / [ I.T2 /; output U

Algorithm 7 .A/ 1: B sI 2: while h.B/ > 0, 3: construct .B/; 4: find a cheapest T 2 .B/I 5: B B [ I.T /; 6: output a spanning s-arborescence of GhBi

its in-neighbors. Clearly, there must be at least one node in the selected S in every SCDS. Next, they will show their .1:5H.n  1/  0:5/-approximation algorithm A (Algorithm 7 ) for SAFIN where H is the harmonic function. For such A, the algorithm SCDSA is a .3H.n1/1/-approximation algorithm for minimum SCDS by Theorem 9. They introduce some notations and terminologies for algorithm A(Algorithm 7). They assigned each node a unique ID. And s is the source node. GhBi is the spanning subgraph of G whose arc set consists of all arcs of G leaving from the nodes in B for any B  V containing s. If a strongly connected component of GhBi neither contains the source s nor has an incoming arc, it is an orphan with respect to B . The node with the smallest ID in this component is its head for each orphan component. The number of heads (equivalently, the number of orphans) with respect to B is denoted as h.B/. Clearly, a spanning s-arborescence is in GhBi if and only if h.B/ D 0. They use .B/ to denote the set of candidates. The construction of .B/ is described can be found in [55]. Theorem 10 (Li et al. [55]) The approximation ratio of the Algorithm 7 for SAFIN is at most 1:5H.n  1/  0:5.

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Weakly Connected Domination Set

In ad hoc networks, topology changes very often, and most of the changes are localized which means it is in a small area of the network. However, they do not want this kind of change effect the whole networks. Therefore, they need a new network structure such that the local changes will not be seen by the entire network which is cluster. Like the virtual backbone in wireless networks, they would like to control less nodes. In cluster, such node is called clusterhead . One method used in cluster is based on domination in graphs which they already introduced in the previous section. Of course, they wish to find the smallest number of clusterhead which is a minimum Dominating Set. So they can simplify the network structure as much as possible. Unfortunately, it is a NP-complete problem to find a minimum size Dominating Set in a general graph. In order to solve this problem, they can ensure that the clusterheads are adjacent to each other or at least reasonably close to each other. Hence, they can try to use CDS. However, minimum CDS problem in general graph is also NP-complete. Although a CDS provides an obvious routing, the connectivity requirement causes a very large number of clusterhead . Based on this, a Weakly Connected Dominating Set is proposed. Definition 14 (Weakly Connected Domination Set (WCDS)) The subgraph weakly induced by W is Sw =.N ŒW ; E \ .W N ŒW //, where N ŒW  is a set of one-hop neighbor of W and W . The set W is a Weakly Connected Dominating Set (WCDS) if Sw is connected and W is a DS. According to the definition of WCDS, the size of WCDS is less or equal to the size of CDS. Thus, they feel that WCDS is a better method for clustering. An approximation algorithm with ratio O.ln/ was proposed by Chen [11], where  is the maximum degree of the input network. Alzoubi et al. [4] present two distributed algorithms with the ratio 5 and 122.5 in 2003. Dubhashi [33] gave another approximation algorithm with the same ratio as Chen’s in 2003. Du et al. [30] proved that: Theorem 11 (Du et al. [30]) The least upper bound for the ratio of sizes between MIS and the minimum WCDS for UDG is five. To prove Theorem 11, they first proved that five is an upper bound. They considered an MIS M and a minimum WCDS D. For any vertex u, the disk with radius one and center u is called the dominating disk of u. Suppose u 2 D and the dominating disk of u contains n elements which is M , a1 , . . . , an , and order them in clockwise. Connect ua1 , ua2 , . . . , uan (Fig. 2). Because a1 , a2 , . . . , an are independent set, they have ja1 a2 j > 1, . . . , jan a1 j > 1. Hence, they can get †a1 ua2 > 3 , . . . , †an ua1 > 3 . Therefore, n  5. So five is an upper bound is proven.

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ei ai

ai+1

di

xi

bi

xi+1

yi

di+1

yi+1

bi+1 ci

ci+1

Fig. 6 An example for jM j D 5jDj

They constructed an example to show that five is the least upper bound. They showed that jM j D 5jDj where M is an MIS and D is a minimum WCDS. They drawn the example as follows: 1. Construct a sequence of points x1 ; y1 ; x2 ; y2 ; : : : ; xn ; yn on a line ` such that jx1 y1 j D jy1 x2 j D jx2 y2 j D    D jxn yn j D 1, as shown in (Fig. 6). 2. Construct four points ai ; bi ; ci ; ei on the boundary of the dominating disk of each xi such that segment ai bi is perpendicular to line ` and †ei xi ai D †ai xi bi D †bi xi ci D 3 C " where " > 0 is a small constant. 3. Construct the fifth point di in dominating disk of xi such that di is on line ` and at the middle between segments ai bi and ai C1 bi C1 . They study the UDG G with vertex set fai ; bi ; ci ; di ; ei ; xi ; yi j i D 1; 2; : : : ; ng. By calculating the distance of every pair of MIS, all the distance are larger than 1. Therefore, 5 is the least upper bound.

6

Fault-Tolerant Virtual Backbone in Wireless Network

Due to many factors such as mobility of nodes and limitation of energy supply, the topology of a wireless network is expected to be dynamic. However, most of the existing virtual backbone computation algorithms/protocols assume a static network topology. Naturally, even a slight topology change can make current virtual backbone structure obsolete. In such case, the backbone structure should be completely reconstructed. As a result, in a highly dynamic wireless network, those algorithms could work very inefficiently since they may need to reconstruct

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virtual backbone very frequently. Motivated by this issue, the problem of computing fault-tolerant virtual backbone is introduced [18]. In graph theory, a graph is said to be disconnected only if there is a pair of nodes having no path between them in the graph. A graph is k-connected if the graph is still connected after removing any k  1 nodes in the graph. They say a node dominates a set of nodes, if the node is adjacent to all of the nodes in the set in the graph. Similarly, a subset of the nodes in a graph dominates the other nodes outside the subset only if each of the nodes outside the subset has at least one neighbor in the subset. In addition, they say the subset m-dominates the nodes outside the subset only if each of the nodes outside the subset has at least m neighbors in the subset. Note that the CDS computation algorithms by Guha and Kuller and many conventional CDS computation algorithms are targeted to produce 1-connected 1-Dominating Set. Let G D .V; E/ denote a connected graph representing a wireless network and C be a CDS of G, which can serve as a virtual backbone of the wireless network. In [79], the additional requirements for C to be a fault-tolerant virtual backbone is formally defined as follow: • k-connectivity: C has to be k-connected so that the virtual backbone can survive after k  1 backbone nodes fail. • m-domination: each node x 2 .V n C / has to adjacent to at least m nodes in C so that x can be connected even after m  1 adjacent backbone nodes fail. Here, k and m determine the level of fault tolerance. By enforcing the two new requirements on C , any pair of nodes can communicate with each other via the virtual backbone even after any minfk; mg nodes fails. The rest of this chapter is devoted to the survey of the recent advances in computing fault-tolerant CDS in wireless network.

6.1

Preliminaries

They first introduce several notations and definitions to simplify further discussion. G D .V; E/ is a connected graph. V D V .G/ and E D E.G/ are the set of vertices and edges in G, respectively. For a pair of nodes u; v 2 V .G/, Eucd i st.u; v/ is the Euclidean distance between them. Let N.u/ be the set of nodes adjacent to u, and N Œu D fug [ N.u/. There are two common graph models to abstract wireless network, unit disk graph (UDG) and disk graph (DG). Unlike UDG, DG is used to abstract nonhomogeneous wireless network, in which nodes may have different maximum transmission range. As a result, DG has directional edges. Clearly, UDG is a special case of DG, and thus, every NP-hard problem in UDG is still NP-hard in DG. In addition, every algorithm working in DG also works in UDG. Definition 15 (Disk Graphs(DG)) The nodes in V are located in the twodimensional Euclidean plane, and each node vi 2 V has a transmission range

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ri 2 Œrmin ; rmax . A directed edge .vi ; vj / 2 E if and only if d.vi ; vj / 2 E, where d.vi ; vj / denotes the Euclidean distance between vi and vj . Such graphs are called disk graphs (DG). An edge .vi ; vj / is bidirectional if both .vj ; vi / and .vi ; vj / are in E, i.e., d.vi ; vj /  min ri ; rj . Usually, they only study the CDS problem in disk graphs where all the edges in the network are bidirectional, called disk graphs with, bidirectional links (DGB). In this case, G is undirected. Definition 16 (Disk graph with bidirectional links (DGB)) A DGB of a graph is a DG of the graph without all of its directional links. Definition 17 (m-Dominating Set (m-DS)) D  V is an m-Dominating Set (m-DS) of G if 8u 2 .V n D/, u is adjacent to at least m nodes in D. Definition 18 (k vertex connectivity) A graph G is k vertex connected if G is still connected after any k  1 nodes are removed from G. In the rest of this section, they will use “k vertex connected” and “k-connected,” interchangeably. Definition 19 (k-Connected m-Dominating Set (.k; m/-CDS)) C is a k-Connected m-Dominating Set, .k; m/-CDS, if (1) a graph induced by C is k-connected and (2) C is a m-DS. For simplicity, they will call .k; k/-CDS by k-CDS. Definition 20 (Cut-vertex) A vertex v 2 V is a cut-vertex if and only if the induced graph by V n fvg is disconnected. Definition 21 (Block) A subset B  V is a block if and only if B is a maximal connected subgraph of G D .V; E/ without any cut-vertex. Definition 22 (Leaf block) A block of G is called a leaf block if it contains only one cut-vertex.

6.2

Approximations for Computing .k; m/-CDS for k  3 and m  1

The minimum CDS problem, which is a problem of computing minimum (1,1)CDS, of a graph, is a well-known NP-hard problem. Therefore, its generalized version, the problem of computing minimum .k; m/-CDS is also NP-hard. As a result, many researches have been conducted to find an approximation algorithm for this problem. In this section, they introduce approximation algorithms for computing .2; 1/-CDS, .1; m/-CDS, .2; m/-CDS, and .3; m/-CDS for any m  1.

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6.2.1 A Constant Approximation for .2 ; 1/-CDS The CDS augmentation algorithm (CDSA) in [79] is the first approximation algorithm to compute multi-connected CDS. Given a 2-connected UDG G, CDSA first utilizes an existing approximation algorithm to compute 1-CDS of G, denoted by C1;1 . Once this is done, the algorithm augments C1;1 to increase its connectivity and generate a (2,1)-CDS, denoted by C2;1 as follows: 1. Let L D ; and C D C1;1 . Find every block in C1;1 and put all of them in L. 2. Find a leaf block B 2 L and find the shortest path P from v 2 B to u 2 C1;1 n B such that v and u are not cut-vertices. Add all nodes in P to C . If there is more than one block in L, repeat this step. Figure 7 illustrates an example showing how this algorithm works. It is always true that there is at least one block in any 1-CDS of any graph with more than one vertex. Starting from a block of C1;1 (a maximal 2-connected subgraph of C1;1 ), the algorithm finds a shortest path P from a node u in the block to another node v in C1;1 , which is outside the block such that both of u and v are not cut-vertices in C1;1 . As a result, the union of the block, P , and some additional nodes from the original 1-CDS form a larger 2-connected subgraph C , which can be considered as a larger leaf block. By repeating such procedure until the subgraph includes all nodes in C1;1 , the subgraph, which is originally a 2-connected subgraph of C1;1 , grows into a .2; 1/-CDS. Using geometry, the authors proved that each path P includes at most 8 new nodes. Also, each time P is added and a 2-connected subgraph is augmented, at least one node in the original C1;1 joins to the augmented 2-connected subgraph. As a result, they have 2CDS D 1CDS C 8 1CDS;

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Fig. 7 These figures illustrate how a (2,1)-CDS is computed. In figure (a), the node set fa; b; c; d; eg forms a 1-CDS, C1;1 . In figure (b), the node set fb; cg is a leaf block of C1;1 . In figure (c), two blocks fb; cg and fa; cg sharing a cut-vertex c of C1;1 are merged into a larger 2-connected subgraph fa; b; c; f g by adding a path fb; f; ag. In figure (d), by adding a new path from node a in the 2-connected subgraph to another node d in C1;1 outside the subgraph, they have a larger 2-connected subgraph, fa; b; c; d; f; gg. It is important that both a and d are not cut-vertices. In figure (e), one more node h is added to the subgraph, and finally the augmented subgraph contains C1;1 . Therefore, this subgraph is a (2,1)-CDS

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where 2CDS is an output of CDSA, 1CDS is the original 1-CDS, and 8 is the maximum length of intermediate nodes in each path P . Based on the existence of a 6.91-approximation algorithm for the minimum CDS problem, the authors originally showed that the approximation ratio of the proposed algorithm is 62.19. Recently, a 6.075-approximation algorithm for the minimum CDS problem has been introduced [57]. By relying on this new result, it can be easily shown that the approximation ratio of CDSA is 54.675.

6.2.2 An Approximation for .1; m/-CDS In this section, they introduce an approximation algorithm to compute .1; m/-CDS in [70, 74]. Roughly speaking, this method relies on an existing approximation algorithm for the minimum CDS problem. The algorithm consists of following three steps: 1. Compute a C1;1 of G D .V; E/ using an existing approximation algorithm, which computes an MIS M0 of G first and connects the nodes in M0 by adding some nodes B in V n M0 . Then, C1;1 D M0 [ B. 2. Set C D C1;1 . Repeat following from i D 1 to m  1: compute an MIS Mi of .V n .B [ M0 [    [ Mi 1 // and add it to C . 3. Return C . The correctness of this algorithm can be proven as follow. In the first step, a C1;1 is computed. Then, each node in V n C1;1 must have at least one neighbor in C1;1 since C1;1 includes M0 , which is an MIS of G and thus is also a DS of G. Similarly, after M1 is computed, each node in V n .C1;1 [ M1 / must have at least one neighbor in M1 . This implies that each node in V n .C1;1 [ M1 / must have at least two neighbors in .C1;1 [ M1 /, at least one in C1;1 , and at least another in M1 . Therefore, C D C1;1 [ M1 is a (1,2)-CDS of G. As a result, the output C D B [ M0 [ M1 [   [ Mm1 of the algorithm is a .1; m/-CDS, which is denoted by C1;m . Now, they describe the approximation ratio analysis of this algorithm.  j, where M is an MIS Lemma 17 ([70]) Given G D .V; E/, jM j  maxf m5 ; 1gjDm  of G and Dm is a minimum m-DS of G.

Theorem 12 ([70]) The approximation ratio of the strategy in this section for the minimum .1; m/-CDS problem is .5 C m5 / for m  5 and 7 for m > 5. T  Proof LetTSi D Mi Dm . In the algorithm, each Mi n Si is an IS and   .Mi n Si / Dm D ;. From Lemma 17, they have jMi n Si j  m5 jDm n Si j, and

jM0 [    [ Mm1 j D D

m1 X i D0 m1 X i D0

jSi j C jSi j C

m1 X

jMi n Si j 

i D0 m1 X

5 m

m1 X i D0

jSi j C

m1 X i D0

 .jDm j  jSi j/ D .1 

i D0

5  jD n Si j m m

m1 5 X  jSi j C 5jDm j: / m i D0

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 From this inequality, they can observe jM0 [   P [ Mm1 j  5jDm j for m  5 m   and jM0 [    [ Mm1 j  6jDm j for m > 5 since i D1 jSi j  jDm j. In addition, from Lemma 17, they have

 5  ; 1 jDm jBj  jM0 j  max j; m 

since jBj  jM0 j is clearly true if a coloring is used to find M0 . Therefore, they can conclude that the size of jC j, which is an output of the algorithm, is bounded by   .5 C m5 /jDm j for m  5 and 7jDm j for k > 5, and this theorem is proved. 

6.2.3 An Approximation for .2 ; m/-CDS In Sects. 6.2.1 and 6.2.2, they introduced two approximation algorithms, one for computing .2; 1/-CDS and another for computing .1; m/-CDS. In [70], an approximation for computing .2; m/-CDS can be obtained by combining the two strategies as below: 1. Given a UDG G, compute a C1;m of G using an existing approximation algorithm for computing .1; m/-CDS. Let C D C1;m . 2. Apply a variation of Wang et al.’s strategy for .2; 1/-CDS in Sect. 6.2.1 on C such that only a path P with length at most 3 will be identified and used to merge a leaf block with another block in C1;m . 3. Return C . It is easy to see that an output of this algorithm C is a .2; m/-CDS. Now, they describe the approximation ratio analysis of this algorithm. Theorem 13 ([70]) The approximation ratio of the strategy for the minimum .2; m/-CDS problem is .5 C 25 m / for 2  m  5 and 11 for m > 5.   and C2;m be a minimum .1; m/-CDS of G and a minimum .2; m/Proof Let C1;m   CDS of G, respectively. Clearly, jC1;m j  jC2;m j. Largely, this proof depends on the following facts: 1. C2;m , a .2; m/-CDS of G, can be obtained from C1;m , a .1; m/-CDS, by adding at most 2jC1;m j nodes from V n C1;m . This is possible based on following two facts: • Suppose they have an MIS M0 of G D .V; E/ and a CDS C D M0 [ B of G. Let D be a 2-connected subgraph of G which is a union of a subset of C and some other nodes from VTn C . Then, there has to be a path with T length at most 3 from a node u in D M0 to another node v in .C n D/ M0 such that both of u and v are not cut-vertices of C1;m . Otherwise, there has to be a node in V which is not dominated by any node in M0 , which makes M0 not maximal. • The number of blocks in C1;m is to at most jC1;m j. Furthermore, each time they add a path with length at most 3 from u to v, at least one block will be merged into the 2-connected subgraph which is being augmented. 2. Due to its construction, jC1;1 j D jM0 j C jBj.

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3. jBj  jM0 j if coloring is for M0 .  4. jM0 j  maxf m5 ; 1gjC1;1 j (from Lemma 17). From the second and third facts, they have jC1;1 j  2jM0 j, and by combining this inequality with the first and last ones, they can conclude jC2;m j  jC1;m j C 2jC1;1 j  jC1;m j C 4jM0 j  jC1;m j C 4 maxf

5  ; 1gjC1;1 j: m

25  /jC2;m j for 2  m  5 m  and jC2;m j  11jC2;m j for m > 5, and this theorem is proved.  Finally, from Theorem 12, they have jC2;m j  .5 C

6.2.4 An Approximation for .3; m/-CDS In [52], the authors introduced an approximation for computing .3; m/-CDS. The core part of this work is about on how to approximate the minimum .3; 3/CDS problem. After proposing a constant-approximation algorithm (3,3)-CDS, the authors also showed how to approximate .3; m/-CDS for any m  1. Now, they briefly explain how the algorithm works. In the algorithm, each node u in a (2,3)-CDS, C2;3 is either (1) a good-point if C2;3 n fug is still 2-connected or (2) a bad-point if C2;3 n fug is 1-connected. Clearly, if all nodes in C2;3 are good-points, then C2;3 is a .3; 3/-CDS. Note that some of two bad-points u; v 2 C2;3 can make C2;3 n fu; vg completely disconnected. Denote such node pair by a separator of C2;3 . A separator fu; vg of C2;3 can be easily identified using min-max flow algorithm and takes polynomial time. Now, they introduce the concept of a block graph of a graph G D .V; E/: remind that a block is a maximal 2-connected subgraph, and a vertex u 2 V is a cut-vertex if G n fug is disconnected. Then, a block graph G 0 is constructed as follows: 1. For each block B of G, there is a corresponding node vB in G 0 . 2. For each cut-vertex u in G, there is a node u in G 0 . 3. At last, for each pair of vB (representing a block in G) and u (representing a cutvertex in G) in G 0 , establish an edge between them only if u 2 B in G, where B is a block represented by vB in G 0 . Suppose TB is a block tree of C2;3 n fug rooted at v. Then, v and a cut-vertex w of TB forms a separator of C2;3 . Note that w is also a bad-point of C2;3 . Then, the algorithm adds a path P with a bounded length for w such that w becomes a goodpoint of C2;3 [ fP g. Clearly, the number of paths that must be added to make all of the bad-points in C2;3 to be good-points is bounded by the size of C2;3 multiplied by the number of bad-points. In [52], the authors showed the approximation ratio of this approach is 520 3 for (3,3)-CDS.

6.3

Approximations for General .k; m/-CDS

So far, several approximation algorithms have been proposed for the minimum .k; m/-CDS problem [59, 74, 83, 85]. Recently, Kim et al. proved that each of them is flawed in a sense that either it does not produce a .k; m/-CDS or its

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approximation ratio analysis is not correct [52]. As a result, it is still an open problem to approximate the minimum .k; m/-CDS problem for k > 3 and m  1.

6.3.1 Centralized Approximation Algorithms for .k; m/-CDS CDSAN [74] is an approximation algorithm for computing k-CDS. To calculate a k-CDS, CDSAN first computes a .1; k/-CDS and gradually increases its connectivity using a strategy similar to Wang et al.’s in Sect. 6.2.1. In detail, for some l such that 1 < l < k and Cl;k (note that C1;k can be computed using the algorithm in Sect. 6.2.2), CDSAN repeats (1) identifying a .l C 1/-connected subgraph of Cl;k (e.g., leaf block) and (2) adding a path, from a node of Cl;k in the subgraph to another node of Cl;k outside the subgraph, to the subgraph so that the .l C 1/-connected subgraph can be augmented into a larger .l C1/-connected subgraph. This repeating is terminated once a .l C 1/-connected subgraph contains Cl;k . By performing such augmentation for l D 2;    ; k  1, CDSAN generates a k-connected subgraph containing C1;k . CDSAN works well for computing C2;k since there is always a block (maximal 2-connected subgraph) in any C1;k if jC1;k j  2. However, it is possible that there is no 3-connected subgraph in some C2;k of G even though there exists a .3; k/-CDS of G [52]. As a result, CDSAN will fail in some problem instances since it cannot find any .l C 1/-connected subgraph in a .l; k/-CDS. ICGA [83] is a centralized approximation algorithm to compute .k; m/-CDS in UDG. In the algorithm, a .1; m/-CDS is computed and is evolved into a .k; m/-CDS using following lemma. Lemma 18 Given a k-connected T subgraph H of G D .V; E/ and a connected subgraph F of G such that H F D ; and F does k-dominate H (each node in F has k neighbors in H ), the graph induced by V .H / [ V .F / is .k C 1/-connected. Roughly speaking, ICGA first computes a .1; m/-CDS, C1;m . Then, it identifies (1) a 2-connected subgraph of C1;m and (2) a subset F which does k-dominate the subgraph. By the lemma, the union of the subgraph and F should be a larger 3-connected subgraph. If the lemma is true, by repeating the augmentation procedure, they may eventually have a .k; m/-CDS. However, the lemma is not always true since there is a 3-connected graph with no such H and F [52]. CSAA [59] is another centralized algorithm for .k; m/-CDS. In detail, given a C D C1;m , CSAA first finds a set of i -cut Spi D fvx1 ;    ; vxi g such that C n Spi is disconnected, and adds more nodes to C to remove the i -cuts. However, it is likely to happen that C is a kind of complete graph, Gk1 , while they are looking for a .k; k/-CDS. Then, Spi will be an empty set, but C is not k-connected, and therefore this algorithm does not work any further.

6.3.2 Distributed Approximation Algorithms .k; m/-CDS DDA [59, 85] is a distributed approximation algorithm to compute .k; m/-CDS in UDG and consists of following three steps: 1. Compute a (1,1)-CDS, C1;1 using an existing distributed approximation [54]. 2. Augment C1;1 to a .1; m/-CDS, C1;m by exploiting the strategy in Sect. 6.2.2.

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a R

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Fig. 8 These figures illustrate an example where DDA does not work correctly. Suppose they want to find a (2,1)-CDS. In figure (a), the set of black nodes forms a (1,1)-CDS. In figure (b), the root finds a 2-connected subgraph with its local information (the set of black nodes). Now, among the gray nodes (which is a member of the (1,1)-CDS other than the black nodes), there is no gray node which can have two vertex independent paths with length at most three from itself to a member of the 2-connected subgraph (one of the black node)

3. Augment C1;m into a .k; m/-CDS, Ck;m . This can be done by selecting a node using the well-known Expansion Lemma in Graph Theory (below) one by one. Lemma 19 (Expansion Lemma) Given a k-connected subgraph H  V and v … H such that v has at least k neighbors in H , H 0 D H [ fvg is a k-connected subgraph of G D .V; E/. Now, they describe an brief idea of Step 3. At the beginning of Step 3, DDA first selects a random root node r from C1;m and try to identify a 2-connected subgraph H of C1;m which consists of r and some or all of its two-hop neighbors. Then, it finds another node v 2 .C1;m nH / such that there is k vertex independent paths from v to H with length at most 3. The algorithm claims if they can find such paths the union of H , fvg, and the nodes on the paths will be a k-connected subgraph which includes H . As they can see in Fig. 8, there are some cases where DDA fails [52], which is also admitted by the authors of [59]. LDA [83] is a localized approximation algorithm for computing .k; m/-CDS in UDG and consists of following steps. Initially, the algorithm assumes that all nodes are colored white: 1. Compute a C1;m using an idea used in DDA and color all nodes in C1;m black. 2. C1;1 in C1;m can be considered as a tree rooted at some node. For each parent and child pair u; v 2 C1;1  C1;m , color its common neighbors black so that they can share at least k common black neighbors. 3. For each (black) node in C1;1 , it forms a local k-connected graph, which includes all the neighboring black nodes from Steps 1 and 2. At the end of the steps, the union of all black nodes is a .k; m/-CDS. The main idea here is that if every parent and child pair in C1;1 shares k black neighbors, then

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Fig. 9 An example where LDA fails. Suppose they are computing a (2,1)-CDS and the set of black nodes above is C.1;1/ . During the negotiation of each pair of neighboring black node u; v, no new black node will be introduced. At the end, the union of black nodes is still 1-connected. They can have a feasible solution, a (2,1)-CDS, by selecting all nodes

the union of them can be k-connected. However, there are some graph instances G (see Fig. 9), in which some node u 2 G1;1 of G cannot induce a local k-connected graph, and the union of the all black nodes is not k-connected, but G itself contains a feasible solution, a .k; m/-CDS, and therefore LDA fails [52]. Generally, finding a k-connected subgraph where k  2 needs a global knowledge. Therefore, they believe that it will be very interesting and challenging question to design a localized distributed algorithm for a .k; m/-CDS where k  2.

7

Conclusion

Virtual backbone can help reduce the searching space for the routing problem from the whole networks to the virtual backbones so that the routing path searching time will be reduced a lot. Due to the advantages of the virtual backbone, it is widely used in broadcasting and routing in wireless networks. Connected Dominating Set is a good choice for virtual backbone. In this book chapter, the previous literatures on Connected Dominating Set are recalled.

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Connections Between Continuous and Discrete Extremum Problems, Generalized Systems, and Variational Inequalities Carla Antoni, Franco Giannessi and Fabio Tardella

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Equivalence Through Relaxation–Penalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 General Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equivalence Between Mixed-Integer and Real Optimization. . . . . . . . . . . . . . . . . . . . . . . . . 2.3 On the Estimate of Penalization Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 On the Equivalence of Lagrangian Duality and Penalization. . . . . . . . . . . . . . . . . . . . . . . . . 3 Generalized Concavity and Extreme Point Minimizers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Global Minimizers at Extreme Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Case of Nonconvex Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Case of Polyhedral Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Pseudo-Boolean Optimization and Optimization with Box Constraints. . . . . . . . . . . . . . 3.5 Convex–Concave Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Max Clique and Optimization Over Simplices and Polymatroids. . . . . . . . . . . . . . . . . . . . 4 Piecewise Concavity and Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Piecewise Concavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Minimax Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Location Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

836 838 838 842 846 850 852 852 854 856 863 867 868 872 872 874 875

 Section 1 is due to Giannessi; Sects. 2, 5, and 6 are due to Antoni; and Sects. 3 and 4 are due to Tardella.

C. Antoni Naval Academy, Livorno, Italy e-mail: [email protected] F. Giannessi University of Pisa, Pisa, Italy e-mail: [email protected] F. Tardella Sapienza University of Rome, Rome, Italy e-mail: [email protected]

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5 Generalized Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 General Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Case of a Discrete Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Discrete Vector Optimization and Variational Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This writing aims at surveying what has been done to analyze some connections between continuous and discrete extremum problems and related models, like generalized systems and variational inequalities. Keywords

Continuous Optimization • Discrete Optimization • Generalized Convexity • Relaxation • Penalization • Variational Inequalities

AMS Classification: 65K05; 65K10; 65K15; 90C05; 90C11; 90C30;

1

Introduction

Problems defined in a discrete space are, perhaps, the oldest mathematical problems. The birth and great development of infinitesimal analysis threw discrete problems somewhat into the shade. After the Second World War, due to the development of automatic computation and its wide applications, discrete problems have had a new life. In all these strong movements, one thing has remained almost stable: the fields of continuous and discrete problems have had a small intersection; this is true, in particular, for constrained extremum problems. There is a deep convincement that the two fields, which could be called “Continuous Structures of Mathematics” and “Discrete Structures of Mathematics,” should be considered explicitly and connected more and more. The systematic study of such connections should by no way imply that one field can be reduced to the other; on the contrary, the more “spectacles” (of Galilean memory) one has to look at reality, the more insight one can achieve. The analysis of connections is meant in this light. In the last four decades of the last century, in the field of constrained extremum problems, most of the efforts have been devoted to transform discrete problems into continuous ones. Even though this activity has turned out to be useful, it has sometimes led to the wrong convincement that a continuous format be the final destination of every problem. Like every human activity, mathematical research is influenced by a sort of inertia: if a problem has been classified, since the beginning, as a continuous one, almost surely, it will be treated by the methods of continuous mathematics, independently of its deep nature.

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For instance, the analysis of elastoplastic behavior of structures and materials has always been considered a continuous problem; a part of such an analysis leads to search for the minimum of a nonconvex function, which has several local, but not global, minima; this implies that, in the study of stationary points, one meets– without being ready–a combinatorial problem, which often leads the computation to quicksands. This writing aims at surveying what has been done to analyze some connections between continuous and discrete extremum problems and related models, like generalized systems and variational inequalities. As concerns extremum problems, a first trace is contained in [64] and followed in [49]. Independently of [64], and in a slightly more general context, a first analysis of such connections has been faced in [38]. The idea expressed in [64] has been picked up in [36], where, however, the mathematical tools are independent of those in [64], inasmuch as the “ad hoc” analysis of linear algebraic problems is no longer valid for nonlinear ones; indeed, in [36], the equivalence is established between a nonlinear constrained extremum problem in a Euclidean space and a problem that is both a relaxation and a penalization of it; the equivalence between a continuous and a discrete problem is obtained as a corollary of it. The results of [36] stimulated some further research in the field; see, e.g., [2–5, 26, 29, 34, 35, 38, 44, 55–57, 61, 65]. The great development of the theory of constrained extrema and, more recently, of that of variational inequalities, has led to search for models that embody both theories. A possible answer is offered by the so-called generalized systems. Also such theories have received different and, sometimes, almost disjoint developments depending on the kind of space, where the problems have been defined. These kinds of problems will also be considered here. Given a set X , a function f W X ! R and a subset S of X consider the problem: min f .x/

s:t: x 2 S

(1)

Problem (1) is usually called a combinatorial optimization problem when S is finite and a discrete optimization problem when the points of S are isolated in some topology, i.e., every point of S has a neighborhood which does not contain other points of S . Obviously, all combinatorial optimization problems are also discrete optimization problems, but the converse is not true. A simple example is the problem of minimizing a function on the set of integer points contained in an unbounded polyhedron. Even though some discrete and combinatorial optimization problems have been studied since ancient times, the increase of their importance and their development have been very fast only in the last few decades thanks to the possibility of practically solving them with modern computers and because of the several applications that they have found in many fields. In most cases, discrete and combinatorial optimization problems can be formulated as linear or nonlinear integer programming problems and are solved by means of methods that exploit in a crucial way the finiteness or discreteness of the feasible region.

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The linear programming formulation of most discrete and combinatorial optimization problems is typically based on the Fundamental Theorem of Linear Programming and on polyhedral theory. This subject, which is very well illustrated in the recent books [69] by Schrijver, is not covered here. In the case where the feasible set S is defined by a family of equations and inequalities in Rn and at least one of the constraints or objective functions is nonlinear, problem (1) is called a nonlinear programming problem or a nonlinear program. Problems of this kind have been studied for at least four centuries with tools that exploit the differential, geometric, and topological properties of the functions and sets involved. Although the methods employed in integer and nonlinear programming are quite different in general, they can be complementary in several cases. In fact, it is often possible to restrict the search for a global solution of a nonlinear program to a finite set of points. On the other hand, most discrete optimization problems can be reformulated equivalently as nonlinear programs. Clearly, the restriction to a finite set of points or the nonlinear reformulation does not always lead to practical solution methods. Nevertheless, these approaches provide useful tools for many important classes of problems, as will be shown in the sequel. This chapter is not meant to give an exhaustive survey of all connections between discrete optimization and nonlinear programming. The purpose here is to present the main tools that have been employed in the literature to transform a discrete problem into an equivalent nonlinear one and vice versa. This chapter does not cover, e.g., the many important continuous relaxations of discrete problems that lead to lower or upper bounds on the original problem without being equivalent to it.

2

Equivalence Through Relaxation–Penalization

2.1

General Results

This section contains general results about the equivalence of a constrained extremum problem with another one obtained from the former by enlarging the feasible region and penalizing the objective function. Several known and new results on the equivalence between mixed-integer and (continuous) nonlinear programming problems are derived as special cases of such general results. More formally, one wishes to describe conditions that guarantee the equivalence of the problems

min f .x/

s:t: x 2 W;

(2)

and minŒf .x/ C '.xI /

s:t: x 2 X;

(3)

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where W  X  Rn ; f W Rn ! R, and, for every  2 RC , '.  I / W Rn ! R is a suitable penalty function. A recent general result in this direction is the following theorem, originally stated by Lucidi and Rinaldi with an unnecessary compactness assumption here omitted. Theorem 1 (Lucidi and Rinaldi [56]) Let W  X  Rn , let k  k be any norm in Rn , and make the following assumptions: .A1/ The function f is bounded on X and there exist an open set A  W and real numbers ˛; L > 0 such that, for any x; y 2 A, jf .x/  f .y/j  L k x  y k˛ I

(4)

The function '.  I / satisfies the following conditions: .A2/ for every x; y 2 W , and  2 RC , '.xI / D '.yI /:

(5)

.A3/ There exists a value O such that, for every z 2 W , there is a neighborhood S.z/ of z such that, for every x 2 S.z/ \ .X n W / and   O , '.xI /  '.zI /  LO k x  z k˛ ;

(6)

where LO > L. Furthermore, set S WD [z2W S.z/ and suppose there is a point xN 2 X n S such that lim '.xI N /  '.zI / D C1;

(7)

'.xI /  '.xI N /; 8x 2 X n S:

(8)

!0

Then, there exists a value Q > 0 such that, for all  < Q , (2) and (3) have the same minimum points. Proof First, observe that a minimum point of (3) is also a minimum point of (2). Indeed, for all  > 0, if x  is a minimum point of (3), then f .x  / C '.x  I /  f .x/ C '.xI /; 8x 2 X: Since W  X , it follows that f .x  / C '.x  I /  f .x/ C '.xI /; 8x 2 W: If x  2 W; then Assumption .A2/ ensures that f .x  /  f .z/; 8z 2 W;

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which shows that x  is a minimum point of (2). Now one can prove that there exists a value Q such that, for all  > Q , every minimum point of (3) belongs to W. Let xN and S be, respectively, the point and the open set defined in Assumption .A3/. Hence, by (7), there exists a value N such that, for  > N , the following inequality holds: '.xI N /  '.zI / > sup f .x/  x2W

inf

x2.X nS /

f .x/:

(9)

Then, put Q WD maxfN ; O g;

(10)

where O is defined as in .A3/. Ab absurdo, suppose there exists a  > Q such that x  is a minimum point not in W. Consider two different cases: Case 1: x  2 S . Without any loss of generality, consider S  A. In this case, there exists z 2 W such that x  2 S.z/. Using the definition of O , Assumptions .A1/ and .A3/, one obtains f .z/  f .x  /  L k z  x  k˛ < LO k z  x  k˛  '.x  I /  '.zI /: Thus the contradiction f .z/ C '.zI / < f .x  / C '.x  I /: Case 2: x  2 X n S . In this case f .x  / C '.x  I /  inf f .x/ C '.x  I /: x2X nS

(11)

For all z 2 W , one has f .z/  supx2W f .x/  0; then f .x  / C '.x  I /  f .z/  sup f .x/ C inf f .x/ C '.x  I /: x2W

x2X nS

(12)

By using (8) and Assumption .A3/, one obtains f .x  / C '.x  I /  f .z/  sup f .x/ C inf f .x/ C '.xI N /: x2W

x2X nS

Adding and subtracting '.zI / lead to N /  '.zI / f .x  / C '.x  I / f .z/ C '.zI / C '.xI  sup f .x/ C inf f .x/: x2W

x2X nS

(13)

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Finally, recalling the definition of Q and exploiting (9), one obtains the contradiction f .x  / C '.x  I / > f .z/ C '.zI /: Now, one proves that, for all  > Q , every minimum point of (2) is also a minimum point of (3). Ab absurdo, suppose that there is an  > Q such that f .x  / C '.x  I / < f .z/ C '.zI /; where z is a minimum point of (2) and x  is a minimum point of (3). Recalling the first part of the proof, one obtains that, for every  < Q , the point x  is also a minimum point of (2); hence, using Assumption .A2/, it follows that f .x  / < f .z/; which contradicts the fact that z is a minimum point of (2).



The previous theorem generalizes a similar result in [36] described below. In this case, the problem considered is min f .x/ s:t: x 2 R \ Z;

(14)

where Z and R are subsets of Rn and f W Rn ! R; the result states conditions under which (14) is equivalent to minŒf .x/ C  .x/ s:t: x 2 R \ X; where X Z,

(15)

W Rn ! R is a suitable penalty function and  is a real parameter.

Theorem 2 (Giannessi and Niccolucci [36]) Let R  Rn be closed, let Z  X be compact, and let the following hypotheses hold: .H1 / f W X ! R is bounded, and there exist an open set A  Z and real numbers ˛; L > 0, such that, for every x; y 2 A, f fulfills the following H¨older condition: jf .x/  f .y/j  L k x  y k˛ : .H2 / Let W Rn ! R be such that: (i) is continuous on X . (ii) .x/ D 0; 8x 2 Z, .x/ > 0; 8x 2 X n Z. (iii) 8z 2 Z, there exists a neighborhood S.z/ of z, and a real .z/ > 0 such that .x/  .z/ k x  z k˛ ; 8x 2 S.z/ \ .X n Z/:

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Then, a real 0 exists such that, for every  > 0 , (14) and (15) are equivalent. Proof Apply Theorem 1: now the sets W and X are, respectively, equal to R \ Z and R\X . Furthermore, the penalty function ' of (3) is given by '.xI / D .x/=. Then .H2 /.i i / for implies .A2/ for '. Also, consider any LO with LO > L. Since Z is compact, there is a finite cover S of Z, S D [kiD1 S.zi /, where the neighborhoods S.zi / are those of .H2 /.i i i /. Setting O WD minf .zOi / ; i D 1; : : : ; kg, for all   O L and x 2 .S.zi / \ X / n Z, it holds that O k x  zi k : '.xI /  L Since .R \ X / n S is compact, the continuous function has a minimum point xN on .R \ X / n S ; thanks to .H2 /.i i /, .x/ N > 0, and then (7) and (8) hold. So thanks to Theorem 1, there exists N such that, for  < N , (14) and minŒf .x/ C

.x/= s:t: x 2 R \ X

have the same minimum points. The thesis follows choosing 0 D 1=N and  D 1=. 

2.2

Equivalence Between Mixed-Integer and Real Optimization

A special case of (14), which embraces most combinatorial problems, is that where Z D f0; 1gn;

R D fx 2 Rn W g.x/  0g;

with g W Rn ! Rm . Thus, problem (14) becomes min f .x/ s:t: x 2 fx 2 Rn W g.x/  0g \ f0; 1gn:

(16)

In this case, the set X D XQ WD fx 2 Rn W 0  xi  1; i D 1; : : : ; ng is the natural relaxation of Z. In [36] the function defined by .x/ D

n X

xi .1  xi /;

(17)

i D1

is introduced as a penalty function. In this way, (15) becomes " min f .x/ C 

n X

# xi .1  xi /

s:t: x 2 R \ XQ :

i D1

From Theorem 2 one then derives the following result.

(18)

Connections Between Continuous and Discrete Extremum Problems

843

Theorem 3 ([36]) Let f be a Lipschitz function on XQ . Then, there exists 0 2 R, such that, 8 > 0 , problems (16) and (18) have the same set of global minimum points. Furthermore, if f 2 C 2 .XQ /, then there exists 1 2 R such that, 8 > 1 , f C  is a strictly concave function. Proof One only needs to prove that of (17) satisfies Assumption .H2 / of Theorem 2 with ˛ D 1. Note that .H2 /.i / and .H2 /.i i / are trivially verified. For N in 0; 1; Œ one can prove that fulfills .H2 /.i i i /, with S.z/ D fx 2 Rn Wk x  z k g, N .z/ D 1  . N Indeed, for x 2 S.z/, put  WDk x  z k and u WD .x  z/=. Then .x/ D

n X

.uj C zj /.1  zj  uj /:

(19)

j D1

As z C u D x 2 XQ , if uj > 0, then zj D 0; but if uj < 0, then zj D 1. From (19) it follows .x/ D

X

uj .1  uj / C

uj >0

X

.1 C uj /.uj / D

.x/ D 

n X

juj j.1  juj j/: (20)

j D1

uj 0; and 0 < q < 1.

Proposition 1 ([56]) For every penalty term (21)–(25), there exists a value N such that, for any  20; N Œ, problems (16) and (18) have the same set of global minimum points. A penalty function for mixed-integer problems is introduced in [3]. More precisely, consider the mixed-integer problem: inf f .u; v/ s:t: .u; v/ 2 R;

(26)

where f W Rn Rk ! R, R WD f.u; v/ 2 f0; 1gn Rk W g.u; v/ 2 Dg, g W Rn Rk ! Rm , and D  Rm . For a closed convex set A  Rh , let projA W Rh ! Rh be the function defined by k x  projA .x/ k D min k y  x kI y2A

let … W R R ! R be the canonical projection on Rk : n

k

k

….u; v/ D v:

Connections Between Continuous and Discrete Extremum Problems

845

Proposition 2 (Antoni [3]) Suppose R is closed. If, for all z 2 f0; 1gn, the set Az D f.u; v/ W .u; v/ 2 R; u D zg is a closed convex set, then the function ‰ W Rn Rk R ! R defined by ‰.u; vI / D

X n

 ui .1  ui / C k v  ….projR .u; v// k =

i D1

satisfies Assumptions .A2/ and .A3/ of Theorem 1.

When the equivalence between (16) and (18) holds, properties and methods valid for one of the two problems can be transferred to the other one. As an instance of this, consider the quadratic programming problem h i 1 min hc; xi C hx; C xi s:t: x 2 fAx  bg \ f0; 1gn; 2

(27)

where b 2 Rm ; c 2 Rn ; A 2 Rmn , and C 2 Rnn is a symmetric matrix. Theorem 4 ([36]) There is a real 2 such that, for all  > 2 , THE problem (27) and the linear complementarity problem minhc; N i s:t: .; / 2 D;

(28)

where N   0;  0; h; i D 0g N C D b; D WD f.; / 2 R2nCm R2nCm W A 1 C C 2In AT In AN D @ A 0 0 A; 0 0 In 0

cNT D .1=2/.c T C e T C e T ; b T ; e T /; bN T D .c T ; b T ; e T /; have the same set of global minimum points. Proof Because of Theorem 3, whose hypotheses are trivially satisfied, (27) is equivalent to the quadratic problem h i 1 min hc C e; xi C hx; .C  2In /xi s:t: x 2 fAx  bg \ XQ ; 2

(29)

where e D .1; : : : ; 1/ 2 Rn , if  is large enough. The well-known Karush–Kuhn– Tucker necessary condition for (29) is

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c T C e T C .C  2In /x  AT y C t  u D 0 Ax  v D bI x C w D eI x; y; t; u; v; w  0 hx; ui D hy; vi D ht; wi D 0;

(30)

where y; t; u are vectors of multipliers associated to the inequalities Ax  b, x  e, and x  0, respectively, and v; w are slack variables. Solving (29) is equivalent to finding, among the solutions of the complementarity system (30) (stationary points), those which minimize the function in square brackets of (29). Such a function, evaluated at the stationary points, becomes “linear”: 1 Œhc C e; xi C hb; yi C he; ti: 2 Indeed, (30) implies C x  2x D .c C e/ C AT y  t C u; hx; ui D 0; 0 D hy; vi D hy; b  Axi; 0 D ht; wi D ht; e  xi; and therefore hy; bi D hy; Axi; ht; ei D ht; xi: Now, to achieve the thesis, it is sufficient to set  T WD .x T ; y T ; t T /; T WD .uT ; vT ; wT /: This completes the proof.



Note that no assumption has been made on the matrix C , so the convex case, as well the nonconvex one, has been considered. See also [61] for a reduction of a mixed-integer feasibility problem to a linear complementarity system.

2.3

On the Estimate of Penalization Parameters

For stability reasons, it is often useful to be able to choose a penalization parameter  not too large. This section contains some theoretical results on the comparison between the parameters  given in [36, 49]. More precisely, in [49] the following linear optimization problem is considered:

Connections Between Continuous and Discrete Extremum Problems

minŒhc; xi s:t: x 2 R \ Z;

847

(31)

where x; c 2 Rn , R is a polyhedron of Rn , and Z D f0; 1gn is the set of vertices of the hypercube X D fx 2 Rn W 0  xi  1g. Without any loss of generality, suppose c have nonnegative components. Kalantari and Rosen [49] find a solution x0 of minŒhc; xi s:t: x 2 R \ X: If x0 2 Z, then x0 solves (31). Otherwise, they consider the following maximization problem max g.x/ s:t: x 2 FO ;

(32)

P where g W Rn ! R; g.x/ D niD1 .xi 1=2/2 , and FO denotes the set of the vertices of R \ X not in Z. Denoting by xN a solution of (32), they prove the equivalence between (31) and   min f .x/  g.x/ s:t: x 2 R \ X ; when  > R WD

hc; x0 i : n=4  g.x/ N

Observe that the strictly concave function .x/ D

n X

xi .1  xi /

(33)

i D1

is related to g by the relation .x/ D n=4  g.x/: So that the denominator of R is

.x/. N

The parameter R is now compared with the parameter G introduced in [36] and denoted by 0 in Theorem 3, which establishes equivalence between (31) and   min  hc; xi C  .x/ s:t: x 2 R \ X ; for  > G D 0 .

(34)

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The following notation is adopted: q WD .1=2; : : : ; 1=2/ 2 Rn ; d.x; A/ denotes the distance between the vector x and the set A of Rn ; B.y; r/ WD fx 2 Rn W d.x; y/ < rg; for a subset A of Rn , cl A and Ac denote, respectively, the closure and the complement of A. In order to verify the hypotheses of Theorem 3, observe that the objective function x 7! hc; xi is a Lipschitz function with constant L D jjcjj, and satisfies the hypothesis .H2 /. Indeed, let  be a value of 0; 1Œ, and, for every zi 2 R \ Z, let the set S.zi / be a neighborhood of zi with S.zi /  B.zi ; /. Then, satisfies .H2 / for ˛ D 1 and  D 1  . Recall the definition of parameter G : k c k .supx2R\X Œhc; xi/  .infx2R\X Œhc; xi/ G WD max ; 1 minx2X0; .x/

! ;

  X0; WD .X \ R/ n [kiD1 S.zi / :

where

Thanks to Theorem 3, the equivalence between (31) and (34) holds for any  > G . Observe the dependence of G on the radius ; below, to highlight this dependence, the notation G; will be used instead of G . Before giving sufficient conditions to establish a comparison, observe the following: Remark 1 The hypothesis .H2 / of Theorem 3 and the continuity of  < 1.

imply that

For the sake of simplicity, put 0; WD

.supx2R\X Œhc; xi/  .infx2R\X Œhc; xi/ : minx2X0; .x/

The following result gives sufficient conditions assuring that there exist a N > 0 and a neighborhood S.zi /  B.zi ; / N such that G;N  R : Theorem 5 ([5]) Let xN be a solution of (32). Suppose that p n1 d.x; N q/  2 and supx2R\X hc; xi 1  : kck 2

(35)

Connections Between Continuous and Discrete Extremum Problems

849

Then, there exist N 2 0; 1Πand, 8zi 2 Z, a neighborhood S.zi /  B.zi ; / N such that G;N  R :

(36)

Proof Consider the one-dimensional face of X given by F1 WD fx 2 Rn W x D .x1 ; 0; : : : ; 0/; x1 2 Œ0; 1g: p p Since d.x; N q/  n  1=2 and d.q; F1 / D n  1=2, necessarily the intersection A WD cl B.q; d.x; N q// \ F1 is nonempty. Put N D d.0; A/ and denote by x  the n point of A at minimum distance from  the origin  of R . Finally, let S.zi / WD X \ k N Since X0;N D .X \ R/ n [i D1 S.zi / , by construction, B.zi ; /. X0;N  clB.q; d.q; x// N D fx 2 Rn W

.x/ 

.x/gI N

then one has min

x2X0;N

so 0;N 

.x/ 

.x/; N

.supx2R\X Œhc; xi/  .infx2R\X Œhc; xi/ : .x/ N

Moreover, obviously, .supx2R\X Œhc; xi/  .infx2R\X Œhc; xi/  . inf Œhc; xi/ .x/ N D R I x2R\X .x/ N thus 0;N  R :

(37)

kck  R 1  N

(38)

Besides, the inequality

is equivalent to supx2R\X hc; xi .x/ N  : 1  N kck The last inequality is true because .x/ N D .x  / D x1 .1  x1 / D .1 N  /, N and thanks to the inequality in (35); indeed, one has 1 supx2R\X hc; xi .x/ N D N   : 1  N 2 kck So (36) follows from (37) and (38) and the theorem is thus proved.



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2.4

On the Equivalence of Lagrangian Duality and Penalization

Based on results by Larsen and Tind [55], it is shown here that the penalization approach to a constrained optimization problem described in Theorem 1 can be interpreted in terms of Lagrangian duality with zero gap for an associated problem. More precisely, consider the sets W  X  Rn and a (penalty) function g W X ! R such that g.x/  0 for all x 2 X and g.x/ D 0 for all x 2 W . Then, problem (2) can be equivalently restated as min f .x/ s:t: x 2 W D fx 2 X; g.x/  0g:

(39)

Its Lagrangian dual problem is sup w./ s:t:  2 RC ;

(40)

w./ D minŒf .x/ C g.x/ s:t: x 2 X:

(41)

where

Then, requiring the existence of a penalization parameter N such that, for all  > , N problems (2) and (3) have the same solutions is equivalent to requiring that problem (39) has no duality gap and, more precisely, that for all  > N problem (41) has the same optimal solutions as (39). Thus any known result for zero duality gap can be immediately rephrased as an exact penalty result and vice versa. As an example, recall the following results by Larsen and Tind [55], rephrased in the current setting, with the notation B.x; N / WD fx 2 Rn Wk xN  x k< g. Theorem 6 ([55]) Let W  X be nonempty compact sets in Rn and let f and g be continuous functions on X such that g.x/  0 for all x 2 X and g.x/ D 0 for all x 2 W . Assume that there exist constants  > 0 and H > 0 such that f .x/f N .x/  H g.x/ for all solutions xN of (39) and for all x 2 B.x; N /\.X nW /. Then, there exists 0  0 such that, 8  0 , min f .x/ D w./:

x2W

Consider, now, the following facial disjunctive special case of (39): min f .x/ s:t: x 2 W D fx 2 X W ha1h ; xi  b1h _ ha2h ; xi  b2h ; h 2 H g; (42) where a1h ; a2h 2 Rn n f0g; b1h; b2h 2 R for all h 2 H , and H is a finite index set. According to Balas [10], the facial requirement for (42) is .A/

8x 2 X;

Indeed, for the function

ha1h ; xi  b1h ^ ha2h ; xi  b2h ;

8h 2 H:

Connections Between Continuous and Discrete Extremum Problems

g.x/ D

X

.b1h  ha1h ; xi/.ha2h ; xi  b2h /;

851

(43)

h2H

one has g.x/  0 for all x 2 X and g.x/ D 0 for all x 2 W . So the feasible set of (42) has the desired form W = fx 2 X W g.x/  0g and g satisfies the assumptions of Theorem 6. On the feasible set W , the following further assumption is made: .B/

9ı > 0 8xN 2 W 8h 2 H;

b1h  ha1h ; xi ha2h ; xi N N  b2h  ı or  ı: k a1h k k a2h k

Theorem 7 ([55]) Let Assumptions .A/ and .B/ hold for the feasible set of (42) and choose  2 0; ıŒ. Then, the constraint function g in (43) satisfies, for all xN 2 W and x 2 B.x; N / \ .X n W /; the inequality g.x/  ..ı  /=ı 2 / inf

nX

o jha1h ; uijjha2h ; uij W u 2 U.ı/ k x  xN k;

h2H

N k x  xN k; xN 2 W; x 2 X n W g. where U.ı/ WD fu 2 Rn W u D ı.x  x/= Now, the previous results are applied to the 0–1 problem given by min f .x/ s:t: x 2 W D R \ f0; 1gn:

(44)

Obviously it holds that n o W D R \ Œ0; 1n \ \nj D1 .fxj  1g [ fxj  0g/ : Putting X D R \ Œ0; 1n , (44) is a facial disjunctive program satisfying the facial requirement .A/. Moreover it becomes the facial program min f .x/ s:t: x 2 W D fx 2 R \ Œ0; 1n ; g.x/  0g

(45)

using the realization of g given by g.x/ D

n X

xj .1  xj /:

j D1

Therefore the Lagrangian dual of (45) is h sup

min

n

M 0 x2R\Œ0;1

f .x/ C M

n X j D1

i xj .1  xj / :

(46)

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The next proposition, first proved in [36], can be stated as a direct application of Theorem 6. Theorem 8 ([55]) Assume R and W are, respectively, a closed and a nonempty set of Rn ; assume f is a Lipschitz continuous function on R \ Œ0; 1n . Then the duality gap between (45) and (46) is closed with a finite dual multiplier. Proof Problem (44) is a facial disjunctive problem; Assumption .A/ holds with X D R \ Œ0; 1n ; Assumption .B/ holds with ı  1; finally, for all u 2 U.ı/, P h h 2 h2H jha1 ; uijjha2 ; uij D ı . Thus, Theorem 7 implies the inequality g.x/  .ı  / k x  xN k for all x 2 B.x; N / \ .X n W / and xN 2 W . Therefore Theorem 6 can be applied. 

3

Generalized Concavity and Extreme Point Minimizers

3.1

Global Minimizers at Extreme Points

In many cases it is possible to restrict the search for the global solution of a nonlinear program to a finite set of points. One way of doing this consists in considering only the set of points that satisfy some kind of first- or second-order necessary condition for local optimality. This set is often finite, but in general it is not practical to compute all its elements or to minimize the objective function over it. However, there are some interesting cases where this simple idea leads to useful connections between nonlinear and combinatorial problems. This section describes another important case where the search for a solution of a nonlinear program can be restricted to a finite set, namely, when at least one global minimizer is guaranteed to be in the set of extreme points of a polyhedral feasible region. The following definitions of extreme point and of concavity of a function in the general setting on nonconvex domains will be needed in the sequel. Definition 1 (Extreme Point) Let X be a subset of Rn . A point x 2 X is called an extreme point of X iff y; z 2 X , ˛ 2 0; 1Œ, and x D ˛y C .1  ˛/z imply y D z. A point x 2 X is called a convex hull extreme point of X iff it is an extreme point of the convex hull (denoted by conv.X /) of X . Note that the set of extreme points of a polyhedron is finite and coincides with the set of its vertices. Let E.X / denote the set of extreme points of a set X . Then the set of convex hull extreme points is given by E.conv.X //. Since X  conv.X /, it follows from the above definition that E.conv.X //  E.X /  X . It is well known (see, e.g., [66]) that the convex hull of a compact set is compact. Furthermore, the Krein–Millman Theorem establishes the equality X D conv.E.X // for a nonempty convex compact set X . From this it easily follows that a nonempty compact set X has a nonempty set of convex hull extreme points and,

Connections Between Continuous and Discrete Extremum Problems

853

furthermore, that every extreme point of X is in the convex hull of the convex hull extreme points of X . Proposition 3 For a nonempty compact set X , one has E.conv.X // ¤ ; and conv.E.conv.X /// D conv.E.X //: Proof The results follow from Krein–Millman Theorem and from the inclusions conv.X / D conv.E.conv.X ///  conv.E.X //  conv.X /:  Definition 2 (Concavity on Nonconvex Sets) Let X be a subset of Rn . A function f W X ! R is concave (resp., quasi-concave) on X iff, for every x 2 X and for everyPset of points fx 1 ; : : : P ; x m g  X and of coefficients ˛1 ; : : : ; ˛m 20; 1Œ such m i that i D1 ˛i D 1 and x D m i D1 ˛i x , one has f .x/ 

m X i D1

˛i f .x i /

(resp. f .x/  min f .x i //: i

(47)

A function is called strictly concave or strictly quasi-concave iff the above relations are satisfied with a strict inequality whenever x i ¤ x for at least one index i . Note that if the set X coincides with E.conv.X // (like, e.g., in the case of a circle), then every function is strictly concave on X with the above definition. However, when X is convex, Definition 2 is equivalent to the standard definition of concavity and of strict concavity of a function. Several conditions for the validity of the following property will be investigated in this section.

Property E At least one (or all) global minimizer of f on a set X belongs to E.X /. The Fundamental Theorem of linear programming is the most popular result that guarantees the validity of Property E. Theorem 9 (Fundamental Theorem of LP) If a linear function f is bounded below on a polyhedron X , then at least one face of X of minimum dimension must be formed by global minimum points of f . In particular, whenever E.X / ¤ ;, at least one global minimum point of f belongs to E.X /. The Fundamental Theorem of LP establishes an important bridge between a (continuous) Linear Programming problem and many important combinatorial problems that can be viewed as problems of minimizing a linear function on the vertices of some polyhedron.

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It is well known that Property E also holds for classes of feasible sets X and of objective functions f that are considerably larger than the classes of polyhedra and of linear functions. Notable examples are the cases where f is concave and X is a closed convex set, or where f is quasi-concave and X is a compact convex set (see, e.g., [45, 77]). Note, however, that Property E does not hold in general for quasiconcave functions on unbounded closed convex sets. A simple counterexample is provided by the quasi-concave (and convex) function f W X D Œ0; C1Œ! R defined by f .x/ D x; for x 2 Œ0; 1, and f .x/ D 1, for x 2 1; C1Œ.

3.2

The Case of Nonconvex Domains

It will now be shown that Property E still holds (and in a stronger form), for the minimization of concave or quasi-concave functions on nonconvex domains. Definition 3 (Concave Envelope) Let X be a subset of Rn . The concave envelope (or concave hull) of a function f W X ! R, denoted f cv , is the smallest concave function on conv.X / which is minorized by f on X. The convex envelope (or convex hull) of f , denoted f cx , is defined as the greatest convex function on conv.X / which is majorized by f on X. The convex and concave envelopes of f can be obtained as follows (see also [66]): f cx .x/ WD inffy W .x; y/ 2 conv.Epif /g; f cv .x/ WD supfy W .x; y/ 2 conv.Hypof /g;

(48)

where Epi f D f.x; y/ 2 X R W y  f .x/g and Hypo f D f.x; y/ 2 X R W y  f .x/g denote the epigraph and the hypograph of f , respectively. The convex envelope of f satisfies the following properties (see, e.g., [52, 62]): inf f .x/ D

x2X

inf

x2conv.X /

f cx .x/

and arg min f .x/  arg min x2X

x2conv.X /

f cx .x/;

(49)

where arg minx2S g.x/ denotes the set of global minimum points of a function g on a set S . Thanks to properties (49), convex envelopes have been employed by several authors to develop algorithms for the minimization of some classes of nonconvex functions (see [19, Sect. 3.2] for a brief survey of these approaches). When X is any subset of Rn and f is a concave function on X , the concave envelope f cv .x/ satisfies the same properties as the convex envelope. Theorem 10 Let X be a subset of Rn and let f W X ! R be any real-valued function on X . Then f cv .x/ D f cx .x/ D f .x/; 8x 2 E.conv.X //: Furthermore, if f is concave on X , then

Connections Between Continuous and Discrete Extremum Problems

f cv .x/ D f .x/; 8x 2 X; infx2X f .x/ D infx2conv.X / f cv .x/; arg minx2X f .x/  arg minx2conv.X / f cv .x/:

855

(50)

Proof Let x 2 E.conv.X //. Then, as observed earlier, x 2 X . Furthermore, f cv .x/  f .x/ by definition of f cv . Assume, ab absurdo, that f cv .x/ < f .x/. Then, by (48), there exists aPset of pointsPfx 1 ; : : : ; x m g  X P and coefficients m m i i ˛1 ; : : : ; ˛m 2 0; 1Œ such that m ˛ D 1, ˛ x D x, and i D1 i i D1 i i D1 ˛i f .x / < f .x/. However, this contradicts the fact that x is a convex hull extreme point of X , which implies x 1 D x 2 : : : D x m D x. The proof of the equality f cx .x/ D f .x/; 8x 2 E.conv.X //; is analogous. Assume now that f isPconcave, and take any x 2 X . From the concavity of f m i 1 it follows that f .x/  ; : : : ; xm g  X i D1 ˛i f .x / for every Pmset of points fxP i and coefficients ˛1 ; : : : ; ˛m 2 0; 1Œ such that i D1 ˛i D 1 and m i D1 ˛i x D x. This clearly implies f .x/  y for every y such that .x; y/ 2 conv.Hypof / and hence f .x/  f cv .x/ by (48). Since f .x/  f cv .x/ by definition, the equality f .x/ D f cv .x/ follows. Let m D infx2X f .x/. If m D 1, then clearly infx2conv.X / f cv .x/ D 1 since f .x/ D f cv .x/ on X . If m > 1, consider the constant function h.x/ D m defined on X and note that hcv .x/ D m for every x 2 conv.X /. Furthermore, since f .x/  h.x/ for every x 2 X , one has f cv .x/  hcv .x/ on conv.X /. Hence, it follows that infx2conv.X / f cv .x/ D m D infx2X f .x/. The inclusion arg minx2X f .x/  arg minx2conv.X / f cv .x/ is a trivial consequence of the equality infx2conv.X / f cv .x/ D infx2X f .x/ and of the coincidence of f and f cv on X .  It is well known [45] that a concave function that achieves its minimum on a closed convex set X that contains no line has at least one global minimum point among the extreme points of X . This result can be extended to the case where X is not convex. Theorem 11 Let X  Rn and f W X ! R. (i) If either conv.X / is closed and contains no line and f is concave on X , or X is compact and f is quasi-concave on X , then arg min f .x/ ¤ ; , arg min f .x/ \ E.conv.X // ¤ ;: x2X

x2X

(51)

(ii) If either conv.X / is closed and contains no line and f is strictly concave on X , or X is compact and f is strictly quasi-concave on X , then arg min f .x/  E.conv.X //: x2X

(52)

Proof Note that, by Theorem 10, the concave envelope f cv coincides with f on X and the minimum of f cv on conv.X / coincides with the minimum

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f on X . Observe also that if X is compact, then conv.X / is closed. Let x be any point in X  conv.X /. Then, by Theorem 18.5 in [66], x can be expressed as a convex combination of extreme points and extreme directions of conv.X /. Hence, there exist y 2 E.conv.X //, z 2 Rn , and   0 such that  x D y C z and y C z 2 conv.X / for all   0. Thus, taking ˇ D C1 , one has y C . C 1/z 2 conv.X / and x D .1  ˇ/y C ˇ.y C . C 1/z/. Since y 2 E.conv.X // and y C . C 1/z 2 conv.X /, this implies that x can be expressed as a convex combination of points y 1 ; : : : ; y h 2 E.conv.X // and w1 ; : : : ; wk 2 conv.X / with h  1 and k  0 (k D 0 if X is compact). Assume now that x is a global minimum point for f on X . Then: (i) If f is concave on X , then f .y 1 / D : : : D f .y h / D f .w1 / D : : : f .wk / D f .x/ and if f is quasi-concave on X , then f .x/ D mi nff .y 1 /; : : : ; f .y h /g. Hence, at least one point in E.conv.X // is a global minimum point for f on X. (ii) If f is strictly concave or strictly quasi-concave on X , then one must have h D 1 and k D 0, i.e., x must belong to E.conv.X //; otherwise a contradiction would arise. 

3.3

The Case of Polyhedral Domains

In the remainder of this section, the attention is restricted to the minimization of a function on a polyhedron, and in order to avoid trivial cases, it is also assumed that all (unbounded) polyhedra contain no line and thus are pointed, i.e., they have at least one vertex. It is easily seen that the class of quasi-concave functions is the most general class of functions for which Property E holds for every bounded polyhedron S . Indeed, by applying Property E to the special polyhedron that coincides with the line segment joining two points x 1 and x 2 , one trivially obtains the inequality defining quasiconcave functions. For a given polyhedron, or class of polyhedra, it is however possible to find classes of functions that properly include quasi-concave functions and satisfy Property E. Several results of this type that have been obtained by Tardella [72, 73, 75, 76] and by Hwang and Rothblum [47] are described next. One first needs to introduce some definitions and notation. The affine hull of a subset S of Rn is the smallest affine manifold containing S . tng.S / denotes the smallest linear subspace of Rn containing S  fx 0 g D fx  x 0 W x 2 S g, where x 0 is any point of S . Note that tng.S / coincides with the set f˛.x y/ W x; y 2 S; ˛ 2 Rg and can also be viewed as the linear subspace obtained by translating in 0 the affine hull of S . The relative interior rint .S / and the relative boundary rbd .S / of S are the interior and the boundary of S for the topology relative to the affine hull of S , respectively. Given a polyhedron P , a point x in P , and a direction d 2 Rn nf0g, let Fx denote the face of P of minimum dimension containing x, and consider the set

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857

Pd .x/ WD fz 2 P W z D x C d;  2 Rg; obtained intersecting P with the line passing through x with direction d . Since P is pointed, Pd .x/ is either a segment or a half line for every x and d . Consider the following conditions: Pd .x/ is bounded and f is quasi-concave on Pd .x/:

(53)

f is concave and bounded below on Pd .x/:

(54)

f is strictly quasi-concave on Pd .x/:

(55)

A face F of P is said to satisfy Property A, B, or C, if for every x 2 rint F there exists d 2 tng.F / n f0g, such that: Property A: Property B: Property C:

Condition (53) or (54) or (55) holds. Condition (53) or (54) holds. Condition (55) holds.

Let FA , FB , and FC denote the sets of all faces of P that do not satisfy Properties A, B, and C , respectively. By definition, each set FA , FB , and FC contains all the vertices VP of P . Furthermore, if f is concave and bounded below on P , or if P is bounded and f is quasi-concave on P , then FA and FB contain only the vertices of P . If f is strictly quasi-concave on P , then FA and FC contain only the vertices of P . Note also that FA  FB and FA  FC . The following lemma shows that the faces in FB contain all the best candidate points for minimizing f over P . S S Lemma 1 If x 2 P n F 2FB F , then there exists x 0 2 F 2FB F such that f .x 0 /  f .x/. Furthermore, if T1 is the time required to find a direction d satisfying condition (53) or (54) in a face F 62 FB , and T2 is the time required to find the extreme points of Pd .x/ for x in F , then x 0 can be determined in O.n.T1 C T2 // time. S Proof Let x 2 P n F 2FB F and let Fx be the face of P of minimum dimension containing x. Then x 2 rint Fx , and thus one can find, in T1 time, a direction d 2 tng.F / n f0g such that (53) or (54) holds. Note that, since x 2 rint Fx , one has x 2 rint Pd .x/. If Pd .x/ is bounded and y; z are its extreme points, then y; z 2 rbd Fx and f .x/  ff .y/; f .z/g by quasi-concavity of f on Pd .x/. On the other hand, if Pd .x/ is unbounded and y is its only extreme point, then y 2 rbd Fx and f .x/  f .y/. Indeed, if f .x/ < f .y/ and f is concave on Pd .x/, then f must be unbounded on Pd .x/ contradicting (54). Thus, in both cases S one can find, in T2 time, a point x 1 2 rbd F satisfying f .x 1 /  f .x/. If x 2 F 2FB F , then set x 0 D x 1 and stop. Otherwise, consider the face Fx 1 of P of minimum dimension containing x 1 , and iterate. Since dim.Fx 1 / < dim.F Sx /  n and VP  FB , after at most n iterations, one must find a point x 0 2 F 2FB F satisfying f .x 0 /  f .x/. 

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The next theorem shows that the infimum of f on P coincides with the infimum over the union of the faces in FB , and if one infimum is attained, the other is also attained. Furthermore, if the infimum is attained, at least one global minimizer of f on P must belong to a face in FA . Finally, the set of all global minimizers of f on P must be contained in the union of the faces in FC .

Theorem 12 (Existence and Location of Minima) inf f .x/ D min inf f .x/; F 2FB x2F

x2P

(56)

where the first infimum is attained if and only if the second infimum is attained. If the infimum is attained, then min f .x/ D min min f .x/ F 2FA x2F

x2P

and arg min f .x/  x2P

[

F:

(57)

(58)

F 2FC

Proof The inequality infx2P f .x/  minF 2FB infx2F f .x/ is trivial. The converse inequality follows easily from Lemma 1. Furthermore, if f attains its minimum at a S point x 2 P , then by Lemma 1 it also attains its minimum at a point x 0 2 F 2FB F . S Assume now S that all global0 minimum points of f on P belong to F n F 2FB F 2FA F , and let F be a face of minimum dimension in FB n FA containing a minimum point xN for f on P . Since F 0 2 FB n FA , condition (55) must hold at x. N Let d 2 tng.F 0 / n f0g be such that f is strictly quasiconcave on Pd .x/. N Then xN must be an extreme point of Pd .x/. N Otherwise, xN could be obtained as a convex combination of two points x 1 ; x 2 2 Pd .x/ N so that f .x/ N > minff .x 1 /; f .x 2 /g, contradicting the minimality of x. N On the other hand, if xN is an extreme point of Pd .x/, N then xN 2 rbd F 0 , S contradicting the 0 minimality of the dimension of F . Hence one must have xN 2 F 2FA F , so that (57) holds. S Let x be a global minimum point of f on P and assume that x 62 F 2FC F . Let Fx be the face of P of minimum dimension containing x. Then x 2 rint Fx and hence x 2 rint Pd .x/ for every d 2 tng.Fx / n f0g. Since Fx 62 FC , there is a direction d 2 tng.Fx / n f0g such that f is strictly quasi-concave on Pd .x/. Furthermore, since x 2 rint Pd .x/, there exist points x 1 ; x 2 2 Pd .x/ such that x D 12 .x 1 C x 2 /. Hence, by strict quasi-concavity, f .x/ > f .x 1 / C f .x 2 /, contradicting the minimality of f .x/.  A further S minima on P at a S extension of the class of functions that attain their point in F 2FA F (and in particular at a vertex of P , when F 2FA F D VP ) can be obtained with the following simple result.

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Theorem 13 Let f; g W X ! R and Y  X satisfy f .x/  g.x/ for all x in X , and f .x/ D g.x/ for all x in Y . Then, if f attains its minimum on X at a point of Y , also g attains its minimum on X at a point of Y . Proof min f .x/ D min g.x/  inf g.x/  min f .x/ D min f .x/: x2Y

x2Y

x2X

x2X

x2Y

t u

Theorem 12 can be used to decompose the problem of minimizing a function f on a polyhedron P into two subproblems: '.F / D min f .x/ x2F

and

min '.F /:

F 2FA

(59)

The first subproblem is similar to the main problem of minimizing f on P . However, such subproblem could be easier to solve due to additional structure of f on some faces of P (like, e.g., convexity). On the other hand, the second subproblem has a combinatorial nature. In Sects. 3.4 and 3.6, it is shown that, in some interesting cases, one or both subproblems can be solved efficiently, thereby providing efficient (polynomial) methods for some classes of problems, or polynomial time reductions from a continuous formulation of a problem to a combinatorial one and vice versa. Equality (57) can also be fruitfully applied in some cases to solve, exactly or approximately, the combinatorial problem minF 2FA '.F / by means of the continuous problem minx2P f .x/. This approach will be used in Sects. 3.4 and 3.6 to obtain continuous formulations for the maximum independent set problem, for the maximum clique problem, and for other combinatorial problems. Besides the Fundamental Theorem of Linear Programming, Theorem 12 extends several well-known results, which can be easily derived from it. A first example is the following existence result in indefinite Quadratic Programming due to Frank and Wolfe [30], for which a number of (more complex) alternative proofs and extensions are already available [12, 27, 50, 58, 63]. Theorem 14 ([30]) If f is a quadratic function that is bounded below on P , then f attains its minimum on P . Indeed, from Theorem 12, one easily derives the following common extension of the Frank–Wolfe Theorem and of the Fundamental Theorem of LP: Theorem 15 If f is a quadratic function that is bounded below on P , then f attains its minimum on P at a point contained in a face of P where f is strictly convex. Proof Note that, in this case, the family of faces FB defined above coincides with the family of faces of P where f is strictly convex. Furthermore, by coercivity, the

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minimum of a strictly convex quadratic function on a closed set always exists. The conclusion then follows from Theorem 12.  Note that when f is linear, the only faces of P where f is strictly convex are the vertices of P . Thus the Fundamental Theorem of LP is a special case of Theorem 15. One difficulty with the direct application of Theorem 12 is due to the hardness of the determination of the classes FA , FB , and FC in general. However, in many cases it is sufficient to use some superclasses of FA , FB , and FC that can be determined more efficiently by using only the directions parallel to the edges of P , as described below. Let D B and D U be two sets of vectors parallel to all the bounded edges and to all the extreme rays (unbounded edges) of P , respectively, and let S B  D B , S U  D U and S  D B [ D U . Lemma 2 Assume that, for every x 2 P , f is quasi-concave on Pd .x/ for all d 2 S B and concave or strictly quasi-concave on Pd .x/ for all d 2 S U . Then FA is contained in the set of faces of P that do not have any edge parallel to a vector in S B [ S U . Furthermore, if, for every x 2 P , f is strictly quasi-concave on Pd .x/ for all d 2 S , then FC is contained in the set of faces of P that do not have any edge parallel to a vector in S . Proof The proof follows from the definition of FA and FC , and from the fact that for every face F of P and every vector d parallel to an edge of F (bounded or unbounded) one has d 2 tng.F / n f0g.  From Lemma 2 and Theorem 12 one can immediately deduce the following extensions of the Fundamental Theorem of Linear Programming. Theorem 16 Assume that, for every x in P , f is quasi-concave on Pd .x/ for all d in S B and concave or strictly quasi-concave on Pd .x/ for all d in S U . Then, if f attains its minimum on P , at least one global minimizer belongs to a face that does not have any edge parallel to a vector in S B [ S U . Furthermore, if, for every x in P , f is strictly quasi-concave on Pd .x/ for all d in S , then all global minimizers belong to a face that does not have any edge parallel to a vector in S . Theorem 17 Assume that, for every x in P , f is quasi-concave on Pd .x/ for all d in D B : (i) If f is concave and bounded below on Pd .x/ for all x in P and d in D U , then f attains its minimum on P at a vertex of P . (ii) If f attains its minimum on P , and f is concave or strictly quasi-concave on Pd .x/ for all x in P and d in D U , then at least one global minimizer of f on P is in a vertex of P . Furthermore, if, for every x in P , f is strictly quasi-concave on Pd .x/ for all d in D B [ D U , then every global minimizer of f on P is in a vertex of P .

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Clearly, in order to apply Theorems 16 and 17, one must have a list of vectors d parallel to some or all the edges of P . Furthermore, one should be able to check if f is concave or (strictly) quasi-concave on Pd .x/ for all x in P . When the polyhedron P is described by a system of m inequalities in n variables, and both m and n are not too large, a list of vectors parallel to all its edges can be obtained computationally in a reasonable time (see, e.g., [8, 32] and references therein). On the other hand, if the polyhedron has a special structure, a list of vectors parallel to all its edges may be readily available or could be obtained by means of theoretical results. The next sections are concerned with optimization problems over some polyhedra for which the vectors parallel to the edges are known: rectangles, simplices, and polymatroids. In the case where the function f is twice continuously differentiable, and, in particular, when it is a quadratic function, some simple necessary and sufficient conditions for its concavity on the sets Pd .x/ are described in the following Remark. Remark 2 When f is twice continuously differentiable, the concavity of f on the sets Pd .x/ for all x 2 P is equivalent to d T Hf .x/d  0 for all x 2 P , where Hf .x/ is the Hessian matrix of f at x. Furthermore, if f .x/ D 12 x T C x C q T x, then the (strict) quasi-concavity of f on Pd .x/ for all x 2 P is equivalent to requiring that at least one of the following conditions hold: (i) d T Cd  0 .< 0/. (ii) minx2P .C x C q/T d  0 .> 0/. (iii) maxx2P .C x C q/T d  0 .< 0/. The equivalence is justified by the property (see, e.g., [9]) that a function is (strictly) quasi-concave on a line if and only if it is either (strictly) monotone, or nondecreasing (increasing) up to a certain point and then nonincreasing (decreasing) on the half line starting that point. When the edge directions of P are not explicitly known, one can still guarantee the validity of Property E by making some more restrictive assumptions on f . Definition 4 Let f be a real function defined on a convex set X Rn and let m  n. f is called m-concave (m-quasi-concave ) on X iff f .˛x 1 C .1  ˛/x 2 //  ˛f .x 1 / C .1  ˛/f .x 2 / (resp. f .˛x 1 C .1  ˛/x 2 //  minff .x 1 /; f .x 2 /g/; for every ˛ 2 Œ0; 1 and for every x 1 ; x 2 2 X such that xi1 D xi2 for at least n  m indices i in f1; : : : ; ng. Assume now that the polyhedron P is expressed in one of the following ways: P D fx 2 Rn W Ax D b; `  x  ug or

(60) P D fx 2 R W Ax  b; `  x  ug; n

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where A 2 Rmn ; b 2 Rm ; ` D .`1 ; : : : ; `n /; u D .u1 ; : : : ; un /; 1  `i < ui  C1 and minfj `i j; j ui jg < C1. Note that the assumptions on ` and u imply that, if P ¤ ;, then P has at least one vertex. Theorem 18 Let P be a polyhedron defined by (60) and assume that A has rank m  1. If f is m-concave on P or f is m-quasi-concave on P and P is bounded, then Property E holds. Proof Let s D .s1 ; : : : ; sn / be any vector parallel to an edge of P . Observe that every edge of P lies in the intersection of n  1 linearly independent hyperplanes taken from among those defining P in (60). Since rank.A/ D m  1, one has si D 0 for at least n  m indices i . From the m(-quasi)-concavity of f one then derives that f is m(-quasi)-concave on the sets Ps .x/ for every x 2 P . Hence, the conclusion follows from Theorem 17.  Some results concerning the location of global minimizers for strictly quasiconvex differentiable functions on a polytope are described next. From these results, one obtains new conditions for the existence of global minimizers at the vertices. It is interesting to observe that these conditions are obtained under assumptions of (quasi) convexity of f rather than concavity. First, recall the following two results by Scozzari and Tardella [70] that establish the existence of a sequence of nested faces with interior minimizers for this type of problems. Theorem 19 Let f be a strictly quasi-convex differentiable function that attains its minimum on a polytope P at an interior point x  of P . Then there exists a facet F of P such that the minimum of f on F is attained at a point in its relative interior. Corollary 1 Let f be a strictly quasi-convex differentiable function that attains its minimum on a polytope P at an interior point x  of P . Then there exists a nested sequence of faces F 1  F 2      F n such that F n D P , dim(F i ) = i , and xF i 2 ri nt.F i / for i D 1; : : : ; n. Using this result, one can easily derive conditions on the location of global minimizers and, in particular, conditions for the existence of vertex minimizers. Theorem 20 Let f be a strictly quasi-convex differentiable function, and let G be a family of faces of P such that: (i) For every face F of P that is not strictly contained in a face of G, and for every nested sequence of faces F 1  F 2      F k D F with dim(F i ) = i , there is an index i for which F i 2 G. (ii) For every face G 2 G, the restriction of f to G does not attain its minimum at a point in the relative interior of G. Then the global minimum of f over P is attained in the relative interior of a face that is strictly contained in a face of G.

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Proof Let x be the (unique) global minimizer of f over P . Clearly, x is in the relative interior of Fx , the smallest face of P containing x. Furthermore, by Corollary 1, there exists a nested sequence of faces F 1  F 2      F k such that F k D Fx , dim(F i ) = i , and the restriction of f to each F i attains its minimum in the relative interior of F i . Thus, by Assumptions (i) and (ii), Fx must be strictly contained in a face of G.  When G is the set of faces of dimension k, Assumption (i) in the previous theorem holds trivially. Thus one obtains the following consequence: Corollary 2 Let f be a strictly quasi-convex differentiable function. If the restriction of f to every face of dimension k does not attain its minimum in the relative interior of such face, then the global minimum of f on P is attained in a face of dimension less than k. In particular, for k D 1, vertex optimality follows. Corollary 3 Let f be a strictly quasi-convex differentiable function. If the restriction of f to every edge of P is minimized at one of its endpoints, then the global minimum of f on P is attained at a vertex of P .

3.4

Pseudo-Boolean Optimization and Optimization with Box Constraints

A simple class of polyhedra to which Theorems 16 and 17 can be easily applied are the (generalized) rectangles (or box constraints) R D Œ`; u D fx 2 Rn W `  x  ug, where one allows the values `i D 1 or ui D C1 (but not `i D 1 and ui D C1). In this case D B [ D U D fe 1 ; : : : ; e n g. Furthermore, a function f is concave or (strictly) quasi-concave on Rei .x/ for all x in R, if and only if the function fi .t/ D f .x C te i / is concave or (strictly) quasi-concave on the interval Œ`i  xi ; ui  xi , respectively. This is equivalent to requiring that f is concave or (strictly) quasi-concave with respect to each variable. In particular, if f is twice continuously differentiable, then f is (strictly) quasi-concave on Rei .x/ if and only if there exists tN 2 Œ`i  xi ; ui  xi  such that fi0 .t/  0 .> 0/ for all t < tN, and fi 0 .t/  0 .tN (see, e.g., [9]). Furthermore, f is concave on Rei .x/ for all x in R if and only if the second partial derivative fxi xi .x/ is nonpositive for every x in R. When f .x/ D 12 x T C x C q T x is a quadratic function, these conditions can be stated as follows: f is (strictly) concave on Rei .x/ for all x 2 R , ci i  0 . m. Then for every 2 H the function g defined in (76) is concave on X WD fx 2 Rn W y .i /  y .i C1/ ; i D 1; : : : ; m C p  1g. Proof Indeed, on X one has jyi  yj j D yi  yj if 1 .i / < 1 .j / and jyi  yj j D yj  yi if 1 .i / > 1 .j /. Hence, on X the functions 'ij .jx i  x j j/ and i j ij .jx  a j/ are concave for all i; j , since they are obtained as the composition of a concave and of a linear function. Therefore, the function g is concave on X because it is a sum of concave functions.  Theorem 31 At least one global minimum point of g on Rn is achieved at a point in A. Proof Note that the sets X are polyhedra with vertices in A and that Rn D [ 2H X . Furthermore, since g is continuous and limt !C1 ij .t/ D C1 for all i and j , there must be a global minimum point of g in a bounded neighborhood of 0. Hence, by Theorem 28, a global minimum point of g must be contained in A.  Another continuous location problem that can be solved by restricting the search to a finite set of candidate solutions is the problem of locating some undesirable facilities at locations x 1 ; : : : ; x m 2 Rn in a way that minimizes the maximum of some decreasing nuisance cost functions of the distances among them and between them and a given set of points b 1 ; : : : ; b p 2 Rn (see [28] for a survey on this type of problems). Formally, one wishes to minimize the function

g.x/ D g.x 1 ; : : : ; x m / WD maxf max ˛ij .kx i  b j k/; max ˇij .kx i  x j k/g; i 2I;j 2J

i ¤j 2I

where ˛ij and ˇij are continuous nonincreasing functions, I D f1; : : : ; mg, J D f1; : : : ; pg, and k  k is any norm in Rn . In this case, the function g is piecewise

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quasi-concave since it is the maximum of a finite family of continuous quasiconcave functions. Clearly, the infimum of g on Rnm is obtained when kx i k ! 1 for all i . Hence, this location problem becomes meaningful only when x is required to be in a compact set X . Under this assumption, Theorem 29 can be applied to establish the following result. Theorem 32 At least one S global minimum point S of g on a compact set X is achieved in the set E D i;j 2I E.conv.Xij // [ i 2I;j 2J E.conv.Yij //, where Xij WD fx 2 X W ˛ij .kx i  b j k/  maxf max ˛hk .kx h  b k k/g; max ˇhk .kx h  x k k/gg; h2I;k2J

h¤k2I

Yij WD fx 2 X W ˇij .kx i  x j k/  maxf max ˛hk .kx h  b k k/g; max ˇhk .kx h  x k k/gg: h2I;k2J

h¤k2I

When only one new undesirable facility has to be established, the objective function g W Rn ! R becomes g.x/ D max ˛j .kx  b j k/: j 2J

In this case, Theorem 32 implies that a global minimum point of g on a compact set X is attained at a point in E D [j 2J E.conv.X \ Xj //, where Xj D fx 2 Rn W ˛j .kx  b j k/  ˛k .kx  b k k/ for all k ¤ i g. The family of sets Xj forms the (generalized) Voronoi diagram with respect to the functions ˛j .kxk/. This geometric structure has received much attention in recent years especially in the case where ˛j .kxk/ D kxk or ˛j .kxk/ D j kxk and k k is the Euclidean norm (see [6] and references therein). In particular, when ˛j .kxk/ D kxk and k  k is the Euclidean norm, the sets Xj are polyhedra described by 

bj C bk Xj D x 2 R W .b  b / x   0 for all k ¤ j : 2 

n

k

j T

In this case, the maximum number N of points in E D following bounds [51]:

S j 2J

E.Xj / satisfies the

n=2Š p n=2  N  2.n=2Š p n=2 / for n even and .dn=2e  1/Š dn=2e p  N  2.dn=2eŠ p dn=2e / for n odd. e

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Furthermore, the points of E can be computed in O.p dn=2eC1 / time in the general case [7], and in time O.p log p/ for n D 2. Thus the minimum of g on a polyhedron X can be found by simple enumeration of all the vertices of the sets Xi \ X . This yields a finite algorithm, while in [71] an algorithm is presented which requires some additional assumptions to solve the same problem in finitely many iterations. Facility location on networks is an important field in location theory where much research has concentrated in identifying a finite set of points that necessarily contain an optimal solution. Good surveys on this topic are [43, 54, 59]. Location problems on networks can be formulated using some definitions and notation taken from [54]. An edge of length ` > 0 is the image of an interval Œ0; ` by a continuous mapping f from Œ0; ` to Rd such that f . / ¤ f . 0 / for any

¤ 0 in Œ0; `. A network is defined as a subset N of Rd that satisfies the following conditions: (a) N is the union of a finite number of edges; (b) any two edges intersect at most at their extremities; (c) N is connected. The vertices of the network are the extremities of the edges defining N and are denoted by V D fv1 ; : : : ; vn g. The set of edges defining the network is denoted by E. For every edge Œvi ; vj  2 E, let fij from Œ0; `ij  to Rd be the defining mapping and denote by ij the inverse of fij which maps Œvi ; vj  into Œ0; `ij . Given two points x 1 ; x 2 2 Œvi ; vj , the subset of points of Œvi ; vj  between and including x 1 ; x 2 is a subedge Œx 1 ; x 2 . The length of Œx 1 ; x 2  is given by j ij .x 1 /  ij .x 2 /j. A path joining two points x 1 2 N and x 2 2 N is a minimal connected subset of N containing x 1 and x 2 . The length of a path is equal to the sum of all its constituent edges and subedges. A metric d on N is defined by setting d.x 1 ; x 2 / equal to the length of a shortest path joining x 1 and x 2 . For any point z 2 N consider the function on Œ0; `ij  defined by d.z; fij . //. Note that, by the definition of the distance, one has d.z; fij . // D minfd.z; vi / C ; d.z; vj / C `ij  g: Hence, d.z; fij . // is the minimum of two linear functions in . Therefore, the following properties hold [54]: (i) d.z; fij . // is continuous and concave on Œ0; `ij . (ii) d.z; fij . // is linearly increasing with slope +1 on Œ0; ij .z/Œ and linearly decreasing with slope 1 on  ij .z/; `ij , where ij .z/ D 12 Œ`ij C d.z; vj /  d.z; vi /. The (single) median problem on the network N consists of finding a point in N that minimizes the function F .x/ D

X

wi ..d.vi ; x///;

vi 2V

where wi .t/ are concave nondecreasing functions from RC to R. In 1964, Hakimi [41] showed that at least one solution of this problem belongs to V when wi .t/ D i t with i  0. This result was extended in [59] to the case where all wi .t/ are concave nondecreasing functions. The proof is based on the remark that if wi .t/ is

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concave nondecreasing, then the function wi ..d.vi ; fij . //// is concave on Œ0; `ij  and hence F .fij . // is also concave on Œ0; `ij . Therefore, the global minimum of F must be attained at one of the points fij .0/ or fij .`ij /, i.e., at a vertex of N . In the (single) center problem on the network N , one seeks to minimize the function G.x/ D max.d.vi ; x//: vi 2V

In this case, the function G.fij . // is not concave on the whole segment Œ0; `ij , but only on subsegments thereof. However, the number of such subsegments is bounded by jV j2 jEj and their extremities can be explicitly determined (see [54] for details). Hence, also the center problem on a network can be solved by enumerating a finite (and polynomially bounded) number of points.

5

Generalized Systems

5.1

Introduction

In the first half of the last century, there have been several important achievements: a great growth of the theory of extrema in the wake of the so-called Peano Function, the parallel development of the Theorems of the Alternative and of the Separation Theorems, the gathering of important results to form Convex Analysis, and a great development of the Calculus of Variations. Due to these achievements and also to some important problems arisen in Mathematical Physics, like the so-called Signorini Contact Problem, at the beginning of the second half of the last century, the theory of variational inequalities and, in parallel, that of Complementarity Systems have been formulated, the former mainly in infinite dimensional spaces, and the latter in finite dimensional spaces. All these great developments have lead to search for models, which embody parallel theories. In particular, this has happened in the case of extremum problems, variational inequalities, and generalized systems. A possible unification has been found with the formulation of the so-called generalized systems. As concerns constrained extremum problems, the connection with a system is, nowaday, clear: while “optimization” is merely a language for looking at real problems, the mathematical core lies in stating whether or not a system be impossible. The results contained in the following Sections are based on [2].

5.2

General Results

Given positive integers n and l, a convex cone C  Rl , sets R; Z  Rn , and a vector-valued function F W Rn Rn ! Rl consider the problem P which consists in finding y 2 R \ Z such that the system, in the unknown x, F .xI y/ 2 C; be impossible.

x 2R\Z

(77)

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Given a set X  Rn such that Z  X , the replacement of R \ Z with R \ X represents a relaxation of the domain of (77); of course, this may change the set of solutions of P. This drawback can be overcome through a suitable change of the system. To this end, consider a vector-valued function ˆ W Rn Rn ! Rl and the family fP./g2R of problems where P./ consists in finding y 2 R \ X such that the system (in the unknown x) F .xI y/ C ˆ.xI y/ 2 C;

x 2 R\X

(78)

be impossible. System (78) shows, with respect to (77), a relaxation of the domain and a penalization of F . In [2] conditions are given under which P and P./ are equivalent in the sense that they have the same (possibly empty) set of solutions. To this end, one needs a preliminary proposition. As concern notation, for every cone C  Rl and for all x; y 2 C, let x C y ” x  y 2 C; x6 C y ” x  y … C; x> C y ” x  y 2 int C;

where int C denotes the interior of C. The notation , 6, and < is defined in analogous way. Finally, for the sake of simplicity, and without any fear of confusion, k  k denotes either a norm in Rl or in Rn , or a matrix norm. Lemma 4 Let C, C 0  Rl be cones such that C 0 be closed and f0g ¤ C 0  int C. Let B D fx 2 Rl Wk x k g with   0, and U D fx 2 Rl Wk x k D 1g. Then, there exists 0 such that V1 C V2 2 int C; 8 > 0 ; 8V1 2 B ; 8V2 2 C 0 \ U

(79)

Proof Since k V2 k D 1, 8V2 2 C 0 \ U , 8 > , one has 1

hV1 C V2 ; V2 i hV1 ; V2 i C k V2 k2  C   : k V1 C V2 k k V1 kk V2 k C k V2 k C

Since the scalar product of vectors of unitary norm is 1 if and only if they coincide, passing to the limit as ! C1, one obtains V1 C V2 D V2 : !C1 k V1 C V2 k lim

Connections Between Continuous and Discrete Extremum Problems

881

Since C 0 \ U is a compact set included in int C, then U n C and C 0 \ U have positive distance (induced by the norm considered). Hence (79) follows. This completes the proof.  Note that, when l D 1, the assumptions of Lemma (4) are fulfilled only by C D RC and by C D RC n f0g (and, of course, by their opposites); in both cases C 0 D RC necessarily (or C 0 D R ). In the sequel, 8x 2 X , p.x/ will denote a vector belonging to the set projZ .x/; where projZ W X ! Z is the multivalued function which projects x on the compact set Z. Theorem 33 Let R  Rn be a closed set, Z  X  Rn , Z and X be compact sets, and let F; ˆ W X X ! Rl satisfy the following assumptions: .H1 / F is bounded on X X , and there exist positive reals L and ˛, and an open set   Z, such that k F .p.x/I x/ k L k x  p.x/ k˛ ; 8x 2  \ X; 8p.x/ 2 projZ .x/I .H2 /

(i) ˆ is continuous on X X . (ii) 8x; y 2 Z; ˆ.x; y/ D 0. (3i) there exists a closed cone C C with ; ¤ .C C n f0g/  int C, such that ˆ.x; y/ 2 .C C n f0g/; 8x 2 Z; 8y 2 X n Z: (4i) 8z 2 Z there exists a neighborhood S.z/ of z, and a real .z/ such that

k ˆ.p.x/I x/ k .z/ k xp.x/ k; 8x 2 S.z/\.X nZ/; 8p.x/ 2 projZ .x/: Then a real 1 exists, such that, 8 > 1 , a solution of P./ is a solution of P. Proof To prove the thesis it is sufficient to show that 90 2 R such that, 8 > 0 , a solution of P./ is achieved necessarily at point z 2 R \ Z. Because of .H2 /.i i /, this claim assures that a solution of P./ is also a solution of P. Consider the sets XO D R \ X; ZO D R \ Z; and SO .z/ D  \ S.z/, where S.z/ is precisely that O is obviously a cover of Z. O Since ZO is a of .H2 /.4i /. The family fSO .z/; z 2 Zg O Set closed subset of Z, there is a finite subcover, say fSO .zi /; i D 1; : : : ; kg, of Z. S D [kiD1 SO .zi / and let  D maxfL=.zi /; i D 1; : : : ; kg. Because of .H1 / and .H2 /.4i /, it holds k F .p.x/I x/ k O  ; 8x 2 S \ .XO n Z/: k ˆ.p.x/I x/ k

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Consider the set U D fx 2 Rl Wk x k D 1g; because of .H2 /.3i / one has ˆ.p.x/I x/ O 2 C C \ U; 8x 2 XO n Z: k ˆ.p.x/I x/ k

(80)

Apply Lemma 4, where V1 ; V2 ; C 0 , and C are identified with F .p.x/I x/= k ˆ.p.x/I x/ k, ˆ.p.x/I x/= k ˆ.p.x/I x/ k; C C , and C, respectively. The assumptions of Lemma 4 being fulfilled, one obtains the existence of a O real 1 , such that (79) holds, namely, for all > 1 and x 2 S \ .XO n Z/ ˆ.p.x/I x/ F .p.x/I x/ C 2 C: k ˆ.p.x/I x/ k k ˆ.p.x/I x/ k

(81)

O It follows that, 8 > 1 , P./ cannot have solutions in S \ .XO n Z/. O O Because of Now, consider the compact set X0 D X n S , and fix zO 2 Z. .H2 /.i; 3i /, ˆ is continuous and different from the null vector on the compact set fOzg X0 . Thus Mˆ D min k ˆ.OzI x/ k > 0: x2X0

Apply again Lemma 4, where  D MF =Mˆ , and V1 ; V2 , C 0 , and C are identified with F .OzI x/= k ˆ.OzI x/ k; ˆ.OzI x/= k ˆ.OzI x/ k; C C , and C, respectively, and MF D sup.x;y/2X X k F .xI y/ k. Then, the hypotheses of Lemma 4 being satisfied, one obtains the existence of 2 , such that, for all > 2 and x 2 X0 , ˆ.OzI x/ F .OzI x/ C 2 C: k ˆ.OzI x/ k k ˆ.OzI x/ k

(82)

Hence, 8 > 2 , P./ cannot have solutions in X0 . If  > 1 D maxf 1 ; 2 g, O This completes account taken of (81) and (82), P./ cannot have solutions in XO n Z. the proof.  Remark 5 Consider the special case where F .xI y/ D f .y/  f .x/;

(83)

with f W X ! Rl bounded. It is trivial to check that, if f fulfills the Holder R condition on the set , i.e., there exist L; ˛ > 0 such that k f .y/  f .x/ k L k x  y k˛ ; 8x; y 2  \ X;

(84)

then the function F defined in (83) satisfies .H1 /. The converse is not true, as shown by the Example 2.

Connections Between Continuous and Discrete Extremum Problems

883

Note that, for F given by (83), y is a solution of P if and only if f is a solution of the vector minimum problem min f .x/ s:t: x 2 R \ Z; C

(85)

where C defines now the partial order and minC marks vector minimum: the impossibility of (77) defines y as a solution of (85); y is called vector minimum point, in short VMP. Example 2 Let n D l D 1, R D R; Z D Œ0; 1, and X D Œ1; 1. Consider (83) with f .x/ D x sin.1=x/ if x ¤ 0 and f .0/ D 0. .H1 / is satisfied at ˛ D 1, L D 1, and  D Œ2; 2. In fact, if x 2 Z so that p.x/ D x, then F .p.x/I x/ D 0; if x 2 . \ X / n Z D Œ1; 0Œ, then jF .0I x/j D jf .x/j D jx sin.1=x/j  jxj. Thus .H1 / holds. Obviously f does not fulfill (84). Remark 6 Consider two special cases. For this, introduce G W Rn ! Rln , and let hG.u/; vil denote the vector whose i th component is the scalar product between the i th row of the matrix G.u/ and the vector v. The former special case is F .xI y/ D hG.y/; y  xil I

(86)

F .xI y/ D hG.x/; y  xil :

(87)

the second case is

If G is bounded on Rn , then both functions F in (86) and (87) fulfill .H1 / at ˛ D 1 and L D k G k D supkxkD1 k G.x/ k, as it is easy to check. Example 3 shows that the converse is not true. Note that y is a solution of P in case (86) (or (87)) if and only if it is a solution of the Vector Inequality of Stampacchia type [34]: find y 2 R \ Z such that hG.y/; x  yil —C 0;

8x 2 R \ Z;

(88)

or of the vector variational inequality of Minty type [34]: find y 2 R \ Z such that hG.x/; y  xil C 0;

8x 2 R \ Z:

(89)

Example 3 Let n D l D 1, C D RC np f0g, R D R, Z D Œ0; 1; and X D Œ0; 2. Let F be defined by (86) with G.y/ D 1= j1  yj, with y ¤ 1 and G.1/ D 0; it holds

jF .p.x/I x/j  jx  p.x/j1=2 ; 8x 21; 2ΠI

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indeed, if x 2 Z, then p.x/ D px so that F .p.x/I x/ D 0; if x 2 1; 2Œ, then p.x/ D 1 so that jF .p.x/I x/j D x  1. p jx  yj.x  1/; Example 4 Let F W R R ! R; F .xI y/ D Z D Œ0; 1; and X D Œ0; 2. In this case a constant L and an open set   Z does not exist such that jF .xI y/j  jx  yj; 8x 2  \ X; 8y 2 Z: However, the hypothesis .H1 / of Theorem 33 holds with  D R, L D ˛ D 3=2.

p 2, and

Example 5 Let n D l D 1, C D RC n f0g, R D R, Z D Œ0; 1 and X D Œ0; 2, F W X X ! R, and F .xI y/ D .x  y/2 .1  y/.x  1/. Then F fulfills .H1 / with  D   1; 2Œ, L D 1, ˛ D 2. In fact, F is bounded; moreover, if x 2 1; 2Œ, p.x/ D 1 and F .1I x/ D 0; if x 2 Z, p.x/ D x and F .xI x/ D 0. Each y 2 Œ0; 1 is a solution of the following problem P: find y 2 Z such that F .xI y/ 2 C;

x 2 Œ0; 1

be impossible. Let ˆ W X X ! R be a penalty function defined as follows: 8 ˆ .1  x/2 ; ˆ ˆ ˆ ˆ ˆ ˆ x

Then ˆ fulfills condition .H2 / choosing C C D RC , .z/ D 1, 8z 2 Z, ˛ D 2, and L D 1. Note that, 8 2 RC , y 2 Œ0; 1 is not a solution for P./. In fact, 8x 2 1; 2 F .xI y/ C ˆ.xI y/ D .x  1/Œ.x  y/2 .1  y/  .x  1/: Observe that limx#1 .x  y/2 .1  y/  .x  1/ D .1  y/3 > 0. Thus y 2 Œ0; 1 is not a solution of P./. Furthermore, F does not fulfill .H3 / of the following Theorem 34 at ˛ D 2, L D 1. In fact, p.x/ D 1 for x  1 and the inequality jF .p.x/I y/  F .xI y/j  .x  1/2 ;

8x 2 X \ ; and 8p.x/ 2 projZ .x/

holds if and only if .x  y/2 .1  y/  .x  1/; 8x > 1; 8y 2 Œ0; 1:

Connections Between Continuous and Discrete Extremum Problems

885

This is impossible for x D 5=4; y D 1=4. Moreover, observe that P./ has no solution: for each y 2 Œ1; 2, x 2 Œ0; 1, and >0 F .xI y/ C ˆ.xI y/ 2 C: If Z is finite, then the inequality in .H2 /.4i / can be equivalently replaced (in the sense that the thesis of the Theorem 33 is still valid and the class of the penalty functions ˆ which satisfy it is nonempty) by the following condition: 8z 2 Z there exist a neighborhood S.z/ and a real .z/ > 0, such that k ˆ.zI x/ k .z/ k x  z k; 8x 2 S.z/ \ .X n Z/: In fact, choosing a suitable neighborhood S.z/ of z, one has p.x/ D z. If Z is not finite, then the above condition might be in contrast with Assumptions .H2 /.i; i i /. In this case, the latter is replaced by the following one: .H2 /0 There exists a vector-valued function ' W X ! Rl , such that: .i / ' is continuous on X . .i i / 8x 2 Z; '.x/ D 0. .3i / There exists a closed cone C C with f0g ¤ C C  int C, such that '.x/ 2 C C n f0g; 8x 2 X n Z: .4i / 8z 2 Z; there exist a neighborhood S.z/ and a real .z/ > 0 such that k '.x/ k  .z/ k x p.x/ k˛ ; 8x 2 S.z/\.X nZ/; and 8p.x/ 2 projZ .x/: Note that if ˆ.xI y/ D '.x/  '.y/, 8x; y 2 X , then .H2 / of Theorem 33 is fulfilled if .H2 /0 is fulfilled. The following theorem deals with the special class of problem P where the cone C is equal to RlC ; it gives a condition guaranteeing that a solution of P is also a solution of a suitable problem P./, for  large enough. Theorem 34 Let R  Rn be a closed set, Z  X  Rn , Z, and X be compact set, and let F; ˆ W X X ! Rl satisfy the following hypotheses: .H3 / F is bounded on X X and there exist positive reals L; ˛ and an open set   Z, such that k F .xI y/  F .p.x/I y/ k L k x  p.x/ k˛ ; 8x; y 2  \ X; 8p.x/ 2 projZ .x/ .H4 / .i / ˆ is continuous on X X . .i i / 8x; y 2 Z; ˆ.xI y/ D 0; 8x 2 X; ˆ.xI  / is constant on Z. .3i / There exists a closed cone C  , with f0g ¤ C   int .C/, such that ˆ.xI y/ 2 C  n f0g; 8x 2 X n Z; 8y 2 Z:

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.4i / 8z 2 Z; there exists a neighborhood S.z/ of z and a positive real .z/ such that k ˆ.xI p.x// k .z/ k x  p.x/ k˛ ; 8x; y 2 S.z/ \ .X n Z/; 8p.x/ 2 projZ .x/: Then, there exists 2 2 R, such that, 8 > 2 , a solution of P is a solution of P./. Note that hypothesis .H3 / of Theorem 34 can be weakened by replacing the requirement x; y 2  \ X with x 2  \ X; y 2 Z.

5.3

The Case of a Discrete Domain

This section deals with a special but very important case, namely, that of a discrete problem. More precisely, consider the case Z D Bn ; C D RlC n f0g; C C D fv 2 RlC W v1 D    D vl g; C  D C C ; (90) where B D f0; 1g; the case where Z is a bounded subset of Zn can be reduced to the above one by well-known transformations [61]. The relaxation of Z D Bn is X D XQ D fx 2 Rn W 0  xi  1; i D 1; : : : ; ng and the penalty term ˆ W X X ! Rl is defined by 0

1 '.y/  '.x/ B C :: ˆ.xI y/ D @ A :

(91)

'.y/  '.x/ where ' W Rn ! R; '.x/ D hx; e  xi, with e D .1; : : : ; 1/ 2 Rn . Under Assumptions (90), the function defined in (91) fulfills, with ˛ D 1, the conditions .H2 / of Theorem 33 and .H4 / of Theorem 34. In fact, .H2 /.i /; .i i /; .3i /; .H4 /.i /; .i i /; .3i / are obvious. As concerns .H2 /.4i / and .H4 /.4i /, note that if S.z/ is small enough so that p.x/ D z, then, as proved in [36], ' satisfies the following conditions: 8z 2 Z there is a neighborhood S.z/ of z and a real O .z/, such that '.x/  O k x  z k;

8x 2 S.z/ \ .X n Z/:

Therefore for all z 2 Z and x 2 S.z/ \ .X n Z/ 0

1 '.x/ p B C p k ˆ.p.x/I x/ kDk ˆ.xI p.x// k D @ ::: A D lj'.x/j  lO .z/ k x  z k; '.x/

Connections Between Continuous and Discrete Extremum Problems

which proves .H2 /.4i / and .H4 /.4i / by setting .z/ D following result has been established:

887

p lO .z/. Hence, the

Theorem 35 For the case (90)–(91), let the function F verify with ˛ D 1 the hypothesis .H3 / of Theorem 34, and assume that F .xI x/ D 0 for all x 2 . Then, there exists 3 2 R such that, 8 > 3 , P and P./ have the same set of solutions. Remark 7 Any function F W X X ! Rl of the form F .xI y/ D f .x/  f .y/; where f W X ! Rl , or F .xI y/ D hG.y/; y  xi; where G W Rn ! Rln , fulfills the condition F .xI x/ D 0 for all x 2 . In Theorem 35 the hypothesis .H3 / with ˛ D 1 was sufficient for the special ˆ chosen. For ˛ D 1, the above theorem is still valid for all strictly concave functions '1 ; : : : ; 'l W Rn ! R such that 'i .x/ D 0 for all i D 1; : : : ; l and x 2 Z, and there exist a neighborhood Si .y/ and a real constant i .y/ such that j'i .x/j  i .y/ k x  y k;

8x 2 Si .y/ \ .X n Z/:

Note that for i D 1; : : : ; l, the above condition is a slight generalization of the condition on the function ' in [36], and it is equivalent to .H2 / of Theorem 33 when l D 1. Then one can put h D .'1 ; : : : ; 'l /, and, 8x; y 2 X , ˆ.xI y/ D h.y/  h.x/:

(92)

l Condition p .H2 /.4i / for ˆ follows choosing 8z 2 Z, S.z/ D \i D1 Si .z/, and .z/ D l minfi .z/; i D 1; : : : ; lg.

The following theorem gives a condition which assures that, 8y 2 X ; the function F .  I y/Cˆ.  I y/ is component-wise strictly convex. This is a straightforward extension of Theorem 3.2 of [36] and Theorem 2 of [37]. Theorem 36 In the case (90)–(91), let F W X X ! Rl fulfill the hypotheses of Theorem 35 with ˛ D 1. If F 2 C 2 .X X /, then there exists a real 4 such that, 8 > 4 , P and P./ have the same solutions, and 8y 2 X , the function F .  I y/ C ˆ.  I y/ is component-wise strictly convex.

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Proof 8y 2 X , let Hi .xI y/ and HO i .xI y/ be the Hessian matrices at x, of the i -th component of F .  I y/ and of F .  I y/ C ˆ.  I y/ respectively. Hence, for i D 1; : : : ; l HO i .xI y/ D Hi .xI y/ C 2In ; where In denotes the n n identity matrix. 8i 2 f1; : : : ; lg, 8r 2 f1; : : : ; ng, let i r W X X ! R be the function where i r .xI y/, r D 1; : : : ; n are the eigenvalues of the Hessian Hi .xI y/. Because of the continuity of F , i r is continuous. Hence 4 WD max max ji r .xI y/j < C1: i;r

X X

Moreover, observe that, 8x; y; i .xI y/ is an eigenvalue of HO i .xI y/ if and only if i .xI y/  2 is an eigenvalue of Hi .xI y/. Then, if  > 4 D

1 maxf1 ; 2 ; 4 g 2

it follows that, for all .x; y/ 2 X X , i r .xI y/ D i r .xI y/ C 2 is an eigenvalue of HO i .xI y/ and it results i r .xI y/ > 0. This completes the proof. 

5.4

Discrete Vector Optimization and Variational Inequalities

In this section, equivalence results for vector optimization problems and vector variational inequalities are obtained as a direct application of the preceding theorems. Indeed, consider the special case of (85), where Z D Bn and C D RlC n f0g, namely, consider the vector minimization problem: min f .x/ s:t: x 2 R \ Bn : C

(93)

It is well known that a point y is a solution of (93) if and only if the system f .y/  f .x/ 2 C;

x 2 R \ Bn ;

(94)

is impossible. (When l D 1 and C D .0; C1/ the problem (93) becomes a scalar 0–1 optimization problem, and the impossibility of (94) becomes the impossibility of the inequality f .y/  f .x/ > 0; x 2 R \ Bn .) Then, consider the vector minimum problem: minŒf .x/ C  .x/ s:t: x 2 R \ XQ ; C

(95)

Connections Between Continuous and Discrete Extremum Problems

889

where W Rn ! Rl ; D .'; : : : ; '/; '.x/ D hx; e  xi, with e D .1; : : : ; 1/,  2 R, XQ D fx 2 Rn W 0  xi  1; i D 1; : : : ; ng. Corollary 7 Let the following hypotheses be satisfied: .H5 / f W Rn ! Rl is bounded on XQ , and there exist a positive real L and an open set   Bn such that jfi .x/  fi .y/j  L k x  y k; 8x; y 2  \ XQ ; i D 1; : : : ; l; where fi denotes the i -th component of f . Then, there exists a real 5 2 R, such that, 8 > 5 , (93) and (95) have the same solutions. If, in addition, f 2 C 2 .X /, then there exists 6 2 R such that, 8 > 7 D maxf5 ; 6 g, f C  is component-wise strictly concave. Proof Put ˆ W X X ! Rl ; ˆ.xI y/ D .y/  .x/. According to Remark 5, under Assumptions .H5 /, the function F .xI y/ D f .y/  f .x/ satisfies .H1 / of Theorem 33 and .H3 / of Theorem 34. Moreover, Assumption .H2 / of Theorem 33 and .H4 / of Theorem 34 is fulfilled by the present ˆ given by (91), as shown in Sect. 5.3. As concerns the second part of the thesis, it is enough to note that (93) and (95) are equivalent to P and P./, respectively. Hence, Theorem 36 can be applied. This completes the proof.  In the special–but important–case where R is a polyhedron, Corollary 7 shows that the class of vector minimization problems with component-wise strictly concave objective function (95) has all vector minimum points necessarily at the vertices of the feasible region. This is not true in general. Indeed, while the next theorem shows that component-wise quasi-concavity guarantees the existence of a vector minimum point at a vertex of a polytope, the following Examples 6 and 7 show that component-wise strict concavity need not imply that all vector minimum points must be among the vertices of a polytope. Theorem 37 Let P  Rn be a nonempty polytope and let g WD .g1 ; : : : ; gl / W Rn ! Rl be such that either gi is quasi-concave for all i (component-wise quasiconcavity) or there exists an index i0 for which gi0 is strictly quasi-concave. Then, at least one vector minimum point of the problem min g.x/ s:t: x 2 P C

(96)

is attained at a vertex of P . Proof Without loss of generality assume that i0 D 1. For i D 1; : : : ; l; consider the problem ai WD min gi .x/ s:t: x 2 Si 1 ;

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where S0 D P; Si WD argminx2Si 1 gi .x/; i D 1; : : : ; l: Note that Si  Si 1 ; i D 1; : : : ; l: If gi is quasi-concave for all i , then the sets S1 ; : : : ; Sl are unions of faces of P . On the other hand, if g1 is strictly concave, then S1 , and thus also every S2 ; : : : ; Sl , is a subset of the vertices of P . It will be now shown that each element of Sl is a VMP of (96) so that the thesis will follow. Consider any x 0 2 Sl . Ab absurdo, suppose that x 0 be not solution of (96). Then, 9yx 0 2 P , such that g.yx 0 / C g.x 0 /; so that 9ix 0 2 fi D 1; : : : ; lg such that gix0 .yx 0 / < gix0 .x 0 / D aix0 ;

(97)

gix0 .yx 0 /  gi .x 0 /; 8i D 1; : : : ; l; i ¤ ix 0 :

(98)

Then yx 0 must belong to Sl . Otherwise, 8i D 0; : : : ; l  1, yx 0 2 Si n Si C1 H) gi C1 .yx 0 / > gi C1 .x 0 / D ai C1 ; which contradicts (97) and (98). Since g.x/ D .a1 ; : : : ; al /; 8x 2 Sl ; it follows that g.yx 0 / D .a1 ; : : : ; al /, which contradicts (97). This completes the proof. 

Remark 8 If, for any k 2 f1; : : : ; lg, Sk is a singleton, then obviously its (unique) element is a VMP of (96), and the subsequent Si coincide with Sk . This observation suggests a method for finding a solution of (96). However, this method does not necessarily find all VMPs; for instance, even if g is component-wise strictly concave, the method cannot find the VMPs (if any) which fall in int P whatever the ordering of the component of g may be. The following examples show some strictly concave cases where there are non-vertex solutions even under the further assumption that the (global) maximum points of all gi fall into the interior of P . Example 6 In (96), put n D 1; P D Œ0; 1; C D RC n f0g; g1 .x/ D 1  x 2 , and g2 .x/ D x.2  x/; it is easy to check that every element of P is a VMP. Example 7 In (96), put n D 1; P D Œ3; 3; C D RC n f0g; g1 .x/ D .x C 3/ .7  x/, and g2 .x/ D .3  x/.x C 7/. It is easy to check that the VMP are now ˙3 and all the elements of   1; 1Œ.

Connections Between Continuous and Discrete Extremum Problems

891

Now, consider the vector variational inequality find y 2 R \ Z s:t: hG.y/; x  yil —C 0; 8x 2 R \ Z;

(99)

where Z D Bn and C D RlC n f0g. As noted in Remark 6, y solves (Eq. 99) if and only if it solves the problem P given by the following: find y 2 R \ Z such that hG.y/; y  xil 2 C; x 2 R \ Z is impossible. Similarly, y solves the vector variational inequality: find y 2 R \ XQ s:t: hG.y/; x  yil  ˆ.xI y/ —C 0; 8x 2 R \ XQ ; (100) where ˆ is the function in (91) and  2 RC , if and only if y solves the problem P./ given by the following: find y 2 R \ XQ such that hG.y/; y  xil C ˆ.xI y/ 2 C; x 2 R \ XQ is impossible. So, next result is a direct application of Theorems 33 and 34. Corollary 8 Let G W Rn ! Rln be bounded on XQ . Then, there exists a real 8 2 R, such that, 8 > 8 , (99) and (100) have the same set of solutions. Proof The function F defined by F .xI y/ D hG.y/; y  xi fulfills .H1 / of Theorem 33 according to Remark 6. Moreover, Assumption .H3 / of Theorem 34 holds, since k F .x1 I y/  F .x2 I y/ k D k hG.y/; y  x1 i  hG.y/; y  x2 i k  k G kk x1  x2 k; 8x1 ; x2 ; y 2 XQ : The present ˆ fulfills .H2 / of Theorem 33 and .H4 / of Theorem 34 as shown in Sect. 5.3. Hence, Theorems 33 and 34 guarantee the existence of a real 8 2 R such that P and P./, and then (99) and (100) have the same set of solutions.  In the special–but important–case where R is a polyhedron, Corollary 8 provides a class of vector variational inequalities with bounded operator, i.e., (100), having– because of the equivalence with (99)–a solution necessarily at a vertex of the domain. This is not true in general, as simple examples show.

6

Conclusion

The developments of the previous sections suggest some interesting topics that should be further investigated. A first question concerns the identification of a class of constrained extremum problems, or Complementarity Systems, or generalized

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systems, or Variational Inequalities for which the exact penalty parameter can be improved. Some results on this topic have been presented in Sect. 2.3. Some of the assumptions of Theorems 2 or 3 might be weakened; in the general case, Rn might be replaced by a metric space. For instance, isoperimetric problems or optimal control problems, where the feasible region has only a discrete or even finite number of curves, might be equivalently transformed into a classic differentiable problems which admits Euler equation as necessary condition. In Sect. 2.2, it has been observed that integer programs can be reduced to (16). However, such a reduction implies, in general, an enormous increase in the number of variables. Hence, it is interesting to extend the analysis to the case where f0; 1gn is QLj P P .xj s/2 , replaced by Zn . In this case, functions like nj D1 sin. xj / or nj D1 sD0 with Lj upper bound on x, might be used as penalty functions. Furthermore, for problem (16)–or, more generally, when Z is finite–it is interesting to investigate other types of penalty functions, e.g., in the field of gauge functions. The idea of resolvent for a Boolean problem [42] might be transferred to a concave minimization problem through the equivalence established in Sects. 2.2 and 3.4. Theorem 4 states a connection between 0–1 programming problems and complementarity problems, which are a special type (when the domain is a cone) of variational inequalities. Hence, one has the possibility of connecting two fields very far from each other. For instance, fixed-point methods or gap function methods, which have been designed for variational inequalities, may be adapted to 0–1 problems. On the other hand, cutting plane methods and the related group theory, as well as other methods conceived for integer programs, may be transferred to variational inequalities. It is well known that the concept of saddle of the Lagrangian function associated to a problem like (18) can be used to achieve sufficient optimality conditions (beside necessary ones). Through the equivalence such a concept can be introduced for 0–1 programs. The equivalence between nonlinear and 0–1 programs can be used also in the context of duality. For instance, it can be used to close the duality gap when Lagrangian duality is applied to facial constraint [55]. Since the optimality of a constrained extremum problem can be put in terms of the impossibility of a suitable system of inequalities, it might be useful to extend the connections between continuous and discrete problems to a system of inequalities, recovering constrained extremum problems as special cases. This could also embrace vector extremum problems. General classes of functions that attain their minimum at the vertices of a polyhedron have been described in Sect. 3. It is interesting to investigate the possibility of extending the methods employed for concave minimization problems [11, 46, 62] to this more general case. Consider a special case of (85), where R D Rn , Z D [rkD1 Zk , with Zk convex and compact, f W Rn ! Rl component-wise convex, and C D RlC n f0g, namely, the problem

Connections Between Continuous and Discrete Extremum Problems

min f .x/; s:t: x 2 [rkD1 Zk I C

893

(101)

y is a solution of (101) if and only if the system (in the unknown x) f .y/  f .x/ 2 C;

x 2 [rkD1 Zk

(102)

is impossible. Consider compact sets Xk Zk and functions hk W [rkD1 Xk ! R, k D 1; : : : ; r, such that there is an ˛ 2 R for which each hk fulfills .H2 /0 of Sect. 5.2 with l D 1, C C D RC . It is easily seen that H.x/ WD

r Y

hk .x/˛=r

(103)

kD1

fulfills .H2 /0 . In fact, it is trivial to verify .H2 /0 .i /; .i i /; and .3i /. In the following formulas, k as index will denote that one is referring to Zk and Xk instead of Z and X . H fulfills also .4i /: H.x/ D

r Y

hk .x/˛=r 

kD1

D

r Y kD1

r Y

Œk .z/˛=r k x  pk .x/ k˛=r

kD1

k .z/˛=r

r Y

k x  pk .x/ k˛=r 

kD1

D Q .z/ k x  p.x/ k˛ ;

r Y

k .z/˛=r k x  p.x/ k˛

kD1

Q .z/ D

r Y

k .z/˛=r ;

kD1

where the first inequality comes from .H2 /0 at ' D hk and the second inequality is due to the fact that k x  p.x/ k D minfk x  pk .x/ k; k D 1; : : : ; rg: The function ˆ W X X ! Rl defined by 0

1 H.y/  H.x/ B C :: ˆ.xI y/ D @ A :

(104)

H.y/  H.x/ can be chosen as “penalty term.” The decomposition (103) may help in setting up H and in conceiving solution methods. For instance, if Zk is a polytope defined by Zk D fx 2 Rn I Ak x  b k g;

k D 1; : : : ; r;

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where Ak 2 Rmk n ; b k 2 Rmk , then one can set 9 8 

n = < X hk .x/ D max 0; exp  ˛ik aijk xj  bik  1; i D 1; : : : ; mk ; ; : j D1

k T / ; ˛ik being where Ak D .aijk ; i D 1; : : : ; mk ; j D 1; : : : ; n/; b k D .b1k ; : : : ; bm k positive parameters. A particular but interesting case is that where Z is not convex, while the sets Zi , Xi , and [riD1 Xi are all convex. Corollary 7 suggests a method for solving (93) that is based on the theory introduced in [46, 77] and that will be shortly outlined. To this end, consider the special, but common, case where R is a polytope. Because of Corollary 7, the combinatorial vector problem (93) can be replaced by the continuous vector problem (95). If  is large enough (i.e.,  > 7 ), then, because of Theorem 37 (see Remark 8), a VMP of (95) is a vertex of P \ XQ . Hence, a method could be the following. The first step must consist in finding a local VMP of (95). Then, consider the family of strictly concave (scalar) problems:

min gi .xI /;

x 2 P \ XQ I i D 1; : : : ; l;

(105)

where gi .xI / WD fi .x/ C '.x/. If i is such that x 0 is a local1 (scalar) minimum point of (105), then Tuy’s Theory [46, 77] provides a “cutting halfspace,” say Hi , such that  x 2 P \ XQ x 0 … Hi I (106) H) x 2 .P \ Hi / \ XQ : gi .xI / < gi .x 0 I / This condition means that, if there exists an x at which gi takes a value less than gi .x 0 I /, then x must belong to Hi . For all other indices i Tuy’s “cutting halfspace” collapses to a supporting halfspace of P \XQ ; this will be denoted again by Hi . The former (latter) set of indexes will be denoted by I C (respectively I  ). If I C ¤ ;, then from (106) one can easily deduce that 

  x 2 P \ XQ : H) x 2 P \ \i 2I C Hi \ XQ : 0 g.xI / th1 , light > th2 , and smoke > th3 , where “th” denotes a threshold for the corresponding attribute. That is, f i re D .te mperature > th1 / ^ .light > th2 / ^ .smoke > th3 /. It is more accurate to report the fire when all these atomic events occur, instead of the case when only one attribute is above the threshold.

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The definitions for k-watching an atomic event and a composite event are as follows: • k-Watched Atomic Event: An atomic event is k-watched by a set of sensors if at any time this event occurs at any point within the interested area, at least k sensors in the network can detect this occurrence. • k-Watched Composite Event: A composite event is k-watched by a set of sensors if every atomic event part of the composite event is k-watched by the set of sensors. The problem studied in [27] is defined as follows: given a set of N sensors with different sensing capabilities, a monitored area A, a composite event which is a combination of M atomic events involving sensing components x1 , x2 , . . . ., xM , and the energy constraint of each sensor Einit design a sensor scheduling mechanism such that (1) the composite event is k-watched in the area A, (2) the set of active sensor nodes is connected, and (3) network lifetime is maximized. The mechanism proposes using sensor scheduling to achieve energy efficiency. Network activity is organized in rounds. Each round has two phases: initialization phase and event detection phase. In the initialization phase, sensor nodes decide if they will be active or if they go to sleep during the next round. The active sensor set must provide a connected topology and must ensure k-watching property across the deployment area. Each node u has a priority p.u/ D .E.u/; ID.u//, where E(u) is the node u’s residual energy and ID.u/ is the node u’s ID. A node with higher residual energy has a higher priority. Note that a node priority change over time: if the node is active, then its residual energy decreases, so it has a lower priority in the next round. The h-hop neighborhood of a node u is defined as Nh .u/ D fvjdistance.u; v/  h hopsg. The set of nodes within the h-hop neighborhood of u that have higher priority is defined as N0h .u/ D fv 2 Nh .u/jp.v/ > p.u/g. A node u decides its status (active/sleep) for the next round by using the following rule: The default status of a sensor node is active. A sensor u is in sleep mode if the following two conditions hold: 1. (k-Watching property) the set of nodes N0h .u/ provides the k-watching of each of u’s sensing components. 2. (Connectivity property) any two of u’s neighbors in N1 .u/, w and v, are connected by a path with all intermediate nodes in N0h .u/. The intuition behind this rule is that a sensor u can go to sleep if the nodes with higher priority in its h-hop neighborhood can provide the k-watching property and the connectivity property on behalf of u. Note that a node with higher priority can also go to sleep if the rule holds true in its h-hop neighborhood. To avoid inconsistencies, a mechanism that assigns global priorities to the nodes is used. If a node u has only one neighbor, then the connectivity requirement is fulfilled, and u goes to sleep if the k-watching property holds. Work [35] proposes a centralized approach that builds the detection sets (or data collection trees) using the breadth-first search algorithm [15] starting from a gateway, which can be any sensor node with richer energy resources. One drawback of the proposed approach is the global knowledge required by the algorithm constructing the detection sets. The sets are computed by the gateway node, and

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the decision if an event (simple or composite) occurs is also made by the gateway node. Simulation results show that the centralized tree-based approach has a longer network lifetime (more detection sets), but the trade-off consists in using a single point of failure and large overhead for a large network size.

3.5

Adaptive Sensor Scheduling for Area Coverage

Article [41] considers a publish-subscribe scenario similar to [21]. The sink sends a query as interest toward sensors in the monitored area. When sensors receive a query, if they can satisfy the interests and meet the query, then they activate to perform the sensing task. The intermediate sensors help route sensed data toward the sink, and the results are finally reported to the sink. For example, assume that the sink is interested in whether a fire will happen in the next 2h; then it sends the interest query: type D te mperature ^ smoke i nterval D 1 mi n rect D Œ.50; 70/; .90; 90/ ti mestamp D 01 W 30 W 00 expi resAt D 03 W 30 W 00 The sensors that can sense the temperature and/or smoke and are within the rectangle with the lower left end (50, 70) and the upper right end (90, 90) are activated and report data every 1 min from time 01: 30: 00 to 03: 30: 00. In general, the type field in the interest query is the combination of one or more sensing components, that is, x1 ^ x2 ^ : : : ^ xl . More than one query can be sent to the monitored area. For example, another query can be issued as follows: type D te mperature i nterval D 2 mi n rect D Œ.30; 60/; .70; 80/ ti mestamp D 03 W 00 W 00 expi resAt D 04 W 00 W 00 The areas of interest for the two queries intersect; thus, some sensors might report data for both queries. The active sensors have to satisfy both the coverage and the global connectivity requirements. The objective in [41] is to design a sensor scheduling algorithm that allows sensors not actively participating in sensing or data relaying to go to sleep in order to conserve energy and to prolong the network lifetime. The sensor scheduling mechanism adaptively decides the set of active sensor nodes such that both the coverage and the connectivity requirements are met: • Coverage requirement: as queries propagate in the monitored area, related sensors in the area of interest are activated for the sensing tasks. This is an adaptive mechanism: as new query requests arrive, new sensors are activated as

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needed; as queries expire, sensors in the area of interest go to sleep if they are not sensing any other request. • Connectivity requirement: the set of active sensors must be connected all the time. This is needed for the communication between sensors and the sink in operations such as data reporting, query propagation, and forwarding of control messages. The problem studied in [41] is as follows: Given a WSN with sensors equipped with one or more sensing components (attributes) from the set fx1 ; x2 ; : : : ; xM g, design an adaptive sensor scheduling mechanism such that the set of active sensors change over time such that to satisfy the coverage and connectivity requirements, and the WSN lifetime is maximized. The adaptive sensor scheduling in WSNs (ASW) localized mechanism proposed in [41] forms an adaptive connected dominating set (CDS) that change over time as new requests arrive or expire. When a new request regarding sensing a particular rectangular area is received/expires, sensors adaptively update the active nodes in the required area to satisfy both coverage requirement and global connectivity. Initially, a CDS backbone is selected so that the requests can be propagated to the monitored area along the backbone. The CDS is updated periodically after each time interval T . Before rerunning the CDS, sensors update their priority based on the remaining energy. Then for a time T , priority and the CDS stays unchanged. All the nodes participate in the new CDS selection and update their fields accordingly depending on whether they will be part of the CDS or not. The nodes serving queries will continue to be active and to perform their sensing tasks. Nodes which are not in the CDS or serving queries go to sleep. Every sensor u decides whether it will be a CDS node or not during the next period T as follows. Initially, a node u sets CDS(u) = TRUE. Then sensor u sets CDS(u) = FALSE if the following condition holds: for any two of u’s neighbors in N1 .u/, w and v, w and v are connected by a path with all intermediate nodes in N0h .u/. See Sect. 3.4 for a definition of Nh .u/ and N0h .u/. Consider that the following query is sent by the BS: Query(type, interval, area R, Tstart, Tend). The field type contains the attributes of interest that have to be reported, for example, type D x1 ^ x3 . The CDS nodes awake the 1-hop sleeping nodes in area R and forward the query information. An awaken node checks if it has at least one of the sensing components from the query type field, then it remains active for the duration of the query and will report data according to the reporting interval in the query. When a query time request ends, the reporting sensors check their role field. If a sensor is part of the CDS or is serving another query, then it remains active. Otherwise, it returns to sleep. One case that can occur regards overlapping queries. A sensor may serve multiple queries if their reporting areas intersect. The frequency of reporting is done according to the requesting queries, and this field updates as queries expire or new queries arrive. Compared with directed diffusion [21], ASW is more energy efficient, it generates less overhead; however, it may have a longer delivery path. This is because in the directed diffusion, all sensors are active and they may form shorter data delivery paths, while in ASW, only part of the sensors are active.

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Point Coverage

In the point coverage problem, the objective is to cover a set of points. Figure 1b shows an example of a set of sensors randomly deployed to cover a set of points (star nodes). The connected black nodes form the set of active sensors, the result of a scheduling mechanism. Some important cases discussed in this section are energyefficiency, sensors with adjustable sensing ranges, multiple point coverage, connectivity, heterogeneous WSNs, and coverage breach.

4.1

Energy-Efficient Coverage

Article [5] addresses the point coverage problem in which a limited number of points (targets) with known locations need to be monitored. A large number of sensors are dispersed randomly in close proximity of the targets and send the monitored information to a central processing node. The requirement is that every target is monitored at all times by at least one sensor, assuming that every sensor is able to monitor all targets within its sensing range. One method for extending sensor network lifetime through energy efficiency is to divide the set of sensors into disjoint sets such that every set completely covers all targets. These disjoint sets are activated successively, such that at any moment in time, only one set is active. Using this approach, all targets are monitored by every sensor set. The goal of this approach is to determine a maximum number of disjoint sets, such that the time interval between two activations for any given sensor is longer. By decreasing the fraction of time, a sensor is active, the overall time until power runs out for all sensors is increased, and the application lifetime is extended proportionally by a factor equal to the number of disjoint sets. A solution for this application is proposed in [5], where disjoint sets are modeled as disjoint set covers, such that every cover completely monitors all the target points. Article [5] shows that the disjoint set coverage problem is NP-complete and any polynomial-time approximation algorithm has a lower bound of 2. The disjoint set cover problem is reduced to a maximum flow problem, which is modeled as a mixed integer programming. Simulations show better results in terms of the number of disjoint set covers computed compared with most-constrained least-constraining algorithm [31], where every target is modeled as a field. In [7], the scheduling mechanism also forms sets of sensor nodes that monitor the targets successively. In contrast with the previous approach, a sensor can be part of multiple sets (e.g., the sets are not necessarily disjoint) and the sets can be operational for different time intervals. The objective of the maximum set covers (MSC) problem is to organize the sensors in sets and to determine the time interval of each set such that the sum of all time intervals is maximized (e.g., maximum network lifetime), subject to the constraints that each sensor cover monitors all targets and the time a sensor is active is less than the sensor lifetime. The paper assumes all sensors can be active for the same amount of time (e.g., they have the same starting energy). Article [7] shows that the MSC problem is NP-hard by reduction from the 3-SAT problem.

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Two heuristic solutions are proposed. The first one formulates the MSC problem using integer programming (IP) and then uses the relaxation technique to design a linear programming (LP)-based heuristic. The total complexity is O.p 3 n3 /, where p is an upperbound on the number of covers and n is the number of sensor nodes. The second heuristic builds the covers successively. When building a new cover, sensors with greatest contribution (e.g., covers more uncovered targets and has more residual energy) are added first. Sensors are added to a cover until all the targets are covered. The complexity is O.d m2 n/, where d is the number of sensors that cover the most sparsely covered target, m is the number of targets, and n is the number of sensors. Simulation results show that the IP approach returns a longer network lifetime but has a larger complexity than the greedy mechanism.

4.2

Point Coverage in Sensor Networks with Adjustable Sensing Ranges

Article [8] addresses the case when sensor nodes can adjust their sensing range. In the adjustable range set covers (AR-SC) problem, a set of points (or targets) and a set of sensors with adjustable sensing ranges are given. The objective is to find a family of set covers such that (1) the number of covers is maximized, (2) each set monitors all targets, and (3) a sensor consumes at most E energy. The problem assumes a discrete set of sensing ranges for the sensors. The first approach uses IP to determine the set covers c1 ; c2 . . . cK and the sensors in each cover xi kp . Given: • N sensor nodes s1 ,. . . , sN . • M targets t1 , t2 ,. . . , tM . • P sensing ranges r1 , r2 ,. . . , rP and the corresponding energy consumption e1 , e2 ,. . . , eP . • Initial sensor energy E. • The coefficients showing the relationship between sensor, radius, and target: aipj D 1 if sensor si with radius rp covers the target tj . Maximize c1 C : : : C cK subject to PK PP . pD1 xi kp ep /  E for all i D 1 : : : N PkD1 P xi kp  ck for all i D 1::N, k D 1 : : : K PpD1 PP N . x  a /  c for all k D 1 : : : K, j D 1 : : : M i kp ipj k i D1 pD1 xi kp 2 f0; 1g and ck 2 f0; 1g

(1)

The variables used are: • ck , boolean variable, for kD1..K; ck D 1 if this subset is a set cover; otherwise, ck D 0. • xi kp , boolean variable, for i D 1::N, kD 1::K, p D 1::P; xi kp D 1 if sensor i with range rp is in cover k; otherwise, xi kp D 0.

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The first constraint guarantees that the energy consumed by each sensor i is less than or equal to E, which is the starting energy of each sensor. The second constraint ensures that if sensor i is part of the cover k, then exactly one of its P sensing ranges is set. The third constraint guarantees that each target tj is covered by each set ck . Since the complexity of the IP is large for large number of sensors, the IP is relaxed to a LP and then rounded to get a solution to the original problem AR-SC. In the relaxed LP, the variables xi kp and ck are relaxed to 0  xi kp  1 and 0  ck  1. To form a set cover ck (taken in decreasing order), sensors are added in decreasing order of xi kp as long as they have enough energy resources and new uncovered targets are monitored. The second solution is a centralized greedy algorithm, which repeatedly constructs set covers and sets sensors’ sensing ranges as long as each target is covered by at least one sensor with enough energy resources. In forming a set cover, sensors are selected based on their contribution. Bip is the incremental contribution of the sensor i when its sensing range is increased to rp . Bip D Tip =ep , where Tip D jTip j  jTi q j and ep D ep  eq . The range rq is the current sensing range of the sensor i , thus rp > rq . Initially, all the sensors have assigned a sensing range r0 D 0, and the corresponding energy is e0 D 0. Sensors with higher contribution are added first (e.g., by increasing their sensing range) until all the targets are covered. Intuitively, a smaller sensing range is preferable as long as the target coverage objective is met, since energy resources are conserved, allowing the sensor to be operational longer. Once the set cover is formed (e.g., all targets are covered by the selected set of sensors), the sensors with a sensing range greater than zero form the set of active sensors, while all other sensors with sensing range r0 D 0 will be in sleep mode. The third method proposed in [8] is a distributed and localized method. In a distributed and localized method, the decision process at each node makes use of only information in a neighborhood within a constant number of hops. A distributed and localized algorithm is desirable in WSN since it adapts better to dynamic and large topologies. The algorithm runs in rounds. Each round begins with an initialization phase, where sensors decide whether they will be in an active or sleep mode during the current round. Each sensor maintains a waiting time, after which it decides its status (sleep or active) and its sensing range, and then it broadcasts the list of targets it covers to its one-hop neighbors. The waiting time  sensor si depends on si ’s contribution,  of each BiP and it is set up initially to Wi D 1  Bmax  W where Bmax is the largest possible contribution, defined as Bmax D M=e1 , where M is the number of targets. Bip is the contribution of sensor i with range rp , defined as Bip D jTip j=ep , where Tip is the set of uncovered targets within the sensing range rp of sensor si . When its waiting time expires, a sensor broadcasts its status (active or sleep) as well as the list of targets it covers. The neighbors receiving the message update their contribution and waiting time accordingly. If a sensor has all the targets in its range already covered, then it sets its range to r0 D 0. In the initialization phase, nodes which are active during the current round are determined. If, after the initialization time, one of the sensors cannot cover a target in their range, then it announces a

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failure message to the base station (BS). Network lifetime is computed as the time elapsed until the BS detects the first failure. Simulation results show that the centralized greedy algorithm performs the best, followed by the LP-based mechanism, and then the distributed greedy one. There is also high improvement in network lifetime when sensors can adjust their sensing ranges. Compared to the base case when sensors have only one sensing range, the most noticeable improvements are for up to four different sensing range values. Between 4 and 6, the network lifetime improvement was not significant.

4.3

Node Coverage as Approximation

This section has two parts. Section 4.3.1 discusses the case when each point has to be monitored by at least one sensor, and Sect. 4.3.2 discusses the case when each point has to be covered by at least k sensors.

4.3.1 Simple Point Coverage When a large and dense sensor network is randomly deployed for area monitoring, the area coverage can be approximated by the coverage of the sensor locations. That is, based on the assumption of a large and dense population of sensors, covering each sensor location approximates the coverage of each point in the given area. One method to assure coverage and connectivity is to design the set of active sensors as a connected dominating set (CDS). A distributed and localized protocol for constructing the CDS is proposed by Wu and Li, using the marking process in [38]. A node is a coverage node if there are two neighbors that are not connected (i.e., not within the transmission range of each other). Coverage nodes (also called gateway nodes) form a CDS. A pruning process can be used to reduce the size of coverage node set while keeping the CDS property. A generalized pruning rule called pruning rule k is proposed in [17]. Basically, a coverage node can be withdrawn if its neighbor set can be collectively covered by those of k coverage nodes. In addition, these k coverage nodes have higher priority and are connected. Pruning rule k ensures a constant approximation ratio. The CDS derived from the marking process with Rule k can be locally maintained, when sensors switchon/off. Wu et al. [40] also discuss the energy-efficient dominating set coverage approach. In general, nodes in the connected dominating set consume more energy in order to handle various bypass traffic than nodes outside the set. To prolong the life span of each node and, hence, the network by balancing the energy consumption in the network, nodes should be alternated in being chosen to form a connected dominating set. A set of power-aware pruning rules is proposed, where coverage nodes are selected based on their energy levels. Simulation results in [40] show that the modified power-aware marking process outperforms several existing approaches in terms of life span of the network. The node coverage problem can be related to the broadcasting problem, where a small set of forwarding nodes is selected [37]. Forwarding set selection in broadcasting is similar to the point coverage problem, where both try to find a small coverage set. Note that directed diffusion [21] also uses this platform to

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collect information through broadcasting. As the interest is propagated through the network, sensor nodes set up reverse gradients to the sink in a decentralized way. The difference is that in the directed diffusion, all sensors forward the data. One difference between node coverage and area coverage is that neighbor set information is sufficient for node coverage, while in area coverage, geometric/directional information is needed.

4.3.2 Multiple Point Coverage Article [42] considers that the sensor network is sufficiently dense so that covering the sensors can approximate area coverage. Thus, area coverage is reduced to the point coverage. The desired level of coverage is defined as a multiple coverage for the purpose of reliability in case of failure or for other applications related to security (e.g., localized intrusion detection) or localization (e.g. triangulation-based positioning). The k-coverage set (k-CS) problem is defined as follows: given a constant k > 0 and an undirected graph G D .V; E/, find a subset of nodes C  V such that (1) each node in V is dominated (covered) by at least k different nodes in C and (2) the number of nodes in C is minimized. The k-connected coverage set (k-CCS) problem is defined similarly, with the additional requirement (3) the nodes in C are connected. First, a centralized IP-based approach with performance ratio  is proposed for the k-CS problem. Given: • n nodes s1 , . . . ,sn • aij , the coefficients showing the coverage relationship between nodes. These coefficients are defined as follows:  1 if node si is covered by node sj aij D 0 otherwise Variables: xj , boolean variable, for j D 1 : : : n:  1 if node sj is selected in the subset C xj D 0 otherwise Integer programming: Minimize x1 C x2 C    C xn subject to

Pn

Pn

j D1 aij xj

xj 2 f0; 1g

k

for all i D 1; : : : ; n

(2)

for i D 1; : : : ; n

The constraint j D1 aij xj  k, for all i D 1; : : : ; n, guarantees that each node in V is covered by at least k nodes in C. Relaxation and rounding technique are used and provide an algorithm with approximation ratio , where  D  C 1 and  the maximum node degree in G. The IP is relaxed to an LP by requiring that 0  xj  1

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for j D 1; : : : ; n. As part of the rounding technique, a sensor sj is added to the set C if its corresponding variable xj  1=. The second approach [42] is called the cluster-based k-CCS/k-CS algorithm: 1. Using a clustering algorithm to select clusterheads, set C1 , and the selected clusterheads are marked and removed from the network. 2. Repeat step 1 for k times; mark and remove Ci , i D 2; : : : ; k. 3. Use a gateway selection approach to select gateways to connect clusterheads in the first set C1 ; denote the gateway set by D. (This step is removed for without connectivity.) 4. For each node in C1 [ C2 [ : : : Ck [ D (clusterhead or gateway), if the number of its marked neighbors t is smaller than k, then it designates k  t unmarked neighbors to be marked. The clustering algorithm divides the network into several clusters; each cluster has a clusterhead and several neighbors of this clusterhead as members. Any two clusterheads are not neighbors, and the clusterhead set is a maximum independent set of the network in addition to a dominating set. Such a clustering algorithm is provided in [39]. The gateway selection can be tree-based, whereby gateways are selected globally to make the connected domination set (CDS) a tree, or mesh-based, whereby each clusterhead is connected to all of its neighboring clusterheads, and thus the CDS is a mesh structure. For coverage without connectivity, the third step, gateway selection, can be removed. The third approach [42] is a completely localized solution, where each node determines its status (marked or unmarked) based on its 2-hop neighborhood information: 1. Each sensor node u is given a unique priority L.u/, and each node u is represented by tuple (L.u/, ID(u)), where ID(u) is the ID of u. 2. Each node broadcasts its neighbor set N(u), where N(u)D fvjv is a neighbor of u g. 3. At node u, build a subset: C.u/ D fvjv 2 N.u/; L.v/ > L.u/g. Node u is unmarked if a. Subset C(u) is connected by nodes with higher priorities than u (This constraint is removed for a solution without connectivity) b. For any node w 2 N.u/, there are k distinct nodes in C(u), say v1 ; v2 ; : : : ; vk , such that w 2 N.vi / After the algorithm terminates, all the marked nodes form the k-CCS/k-CS. Simulation results show that (i) cluster-based algorithm has better performance in terms of the size of the set C of nodes selected to be active, and (ii) cluster and localized algorithms have better scalability than the IP approach, especially when the network is relatively dense.

4.4

Connected Point Coverage

One of the important issues regards sensor connectivity which forms a network that allows the gathering of the sensed data to the sink. Article [24] addresses the adjustable sensing range connected sensor cover (ASR-CSC) problem: given a set of

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targets (or monitoring points) and a set of sensors with adjustable sensing ranges in a WSN, schedule sensors’ sensing ranges such that the WSN lifetime is maximized, under the conditions that both target coverage and network connectivity are satisfied, and each sensor energy consumption is upperbounded by the initial energy E. Two localized algorithms are proposed [24] depending on the order in which connectivity and coverage are ensured. The first algorithm first ensures connectivity, then the target coverage. In the second algorithm, the order is reversed: coverage first, then connectivity. The first algorithm starts by building a virtual backbone consisting of sensor nodes that will be active and participate in data forwarding. It has the property that any sensor node which is not in the backbone is one hope away from a node in the backbone. Thus, if it has to be active for the coverage task, then it can send its sensing data through the backbone. The algorithm executes in rounds, and each round has the following steps: (1) construct a virtual backbone for the WSN; (2) for each sensor in the virtual backbone, set its transmission range dc and calculate its transmission energy consumption; (3) all sensors including inactive sensors (dominatees) together with active sensors in the virtual backbone (dominators) iteratively adjust their sensing ranges based on contribution until a full coverage is found; and (4) each sensor i active for sensing and connectivity consumes ep.i /Cge from its residual energy, while sensors active only for connectivity consume ge . The virtual backbone is constructed by forming a connected dominating set and pruning redundant sensors by applying the Rule k in [17]. If a sensor node meets the rule’s requirement, then it is not part of the backbone in the next round; otherwise, it is part of the backbone for the next round. The coverage mechanism is the same as the one in [8], which means sensors use a waiting time based on the sensor contribution, and when this expires, they decide the coverage responsibilities. The second algorithm starts by deciding the sensing sensors first. These sensors are connected using Rule k, where sensing nodes have the highest priority; thus, it is ensured they will be part of the backbone. The simulation study shows that the two algorithms have comparable network lifetime and that using sensors with adjustable sensing ranges has a good impact on network lifetime. Another connected point-coverage approach is presented in [4]. Here the objective is that the sensing nodes are connected to a base station (BS) which collects sensed data, see Fig. 4. This is different from requiring that two sensor nodes be connected to each other, as discussed in the article [24]. The connected set cover (CSC) problem [4] is defined as follows: given a set of sensors s1 , s2 ,. . . , sN , a base station s0 , and a set of targets r1 , r2 ,. . . , rM , find a family of set covers c1 , c2 ,. . . , cK such that (1) K is maximized, (2) sensors in each set ck (k = 1,. . . ,K) are BS-connected, (3) each sensor set monitors all targets, and (4) each sensor appearing in the sets c1 , c2 ,. . . , cK consumes at most E energy. The first method proposed in [4] is a centralized approach using IP. The target coverage and the energy constraints have been introduced in the previous discussions. The connectivity requirement has been modeled using flows. Each edge has capacity M. Sensing nodes are the nodes in charge with monitoring the targets and they must be able to transmit data to the sink. In Fig. 4, s1 ; s3 ; s5 , and s6 are the

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Fig. 4 Connected point coverage

sensing nodes. Some of the sensing nodes (e.g., s1 ) may not be able to communicate directly with the BS, so some relay nodes must be used to ensure BS connectivity. In the IP, one unit of flow is inserted at each sensing node (which are the flow sources), and the flow is collected at the BS (which is the flow sink). In Fig. 4, one unit of flow is inserted at each of the four sensing nodes, and four units of flow are collected at the BS. Flow conservation property is also ensured for all nodes in the network. Secondly, a centralized greedy algorithm [4] is proposed which is executed at the BS. Once the sensors are deployed, they send their coordinates to the BS. The BS computes the connected set covers and broadcasts back the sensors’ schedule. The algorithm builds each set consecutively, as long as remaining sensor energy resources allow building a new set. The mechanism to build a set is as follows. At each step, a critical target is selected to be covered. This can be, for example, the target most sparsely covered, both in terms of number of sensors as well as with regard to the residual energy of those sensors. The sensors considered as candidates must have sufficient residual energy. Once the critical target has been

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selected, the heuristic selects the sensor with the greatest contribution that covers the critical target. Various sensor contribution functions can be defined. For example, a sensor can be defined to have a greater contribution if it covers a larger number of uncovered targets and if it has more residual energy available. Once a sensor s has been selected, it is added to the current set cover and all additionally targets within range of s are marked as covered. Once the sensing sensors have been determined, the BS connectivity is ensured by running the breadth-first search algorithm [15] starting from the BS. The resulting breadth-first tree is pruned: only the paths from the sensing nodes to the BS are kept. The sensors on these paths become relay nodes for the current round. The third mechanism [4] is a localized approach, which runs in rounds. Each round starts with an initialization phase where sensors decide whether they will be in active mode (sensing or relaying) or sleep mode during the current round. First, the sensing nodes are selected. Each node with uncovered targets has a waiting time which is inverse proportional with the residual energy and the number of uncovered targets. When the timer expires, the node declares itself as a sensing node for the current round and broadcasts this decision and the set of covered targets to its 2-hop neighborhood. In the second part, the relay nodes are selected. Each sensing node builds a virtual tree where the set of vertices is comprised of the BS and the set of targets. The tree is a minimum spanning tree built using Prim’s algorithm [15], where the BS is the root. Each target t has a supervisor sensor, which is the first sensing sensor which covers it. Each supervisor selects relay nodes to allow it to connect with the supervisor of t’s parent in the virtual tree. Simulation results show that the IP algorithm finds the largest number of covers, but it has a long run time and therefore is not feasible for large scenarios. In general, the distributed algorithm performs better than the greedy approach because it employs a virtual tree-based mechanism for finding relay nodes, and this promotes the reuse of already active supervisor nodes, while the greedy one uses a breadth-first tree that results in shorter sensor-BS paths, involving more relay nodes.

4.5

Point Coverage in Heterogeneous Wireless Sensor Networks

Article [1] considers a heterogeneous WSNs that contains two types of wireless devices: resource-constrained wireless sensor nodes deployed randomly in large number and several resource-rich, predeployed supernodes, see Fig. 2b. Sensor nodes transmit and relay measurements. Once data packets encounter a supernode, they are forwarded using fast supernode to supernode communication toward the user application. Additionally, supernodes could process sensor data before forwarding. The heterogeneous connected set covers problem is defined as follows: given a set of points (or targets) t1 ; t2 ; : : : ; tT , a set of supernodes g1 ; g2 ; : : : ; gM , and a set of randomly deployed sensors s1 ; s2 ; : : : ; sN , find a family of sensor set covers

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c1 ; c2 ; : : : ; cP , such that (1) P is maximized, (2) sensors in each set cover cp (p D 1; : : : P) are connected to supernodes, (3) each sensor set monitors all targets, and (4) each sensor appearing in the sets c1 ; c2 ; : : : ; cP consumes at most E energy. Network activity is organized in rounds. Each round has two phases: initialization and data collection. During the initialization phase, a set of active sensors (e.g., the set cover ci ) is established such that conditions (2), (3), and (4) are satisfied. During the data collection phase, sensors in the set cover ci are active, while all other sensors are in the sleep mode for the rest of the round, and they will wake up for the next initialization phase. Building a set cover ci has two steps: (1) choosing the sensing nodes and (2) choosing the relay nodes. To select the sensing nodes, each node defines a waiting time which is smaller for sensors that have a higher residual energy and cover a larger number of uncovered targets. When a timer expires and a sensor node decides to be a sensing node, then it broadcasts this decision to its 2-hop neighborhood. Nodes that receive this message update the waiting time accordingly. Three mechanisms are proposed to select the relay nodes: two of them use a cluster-based approach, and the third one uses a greedy mechanism. For the clustering mechanisms, supernodes serve as clusterheads. Each supernode serves as clusterhead, and it broadcasts a CLUSTER-INIT message containing the supernode id and the number of hops which is initially zero. Each sensor node maintains information about the closest supernode and forwards only messages from which it learns about a closer supernode. Once the clusters have been constructed, there are two algorithms for relay nodes selection: the shortest-path mechanism and the Rule k mechanism. In the shortest-path mechanism, each sensing node broadcasts a special control message RELAY-REQ to the supernode of the cluster where it belongs to. The message is sent along the parent path which was set up during cluster formation. All nodes that participate in relaying the RELAY-REQ message become relay nodes. In the Rule k mechanism, each cluster selects first a backbone using the Rule k algorithm [37]. Then, similar with the previous mechanism, a RELAY-REQ message is sent by the supernode, but this time, the message is forwarded only by the backbone nodes. Sensing nodes send back RELAY-REQ messages that help set up the relay nodes. In the greedy approach, one or more connected components are formed such that the sensing nodes in each component are connected through relay nodes to a supernode. The goal is to activate a minimum number of relay nodes in order to satisfy the supernode connectivity requirement. Components are built in a greedy fashion, similar to Kruskal’s algorithm [15], by successively merging two components connected by a minimum number of relay nodes. Components merge successively until each component contains one supernode. More details are presented in [1]. Simulation results show that the greedy algorithm performs better than the cluster Rule k which performs better than cluster shortest-path. Still the complexity of the greedy is higher since it operates over the whole network. Greedy gets the best

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results since it minimizes the number of relay nodes added on the whole network, while the two other algorithms minimize the number of relay nodes added per cluster.

4.6

Minimum Coverage Breach Under Bandwidth Constraints

An efficient method used for data reporting is to divide sensor nodes in sets; then only one set is active in each round, while all other sensors are in sleep mode to conserve their energy resources. Consider the case when the BS is within direct communication of all sensors. If each active sensor sends the sensed data to the BS, this requires that there must be sufficient bandwidths for this activity. The bandwidth could be the total number of time slots if a time division scheme is used on a single shared channel or the total number of channels if multiple channels are available. The issue occurs when there are W channels available and a subset has more than W sensors. Then some sensors cannot have channel access for data transmission. To properly allocate time slots or channels to sensors, the scheduling techniques must select sensors in each set such that they satisfy the coverage requirement while being fully restricted by the bandwidth constraints. If a target (or monitor region) is not covered by any active sensor, it is called breached. The objective is to minimize the total breach of all targets. The minimum breach problem has been introduced in [13]: given a set A of fixed points, and a set S of sensor nodes, organize sensor nodes into disjoint subsets Ci D fsi1 ; si2 ; : : : g, i D 1; : : : K, where each subset jCi j  W and the overall breach is minimized. Article [13] shows that the decision version of the minimum breach problem is NP-complete. Then two solutions are proposed using IP and LP approaches. Simulation results show that to improve the coverage performance in WSNs, the bandwidth also needs to be increased; in bandwidth constrained networks, increasing the number of sensors alone does not always improve the coverage results.

5

Conclusion

This chapter categorizes and describes recent area and point coverage problems proposed in literature, their formulations, assumptions, and proposed solution. Sensor coverage is an important element of QoS in applications with WSNs. Coverage is, in general, associated with energyefficiency and network connectivity, two important properties of a WSNs. To accommodate a large WSN with limited resources and a dynamic topology, coverage control algorithms and protocols perform best if they are distributed and localized. Various interesting formulations for sensor coverage have been proposed in literature. Scheduling sensor nodes to alternate between sleep and active mode is an important method to conserve energy resources. Such mechanisms that efficiently organize or schedule the sensor activity after deployment are very efficient and have a direct impact on prolonging the

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network lifetime. Sensor coverage with various design choices, such as breach coverage, composite event coverage, heterogeneous WSNs, and so on, poses interesting research problems. The literature on coverage problems is more vast than the papers surveyed here, and the readers are invited to explore them in the research literature. Acknowledgements This work was supported in part by NSF grant CCF 0545488.

Cross-References Connected Dominating Set in Wireless Networks

Recommended Reading 1. W. Awada, M. Cardei, Energy-efficient data gathering in heterogeneous wireless sensor networks, in IEEE International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob’06), Montreal, 2006 2. Y. Bi, L. Sun, J. Ma, N. Li, I.A. Khan, C. Chen, HUMS: an autonomous moving strategy for mobile sinks in data-gathering sensor networks. EURASIP J. Wirel. Commun. Netw. 2007, 1–16 (2007) 3. J. Butler, Robotics and microelectronics: mobile robots as gateways into wireless sensor networks, in Technology@Intel Magazine, 2003 4. I. Cardei, M. Cardei, Energy-efficient connected-coverage in wireless sensor networks. Int. J. Sens. Netw. 3(3), 201–210 (2008) 5. M. Cardei, D.-Z. Du, Improving wireless sensor network lifetime through power aware organization. ACM Wirel. Netw. 11(3), 333–340 (2005) 6. M. Cardei, D. MacCallum, X. Cheng, M. Min, X. Jia, D. Li, D.-Z. Du, Wireless sensor networks with energy efficient organization. J. Interconnect. Netw. 3(3–4), 213–229 (2002) 7. M. Cardei, M. Thai, Y. Li, W. Wu, Energy-efficient target coverage in wireless sensor networks, in IEEE INFOCOM, Miami, USA, 2005 8. M. Cardei, J. Wu, M. Lu, Improving network lifetime using sensors with adjustable sensing ranges. Int. J. Sens. Netw. 1(1/2), 41–49 (2006) 9. J. Carle, D. Simplot, Energy efficient area monitoring by sensor networks. IEEE Comput. 37(2), 40–46 (2004) 10. A. Chakrabarti, A. Sabharwal, B. Aazhang, Using predictable observer mobility for power efficient design of sensor networks, in Proceedings of the 2nd IEEE IPSN, Palo Alto, 2003 11. S. Chellappan, X. Bai, B. Ma, D. Xuan, C. Xu, Mobility limited flip-based sensor networks deployment. IEEE Trans. Parallel Distrib. Syst. 18(2), 199–211 (2007) 12. B. Chen, K. Jamieson, H. Balakrishnan, R. Morris, Span: an energy-efficient coordination algorithm for topology maintenance in Ad Hoc wireless networks, in ACM/IEEE International Conference on Mobile Computing and Networking, Rome, 2001, pp. 85–96 13. M.X. Cheng, L. Ruan, W. Wu, Achieving minimum coverage breach under bandwidth constraints in wireless sensor networks, in IEEE INFOCOM, Miami, 2005, pp. 2638–2645 14. C.-Y. Chong, S.P. Kumar, Sensor networks: evolution, opportunities, and challenges. Proc. IEEE 91(8), 1247–1256 (2003) 15. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 3rd edn. (MIT, Cambridge, 2009). ISBN:0262033844 16. Crossbow Technology (2011), http://www.xbow.com:81/Products/wproductsoverview.aspx. Accessed Dec 2011

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17. F. Dai, J. Wu, Distributed dominant pruning in Ad Hoc wireless networks, in Proceedings of IEEE International Conference on Communications, Anchorage, 2003 18. K. Dantu, M.H. Rahimi, H. Shah, S. Babel, A. Dhariwal, G.S. Sukhatme, Robomote: enabling mobility in sensor networks, in IPSN, Los Angeles, 2005, pp. 404–409 19. P.B. Gibbons, B. Karp, Y. Ke, S. Nath, S. Seshan, IrisNet: an architecture for a worldwide sensor web. IEEE Pervasive Comput. 2, 22–33 (2003) 20. W.R. Heinzelman, A. Chandrakasan, H. Balakrishnan, Energy-efficient communication protocols for wireless microsensor networks, in Proceedings of HICSS, Maui, 2000 21. C. Intanagonwiwat, R. Govindan, D. Estrin, Directed diffusion: a scalable and robust communication paradigm for sensor networks, in Proceedings of ACM MobiCom, Boston, 2000, pp. 56–67 22. A. LaMarca, W. Brunette, D. Koizumi, M. Lease, S. Sigurdsson, K. Sikorski, D. Fox, G. Borriello, in Making Sensor Networks Practical with Robots, ed. by F. Mattern, M. Naghshineh. Lecture Notes in Computer Science (Springer, New York, 2002), pp. 152–166 23. U. Lee, E.O. Magistretti, B.O. Zhou, M. Gerla, P. Bellavista, A. Corradi, Efficient data harvesting in mobile sensor platforms, in PerCom Workshops, 2006, pp. 352–356 24. M. Lu, J. Wu, M. Cardei, M. Li, Energy-efficient connected coverage of discrete targets in wireless sensor networks. Int. J. Ad Hoc. Ubiquitous Comput. 4(3/4), 137–147 (2009) 25. J. Luo, J.-P. Hubaux, Joint mobility and routing for lifetime elongation in wireless sensor networks, in IEEE INFOCOM, Miami, 2005 26. M. Marta, M. Cardei, Improved sensor network lifetime with multiple mobile sinks. J. Pervasive Mobile Comput. 5(5), 542–555 (2009). Elsevier 27. M. Marta, Y. Yang, M. Cardei, Energy-efficient composite event detection in WSNs, in International Conference on Wireless Algorithms, Systems and Applications (WASA’09), Boston, 2009 28. S. Meguerdichian, F. Koushanfar, M. Potkonjak, M. Srivastava, Coverage problems in wireless Ad-Hoc sensor networks, in IEEE INFOCOM, Anchorage, 2001, pp. 1380–1387 29. V. Raghunathan, C. Schurgers, S. Park, M.B. Srivastava, Energy-aware wireless microsensor networks. IEEE Signal Process. Mag. 19, 40–50 (2002) 30. R.C. Shah, S. Roy, S. Jain, W. Brunette, Data MULEs: modeling a three-tier architecture for sparse sensor networks, in Proceedings of the 1st IEEE International Workshop on Sensor Network Protocols and Applications (SNPA’03), Anchorage, 2003 31. S. Slijepcevic, M. Potkonjak, Power efficient organization of wireless sensor networks, in Proceedings of IEEE International Conference on Communications, Kyoto, 2001, pp. 472–476 32. A.A. Somasundara, A. Kansal, D.D. Jea, D. Estrin, M.B. Srivastava, Controllably mobile infrastructure for low energy embedded networks. IEEE Trans. Mobile Comput. 5, 958–973 (2006) 33. D. Tian, N.D. Georganas, A coverage-preserving node scheduling scheme for large wireless sensor networks, in Proceedings of the 1st ACM Workshop on Wireless Sensor Networks and Applications, Atlanta, 2002 34. L. Tong, Q. Zhao, S. Adireddy, Sensor networks with mobile agents, in Proceedings of IEEE Military Communications Conference (MILCOM’03), Boston, 2003 35. C.T. Vu, R.A. Beyah, Y. Li, Composite event detection in wireless sensor networks, in IEEE International Performance, Computing, and Communications Conference, New Orleans, 2007 36. X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless, C.D. Gill, Integrated coverage and connectivity configuration in wireless sensor networks, in ACM Conference on Embedded Networked Sensor Systems, 2003 37. J. Wu, F. Dai, Broadcasting in Ad Hoc networks based on self-pruning, in Proceedings of the 22nd Annual Joint Conference of IEEE Communication and Computer Society, San Franciso, 2003

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38. J. Wu, H. Li, On calculating connected dominating set for efficient routing in Ad Hoc wireless networks, in Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, Seattle, 1999, pp. 7–14 39. J. Wu, W. Lou, Forward node set based broadcast in clustered mobile ad hoc networks. Wirel. Commun. Mobile Comput. 3(2), 141–154 (2003) 40. J. Wu, F. Dai, M. Gao, I. Stojmenovic, On calculating power-aware connected dominating sets for efficient routing in Ad Hoc wireless networks. J. Commun. Netw. 1, 1–12 (2002) 41. Y. Yang, M. Cardei, Adaptive energy efficient sensor scheduling for wireless sensor networks. Optim. Lett. 4(3), 359–369 (2010). Springer, ISSN:1862–4472 42. S. Yang, F. Dai, M. Cardei, J. Wu, F. Patterson, On connected multiple point coverage in wireless sensor networks. Int. J. Wirel. Inf. Netw. 13(4), 289–301 (2006) 43. F. Ye, G. Zhong, S. Lu, L. Zhang, Energy efficient robust sensing coverage in large sensor networks. Technical Report UCLA, 2002 44. H. Zhang, J.C. Hou, Maintaining sensing coverage and connectivity in large sensor networks. Technical Report UIUC, UIUCDCS-R-2003-2351, 2003

Data Correcting Approach for Routing and Location in Networks Boris Goldengorin

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Asymmetric Traveling Salesman Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Data-Correcting Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Computational Experience with ATSP Instances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Simple Plant Location Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Pseudo-Boolean Formulation of the SPLP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preprocessing SPLP Instances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Data-Correcting Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Computational Experience with SPLP Instances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The p-Median Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Brief Overview of PMP Formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Mixed Boolean Pseudo-Boolean Model (MBpBM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Reduction Tests for the p-Median Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Computational Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary and Future Research Directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Maximization of General Submodular Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A Simple Data-Correcting Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Data-Correcting Algorithm Based on Multi-level Search. . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Computational Experience with Quadratic Cost Partition Instances. . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B. Goldengorin () Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA e-mail: [email protected]; [email protected] 929 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 84, © Springer Science+Business Media New York 2013

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Abstract

A data-correcting (DC) algorithm is a branch-and-bound-type algorithm, in which the data of a given problem is “heuristically corrected” at the various stages in such a way that the new instance will be polynomially solvable and its optimal solution is within a prespecified deviation (called prescribed accuracy) from the optimal solution to the original problem. The DC approach is applied to determining exact and approximate global optima of NP-hard problems. DC algorithms are designed for various classes of NP-hard problems including the quadratic cost partition (QCP), simple plant location (SPL), and traveling salesman problems based on the algorithmically defined polynomially solvable special cases. Results of computational experiments on the publicly available benchmark instances as well as on random instances are presented. The striking computational result is the ability of DC algorithms to find exact solutions for many relatively difficult instances within fractions of a second. For example, an exact global optimum of the QCP problem with 80 vertices and 100 % density was found within 0.22 s on a PC with 133-Mhz processor, and for the SPL problem with 200 sites and 200 clients, within 0.2 s on a PC with 733-Mhz processor.

1

Introduction

Let us consider the following combinatorial minimization problem (CMP) .E; C; D; fC / which is the problem of finding S  2 arg minffC .S / j S 2 Dg; where C W E ! < is the given instance of the problem with a ground set E satisfying jEj D m .m  1/, D  2E is the set of feasible solutions, and fC W 2E ! < is the objective function of the problem. By D  D arg minffC .S / j S 2 Dg, the set of optimal solutions is denoted. It is assumed that D  ¤ ; and that S ¤ ; for some S 2 D. Classic examples of routing and location problems include, among others, the following: (A) The asymmetric traveling salesman problem (ATSP) [29]: it is given n  3 cities and distances C D jjc.i; j /jj between i -th and j -th cities, find a Hamiltonian cycle of minimum length, which enters and leaves each city exactly once. Here in the CMP .E; C; D; fC /, E D A is the set of arcs in a directed weighted simple graph G D .V; A; C / with the set of vertices V D f1; 2; : : : ; ng, and A  V  V is the set of arcs .i; j / 2 A with nonnegative weights (distances) c.i; j / connecting vertices i and j ; D is the set of all Hamiltonian cycles, P and the objective function of ATSP fC is given by fC .S / D .i;j /2S c.i; j / for S 2 D.

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(B) The assignment problem (AP): it is given a directed graph G D .V; A; C / as above and a nonnegative Pcost function C on E D A, find a vertex permutation a W V ! V such that niD1 c.i; a .i // is minimized on the set of all possible permutations of V . An optimal solution to the AP is denoted by a and its optimal value fC .a /. Note that a set S  A is a feasible solution of the AP if it is of the form S D f.1; .1//; .2; .2//; : : : ; .n; .n//g for some permutation  of V . Clearly, a Hamiltonian cycle corresponds to the cyclic permutation c of V and vice versa. (C) The simple plant location problem (SPLP): it is given a weighted bipartite graph G D .V; A; C; F / with the set of vertices V D I [J , such that I D f1; : : : ; mg is the set of sites in which plants can be located and J D f1; : : : ; ng is the set of clients, each having a unit demand. Thus, A  I  J is the set of arcs, F D .fi / are fixed costs for setting up plants at sites i 2 I , and C D jjc.i; j /jj is a matrix of transportation costs from i 2 I to j 2 J as input. The SPLP computes a set S  W ;  S   I , at which plants can be located so that the total costs of satisfying all client demands are minimal. The costs involved in meeting the client demands include the fixed costs of setting up plants and the transportation costs of supplying clients from the plants that are set up. An instance of the problem is described by an m-vector F D .fi / and an m  n matrix C D jjc.i; j /jj. It is assumed that F and C are nonnegative, that mn is, F 2 3 [44]. In [4, 40], it has been also shown that the lower bound for the performance ratio

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of the minimum-length spanning tree is given by the d -dimensional kissing number nd , which is the maximum number of nonoverlapping unit balls touching a unit ball. For d D 3, nd D 12. In general, nd D 2cd.1Co.1// where 0:2075  c  0:401 for large d [48]; determination of exact value of nd for large d is quite hard [49]. Is it possible to have an approximation with performance better than that of the minimum-length spanning tree for MIN-POWER BROADCAST? The answer is positive. Wan et al. [4] indicated that the BIP (Broadcasting Incremental Power) is a possible candidate because its performance ratio is between 13/3 and 12 while the minimum-length spanning tree has performance ratio between 6 and 12 (note: in both cases, upper bound 12 has been improved to 6 by Amb¨uhl [42]). To construct a BIP tree, starting from the source node, at each iteration, a new node is reached with smallest increasing power. Note that this increasing may have two ways. One is to assign power to an unassigned node, and the other one is to increase the power to a node with power assigned in previous iteration. Caragiannis et al. [50] gave a greedy approximation with performance ratio 4.2 which is definitely better than both BIP and the minimum-length spanning tree. The result of Caragiannis et al. [50] is for any metric space that if the minimumlength spanning tree induces a -approximation, then their greedy approximation has performance ratio 2 ln   2 ln 2 C 2. It is 4.2 for d D 2, 6.49 for d D 3, and 2:20d C 0:61 for d  4. This greedy approximation is quite interesting, which is designed using the following properties of spanning tree. Lemma 10 Let T1 and T2 be two rooted spanning trees over the same set of nodes V . There exists a one-to-one mapping  W E.T1 / ! E.T2 / such that for any node u and its children v1 ; : : : ; vk in T1 , .T1 [ f.u; v1 /; : : : ; .u; vk /g/  f..u; v1 //; : : : ; ..u; vk //g is a spanning tree.

Lemma 11 Let T be a spanning tree. Consider k edges .u; v1 /; : : : ; .u; vk /. Suppose .T [ f.u; v1 /; : : : ; .u; vk /g/  f.w1 ; y1 /; : : : ; .wk ; yk /g is a spanning tree. Then, for f.u; z1 /; : : : ; .u; zh /g, there exists a subset F of h edges of T such that .T [ f.u; v1 /; : : : ; .u; vk /; .u; z1 /; : : : ; .u; zh /g/  .f.w1 ; y1 /; : : : ; .wk ; yk /g [ F / is a spanning tree. There is no proof for above two lemmas in [50]. They can be derivated from the following. Lemma 12 Let T1 and T2 be two spanning trees. There exists a one-to-one onto mapping  W E.T1 / ! E.T2 / such that for any A  E.T1 /, .E.T2 / [ A/  .A/ is the edge set of a spanning tree. However, it is an open problem whether Lemma 12 holds or not, which is an interesting open problem. Next, an application of above lemmas will be presented.

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Consider a spanning tree T of a graph G D .V; E/ and an edge subset F of E. A swap set for T with F is an edge subset A of E.T / such that .E.T / [ F /  A is the edge set of a spanning tree. Let E.u; p/ D f.u; v/ 2 E.G/ j w.u; v/  pg. A.u; p/ denotes a swap of T with E.u; p/. The greedy algorithm can be described as follows. CFM Algorithm T the minimum-length spanning tree of G. ok 1; A ;; while ok D 1 do begin 0 ;p 0 / E.u; p/ D argmaxE.u0 ;p0 / w.A.u ; p0 > 2 if w.A.u;p/ p then E E [ fE.u; p/g .T [ E.u; p// n A.u; p/ assign all edges in E.u; p/ with weight 0 else ok 0; output E and T . The following shows an upper bound for power-cost of broadcasting tree constructed from output of CFM Algorithm. Lemma 13 With output E and T , a broadcasting tree can be constructed with powercost at most X 2 p C w.T /: E.u;p/2E

Proof Suppose s is the source node. Turn tree T into a tree rooted at s and turn each edge into an arc by giving the direction from parent to child. This rooted tree (or outarborescence) can be realized with the following power assignment. Consider an arc .x; y/ in rooted tree T . If edge .x; y/ is not in [E.u;p/2E E.u; p/, then assign p.x/ D w.x; y/. If .x; y/ 2 E.u; p/ for some E.u; p/ 2 E, then assign p.x/ D p. An important observation is that for each E.u; p/, there is at most one edge .u; v/ in E.u; p/ such that arc .v; u/ is in rooted tree T . In fact, if .v; u/ is in the tree T with root s, then for every edge .u; v0 / 2 E.u; p/, other than .u; v/, one can reach v0 from u through arc .u; v0 /, that is, if .v0 ; u/ cannot be in T . In above assignment, the total power is at most 2

X

p C w.T /:

E.u;p/2E

 The performance of CFM is given in the following theorem.

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Theorem 8 Suppose  > 2 is the performance ratio of the minimum-length spanning tree as an approximation for MIN-POWER BROADCAST. Then, the performance ratio of approximation solution obtained from CFM Algorithm is at most 2 ln   2 ln 2 C 2. Proof Let T0 denote the minimum spanning tree at the initial step and Ti for i D 1; : : : ; k the tree obtained in the i th iteration. Let E.ui ; pi / be selected at the i th iteration. Clearly, w.A.ui ; pi // D w.Ti 1 /  w.Ti /: First, it can be observed that 

w.Ti 1 /  w.Ti /  opt: w.Ti 1 /

where opt is objective function value of optimal solution for MIN-POWER BROADCAST . Suppose T  is the optimal broadcast tree. Let I N.T  / denote the set of internal nodes of T  . Denote by B.u/ the set of arcs going out from u. Then, X

p.T  / D

u2IN.T  /

max w.u; v/:

.u;v/2B.u/

By Lemma 12, there exists a one-to-one onto mapping  W E.T  / ! E.Ti 1 / such that .B.u// is a swap set for Ti 1 with B.u/. Denote u D max.u;v/2B.u/ w.u; v/. Then, P w.Ti 1 / w..B.u/// u2IN.T  / w..B.u/// P :  D u  opt  u2IN.T / u Therefore, there exists u 2 I N.T  / such that w..B.u/// w.Ti 1 / :  u opt Since .B.u//  A.u; u /, it follows that w.Ti 1 / w.A.u; u // :  u opt It follows from greedy principle that w.A.ui ; i / w.A.u; u // w.Ti 1 / ;   i u opt that is, i 

w.Ti 1 /  w.Ti /  opt: w.Ti 1 /

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Now, by Lemma 13, GFM Algorithm produces a broadcast tree with total power at most

2

k X w.Ti 1 /  w.Ti / i D1

2

w.Ti 1 /

k1 X w.Ti 1 /  w.Ti / i D1

w.Ti 1 /

 opt C w.Tk /

 opt C 2k C w.Tk /:

Denote ı D 2opt  w.Tk /. Note that w.A.uk ; k // > 2: k Thus, 2k C w.Tk / D 2 

w.A.uk ; k //  ı C 2opt w.A.uk ; k //=k

 2

w.A.uk ; k //  ı C 2opt w.A.uk ; k //=k

 2

w.A.uk ; k //  ı C 2opt w.Tk1 /=opt 

D 2opt 

./

 w.A.uk ; k //  ı C1 ; w.Tk1 /

Therefore, GFM Algorithm produces a broadcast tree with total power at most

2opt

k1 X w.Ti 1 /  w.Ti / i D1

 2opt

k1 Z X i D1

Z

w.Ti 1 / w.T0 /w.Ti / w.T0 /w.Ti 1 /

w.T0 /2opt

w.Tk1 /  2opt C C1 w.Tk1 /

dt C w.T0 /  t !

Z

w.T0 /2opt w.T0 /w.Tk1 /

!

dt C1 w.T0 /  t

!

dt C1 w.T 0/  t 0   w.T0 / D 2opt ln  ln 2 C 1 opt

D 2opt

 opt.2 ln   2 ln 2 C 2/:



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The above proof contains a bug. The inequality sign at line ./ holds only if w.A.uk ; k //  ı  0, that is, w.Tk1 /  2opt. If w.Tk1 / < 2opt, is there some way to fix the proof? So far, nobody knows. Dai and Wu [51], Li et al. [52, 53], and Guo and Yang [54] minimize the total energy cost of broadcast in wireless networks with directional antennas or adaptive antennas. Ghosh [55] designs distributed algorithm and Wu and Dai [56] design an broadcast algorithm with self-pruning technique. Agarwal et al. [57] study energyefficient broadcast in wireless ad hoc network with hitch-hiking.

2.4

Asymmetric Power Requirement

In wireless networks studied in [58], each node x has its own rate r.x/ for power requirement to establish a link .x; y/, i.e., the power-cost for the existence of .x; y/ is p.x; y/ D ck.x; y/k˛ =r.x/ instead of ck.x; y/k˛ . When r.x/ ¤ r.y/, it has the following that p.x; y/ ¤ p.y; x/. Althaus et al. [20] also considered the case that arcs p.u; v/ ¤ p.v; u/ and studied the following problem: M IN -POWER SYMMETRIC CONNECTIVITY WITH ASYMMETRIC POWER REQUIREMENT: Given a complete graph G D .V; E/ and power requirement p W V  V ! RC , find a minimum power-cost spanning tree where an (undirected) edge .u; v/ exists iff the power at u is at least p.u; v/ and the power at v is at least p.v; u/.

For this problem, they showed that the minimum spanning tree with respect to edge weight p.u; v/ C p.v; u/ is a 2-approximation. However, the proof contains a flaw because the power-cost of an out-arborescence has not been proved to be lowerbounded by the sum of its edge power requirements. In fact, Lemma 7 indicates that the power-cost of an out-arborescence can be lower bounded by one-sixth of the sum of edge power requirements of a minimum spanning tree which is not the out-arborescence itself. Therefore, it is still an open problem whether MIN-POWER SYMMETRIC CONNECTIVITY WITH ASYMMETRIC POWER REQUIREMENT has a polynomial-time constant approximation or not. Calinescu et al. [59] gave a positive answer in the case that p.x; y/ D cd.x; y/˛ =r.x/; where d.x; y/ is the distance between x and y in a metric space. Theorem 9 The minimum spanning tree with respect to edge weight w.u; v/ D p.u; v/ C p.v; u/ induces an approximation solution within a factor of  min 2 max 2˛ C .r C 1/˛ ; r>1

r˛ ˛ r 1



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from optimal for MIN-POWER SYMMETRIC CONNECTIVITY WITH ASYMMETRIC POWER REQUIREMENT. This theorem follows immediately from the following lemma. Lemma 14 [Rooted-Spanning Tree Lemma] For any rooted spanning tree T and a number r > 1, there exists a rooted spanning tree Tr such that  w.Tr /  2 max 2˛ C .r C 1/˛ ; where w.Tr / D

P

.u;v/2E.Tr / .p.u; v/

 r˛ g  P .T / r˛  1

C p.v; u//.

Proof Let V i .T / be the set of all internal nodes of T . Denote by T .u/ the subtree induced by an internal node u and all its children. Then, X

P .T .u//  2P .T /:

u2V i .T /

For each T .u/, a tree Tr .u/ is constructed by modifying T .u/ in the following way: • Sort all children x1 ; x2 ; : : : ; xk of u in ordering of nonincreasing in d.u; xi /, that is, d.u; x1 /  d.u; x2 /      d.u; xk /. • For each i D 1; : : : ; k  1, if .d.u; xi /  rd.u; xi C1 /, then replace edge .u; xi / by edge .xi ; xi C1 /. Note that for each edge .xi ; xi C1 / in Tr .u/, d.xi ; xi C1 /  d.u; xi / C d.u; xi C1 /  .1 C r/d.u; xi C1 / and d.xi ; xi C1 /  d.u; xi / C d.u; xi C1 /  2d.u; xi /: Thus, p.xi ; xi C1 /  2˛ p.xi ; u/ and p.xi C1 ; xi /  .1 C r/˛ p.xi C1 ; u/: Therefore, w.xi ; xi C1 /  .1 C r/˛ p.xi ; u/ C 2˛ p.xi C1 ; u//:

(1)

Let .u; xi1 /, .u; xi2 /, . . . , .u; xih / be those edges of T .u/, which remain in Tr .u/. Then, it has the following that d.u; xi1 /  rd.u; xi2 /  r 2 d.u; xi3 /      r h1 d.u; xih /:

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Therefore, p.u; xi1 / C p.u; xi2 / C    C p.u; xih /  p.u; xi1 /.1 C r ˛ C    C r .h1/˛ 1 1  r ˛ r˛  p.u; x1 / ˛ : r 1  p.u; xi1 /

Combining with (1), it can be obtained X

w.Tr / D

w.u; xi /

.u;xi /2E.Tr .u//

X

C

w.xi ; xi C1 /

.xi ;xi C1 /2E.Tr .u//

 p.u; x1 /

r˛ C r˛  1

X

X

C.1 C r/˛

p.xi ; u/

.u;xi /2E.Tr .u//

.p.xi ; u/ C p.xi C1 ; u//

.xi ;xi C1 /2E.Tr .u//

 r˛ ˛ ˛ ; 2 C .1 C r/ :  P .T .u// max ˛ r 1 

Now, let Tr D [u2V i .T / Tr .u/. Then  w.Tr /  max   max

2.5

r˛ ; 2˛ C .1 C r/˛ ˛ r 1

 X

P .T .u//

u2V i .T /

 r˛ ˛ ˛ ; 2 C .1 C r/  2P .T /: r˛  1



Unicast

Calinescu et al. [15] studied a unicast problem as follows: M IN -POWER UNICAST: Given a set V of points, a source s 2 V , and a destination t 2 V , find a minimum power-cost path between s and t .

First, they note that the minimum power-cost path is different from the minimum cost path. For example, consider three points s, x, and t which are three vertices of an isosceles triangle with a right angle at x. Suppose ˛ D 2 and c D 1. Then, k.s; t/k2 D k.s; x/k2 C k.x; t/k2 . However,

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P ..s; t// D 2k.s; t/k2 ; and

3 k.s; t/k: 2 Indeed, the calculation of power-cost of a path is not a simple sum of edge cost. This fact results in that the algorithm for computing the shortest path does not work on the geometric network on node set V . Calinescu et al. [15] gave the following solution. First, construct an edgeweighted graph G with node set V V and following edges: • For every three nodes u; v; w 2 V , construct an arc ..u; v/; .u; w// with weight max.0; k.u; w/k˛  k.u; v/k˛ . • For every two nodes u; v 2 V , construct two arcs ..u; v/; .v; u// and ..v; u/; .u; v// each with weight k.u; v/k˛ . For example, the graph G on above three points s, x, and t contains nine nodes, and the path .s; x; t/ corresponds to the path ..s; x/; .x; s/; .x; t/; .t; x// with total weight equal to the power-cost P .s; x; t/ of path .s; x; t/. Now, MIN-POWER UNICAST is reduced to the shortest path problem in constructed graph G. Therefore, it has the following that P .s; x:t/ D p.s/ C p.x/ C p.t/ D

Theorem 10 MIN-POWER UNICAST can be solved in polynomial-time.

3

Minimum Energy with Unadjustable Power

In real systems of wireless ad hoc networks, after network is configured, each node is assigned with a transmission power level which would not be changed for any request such as broadcast and unicast. In wireless sensor networks, the communication range of each sensor is often fixed and hence also in the same situation that the minimization of total energy consumption is under given transmission power level at each node. When power level at every node is the same, the total energy consumption can be measured through the total node-weight at transmission nodes. In this section, several optimization problems of this type is presented.

3.1

Operations in Multiradio Multichannel Wireless Networks

In works of Li et al. [7], Cagalj et al. [6], and Liang [5], the total energy consumption of broadcast in static wireless ad hoc networks is minimized under the same given power level at every node. The following is a general formulation. Min Node-Weight Arborescence: Given a node-weighted digraph G D .V; E/ with a source node s, construct an out-arborescence rooted at s with the minimum total weight of internal nodes.

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Liang [5] gave an efficient algorithm evaluated through simulations. Cagalj et al. [6] presented a polynomial-time O.log3 n/-approximation. Li et al. designed a polynomial-time approximation algorithm with performance ratio .1 C 2 ln.n  1//. Those algorithms are presented for Min Node-Weight Arborescence in non-weighted case. However, it is easy to extend them to weighted case, corresponding to that power levels at different nodes may be different. Li et al. [60] and Ma et al. [61] study the broadcast in multiradio multichannel multihop wireless networks. In a multiradio multichannel multihop wireless network (MR-MC network), there are totally C non-overlapping orthogonal frequency channels, 1; 2; : : : ; c. Each node v is equipped with some omni-directional radio interfaces. Suppose the following are done after configuration: (a) Each node v is assigned with a channel subset B.v/. This means that each node v can receive messages through any channel in subset B.v/ and transmit messages using any channel in subset B.v/. (b) The transmission power is fixed at a certain level p.v/ at each node v. Assume that compared with transmission power, energy spending for receiving data at each node can be ignored. Then, energy consumption at each node v is p.v/ multiplying the number of forward channels, that is, channels used for sending messages. The problem studied in [60, 61] is to find a forward scheme, that is, for each node v, find a channel subset F .v/  B.v/ such that data can be broadcasted from a given source node to all wireless nodes and the total energy consumption is minimized. For any broadcasting tree T , let NL.T / be the set of its internal nodes. The following is the formulation. BROADCAST IN MR-MC NETWORK : Consider a digraph G D .V; E/ with a source node s and a set of radio channels C D f1; 2; : : : ; cg. Given each node v a set B.v/ of radio channels and a power p.v/, find an assignment F W V ! 2C to minimize W .T / D

X

jF .v/j  p.v/

v2NL.T /

under the following conditions: (1) F .v/  B.v/ for every v 2 V (2) There exists a broadcast tree T from source s satisfying condition that for each arc .u; v/ 2 T; F .u/ \ B.v/ 6D ;.

While Li et al. [60] assumed that transmission power level at all nodes is the same, Ma et al. [61] did not. Actually, Min Node-Weight Arborescence and BROADCAST IN MR-MC NETWORK are equivalent in terms of approximability. Theorem 11 There is a polynomial-time -approximation for BROADCAST IN MR-MC NETWORK if and only if there is a polynomial-time -approximation for MIN NODE-WEIGHT ARBORESCENCE. Proof MIN NODE-WEIGHT ARBORESCENCE can be seen as a special case of BROADCAST IN MR-MC NETWORK; in such a case, B.v/ contains only one channel which is the same for all v 2 V . Thus, “only if” part is trivial.

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To show “if” part, BROADCAST IN MR-MC NETWORK is transformed to a special case of MIN NODE-WEIGHT ARBORESCENCE. In this transformation, first construct an auxiliary digraph Gaux D .Vaux ; Eaux / based on the input digraph G D .V; E/ of BROADCAST IN MR-MC NETWORK as follows. (1) Create a source node s  with weight p.s  / D 0. (2) For each node v, create jB.v/j nodes vi with weight p.vi / D p.v/ for i 2 B.v/. (3) Connect s  to every si for the input source node s of BROADCAST IN MR-MC NETWORK with arc .s  ; si /. (4) For each arc .u; v/ 2 E, add arcs .ui ; vj / for i 2 B.u/, j 2 B.v/, and i 2 B.v/. Thus, one has Vaux D fs  g [ [v2V fvj j j 2 B.v/g with p.s  / D 0, and p.vj / D p.v/ for j 2 B.v/, and Eaux D [.u;v/2E f.ui ; vj /j.u; v/ 2 E; i 2 B.u/ \ B.v/; j 2 B.v/g: This transformation has the following property: There exists a forward scheme F with cost at most W containing a broadcast tree from node s if and only if Gaxu contains a broadcast tree with weight at most W from node s  . To show “only if” part of this property, suppose the existence of forward scheme F with cost at most W containing a broadcast tree T . Construct a broadcast tree Taux for Gaux as follows: Initially, set Taux D ;. (1) For each i 2 B.s/, add arc .s  ; si / to Taux . (2) For each arc .u; v/ in T , choose a channel i from F .u/ \ B.v/ and add arcs .ui ; vj / to Taux for every j 2 B.v/. From the above construction, it is easy to see that for any path from s to a node v, Taux contains a path from s  to node vj for any j 2 B.v/. Therefore, Taux being a broadcast tree follows from T being a broadcast tree. Moreover, only in case that arc .u; v/ exists (i.e., u is an internal node of T ) and i 2 F .u/, ui can be an internal  node of PTaux . Therefore, the total weight of internal nodes other than s in Taux is at most u2NL.T / jF .u/j  p.u/  W . To show the “if” part of the property, suppose Gaux contains a broadcast tree Taux with weight at most W . For each u 2 V , define F .u/ D fi j .ui ; vj / in Taux g: Denote by T the subgraph induced by the arc set f.u; v/ j .ui ; vj / in Taux g: It follows from the definition of Gaux that for any arc .u; v/ in the arc set, i 2 F .u/ \ B.v/, and hence, F .u/ \ B.v/ ¤ ;. Now, T is claimed to be a broadcast tree.

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In fact, for each node v 2 V , Taux contains a path from s  to vj for any j 2 B.v/ since Taux is a broadcast tree. This path would induce a path from s to v. Finally, it is noticed P that i 2 F .u/ if and only if ui is an internal node of Taux . Therefore, W .F /  .u;i /2NL.Taux / p.u/  W . Now, the theorem can be proved easily from the property. First, it follows immediately from the property that the minimum weight for a forward scheme F to contain a broadcast tree is opt if and only if the minimum cost of an arborescence in G opt is opt. Next, suppose an out-arborescence Taux with cost at most   opt for Gaux can be computed in polynomial-time. Then, from the proof of the property, one may see that a forward scheme F can be constructed in polynomial time to contain a broadcast tree and to have cost at most   opt. This means that if there is a polynomial-time -approximation for MIN NODE-WEIGHT ARBORESCENCE on input Gaux with weight p.ui / D p.u/, then there is a polynomial-time -approximation for BROADCAST IN MR-MC NETWORK.  The above theorem indicates that the study on broadcast in multiradio multichannel wireless networks can be reduced to the study on broadcast in single-radio single-channel wireless networks. Du et al. [62] found that this is true for all network operations, such as gossiping, convergecast, multicast, and unicast. For example, gossiping is another one of most fundamental communication primitives in wireless network. The energy-efficient gossiping in multiradio multichannel wireless networks can be formulated as follows. GOSSIPING IN MR-MC NETWORK : Consider a graph G D .V; E/. Each node v is associated with a set B.v/ of radio channels and a weight p.v/. The problem is to assign each node v with a subset F .v/ of B.v/ such that the directed graph with node set V and arc set f.u; v/ 2 E j F .u/ \ B.v/ ¤ ;g is strongly connected.

Its corresponding problem in single-radio single-channel wireless networks can be described in the following way. M IN -WEIGHT SCDS: Given a directed graph G D .V; E/ with nonnegative node weight c W V ! RC , find the minimum-weight strongly connected dominating set where a subset C of nodes is called a strongly connected dominating set if every node v 2 V  C has an arc .v; u/ ending at u 2 C and also an arc .w; v/ coming from w 2 C .

Du et al. [62] showed their equivalence as follows. Theorem 12 If min-weight SCDS has a polynomial-time -approximation if and only if gossiping in MR-MC networks has a polynomial-time -approximation.

3.2

Energy-Efficient Virtual Backbone

There are several works on energy-efficient virtual backbone, that is, energyefficient connected dominating sets in the literature [63–66]. Since they assume that the communication range at every station has the same radius, the minimization on

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total power is the same as the minimization on the size of connected dominating set, which has been studied extensively. For a current research status, the reader may see [67].

3.3

Connected Sensor Cover

Motivated from study on coverage problem [68, 69] and connected dominating set problem, Gupta , Das, and Gu [70] proposed the connected sensor cover problem as follows. CONNECTED SENSOR COVER (AREA ): Consider a sensor network G D .V; E/ of n sensors in the Euclidean plane and a region A. Each sensor si 2 V is associated with a sensing region Si . The problem is to select the minimum number of sensors such that A is covered by sensing regions of selected sensors and the subgraph induced by selected sensors is connected.

A subregion of A is called a subelement if the maximal region cannot be cut by any sensor’s sensing region. Put a target in each subelement. Then, a set of sensors can cover A if and only if it can cover all targets. This means that the area coverage problem can be reduced to a target coverage problem. CONNECTED SENSOR COVER (TARGET): Consider a sensor network G D .V; E/ of n sensors and a set of m targets. Each sensor si 2 V is associated with a sensing set Si of targets. The problem is to select the minimum number of sensors such that every target appears in the sensing set of at least one selected sensor and the subgraph induced by selected sensors is connected.

If the sensor set and the target set is identical, then this problem is exactly the minimum connected dominating set problem. Thus, CONNECTED SENSOR COVER is NP-hard because the minimum connected dominating set problem is NP-hard. Gupta, Das, and Gu [70] designed a greedy approximation with the following performance. Theorem 13 There is a polynomial-time approximation for CONNECTED SENSOR COVER with performance ratio at most .r  1/.1 C log m/ where r is the link radius of the sensor network, that is, the maximum communication distance between two sensors with overlapping sensing sets. Wang et al. [71] and Zhang and Hou [72] study CONNECTED SENSOR COVER (AREA) in the case that each sensor s is associated with two disks with center s, sensing region and communication region. They found an interesting property specially for area coverage as follows. Theorem 14 Suppose the radius of communication region is at least twice the radius of sensing region. If a connected region is covered by a subset S of sensors, then S induced a connected subgraph. Zhou, Das, and Gupta [73] extended this property to k-coverage.

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Theorem 15 Suppose the radius of communication region is at least twice the radius of sensing region. If a subset of sensors has k-coverage to a connected region, that is, every point in the region receives sensing from at least k sensors, then it induced a k-connected subgraph, that is, for any two sensors there are at least k node-disjoint paths between them. Proof Suppose S is a subset of sensors with k-coverage. Then, for any k1 sensors s1 ; : : : ; sk1 , S  fs1 ; : : : ; sk1 g is still able to cover the target connected region. Therefore, the subgraph induced by S  fs1 ; : : : ; sk1 g is connected.  By above theorem, one need only to pay attention to coverage when design algorithm to construct connected sensor coverage. While many efforts [70, 72, 73] were made on design of greedy approximation with performance ratio O.log n/, Funke et al. [74] showed that greedy approximation cannot have performance ratio better than .log n/. They also proposed a fine grid algorithm with the following performance. Theorem 16 Suppose in a uniform wireless sensor network, the communication radius Rc is at least twice ofpthe sensing radius Rs . Also, suppose with reduced sensing radius Rs0 D .1  = 2/Rs , the target connected region can still be fully covered. Then, the fine grid algorithm can produce an approximation solution for CONNECTED SENSOR COVER with size within a factor of 12 from the size of optimal solution for sensing radius Rs0 . Zhou, Das, and Gupta [75] study CONNECTED SENSOR COVER in sensor networks with variable sensing radius and communication radius. Alam and Hass [76] and Bai et al. [77] study the problem in three-dimensional networks. Yu et al. [78] consider the problem in networks with directional antennas. Related to CONNECTED SENSOR COVER, there is also a sensor placement problem [79–82]. For a nice and complete survey about the problem, the reader may refer to [83].

4

Maximum Lifetime

With a certain resource, adjust usage of resource to maximize the lifetime of the system. This is a classic type of resource management problems. In wireless networks, there are also many optimization problems of this type on energy efficiency. Some examples will be discussed in this section.

4.1

Sensor Coverage

A result of randomly deploying a very large number of sensors into a certain region, possibly by an aircraft, is the existence of redundant sensors. Usually, such a result is utilized to increase the lifetime of network system by scheduling

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Fig. 5 Target subareas are represented by target points

active/sleep time of sensors such that sensor can be organized to work in different period so that the target area (or target points) can be covered in every period. This sensor activation scheduling with coverage of targets is called the sensor coverage problem. According to target type, there are two types problems: area coverage [71, 72, 84–86] and target coverage [8–10, 79, 87]. By target coverage, one usually means that targets are points. AREA -COVERAGE. Given an area and a set of sensors each with a sensing area, find a sensor activation scheduling to maximize the lifetime of the sensor network where the sensor network is alive if and only if the given area is fully covered by active sensors. TARGET-COVERAGE. Given a set of (point) targets and a set of sensors each with a sensing area, find a sensor activation scheduling to maximize the lifetime of the sensor network where the sensor network is alive if and only if the given targets are fully covered by active sensors.

AREA-COVERAGE can be reduced to TARGET-COVERAGE as follows: For every sensor, draw the boundary of its sensing area to divide the target area into small subareas; each subarea is a maximal area such that no boundary point of sensor’s sensing area lies in its interior. Place a point-target in the interior of each subarea (Fig. 5). Then, a subarea is covered if and only if placed point-target is covered. Therefore, the given area is fully covered if and only if all point-targets are fully targets. In the following, studying will be focused on TARGET-COVERAGE. A simple scheduling is to divide sensors into disjoint subsets each of which fully covers all targets, called a sensor cover. For this type of scheduling, the sensor is activated only once, that is, once the sensor is activated, it keeps active until it dies. The specific formulation corresponding this scheduling is as follows.

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DISJOINT SENSOR COVER : Given n targets r1 ; : : : ; rn and m sensors s1 ; : : : ; sm each with a sensing subset of targets, find the maximum number of disjoint sensor covers.

This problem is NP-hard. Various heuristics and approximation algorithms have been given in [85, 87, 88]. Cardei et al. [8] found that it is possible to increase the lifetime if each sensor is allowed to make more changes between active and sleeping. For example, suppose that sensor s1 covers targets r1 and r2 , sensor s2 covers targets r2 and r3 , and sensor s3 covers targets r1 and r3 . Using disjoint-set strategy, the lifetime of this system is 1 unit, which is the lifetime of each sensor, because there do not exist two disjoint sensor covers. However, the following scheduling can obtain lifetime 1.5 unit time: In the first period of 0.5 unit time, sensors s1 and s2 are activated and sensor s3 is put in sleeping; in the second period of 0.5 unit time, sensors s2 keeps being active and s3 is activated and sensor s1 is put in sleeping; in the third period of 0.5 unit time, sensors s1 is activated again and s3 is still active. The specific formulation on scheduling of nondisjoint sensor covers is as follows [8]. SENSOR -COVER -SCHEDULING : Given m sensors s1 ; : : : ; sm each with a sensing subset of target set fr1 ; : : : ; rn g, find a family of sensor cover S1 ; : : : ; Sp with time lengths t1 ; : : : ; tp in Œ0; 1, respectively, to maximize t1 C    C tp subject to every sensor appears in cover sets with total time length at most 1.

This is still an NP-hard problem, which can be easily written as an 0-1 integer programming. Maximize t1 C    C tp subject to

p X

xij tj  1 for i D 1; : : : ; m

j D1

X

xij  1 for k D 1; : : : ; n and j D 1; : : : ; p;

i 2Ck

xij D 0; 1; and tj  0; where for each target rk , Ck is the set of sensors whose sensing subset contains rk . xij is the indicator that sensor si is in sensor cover Sj . The first inequality constraint means that for each sensor si , the total active time length does not exceed 1. The second inequality constraint means that at any time period, active sensors should form a sensor cover. Note that the first inequality constraint together with tj  0 implies tj  1. Therefore, tj  1 does not need to put in constraints explicitly. Above integer programming is not linear since xij tj is not a linear term. However, it can be easily linearized by setting yij D xij tj as follows: Maximize t1 C    C tp

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yij  1 for i D 1; : : : ; m

j D1

X

yij  tj for k D 1; : : : ; n and j D 1; : : : ; p;

i 2Ck

yij D 0; or tj ; and tj  0: Cardei et al. [8] designed a heuristic using LP relaxation as follows. By relaxing the condition yij D 0; tj to 0  yij  tj , the following linear programming can be obtained from the above integer linear programming.

Maximize t1 C    C tp subject to

p X

yij  1 for i D 1; : : : ; m

j D1

X

yij  tj for k D 1; : : : ; n and j D 1; : : : ; p;

i 2Ck

0  yij  tj : The solution .yij ; t  / of this linear programming may not satisfy condition yij D 0 or tj and even not necessarily satisfies tj  1. Therefore, the following rounding technique is employed: Set tj D minkD1;:::;m maxi 2Ck yij . For each k, choose a ik from Ck such that yij  tj and set yik j D tj . For remaining i , set yij D 0. This algorithm is employed iteratedly to P utilize the remaining P senor lifetime. To p p do so, Cardei et al. [8] replace inequality j D1 xij tj  1 by j D1 xij tj  ei where ei is the remaining lifetime of sensor si . No theoretical bound has been established for this heuristic. However, computational experiments show that the heuristic produces good solutions. The nondisjoint model is better than the disjoint model also due to an interesting fact discovered in [89] that putting a sensor alternatively in active and sleeping states in a proper way may double its lifetime since the battery could be recovered in certain level during sleeping. Berman et al. [90, 91] first designed an approximation algorithm for SENSORCOVER-SCHEDULING with theoretical bound. Theorem 17 There exists a polynomial-time approximation for SENSOR-COVERSCHEDULING with performance ratio O.log n/ where n is the number of sensors. Ding et al. [92] found constant approximation for geometric version.

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Theorem 18 Suppose all sensors and targets lie in the Euclidean plane and the sensing area of every sensor is a disk with the same radius. Then, there exists a polynomial-time .4 C /-approximation for SENSOR-COVER-SCHEDULING. Both result of Berman et al. [90, 91] and Ding et al. [92] employ an algorithm of Garg and K¨onemann [93] and their theorem, which will be introduced in next subsection.

4.2

¨ Garg–Konemann Algorithm

The first one who found the application of Garg–K¨onemann Algorithm in the study of lifetime maximization type of problems is Calnescu et al. [59]. Consider a graph G D .V; E/. A function c W V V ! RC is called a power requirement if edge .u; v/ is established if and only if both the power at u and the power at v are not less than c.u; v/. With a power requirement c, G D .V; E; c/ is called a power requirement graph. Calnescu et al. [59] consider the following general form of the minimum power problem and the maximum lifetime problem. POWER ASSIGNMENT: Given a power requirement graph G D .V; E; c/, and a conC nectivity Pconstraint Q, find a power assignment p W V ! R to minimize the total power u2V p.u/ under condition that supported subgraph H satisfies the connectivity constraint Q. NETWORK LIFETIME: Given a power requirement graph G D .V; E; c/, a battery supply Q, find a feasible power schedule P T D b W V ! RC and a connectivity constraint P m f.p Pm1 ; t1 /; .p2 ; t2 /; : : : ; .pm ; tm /g to maximize iD1 ti , where schedule P T is feasible if iD1 pi .u/  b.u/ for every u 2 V .

Suppose S is the set of all subgraphs satisfying constraint Q. For each H 2 S, let pH W V ! RC be the power assignment required by establishing subgraph H . Then, NETWORK LIFETIME can be formulated a linear integer programming as follows: X max tH H 2S

subject to

X

pH .u/tH  b.u/

H 2S

tH  0; 8H 2 S: Now, the sum of coefficients for each column is exactly the total power for establishing H , which is called the length of the column. Computing the minimum column length is exactly the POWER ASSIGNMENT problem. Consider the following weighted version of POWER ASSIGNMENT. W EIGHTED POWER ASSIGNMENT: Given a power requirement graph G D .V; E; c/, a C connectivity constraint Q, and a node weight P y W E ! R , find a power assignment C p W V ! R to minimize the total power u2V p.u/y.u/ under condition that supported subgraph H satisfies the connectivity constraint Q.

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Suppose WEIGHTED POWER ASSIGNMENT has an f -approximation. Then, a .1 C "/f -approximation can be computed for NETWORK LIFETIME. Indeed, the linear programming formulation of NETWORK LIFETIME falls in a general form of linear programming studied by Garg and K¨onemann [93] as follows: maxfc T x j Ax  b; x  0g where A is an m n positive matrix and b and c are m-dimensional and n-dimensional positive vectors, respectively. n can be exponentially larger than m. However, The following Garg and K¨onemann Algorithm [93] assumes that the linear programming is actually implicitly given by vector b. Garg–K¨onemann Algorithm input: A vector b 2 Rm , " > 0, an algorithm F P y.i / for computing f -approximation of minj m i D1 A.i; j / c.j / . Initially, ı D .1 C "/..1 C "/m/1=" . ı for i D 1; : : : ; m, y.i / , b.i / D mı, j D 0; while D < 1 do find a column Aq using f -approximation F / choose index p to minimize Ab.i q .i / j j C1 b.p/ xj Aq .p/ Aj Aq for i D 1; : : : ; m, y.i / y.i /.1 C "  Ab.p/ = b.i / / q .p/ Aq .i / D bTy xj j k output f.A ; 1C" /gj D1 . log1C"

ı

This algorithm would give a .1 C "/f -approximation for NETWORK LIFETIME. It is also proved in [93] that k  m. Theorem 19 Consider a connectivity constraint Q and a power requirement graph G D .V; E; c/. For any f -approximation algorithm F for WEIGHTED POWER ASSIGNMENT, there exists a .1 C "/f -approximation algorithm for the corresponding NETWORK LIFETIME. Garg–K¨onemann Algorithm successfully reduces the lifetime maximization problem to the minimum weighted power assignment problem. This gives a quite useful technique to study approximation of the lifetime maximization. With this technique, several approximation with good performance have been found in the literature [90–92, 94]. This approach may also be used in the problem studied in [95–97] to obtain some new results.

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Coverage Breach

In above sensor coverage problems, the complete coverage is required during the system lifetime period. However, the number of randomly deployed sensors is often very large and the core network may not be able to provide a bandwidth large enough for receiving data from all sensors or from all sensors in a sensor cover. In this situation, a complete coverage is sometimes not available. Therefore, the partial coverage becomes an interesting subject in the study of sensor networks. A target is in breach if it is not monitored by any sensor. There are three coverage breach optimization formulations studied in the literature [9, 10, 85, 90, 91]. For each formulation, there are two models, disjoint model and nondisjoint model. The following are formulations in disjoint model: Max-lifetime coverage with breach (disjoint model): Given a set of sensors, fs1 ; : : : ; sm g, a set of targets, fr1 ; : : : ; rn g, and a nonnegative integer b, the problem is to partition the sensor set into maximum number of subsets such that for each sensor subset, there are at most b breaches. Min coverage breach (disjoint model): Given a set of sensors, fs1 ; : : : ; sm g, a set of targets, fr1 ; : : : ; rn g, and a positive integer k, the problem is to partition the sensor set into k subsets to minimize the total number of breaches. Min-max coverage breach (disjoint model): Given a set of sensors, fs1 ; : : : ; sm g, a set of targets, fr1 ; : : : ; rn g, and a positive integer k, the problem is to partition the sensor set into k subsets such that the maximum number of breaches for each sensor subset is minimized.

Corresponding formulations in disjoint model are as follows. Max-lifetime coverage with breach (nondisjoint model): Given a set of sensors, fs1 ; : : : ; sm g, a set of targets, fr1 ; : : : ; rn g, and a nonnegative integer b, the problem is to find a family of sensor subsets S1 ; : : : ; Sp with time lengths t1 ; : : : ; tp in Œ0; 1, respectively, to maximize t1 C    C tp subject to that every sensor appears in those sensor subsets with total time length at most 1 and for each of those sensor subsets, there are at most b breaches. Min coverage breach (nondisjoint model): Given a set of sensors, fs1 ; : : : ; sm g, a set of targets, fr1 ; : : : ; rn g, and a positive integer k, the problem is to find a family of sensor subsets S1 ; : : : ; Sp with time lengths t1 ; : : : ; tp in Œ0; 1, respectively, to minimize t1 b1 C    C tp bp where bi is the number of breaches for sensor subset Si , subject to that every sensor appears in those sensor subsets with total time length at most 1 and t1 C    C tp D k. Min-max coverage breach (nondisjoint model): Given a set of sensors, fs1 ; : : : ; sm g, a set of targets, fr1 ; : : : ; rn g, and a positive integer k, the problem is to find a family of sensor subsets S1 ; : : : ; Sp with time lengths t1 ; : : : ; tp in Œ0; 1, respectively, to minimize max.b1 ; : : : ; bp / where bi is the number of breaches for sensor subset Si , subject to that every sensor appears in those sensor subsets with total time length at most 1 and t1 C    C tp D k.

For min–max type of problem, there is a variation that the minimization is on the maximum breach interval, where a breach interval is a time period during which there exists a target not covered by any sensor at any time. All above problems are NP-hard. However, not many efforts have been made on design of approximation algorithms [9, 10].

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Maximum Lifetime Routing

The shortest path routing is very popular and a good routing strategy in many cases. However, it may not be good for energy efficiency in wireless networks. In fact, the wireless equipment has limited energy supply, often with batteries. Every message transmission would have energy consumption. If a routing strategy allows repeatedly using some stations, then network lifetime would be shorten. Thus, a maximum lifetime routing problem appeared [98–111]. Let .s; t/ represent a routing request that a message is required to send from sensor s to sensor t. For simplicity of discussion, it is often assumed that all messages have the same length, which require the same amount of time to transmit. Max-lifetime routing: Given a wireless sensor network and a sequence of routing requests .s1 ; t1 /, .s2 ; t2 /, . . . , .sk ; tk /, find a routing strategy to maximize j such that routing requests .s1 ; t1 /, .s2 ; t2 /, . . . , .sk ; tk / are successfully routed.

The problem has an online version [110] as follows. Max-lifetime online routing: Given a wireless sensor network and a sequence of routing requests coming online, find an online routing strategy to maximize j such that first j routing requests can be successfully routed.

More information about the above problems can be found in [112, 119].

5

Conclusion

In this chapter, we have argued many combinatorial optimization problems in the study of power issues in wireless networks. The issues have been discussed into three categories: (1) minimum total power with adjustable-power nodes, (2) minimum total power with unadjustable-power nodes, and (3) maximum lifetime with unadjustable-power nodes. Latest results have been shown in the introduction of these problems.

Cross-References Connected Dominating Set in Wireless Networks Greedy Approximation Algorithms Network Optimization Optimization in Multi-Channel Wireless Networks Optimizing Data Collection Capacity in Wireless Networks

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Equitable Coloring of Graphs Ko-Wei Lih

Contents 1 2 3 4 5 6 7 8 9 10 11 12 13

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bipartite Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Equitable -Coloring Conjecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Split Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outerplanar Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planar Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphs with Low Degeneracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphs of Bounded Treewidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kneser Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interval Graphs and Others. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graph Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Square Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Cross Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Strong Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 List Equitable Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Graph Packing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Equitable -Colorability of Disconnected Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 More on the Hajnal-Szemer´edi Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Related Notions of Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 Equitable Edge-Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Equitable Total-Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Equitable Defective Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Equitable Coloring of Hypergraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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K.-W. Lih Institute of Mathematics, Academia Sinica, Taipei, Taiwan e-mail: [email protected] 1199 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 25, © Springer Science+Business Media New York 2013

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Abstract

If the vertices of a graph G are colored with k colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then G is said to be equitably k-colorable. The equitable chromatic number D .G/ is the smallest integer k such that G is equitably k-colorable. In the first introduction section, results obtained about the equitable chromatic number before 1990 are surveyed. The research on equitable coloring has attracted enough attention only since the early 1990s. In the subsequent sections, positive evidence for the important equitable -coloring conjecture is supplied from graph classes such as forests, split graphs, outerplanar graphs, series-parallel graphs, planar graphs, graphs with low degeneracy, graphs with bounded treewidth, Kneser graphs, and interval graphs. Then three kinds of graph products are investigated. A list version of equitable coloring is introduced. The equitable coloring is further examined in the wider context of graph packing. Appropriate conjectures for equitable -coloring of disconnected graphs are then studied. Variants of the well-known and significant Hajnal and Szemer´edi Theorem are discussed. A brief summary of applications of equitable coloring is given. Related notions, such as equitable edge coloring, equitable total coloring, equitable defective coloring, and equitable coloring of uniform hypergraphs, are touched upon. This chapter ends with a short conclusion section. This survey is an updated version of Lih [102].

1

Introduction

A graph G consists of a vertex set V .G/ and an edge set E.G/. All graphs considered in this chapter are finite, loopless, and without multiple edges. Let jGj and kGk denote the number of vertices, also known as the order, and the number of edges of the graph G, respectively. If the vertices of a graph G can be partitioned into k sets V1 ; V2 ; : : : ; Vk such that each Vi is an independent set (none of its vertices are adjacent), then G is said to be k-colorable and the k sets are called its color classes. Equivalently, a coloring can be viewed as a function  W V .G/ ! f1; 2; : : : ; kg such that adjacent vertices are mapped to distinct numbers. The mapping  is said to be a (proper) k-coloring. All pre-images of a fixed i , 1 6 i 6 k, form a color class. The smallest number k, denoted by .G/, such that G is k-colorable is called the chromatic number of G. The graph G is said to be equitably colored with k colors, or equitably k-colorable, if there is a k-coloring whose color classes satisfy the condition jjVi j  jVj jj 6 1 for every pair Vi and Vj . The smallest integer k for which G is equitably k-colorable, denoted by D .G/, is called the equitable chromatic number of G. Suppose that the graph G of order n is equitably colored with k colors. If n D qk Cr, where 0 6 r < k, then there are exactly r color classes of size q C 1 and exactly k  r color classes of size q. The sizes of the color classes can be enumerated as bn=kc; b.n C 1/=kc;    ; b.n C k  1/=kc in a nondecreasing order. Note that b.n C t  1/=kc D d.n C t  k/=ke for 1 6 t 6 k.

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The notion of an equitable coloring was first introduced in [115] by W. Meyer. His motivation came from Tucker’s paper [138], in which vertices represent garbage collection routes and two such vertices are adjacent when the corresponding routes should not be run on the same day. Meyer thought that it would be desirable to have an approximately equal number of routes run on each of the 6 days in a week. Let degG .v/, or deg.v/ for short, denote the degree of the vertex v in the graph G and .G/ D maxfdeg.v/ j v 2 V .G/g. Let dxe and bxc denote, respectively, the smallest integer not less than x and the largest integer not greater than x. The main result obtained by Meyer was that a tree T can be equitably colored with d.T /=2e C 1 colors. However, there were gaps in his proof. According to Guy’s report [63], Eggleton could extend Meyer’s result to show that a tree T can be equitably colored with k colors, provided k > d.T /=2e C 1. A finer result about trees is the following theorem by Bollob´as and Guy [16]. Theorem 1 A tree T is equitably 3-colorable if jT j > 3.T /  8 or jT j D 3.T /  10. The most interesting contribution made in Meyer’s paper is to propose the following conjecture. It is also called the Equitable Coloring Conjecture (ECC). Let Kn and Cn denote, respectively, a complete graph and a cycle on n vertices. Conjecture 1 Let G be a connected graph. If G is neither a complete graph Kn nor an odd cycle C2nC1 , then D .G/ 6 .G/. Meyer was successful in verifying the ECC only for graphs with six or fewer vertices. Apparently the motivation of the ECC came from the following fundamental Brooks’ Theorem [19]. Theorem 2 Let G be a connected graph. If G is neither a complete graph Kn nor an odd cycle C2nC1 , then .G/ 6 .G/. The well-known Hajnal and Szemer´edi Theorem [64], when rephrased in terms of equitable colorings, had already shown the following before Meyer’s paper. Theorem 3 A graph G (not necessarily connected) is equitably k-colorable if k > .G/ C 1. Let D .G/ denote the smallest integer n such that G is equitably k-colorable for all k > n. Then an equivalent formulation of Theorem 3 is that D .G/ 6 .G/ C 1 holds for any graph G. There is a notable contrast between the equitable colorability and the ordinary colorability: D .G/ may in fact be greater than D .G/. This will be demonstrated later. Therefore, it makes sense to introduce the notion D .G/, and it shall be called the equitable chromatic threshold of G.

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In an entirely different context, de Werra [39] treated color sequences and the majorization order among them. His results have consequences in equitable colorability. A sequence of nonnegative integers h D .h1 ; h2 ; : : : ; hk / is called a color sequence for a given graph G if the following conditions hold: 1. h1 > h2 >    > hk > 0. 2. There is a k-coloring of G such that the color classes V1 ; V2 ; : : : ; Vk satisfy jVi j D hi for 1 6 i 6 k. The majorization order, also known as the dominance order, is a widely used notion in measuring the evenness of distributions. Marshall and Olkin [110] offer a comprehensive treatment of majorization. Let ˛ W a1 > a2 >    > ak > 0 and ˇ W b1 > b2 >    > bk > 0 be two sequences of nonnegative integers. The sequence ˛ is said to be majorized by the sequence ˇ if the following two conditions hold: Pj Pj 1. i D1 ai 6 i D1 bi for any j , 1 6 j < k. Pk P 2. i D1 ai D kiD1 bi . Let Km;n denote the complete bipartite graph whose parts are of size m and size n, respectively. A claw-free graph is a graph containing no K1;3 as an induced subgraph. One of de Werra’s results is the following: Theorem 4 Let G be a claw-free graph and h D .h1 ; h2 ; : : : ; hk / be a color sequence of G. Then any sequence of nonnegative integers h0 D .h01 ; h02 ; : : : ; h0k / is also a color sequence if h0 is majorized by h. Combining Theorems 2 and 4, it follows that, for a claw-free graph G, G is equitably k-colorable for all k > .G/, or equivalently D .G/ D D .G/ D .G/. Although this fact can be shown directly, it was first implicitly implied in de Werra’s paper. It follows immediately that the ECC holds for claw-free graphs. Since every line graph is claw-free, the ECC holds for line graphs in particular. This was also obtained in Wang and Zhang [149]. This ends the history of pre-1990 activities on the equitable coloring of graphs.

2

Bipartite Graphs

A graph is called r-partite if its vertex set can be partitioned into r-independent sets V1 ; V2 ; : : : ; Vr and complete r-partite, denoted by Kn1 ;n2 ;:::;nr , if every vertex in Vi is adjacent to every vertex in Vj whenever i ¤ j and jVi j D ni > 1 for every 1 6 i 6 r. By convention it is always assumed that r > 2 and 1 6 n1 6 n2 6    6 nr . A graph is said to be complete multipartite if it is complete r-partite for some r. A bipartite graph is synonymous with a 2-partite graph. Let In denote the graph consisting of n isolated vertices. Then In is equitably k-colorable with color classes of size bxc or dxe if and only if bxc 6 bn=kc 6 dn=ke 6 dxe or, equivalently, if and only if dn=dxee 6 k 6 bn=bxcc. Lih and Wu [102] first settled the ECC for bipartite graphs.

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Theorem 5 If a connected bipartite graph G is different from any complete bipartite graph Kn;n , then G can be equitably colored with .G/ colors. Theorem 6 The complete bipartite graph Kn;n can be equitably colored with k colors if and only if dn=bk=2ce  bn=dk=2ec 6 1. The proof for the complete bipartite case is straightforward by considering the appropriate sizes of the color classes. One interesting point to note is that, for k D n D .Kn;n /, the difference involved in Theorem 6 is 0 when n is even and is 2 when n is odd. In view of Theorem 5, one can conclude that, except the complete bipartite graphs K2mC1;2mC1 , every connected bipartite graph G can be equitably colored with .G/ colors. Clearly, .K2mC1;2mC1 / D D .K2mC1;2mC1 / D 2, yet D .K2mC1;2mC1 / D 2mC2. There is a gap between the equitable chromatic number and the equitable chromatic threshold. Nevertheless, ECC holds for connected bipartite graphs. In many cases, the equitable chromatic number is below the maximum degree. If additional constraints are imposed upon the graph, a better bound for the equitable chromatic number could be obtained. The following is a result of this type: Theorem 7 Let G D G.X; Y / be a connected bipartite graph with two parts X and Y such that kGk D e. Suppose jX j D m > n D jY j and e < bm=.n C 1/c .m  n/ C 2m. Then D .G/ 6 dm=.n C 1/e C 1. The bound for the equitable chromatic number in the above theorem is indeed better than .G/ when there are at least two edges. The following conjecture was made by B.-L. Chen in a personal communication: Conjecture 2 Let G be a connected bipartite graph. Then D .G/ 6 d.G/=2eC1. Chen proved its validity when the maximum degree is at least 53. It is also trivial to see that the conjecture holds for complete bipartite graphs. The Meyer-Eggleton result about trees gives another positive evidence. Wang and Zhang [149] established the validity of ECC for complete multipartite graphs. The exact value of the equitable chromatic number of a complete multipartite graph Kn1 ;n2 ;:::;nr was determined by Lam et al. [99]. Theorem 8 Let M be the largest natural Pnumber such that ni .mod M / < dni =M e for 1 6 i 6 r. Then D .Kn1 ;n2 ;:::;nr / D riD1 dni =.M C 1/e. Blum et al. [12] also obtained a formula for D .Kn1 ;n2 ;:::;nr /. For r > 2, let Kr.n/ denote the complete multipartite graph K n;n:::;n . Theorem 6 has been generalized „ƒ‚… to Kr.n/ in [104].

r

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Theorem 9 Let m jn > 1kand k > r > 2. Then Kr.n/ is equitably k-colorable l integers n n 6 1. if and only if bk=rc  dk=re Complete multipartite graphs also furnish us with examples to show that the inequalities .G/ 6 D .G/ 6 D .G/ can be strict. For instance [103], let G1 D K 1;1;:::;1 ;2nC1 , G2 D K 2nC1;2nC1;:::;2nC1 , and G3 D K2nC1;2nC1;4nC2;4nC2;:::;4nC2 . „ƒ‚… „ ƒ‚ … „ ƒ‚ … m

m

m

Then 1. .G1 / D m C 1 < D .G1 / D D .G1 / D m C n C 1. 2. .G2 / D D .G2 / D m < D .G2 / D m.n C 1/. 3. .G3 / D m C 2 < D .G3 / D 2.m C 1/ < D .G3 / D .m C 1/.2n C 1/ C 1. As to the gap between the chromatic number and the equitable chromatic number, Wang and Zhang [149] proposed the following: Conjecture 3 For any graph G, D .G/  .G/ 6 b.G/=2c. The upper bound is sharp in the sense that it can be attained by a star K1;2nC1 .

3

Trees

A graph is said to be nontrivial if it contains at least one edge. There is a natural way to regard a nontrivial tree T as a bipartite graph T .X; Y /. The technique used to prove the ECC for connected bipartite graphs can be applied to find the equitable chromatic number of a nontrivial tree when the sizes of the two parts differ by at most one. First try to cut the parts into classes of nearly equal size. If there are vertices remaining, then one can manage to find nonadjacent vertices in the opposite part to form a class of the right size. The following was established in Chen and Lih [26]. Theorem 10 Let T D T .X; Y / be a nontrivial tree satisfying jjX j  jY jj 6 1. Then D .T / D D .T / D 2. When the sizes of the two parts differ by more than one, the determination in [26] for the equitable chromatic number of a tree needs extra notation. For any vertex u of a graph G, an independent set containing u is called a u-independent set. Let ˛u .G/ denote the maximum size of a u-independent set in G. Now suppose that G is partitioned into D .G/ parts of independent sets. Let v be an arbitrary vertex of G. Then the part containing v has size at most ˛v .G/, and other parts have size at most ˛v .G/ C 1. It follows that jGj 6 ˛v .G/ C .D .G/  1/ .˛v .G/ C 1/ D D .G/.˛v .G/ C 1/  1. Lemma 1 Let v be an arbitrary vertex of G, then D .G/ > d.jGj C 1/= .˛v .G/ C 1/e.

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An induction based on Theorem 1 leads to the following: Theorem 11 Let T D T .X; Y / be a tree satisfying jjX jjY jj > 1. Then D .T / D D .T / D maxf3; d.jT j C 1/=.˛u .T / C 1/eg, where u is an arbitrary vertex of maximum degree in T . The following result was proved by Miyata et al. in an unpublished manuscript [116] without using Theorem 1: Theorem 12 Let T be a tree and k > 3 be an integer. Then T is equitably k-colorable if and only if k > maxv2V .T / d.jT j C 1/=.˛v .T / C 1/e. The inequality in the above theorem can be equivalently described as ˛v .T / > bjT j=kc for any vertex v of T . This can be seen from the following equivalences which hold for any integer k and graph G: 

jGj C 1 ˛v .G/ C 1   jGj : > k

k>

 ,k>

jGj C 1 jGj  k C 1 , ˛v .G/ > , ˛v .G/ ˛v .G/ C 1 k

Actually, the difference between characterizations in [26] and [116] is only apparent. In view of the following lemma, Chang [22] gave a simplified and unified proof for the more general case of a forest: Lemma 2 Let u be a vertex of a forest F . If d.jF j C 1/=.˛u .F / C 1/e > 3, then u is the unique vertex of maximum degree in F .

Theorem 13 Let F be a forest and k > 3 be an integer. Then F is equitably k-colorable if and only is ˛v .F / > bjF j=kc for any vertex v of F . To determine when a forest F is equitably 2-colorable needs a bit more work than that of a tree. Without loss of generality, suppose that F has r components such that each component tree Ti consists of two parts Xi and Yi . The P objective is Pto look for a partition of f1; 2; : : : ; rg into two parts A and B such that i 2A jXi jC j 2B jYj j D bjF j=2c. Hansen et al. [66] introduced the notion of an m-bounded coloring of a graph G, i.e., a proper coloring of G such that each color class is of size at most m. The m-bounded chromatic number of G, denoted by m .G/, is the smallest number of colors required for an m-bounded coloring of G. This notion of colorability is closely related to equitable colorability via the following observation:

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Observation. The graph G has an m-bounded coloring using k colors if and only if the graph G 0 obtained from G by adding mk  jGj isolated vertices is equitably k-colorable. The problem of determining the m-bounded chromatic number of a tree was left open in [66]. By modifying the techniques used in [26], Chen and Lih [25] were able to determine the m-bounded chromatic number of a tree. Theorem 14 Let T D T .X; Y / be a nontrivial tree and let u be a vertex of maximum degree. Then one of the following holds: 1. m .T / D 2 when m > maxfjX j; jY jg. 2. m .T / D maxf3; djT j=me; d.jT j  ˛u .T //=me C 1g when m < maxfjX j; jY jg. For a nontrivial tree T D T .X; Y /, suppose that jX j D q1 m C r1 and jY j D q2 m C r2 , where 0 6 r1 ; r2 < m. One can color the part X with q1 C 1 colors and the part Y with q2 C 1 colors. Hence, m .T / 6 q1 C q2 C 2 6 djT j=me C 1 colors. On the other hand, it is obvious that djT j=me 6 m .T /. A tree T is said to be class A if m .T / D djT j=me and class B if m .T / D djT j=me C 1. Jarvis and Zhou [73] gave an explicit characterization when a tree belongs to class B. Their proof can be used to determine m .T / in O.jT j3 / time and produce an optimal coloring even if m is part of the input. To make a comparison, it is known [14] that the problem of determining whether a bipartite graph can be m-bounded colored with three colors is NP-complete when m is part of the input. Bentz and Picouleau [10] studied a variation of m-bounded coloring of trees.

4

The Equitable -Coloring Conjecture

Unlike the ordinary colorability of a graph, the equitable colorability does not satisfy monotonicity, namely, a graph can be equitably k-colorable without being equitably .k C 1/-colorable. Therefore, the ECC does not fully reveal the true nature of the equitable colorability. It seems that the maximum degree plays a crucial role here. For instance, by Theorems 5 and 6 the following conjecture proposed by Chen et al. [28] holds for bipartite graphs. This conjecture is called the equitable -coloring conjecture (ECC). Conjecture 4 Let G be a connected graph. If G is not a complete graph Kn , or an odd cycle C2nC1 , or a complete bipartite graph K2nC1;2nC1 , then G is equitably .G/-colorable. The conclusion of the ECC can be equivalently stated as D .G/ 6 .G/. It is also immediate to see that the ECC implies the ECC. In Chen, Lih, and Wu [28], ECC was settled for graphs whose maximum degree is at least one-half of the order. The following two lemmas supplied the basic tools for the solution. Let G c denote the complement graph of G. Let ı.G/ and ˛ 0 .G/ denote the minimum degree and the edge-independence number of G, respectively.

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c Lemma 3 Let G be a disconnected graph. If G is different from Knc and K2nC1;2nC1 0 for all n > 1, then ˛ .G/ > ı.G/.

Lemma 4 Let G be a connected graph such that jGj > 2ı.G/ C 1. Suppose the vertex set of G cannot be partitioned into a set H of size ı.G/ and an independent set I of size jGj  ı.G/ such that each vertex of I is adjacent to all vertices of H . Then ˛ 0 .G/ > ı.G/. Theorem 15 Let G be a connected graph with .G/ > jGj=2. If G is different from Kn and K2nC1;2nC1 for all n > 1, then G is equitably .G/-colorable. As pointed out by Yap, a close examination of the proof of Theorem 15 in [28] reveals that a stronger result was obtained, namely, D .G/ 6 jGj˛ 0 .G c / 6 .G/. In a similar vein, Yap and Zhang [155] made an analysis of the complement graph and succeeded in verifying the ECC for connected graphs G such that jGj=3 C 1 6 .G/ < jGj=2. By combining Theorem 15, their proof can be modified to establish the following stronger result: Theorem 16 The ECC holds for all connected graphs G such that .G/ > .jGj C 1/=3. Along a different direction, one may try to tackle the ECC for special classes of graphs. By attaching appropriately chosen auxiliary graphs to a nonregular graph, attention may be restricted to regular graphs due to the following lemma [28]: Lemma 5 The ECC holds if it does so for all regular graphs. If the chromatic number of a connected cubic graph G is 2, then the ECC has already been established. It only needs to handle cubic graphs having chromatic number 3 to obtain the following [28]: Theorem 17 The ECC holds for all connected graphs G such that .G/ 6 3. In [85], Kierstead and Kostochka extended the above to .G/ 6 4.

5

Split Graphs

There are interesting results dealing with special families of graphs that provide positive evidence for the ECC. A connected graph G is called a split graph if its vertex set can be partitioned into two nonempty subsets U D fu1 ; u2 ; : : : ; un g and V D fv1 ; v2 ; : : : ; vr g such that U induces a complete graph and V induces an independent set. Denote the split graph G as GŒU I V , and always assume that no vertex in V is adjacent to all vertices in U . A family of bipartite graphs BG.k/, k > 1, can be assigned to the given split graph GŒU I V  in the following way.

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The vertex set of BG.k/ is fuij j 1 6 i 6 n and 1 6 j 6 kg [ V , and fuij ; vt g is defined to be an edge of BG.k/ if and only if ui and vt are nonadjacent in G. Note that BG.k/ is a subgraph of BG.k C 1/. The coloring of a split graph GŒU I V  is closely related to independent edges of the graphs BG.k/. For instance, any given set of independent edges in BG.k/ induces a partial coloring of G in the following standard way. The i-th color is used to color ui and all those vertices in V that are matched by the edges to some uij , 1 6 j 6 k. Chen, Ko, and Lih [29] proved the following: Theorem 18 Let GŒU I V  be a split graph such that jU j D n and jV j D r. Let m D maxfk j ˛ 0 .BG.k// D k ng if the set in question is nonempty; otherwise let m be zero. Then D .GŒU I V / D n C d.r  ˛ 0 .BG.m C 1///=.m C 2/e. Once D .GŒU I V / is known, it is straightforward to verify that split graphs satisfy the ECC. In [29], the m-bounded chromatic number of a split graph was obtained in addition to its equitable chromatic number. Theorem 19 Let G D GŒU I V  be a split graph such that jU j D n and jV j D r. Let m > 1 be a given integer. Then m .GŒU I V / D n C d.r  ˛ 0 .BG.m  1///=me.

6

Outerplanar Graphs

A graph is planar if it can be drawn on the Euclidean plane such that edges only meet each other at points representing the vertices of the graph. An outerplanar graph is a planar graph that has a drawing on the plane such that every vertex lies on the unbounded face. An edge subdivision is the operation of replacing an edge uv by a path uwv of length 2 in which w is a newly added vertex. A subdivision of a graph G is a graph obtained from G by a sequence of edge subdivisions. It is well known [18] that a graph is an outerplanar graph if and only if it has no subgraph that is a subdivision of K4 or K2;3 . Yap and Zhang [156] settled the ECC for outerplanar graphs. Theorem 20 If G is an outerplanar graph with .G/ > 3, then G is equitably .G/-colorable. Kostochka [91] proved the following result which answers a question posed at the end of [156]: Theorem 21 If G is an outerplanar graph with .G/ > 3, then D .G/ 6 .G/=2 C 1. Note that the bound cannot be weakened even for trees because the star K1;2k1 has no equitable k-coloring. An efficient algorithm for equitable k-coloring of

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outerplanar graphs G with maximum degree at least 3 can be extracted from Kostochka’s proof whenever k > .G/=2 C 1. A fundamental phenomenon in equitable colorings can be noticed when one examines the equitable chromatic numbers of trees. Apart from K1;n , “most” trees admit equitable colorings with few colors. This phenomenon happens for outerplanar graphs, too. Pemmaraju [125] showed the following: Theorem 22 An outerplanar graph G is equitably 6-colorable if .G/ 6 jGj=6. The main stepping stone to Pemmaraju’s result involves a special partition of a graph. A partition V .G/ D V1 [ V2 is called an equitable 2-forest partition if jjV1 jjV2 jj 6 1 and each induced subgraph GŒVi  is a forest. The following theorem and conjecture in [125] may have independent interest: Theorem 23 Any outerplanar graph has an equitable 2-forest partition. Conjecture 5 The vertex set of any planar graph can be partitioned into two parts V1 and V2 such that jjV1 jjV2 jj 6 1 and each part induces an outerplanar subgraph. Note that Chartrand et al. [23] have proved that the vertex set of a planar graph can be partitioned into two parts such that each part induces an outerplanar subgraph. They also conjectured that the edge set of a planar graph can be partitioned into two sets such that the subgraph induced by each of the sets is outerplanar. Recently, Gonc¸alves [60] has proved the validity of this conjecture. A graph is called series-parallel if it contains no subgraph that is a subdivision of K4 . This class of graphs can be characterized in a number of equivalent ways [18]. Clearly, outerplanar graphs are series-parallel graphs. Zhang and Wu [159] established the ECC for series-parallel graphs. Theorem 24 If G is a series-parallel graph with .G/ > 3, then G is equitably .G/-colorable. Zhang and Wu also conjectured the following which generalizes Theorem 21: Conjecture 6 If G is a series-parallel graph with .G/ > 3, then D .G/ 6 .G/=2 C 1.

7

Planar Graphs

To determine whether a planar graph with maximum degree 4 is 3-colorable is NP-complete [58]. For a given planar graph G with maximum degree 4, let G 0 be obtained from G by adding 2jGj isolated vertices. Then G is 3-colorable if and only if G 0 is equitably 3-colorable. Therefore, it is NP-complete to determine if a given

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planar graph with maximum vertex degree 4 has an equitable coloring using at most 3 colors. Zhang and Yap [160] proved the following: Theorem 25 A planar graph G is equitably .G/-colorable if .G/ > 13. Nakprasit [117] extended the above result to planar graphs with maximum degree at least 9. Thus, the ECC holds for planar graphs with maximum degree at least 9. One can go further if extra conditions are imposed on the graph. Theorem 26 ([118, 137]) Let G be a C4 -free planar graph. If .G/ > 7, then the ECC holds for G. Zhu and Bu [162] established the following results. Consequently, the ECC holds for planar graphs G that are (i) C3 -free and .G/ > 8 or (ii) C4 -free, C5 -free, and .G/ > 7. Theorem 27 Let G be a C3 -free planar graph. Then D .G/ 6 maxf.G/; 8g. Theorem 28 Let G be a C4 -free and C5 -free planar graph. Then D .G/ 6 maxf.G/; 7g. The girth of a graph G, denoted by g.G/, is defined to be the length of a shortest cycle in G. The girth of a forest is 1 by convention. One may impose conditions on the girth of a planar graph to get tight bound for the equitable chromatic threshold. Wu and Wang [153] first established the following: Theorem 29 Let G be a planar graph with ı.G/ > 2. 1. If g.G/ > 14, then D .G/ 6 4. 2. If g.G/ > 26, then D .G/ 6 3. Luo et al. [108] improved these results further. Theorem 30 Let G be a planar graph with ı.G/ > 2. 1. If g.G/ > 10, then D .G/ 6 4. 2. If g.G/ > 14, then D .G/ 6 3. It remains an open problem to find the best possible girth conditions for 3- or 4-equitable colorability when the planar graph G satisfies ı.G/ > 2. A well-known theorem of Gr¨otzsch [61] states that the chromatic number of any planar graph of girth at least 4 is at most 3. Hence, the above theorem has an immediate consequence. Corollary 1 Let G be a non-bipartite planar graph with ı.G/ > 2. Then .G/ D D .G/ if g.G/ > 14.

Equitable Coloring of Graphs

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1211

Graphs with Low Degeneracy

A graph with small degeneracy can be regarded as “sparse.” A graph G is said to be d -degenerate if every subgraph H of G has a vertex of degree at most d in H . It is well known that graphs without edges are 0-degenerate, forests are exactly the 1-degenerate graphs, outerplanar graphs are 2-degenerate, and planar graphs are 5-degenerate. It follows from the definition that the vertices of every d -degenerate graph can be ordered as v1 ; v2 ; : : : ; vn so that for every i < n, vertex vi has at most d neighbors vj with j > i . The following results were obtained by Zhu and Bu [163]: Theorem 31 Let G be a 2-degenerate graph. Then G is equitably 3-colorable if kGk 6 23 jGj. Theorem 32 Let G be a 2-degenerate graph. Then G is equitably 4-colorable if kGk 6 34 jGj. Kostochka and Nakprasit [92] tried to find bounds on the equitable chromatic thresholds for d -degenerate graphs with a given maximum degree. However, the bound in Theorem 21 on outerplanar graphs does not extend to all 2-degenerate graphs. To see this, consider the graph G.d; / D Kd _ Id C1 , where the join operation _ connects each vertex in one graph to all vertices of the other graph. This graph is d -degenerate and of maximum degree . In every proper coloring of G.d; /, each vertex in Kd forms a single color class. Hence, every equitable coloring of G.d; / uses at least d C d.  d C 1/=2e D d. C d C 1/=2e colors. In particular, G.2; / uses at least d. C 3/=2e colors for an equitable coloring, which is greater than d=2e C 1 for even . Kostochka and Nakprasit showed that d. C d C 1/=2e colors is enough to equitably color a d -degenerate graph G with maximum degree  provided =d is large. Theorem 33 Let 2 6 d 6 =27 and G be a d -degenerate graph with maximum degree at most . Then G is equitably k-colorable if k > . C d C 1/=2. The example G.d; / shows that the bound on k cannot be decreased. The next corollary follows from this theorem since every planar graph is 5-degenerate. Corollary 2 Let  > 135 and the maximum degree of the planar graph G be at most . Then G is equitably k-colorable if k > =2 C 3. Let k > 14d C 1 and the d -degenerate graph G have maximum degree at most k. Then, for  D 2k  1  d , G satisfies the condition of Theorem 33. Corollary 3 Let d > 2. Then every d -degenerate graph with maximum degree at most k is equitably k-colorable if k > 14d C 1.

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In view of Theorem 33, which is also true if d D  by the Hajnal and Szemer´edi Theorem, the following was proposed in [92]: Conjecture 7 Let 2 6 d 6  and G be a d -degenerate graph with maximum degree at most . Then G is equitably k-colorable if k > . C d C 1/=2. A graph with “low degeneracy” is intuitively rather similar to a graph whose every subgraph has a “small average degree.” Kostochka and Nakprasit [93] proved the ECC for graphs that have “small average degree” without restrictions on their subgraphs. The average degree of a graph G is defined to be Ad.G/ D 2kGk=jGj. Theorem 34 Let  > 46 and G be a graph of order at least 46 and maximum degree at most . If Ad.G/ 6 =5 and KC1 is not a subgraph of G, then G is equitably -colorable. An immediate consequence of this result is that the ECC holds for d -degenerate graphs with maximum degree  if d 6 =10. In Kostochka et al. [95], a result similar to Theorem 22 was established for d -degenerate graphs. Theorem 35 Every d -degenerate graph G with maximum degree at most  is equitably k-colorable when k > 16d and  6 jGj=15. If the restriction on  is removed, they proved the following: Theorem 36 Every d -degenerate graph G with maximum degree at most  is equitably k-colorable for any k, k > maxf62d; 31d jGj=.jGj   C 1/g. Corollary 4 Every d -degenerate graph G with maximum degree at most jGj=2 C 1 is equitably k-colorable for any k > 62d . The proof of Theorem 36 is constructive, and by extending their proof method, the following result of algorithmic nature was obtained: Theorem 37 There exists a polynomial time algorithm that produces an equitable k-coloring of G for every equitably m-colorable d -degenerate graph G if k > 31d m. A concept generalizing Pemmaraju’s “equitable 2-forest partition” was also introduced in [95]. An equitable k-partition of a graph G is a collection of subgraphs fGŒV1 ; GŒV2 ; : : : ; GŒVk g of G induced by the vertex partition fV1 ; V2 ; : : : ; Vk g of V .G/ such that jjVi j  jVj jj 6 1 for every pair Vi and Vj . The following provides a tool for obtaining equitable colorings with few colors:

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Theorem 38 Let k > 3 and d > 2. Then every d -degenerate graph has an equitable k-partition into .d  1/-degenerate graphs. This is an extension of Theorem 1 proved by Bollob´as and Guy, which can be restated as follows: Any 1-degenerate graph G with .G/ 6 jGj=3 can be equitably 3-partitioned into 0-degenerate graphs. Pemmaraju et al. [126] gave a direct generalization. Theorem 39 For d > 1, every d -degenerate graph G with .G/ 6 jGj=3 can be equitably 3-partitioned into .d  1/-degenerate graphs. Repeated applications of this theorem can get the following: Theorem 40 For d > 1, every d -degenerate graph G with .G/ 6 jGj=3d can be equitably 3d -colored. In the same paper, the following conjecture was proposed: Conjecture 8 There are functions f .d / D O.d / and g.d / D O.d / such that if G is a d -degenerate graph with .G/ 6 jGj=f .d /, then G can be equitably g.d /colored.

9

Graphs of Bounded Treewidth

A tree decomposition of a graph G is a pair .T; F /, with T a tree and F D fXi  V .G/ S j i 2 V .T /g, that satisfies the following conditions: 1. i 2V .T / Xi D V .G/. 2. For every edge uv of G, there exists an i 2 V .T / such that Xi contains both u and v. 3. For all i1 ; i2 ; i3 2 V .T /, Xi1 \ Xi3  Xi2 if i2 is on the path from i1 to i3 in T . The width of the tree decomposition .T; F / is defined to be maxi 2V .T / jXi j  1; the treewidth of a graph G denoted by tw.G/ is the minimum width among all tree decompositions of G. It is a folklore result that every graph G of treewidth at most k has a vertex of degree at most k and has at most kjGj edges. Hence, every graph of treewidth at most k is k-degenerate. The class of graphs with treewidth at most k can be characterized in terms of partial k-trees. The class of k-trees is defined recursively as follows: 1. The complete graph Kk is a k-tree. 2. A k-tree G with n C 1 vertices (n > k) is constructed from a k-tree H with n vertices by adding a new vertex adjacent to and only to all vertices of a subgraph of H that is a Kk . There are a number of alternative characterizations of k-trees [129]. A graph is called a partial k-tree if it is a subgraph of a k-tree. Forests are partial 1-trees and series-parallel graphs are partial 2-trees.

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Theorem 41 ([139]) A graph G is a partial k-tree if and only if G has treewidth at most k. A corollary of Theorem 36 is the following: Corollary 5 Every graph G with treewidth w and maximum degree at most  is equitably k-colorable for any k, k > maxf62w; 31wjGj=.jGj   C 1/g. The equitable k-coloring problem can be stated as follows. A graph G and an integer k are given, and one asks whether G has an equitable k-coloring. For graphs of bounded treewidth, Bodlaender and Fomin [13] used the above result to establish the threshold for telling when the EQUITABLE k-COLORING problem is trivially solved and when it becomes to be solvable in polynomial time by their dynamic programming approach. It amounts to the following. Theorem 42 The equitable k-coloring problem can be solved in polynomial time on graphs of bounded treewidth. They also showed that such an approach cannot be extended to the equitable k-coloring with precoloring problem: A graph G, an integer k, and a precoloring  of G are given, and one asks whether there exists an equitable k-coloring of G extending . For a graph G, a precoloring  of a subset U of vertices of G in k colors is a mapping  W U ! f1; 2; : : : ; kg. A coloring of G with color classes V1 ; V2 ; : : : ; Vk is said to extend the precoloring  if u 2 V.u/ for every u 2 U . The following was proved in [13]: Theorem 43 The equitable k-coloring with precoloring problem is NP-complete on trees. In the framework of parameterized complexity, e.g., [41, 51], and [119], a parameterized problem with the input size n and a parameter k is called fixed parameter tractable (FPT) if it can be solved in time f .k/  nc , where f is a function only depending on k and c is a constant. The basic complexity class for fixed parameter intractability is WŒ1. The equitable coloring problem was shown by Fellow et al. [47] to be WŒ1-hard, parameterized by the treewidth plus the number of colors. However, the equitable coloring problem is FPT when parameterized by the vertex cover number as shown by Fiala et al. [48]. The vertex cover number of a graph G is the minimum size of a set X  V .G/ such that V .G/ n X is an independent set.

10

Kneser Graphs

For integers i 6 j , let Œi; j  D fi; i C 1; : : : ; j g and  Œn D Œ1; n. If X is a set, then the collection of all k-subsets of X is denoted by Xk . The Kneser graph KG.n; k/   as its vertex set. Two distinct vertices are adjacent in KG.n; k/ if they has Œn k

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have empty intersection as subsets. To exclude trivialities, it is always assumed that n > 2k in KG.n; k/. The order of KG.n; k/ is clearly nk . The odd graph Ok is the Kneser graph KG.2k C 1; k/. Since it is easy to see that KG.n; 1/ D Kn and .KG.n; 1// D D .KG.n; 1// D D .KG.n; 1// D n, it is assumed that k > 2 throughout this section. The following is a much celebrated result of Lov´asz [107] proved by topological method: Theorem 44 The chromatic number of KG.n; k/ is equal to n  2k C 2.   Œn For i 2 Œn, an i -flower F of Œn k is a subcollection of k such that all k-subsets in F have i as a common element. It is clear that every i -flower is an independent set of KG.n;   k/. Any independent set F of KG.n; k/, also called an intersecting family of Œn k , satisfies A \ B ¤ ; for all A and B in F . The independence number ˛.KG.n; k// of KG.n; k/ was obtained by Erd˝os et al. [44]. Theorem 45 Suppose F is an intersecting family of

Œn . Then k

! n1 jF j 6 : k1 Moreover, the equality   holds if and only if F is an i -flower for some i 2 Œn. Hence, ˛.KG.n; k// D n1 k1 . There are independent sets of KG.n; k/ which are not flowers. Denote by ˛2 .KG.n; k//, or simply T ˛2 .n; k/, the maximum size of independent sets H of KG.n; k/ satisfying A2H A D ; and ˛2 .n; k/ was obtained by Hilton and Milner [69]. Theorem 46 Suppose H is an intersecting family of

Œn k

with

T A2H

A D ;. Then

! ! n1 nk1 jHj 6  C 1: k1 k1   Moreover, the equality holds if H is the family fA 2 Œn 3 j jA \ Œ1; 3j > 2g when Œn k=3 or H is the family fA 2 k j 1 2 A; jA \ Œ2; k C 1j > 1g [ fŒ2; k C 1g. n1 nk1  k1 C 1. Hence, ˛2 .n; k/ D k1   Since every flower of Œn k is an independent set of KG.n; k/, it is natural to partition flowers to form an equitable coloring of KG.n; k/. If this is successful, then every k-subset of Œn is in some flower. Hence, if f is an equitable m-coloring of KG.n; k/ such that every color class of f is contained in some flower, then m > n  k C 1. Otherwise, suppose m 6 n  k and each color class f 1 .i / is contained in some ti -flower for 1 6 i 6 m. Since jŒn n ft1 ; t2 ; : : : ; tm gj > n  m > k, one may

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choose a k-subset A  Œn n ft1 ; t2 ; : : : ; tm g. Since f is an equitable m-coloring, A 2 f 1 .i / for some i , i.e., ti 2 A. Thus, a contradiction is obtained. In [24], Chen and Huang tried to show that KG.n;  k/ is equitably m-colorable for all m  n  k C 1 by partitioning flowers of Œn into m-independent sets whose k sizes are as even as possible. They succeeded in establishing the following: Theorem 47 Suppose that m > nk C1. Then KG.n; k/ is equitably m-colorable, i.e., D .KG.n; k// 6 D .KG.n; k// 6 n  k C 1.   ˘ Lemma 6 Suppose that m 6 n  k and nk =m > ˛2 .n; k/. Then KG.n; k/ is not equitably r-colorable for all r 6 m, i.e., D .KG.n; k// > D .KG.n; k// > m C 1. Theorem 48 If n  k C 1.

n ˘ =.n  k/ > ˛2 .n; k/, then D .KG.n; k// D D .KG.n; k// D k

  Observe that kn =.n  k/ D O.nk1 / and ˛2 .n; k/ D O.nk2 /. Hence, the following is true. Corollary 6 Let k be fixed. Then D .KG.n; k// D D .KG.n; k// D n  k C 1 when n is sufficiently large. Finally, D .KG.n; 2//, D .KG.n; 3//, and .Ok / were completely determined in [24]. Theorem 49 Assume n > 5. Then D .KG.n; 2// D D .KG.n; 2// D



n  2 if n D 5 or 6, n  1 if n > 7.

Theorem 50 Assume n > 7. Then 8 16.

Theorem 51 For k > 1, the odd graph Ok satisfies .Ok / D D .Ok / D D .Ok / D 3. Chen and Huang concluded their paper [24] by proposing the following: Conjecture 9 If n > 2k > 4, then D .KG.n; k// D D .KG.n; k//. All results about Kneser graphs in this section were also independently obtained by Fidytek et al. [49].

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11

1217

Interval Graphs and Others

A graph G is called an interval graph if there exists a family I D fIv j v 2 V .G/g of intervals on the real line such that u and v are adjacent vertices if and only if Iu \ Iv ¤ ;. Such a family I is commonly referred to as an interval representation of G. Instead of intervals of real numbers, these intervals may be replaced by finite intervals on a linearly ordered set. A clique of a graph G is a complete subgraph Q of G. A clique is called maximal if it is maximal in the set-inclusion order. For an interval graph G, Gillmore and Hoffman [59] showed that its maximal cliques can be linearly ordered as Q0 < Q1 <    < Qm so that for every vertex v of G, the maximal cliques containing v occur consecutively. The finite interval Iv D ŒQi ; Qj  in this linear order is assigned to the vertex v if all the maximal cliques containing v are precisely Qi ; Qi C1 ; : : : ; Qj . Again u and v are adjacent if and only if Iu \ Iv ¤ ;. This representation of G is called a clique path representation of G. Conversely, the existence of a clique path representation implies that the graph is an interval graph. Once a clique path representation is given, let left.v/ and right.v/ stand for the left and right endpoint, respectively, of the interval Iv . Then the following linear order on the vertices of G can be defined. Let u < v if (left.u/ < left.v/) or (left.u/ = left.v/ and right.u/ < right.v/). If u and v have the same left and right endpoints, choose u < v arbitrarily. For any three vertices u, v, and w of G, this linear order satisfies the following condition. If u < v < w and uw 2 E.G/, then uv 2 E.G/. The existence of a linear order satisfying this condition characterizes interval graphs [120]. Chen et al. [31] utilized this linear order to obtain the following: Theorem 52 Let G be a connected interval graph. If G is not a complete graph, then G is equitably .G/-colorable. And they proceeded further to show the following: Theorem 53 Let G be an interval graph. Then D .G/ D D .G/. A few other classes of special graphs have been investigated for their equitable colorability. For instance, central graphs and total graphs were studied in [2, 140]. Thorny graphs were studied in [56]. Additional examples can be found in [57, 76–78]. For a given graph G, the so-called central graph C.G/ of G is obtained from G by inserting a new vertex to each edge of G and then joining each pair of vertices of G which were nonadjacent in G. The total graph T .G/ of G has vertex set V .G/ [ E.G/ and edges joining all elements of this vertex set which are adjacent or incident in G. The notation Pn represents a path on n vertices. Results obtained in [2, 140] are listed as follows: 1. D .C.K1;n // D n. 2. D .C.Kn;n // > n if n > 3. 3. D .C.Kn // D 3.

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8 ˆ 1 if n D 1; ˆ ˆ ˆ ˆ 2 if n D 2; ˆ ˆ < 3 if n D 3; 4. D .C.Pn // D ˆ 3 if n D 4; ˆ ˆ ˆ ˆ n=2 if n > 5 is even; ˆ ˆ : .n C 1/=2 if n > 5 is odd: 8 2 if n D 3; ˆ ˆ < 3 if n D 4; 5. D .C.Cn // D ˆ n=2 if n > 5 is even; ˆ : .n C 1/=2 if n > 5 is odd: n C 1 if m < n; 6. D .T .Km;n // D n C 2 if m D n; 7. D .T .Pn // D 3. 8. D .T .Cn // D 3 if n is a multiple of 3. An edge in a graph is called an pendant edge if it is incident with a leaf, i.e., a vertex of degree 1. Trees are the smallest set of graphs that contains single vertex and is closed under the operation of attaching a pendant edge to a vertex. By analogy to this recursive definition of trees, graphs called edge-cacti, cacti, and thorny graphs can be defined as follows. Edge-cacti constitute the smallest set of graphs that includes all cycles and is closed under the operation of attaching a cycle to a single edge, i.e., identifying this edge with some edge of the attached cycle. Cacti constitute the smallest set of graphs that contains single vertex and is closed under the operation of attaching a pendant edge or cycle to a vertex. Thorny graphs constitute the smallest set of graphs that includes single vertex and is closed under the following operations: 1. Attaching a pendant edge to a vertex 2. Attaching a cycle to a vertex 3. Attaching a cycle to an edge Every thorny graph is connected, planar, and tripartite. All cacti, edge-cacti, and connected outerplanar graphs are thorny graphs. The following results were established in [56]: Theorem 54 Any thorny graph without leaves and C3 or C5 is equitably 3-colorable. Theorem 55 Any thorny graph without leaves and C3 is equitably k-colorable for all k > 4. Corollary 7 The following statements are true: 1. Any edge-cactus without C3 is equitably k-colorable for all k > 3. Furthermore, if an edge-cactus G is bipartite, then D .G/ D 2. 2. Any cactus without leaves and C3 or C5 is equitably k-colorable for all k > 3. 3. Any bipartite cactus without leaves is equitably k-colorable for all k > 3. 4. Any cactus without leaves and C3 is equitably k-colorable for all k > 4.

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1219

Random Graphs

Let G.n; p/ denote the probability space of all labeled graphs of order n such that every edge appears randomly and independently with probability p D p.n/. The space G.n; p/ is said to possess a property P almost surely if the probability that G.n; p/ satisfies P tends to 1 as n tends to infinity. In [98], Krivelevich and Patk´os conjectured the following: Conjecture 10 There exists a constant C such that if C =n < p < 0:99, then almost surely D .G.n; p// D .1 C o.1//.G.n; p// holds. Partial results proved by them included the following: 1. If n1=5C < p < 0:99 for some positive , then almost surely D .G.n; p// 6 .1 C o.1//.G.n; p// holds. 2. There exists a constant C such that if C =n < p < 0:99, then almost surely D .G.n; p// 6 .2 C o.1//.G.n; p// holds. 3. If n.1/ < p < 0:99 for some positive , then almost surely D .G.n; p// 6 .2 C o.1//.G.n; p// holds. 4. If .log1C n/=n < p < 0:99 for some positive , then almost surely D .G.n; p// D O ..G.n; p/// holds.

13

Graph Products

Given two graphs G1 and G2 , it is natural to use the Cartesian product V .G1 /  V .G2 / of the two vertex sets to be the vertex set of a new graph. There are several ways to define the edge set of such a product graph. Results on three different products will be surveyed. Their edge sets are defined as follows: 1. The square product G1 G2 , also known as the Cartesian product: E.G1 G2 / D f.u; x/.v; y/ j .u D v and xy 2 E.G2 // or .x D y and uv 2 E.G1 //g. 2. The cross product G1  G2 , also known as the Kronecker, direct, tensor, weak tensor, or categorical product: E.G1  G2 / D f.u; x/.v; y/ j uv 2 E.G1 / and xy 2 E.G2 /g. 3. The strong product G1  G2 , also known as the strong tensor product: E.G1  G2 / D f.u; x/.v; y/ j .u D v and xy 2 E.G2 // or .uv 2 E.G1 / and x D y/ or .uv 2 E.G1 / and xy 2 E.G2 //g. Note that square and cross products are so named because the products of two single edges are a square and a cross, respectively, and G1  G2 D .G1 G2 / [ .G1  G2 /.

13.1

Square Product

For the ordinary chromatic number, Sabidussi [130] proved a product theorem.

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Theorem 56 For graphs G1 and G2 , .G1 G2 / D maxf.G1 /; .G2 /g. In Chen et al. [31], the following results were obtained: Theorem 57 If G1 and G2 are equitably k-colorable, so is G1 G2 . Corollary 8 For graphs G1 and G2 , D .G1 G2 / 6 maxfD .G1 /; D .G2 /g. Corollary 9 If .G1 / D D .G1 / and .G2 / D D .G2 /, then .G1 G2 / D D .G1 G2 / D D .G1 G2 / D maxf.G1 /; .G2 /g. Corollary 10 Let G D G1 G2     Gn , where each Gi is a path, a cycle, or a complete graph. Then .G/ D D .G/ D D .G/ D maxf.Gi / j 1 6 i 6 ng. Corollary 11 Suppose that G1 and G2 are nontrivial graphs. Then G1 G2 is equitably .G1 G2 /-colorable, i.e., the ECC holds for the square product of two graphs. Even if .G1 / D D .G2 / D k, the product G1 G2 may not be equitably k-colorable. An example is the product K1;5 P3 . If it is assumed that D .G1 / D D .G2 / D k, it may not lead to the conclusion D .G1 G2 / D k. An example is K1;2n K1;2n . If G1 D K3;3 and G2 D K1;1;2 , then D .G1 G2 / 6 maxfD .G1 /; D .G2 /g is false. Exact values for paths, cycles, and complete graphs are also determined in [31]. Theorem 58 The following hold for positive integers m and n: .Pm Pn / D D .Pm Pn / D D .Pm Pn / D 2. 2 if m and n are even, .Cm Cn / D D .Cm Cn / D D .Cm Cn / D 3 otherwise. .Km Kn / D D .Km Kn / D D .Km Kn / D maxfm; ng. Furma´nczyk [54] also obtained a number of exact values of square products between cycles, paths, and hypercubes Qn D K2 K2    K2 . „ ƒ‚ … n

Theorem 59 Let k; m; n, and r be positive integers. Then the following graphs have their equitable chromatic numbers all equal 2: Qr P2n , Qr C2n , and Qr Qr . Besides D .Km1 ;n1 Km2 ;n2 / 6 4, Lin in his Ph.D. dissertation [103] (also in [105]) determined more exact values. They are listed as follows, in which m; n; and r are assumed to be positive integers. 1. D .P2r Km;n / D D .P2r Km;n / D 2 except D .P2 Km;n / D 4 when m C n C 2 < 3 minfm; ng. 2 if jm  nj 6 1, 2. D .P2rC1 Km;n / D D .P2rC1 Km;n / D 3 otherwise.

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3. D .C2r Km;n / D D .C2r Km;n / D 2 except D .C4 Km;n / D 4 when m C n C 2 < 3 minfm; ng. 4. D .C2rC1 Km;n / D D .C2rC1 Km;n / D 3. 4 if .m  2/.n  2/ > 5; 5. D .K1;m K1;n / D D .K1;m K1;n / D 3 otherwise. Lin also proposed some interesting conjectures: Conjecture 11 If G1 and G2 are bipartite graphs, then D .G1 G2 / 6 4. Conjecture 12 If G1 and G2 are connected graphs, then D .G1 G2 / .G1 /.G2 /.

6

The connectedness is essential in the above conjecture because D .K1;3 3K1 / D D .K1;3 3K1 / D 3 > 2 D .K1;3 /.3K1 /.

13.2

Cross Product

The most well-known conjecture for cross product is the one proposed by Hedetniemi [67]. Conjecture 13 For graphs G1 and G2 , .G1  G2 / D minfjG1 j; jG2 jg. This has been established for complete graphs in [42]. For two recent surveys on Hedetniemi’s conjecture, the reader is referred to Sauer [131] and Zhu [161]. Furma´nczyk [54] showed that D .P3 P3 / D 3 > 2 D D .P3 /. Thus, D .G1 G2 / 6 minfD .G1 /; D .G2 /g does not hold in general. However, Chen et al. [31] gave the following upper bound. Theorem 60 For graphs G1 and G2 , D .G1  G2 / 6 minfjG1 j; jG2 jg. The upper bound is sharp in the case D .Km  Kn / D minfm; ng. In general, minfjG1 j; jG2 jg is not an upper bound for D .G1  G2 /. An example was given in [31] to show that K2  Kn is not equitably .n C 1/=2-colorable when n > 1 and n  1 .mod 4/. Even D .G1  G2 / 6 maxfD .G1 /; D .G2 /g fails in general. Examples are D .K2;3  K2;3 / D 3 > 2 D D .K2;3 / [31] and D .P3  P3 / D 3 > 2 D D .P3 / [54]. The following result was conjectured in [31] and later proved to be true in [103]: Theorem 61 For graphs G1 and G2 , D .G1  G2 / 6 maxfjG1 j; jG2 jg. It suffices to prove the above theorem for the case Km  Kn . A slightly better upper bound was actually established in [103].

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Theorem 62 For positive integers m and n,

D .Km  Kn / 6



 mn : minfm; ng C 1

As to exact values, the following were established in [31] and [54], respectively: 2 if mn is even, 1. D .Cm  Cn / D D .Cm  Cn / D 3 otherwise. 2 if m is even or n D 1, 2. D .Pm  K1;n / D 3 otherwise. Lin in his Ph.D. dissertation [103] determined more exact values. They are listed as follows. Also see [104]. 1. .Pm  K2 / D D .Pm  K2 / D D .Pm  K2 / D 2, where m > 2. 2. .P2mC1 Kn / D 2 < D .P2mC1 Kn / D D .P2mC1 Kn / D 3, where n > 3, ˙ ˙

˙ 3n  ˚˙ ˙

except D .P3  Kn / D max 32 2n j s − 2n and 32 2n 6 4 . s s 3. .P2m  Kn / D D .P2m  Kn / D D .P2m  Kn / D 2, where m > 1 and n > 3. ˙ ˙  ˚ ˙

, where n > 3. 4. D .P2  Kn / D max 2 ns j s − n and 2 n2 6 2n 3 5. .Cm  K2 / D D .Cm  K2 / D D .Cm  K2 / D 2, where m > 3. 6. .C2mC1  Kn / D D .C2mC1  Kn / D D .C2mC1  Kn / D 3, where m > 1 and n > 3, except D .C5  Kn / D D .C5  Kn / D 4, when n > 5. ˙ 3n

if n  2 .mod 4/;  4 ˙ ˙

˙

7. D .C3  Kn / D maxf3 ns j s − n and 3 ns 6 3n 4 g otherwise, where n > 3. 8. .C2m  Kn / D D .C2m  Kn / D D .C2m  Kn / D 2, where m > 1 and n > 3. ˙ ˙ 4n  ˚ ˙

j s − 2n and 2 2n 6 5 , where n > 3. 9. D .C4  Kn / D max 2 2n s s nl m l mo 10. D .Km1 ;n1 ; Km2 ;n2 / D D .Km1 ;n1 ; Km2 ;n2 / D min mn11 ; mn22 C 1, where m1 6 n1 and m2 6 n2 .

13.3

Strong Product

The upper bound for the inequality .G1  G2 / 6 .G1 /.G2 / is exact when both factors are complete graphs. As to lower bounds, there are those established by Vesztergombi [141] and Jha [74], respectively. Theorem 63 For nontrivial graphs G1 and G2 , .G1  G2 / > maxf.G1 /; .G2 /g C 2.

Equitable Coloring of Graphs

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Theorem 64 For nontrivial graphs G1 and G2 , .G1  G2 / > .G1 / C !.G2 /, where !.G/ denotes the clique number of the graph G, i.e., the largest order of a complete subgraph in G. Furma´nczyk [54] first investigated the equitable chromatic number of a strong product. Her results have been subsumed by those in [103]. Again, Lin’s results are listed as follows: l m mn , where m > 3 and 1. .Cm  Kn / D D .Cm  Kn / D D .Cm  Kn / D bm=2c n > 2. 2. .Pm  Kn / D D .Pm  Kn / D D .Pm  Kn / D 2n, where m > 2 and n > 2. 4 if m and n are even,  3. .Cm  Cn / D D .Cm  Cn / D D .Cm  Cn / D 5 otherwise, where m; n > 4. 4 if m is even,  4. .Cm  Pn / D D .Cm  Pn / D D .Cm  Pn / D where 5 otherwise, m > 4 and n > 3. 4 if mn is even,  5. .Pm  Pn / D D .Pm  Pn / D D .Pm  Pn / D where 5 otherwise, m > 3 and n > 3. Conjecture 14 Suppose that G1 has at least one edge. Then D .G1  G2 / > D .G1 / C 2!.G2 /  2 and D .G1  G2 / > D .G1 / C 2!.G2 /  2. The conclusions of the above conjecture hold if the equitable chromatic number of threshold is replaced by the ordinary chromatic number [103].

14

List Equitable Coloring

A mapping L is said to be a list assignment for the graph G if it assigns a finite list L.v/ of possible colors, usually regarded as positive integers, to each vertex v of G. A list assignment L for G is k-uniform if jL.v/j D k for all v 2 V .G/. If G has a proper coloring  such that .v/ 2 L.v/ for all vertices v, then G is said to be L-colorable or  is an L-coloring of G. The graph G is said to be k-choosable if it is L-colorable for every k-uniform list assignment L, equitably L-colorable if it has a djGj=ke-bounded L-coloring for a k-uniform list assignment L, and equitably list k-colorable or equitably k-choosable if it is equitably L-colorable for every k-uniform list assignment L. The concept of list-coloring was introduced by Vizing [144] and independently by Erd˝os et al. [45]. However, it is not appropriate to generalize the ordinary equitable coloring to this list context. To see this, let every vertex except one of a graph G be assigned the list Œk and the remaining vertex v be assigned the list Œk C 1; 2k. Unless jGj 6 k C 1, in every proper coloring, some colors are not used, the color on v appears once, and some other color appears at least d.jGj  1/=ke times.

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This explains why the list version of an equitable coloring is defined in terms of boundedness of color classes. This notion was first introduced by Kostochka et al. [94], and they proposed two conjectures that are analogue to the Hajn´alSzemer´edi Theorem and the ECC. Conjecture 15 Every graph G is equitably k-choosable whenever k > .G/. Conjecture 16 If G is a connected graph with maximum degree at least 3, then G is equitably .G/-choosable, unless G is a complete graph or is Kr;r for some odd r. When .G/ D 2, it is easy to see the validity of Conjecture 16. Conjecture 15 has been proved for .G/ 6 3 independently by Pelsmajer [122] and Wang and Lih [146]. In [122], a graph G was shown to be equitably k-choosable when k > 2 C .G/..G/  1/=2. In [94], Kostochka et al. gave the following partial results to Conjecture 15 and 16: Theorem 65 If k > maxf.G/; jGj=2g, then G is equitably k-choosable unless G contains KkC1 or Kk;k (with k odd in the latter case). Theorem 66 If G is a forest and k > .G/=2C1, then G is equitably k-choosable. Moreover, for all m there is a tree with maximum degree at most m that is not equitable dm=2e-choosable. Theorem 67 If G is a connected interval graph and k > .G/, then G is equitably k-choosable unless G D KkC1 . Theorem 68 If G is a 2-degenerate graph and k > maxf.G/; 5g, then G is equitably k-choosable. Pelsmajer [123] provided more partial results. Theorem 69 Let G be a graph with treewidth w and k > 3w  1. Then G is equitably k-choosable if: 1. w 6 5 and k > .G/ C 1, or 2. w > 5 and k > .G/ C w  4. A graph is said to be chordal if it has no induced cycle of length greater than three. Corollary 12 Let G be a chordal graph with maximum degree at most . Then G is equitably k-choosable for k > maxf3  4;  C 1g. If vertices of degree 1 are removed recursively from a graph G, then the final graph has no vertices of degree 1 and is called the core of G. A graph is called a 2;2;p -graph if it consists of two vertices x and y and three internally disjoint paths

Equitable Coloring of Graphs

1225

of lengths 2; 2; and p joining x and y. Erd˝os et al. [45] characterized 2-choosable graphs in terms of these concepts. Analogous to their result, Wang and Lih [146] gave the following characterization: Theorem 70 A connected graph G is equitably 2-choosable if and only if G is a bipartite graph satisfying the following two conditions: 1. The core of G is either a K1 , an even cycle, or a 2;2;2r -graph, where r > 1. 2. G has two parts X and Y such that jjX j  jY jj 6 1. Bu and his collaborators have established a series of partial results for Conjecture 15 and 16 in the class of planar graphs as follows [100, 162, 163]: Theorem 71 Let G be a C4 -free and C6 -free planar graph. Then G is equitably k-choosable when k > maxf.G/; 6g. Theorem 72 Let G be a C3 -free planar graph. Then G is equitably k-choosable when k > maxf.G/; 8g. Theorem 73 Let G be a C4 -free and C5 -free planar graph. Then G is equitably k-choosable when k > maxf.G/; 7g. Theorem 74 Let G be an outerplanar graph. Then G is equitably k-choosable when k > maxf.G/; 4g. Theorem 75 Let G be an outerplanar graph of maximum degree 3. Then G is equitably 3-choosable. Recently, Theorems 74 and 75 have been generalized to series-parallel graphs by Zhang and Wu [159]. The following result confirms Conjecture 15 and 16 for series-parallel graphs: Theorem 76 If G is a series-parallel graph with .G/ > 3, then G is equitably k-choosable if k > .G/. In the framework of parameterized complexity, the list equitable coloring problem was shown by Fellow et al. [47] to be WŒ1-hard even for forests, parameterized by the number of colors on the lists.

15

Graph Packing

The equitable coloring problem can be stated in the language of graph packing; hence it can be studied in a wider context. Two graphs G1 and G2 of the same order are said to pack if G1 is isomorphic to a subgraph of the complement G2c of G2 , or, equivalently, G2 is isomorphic to a subgraph of the complement G1c of G1 .

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This definition may be extended to n graphs G1 ; G2 ; : : : ; Gn of the same order so that they pack if every pair of them pack: Because the problem of whether G c packs with the cycle CjGj is equivalent to the existence of a Hamiltonian cycle in G, two well-known sufficient conditions for the existence of a Hamiltonian cycle, due to Dirac [40] and Ore [121], can be cast in terms of graph packings. Theorem 77 If .G/ 6 jGj=2  1, then G packs with CjGj . Theorem 78 If deg.u/ C deg.v/ 6 jGj  2 for every edge uv in G, then G packs with CjGj . A graph G is k-colorable if and only if G packs with a graph of the same order that is the union of k-cliques. Let H.n; k/ denote the graph of order n such that it has k components each of which is a clique of order bn=kc or dn=ke. This graph is the complement of the well-known Tur´an graph in extremal graph theory. A graph G is equitably k-colorable if and only if G packs with H.jGj; k/. The Brooks’ Theorem and the Hajnal and Szemer´edi Theorem can now be stated as follows: Theorem 79 If r > 3, G is a connected graph with .G/ 6 r and G does not pack with the complement of any r-partite graph, then G D KrC1 . Theorem 80 Let G satisfy .G/ 6 r. Then G packs with H.jGj; r C 1/. In view of Ore’s theorem, the Ore degree of an edge uv, denoted by .uv/, is defined to be deg.u/ C deg.v/, and the Ore degree of a graph G is .G/ D maxf.uv/ j uv 2 E.G/g. Following [81], those upper bounds in terms of Ore degree giving sufficient conditions for packing graphs are called Ore-type bounds. Those in terms of maximum degree are called Dirac type. An obvious Dirac-type bound on the chromatic number of a graph G is .G/ 6 .G/ C 1. The Brooks’ Theorem characterizes the conditions for equality to hold: either G contains K.G/C1 or .G/ D 2 and G contains an odd cycle. An Ore-type bound on .G/ can be obtained easily such that .G/ 6 b.G/=2c C 1. The bound is also attained at complete graphs. However, for small odd .G/, there are more connected extremal graphs. Theorem 81 If 6 6 k D .G/ D b.G/=2c C 1, then G contains Kk . Kierstead and Kostochka [83] proposed the above as a conjecture and proved the statement when the lower bound 6 was replaced by 7. The theorem has been established recently by Rabern [127]. See [96] for further generalizations. The above theorem can be equivalently stated as follows: for k > 6, Kk is the only k-critical graph with maximum degree at most k whose vertices of degree k form an independent set. A k-critical graph is one that has chromatic number k, and any of its proper subgraphs has chromatic number less than k.

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Fig. 1 The graph G with .G/ D 9 and .G/ D 5

x

y

The bound 6 is sharp as Fig. 1 gives a graph G with .G/ D 9 and .G/ D 5. This graph is adapted from [83]. Note that every 4-coloring of the subgraph induced by x and y and the upper three vertices assigns x and y the same color since the upper three vertices form a K3 . On the other hand, every 4-coloring of the subgraph induced by x and y and the lower four vertices assigns x and y different colors since the lower four vertices form a K4 . It follows that .G/ > 4. Kierstead and Kostochka also constructed infinite families of connected graphs H with .H / 6 7 and .H / D 4. Let G be a graph with .G/ 6 7 and .G/ D 4. An example for such a graph G is illustrated in Fig. 2a. A graph G 0 with .G 0 / 6 7 and .G 0 / D 4 can be constructed as follows. Choose a vertex v of G that has no neighbor of degree 4. Split v into two vertices v1 and v2 of degree at most two. Add two new adjacent vertices xv and yv , each of which is joined to both v1 and v2 . In this example, G 0 is depicted in Fig. 2b. By construction, .G 0 / D 7. Since v1 and v2 are adjacent to xv and yv , any 3-coloring of G 0 will assign the same color to v1 and v2 . But then a 3-coloring of G can be produced, contrary to the assumption .G/ D 4. Kierstead and Kostochka [81] proved a generalization of the Hajnal and Szemer´edi Theorem involving the Ore degree. Theorem 82 For every r > 3, each graph G with .G/ 6 2r C 1 has an equitable .r C 1/-coloring. This implies that the ECC holds for graphs in which vertices of maximum degree form an independent set. In addition to KrC1 , the extremal graphs for the above theorem are Km;2rm for every odd 0 < m 6 r. The following Ore-type analogue of the ECC was also proposed in [81] and, its truth for r D 3 was established. Conjecture 17 Let r > 3 and G be a connected graph with .G/ 6 2r. If G differs from KrC1 and Km;2rm for all odd m, then G is equitably r-colorable. A conjecture in the flavor of Conjecture 15 was proposed in [86], and positive evidence was provided for small .

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a

b

v1

v v2 G

G

Fig. 2 A transformation from G to G 0

Conjecture 18 Every graph G is equitably .d.G/=2e C 1/-choosable. Theorem 83 If .G/ 6 6, then G is equitably 4-choosable. In the graph packing area, a major outstanding conjecture was made independently by Bollob´as and Eldridge [15] and Catlin [20, 21]. Conjecture 19 Let G1 and G2 be two graphs of the same order n. If ..G1 / C 1/ ..G2 / C 1/ 6 n C 1, then G1 and G2 pack. The Hajnal-Szemer´edi Theorem verifies the conjecture in the case when G2 is the disjoint union of copies of a clique. Aigner and Brandt [1] and independently (for huge n) Alon and Fischer [4] settled the case .G1 / 6 2. Csaba et al. [36] proved the case for .G1 / D 3 and huge n. Bollobas et al. [17] proved it in case that for d > 2, G1 is d -degenerate, .G1 / > 40d , and .G2 / > 215. Kaul et al. [86] showed that for .G1 /; .G2 / > 300, if ..G1 / C 1/..G2 / C 1/ < 3n=5, then G1 and G2 pack. Recently, Kun has announced a proof of the conjecture for graphs with at least 108 vertices. The reader is referred to Wozniak [151] for a general survey on the graph packing area and to Kierstead et al. [79] for recent progress in Ore-type and Dirac-type bounds for graph packing problems.

16

Equitable -Colorability of Disconnected Graphs

If a disconnected graph G is .G/-colorable, then the conditions for G to be equitably .G/-colorable are quite different. For an odd integer r > 3, G is equitably .G/-colorable if G D Kr;r [ Kr;r . However, G is not equitably

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.G/-colorable if G D Kr;r [ Kr . The latter example can be generalized. A graph H is said to be r-equitable if r divides jH j, H is r-colorable, and every r-coloring of H is equitable. For an odd integer r > 3, if H D Kr;r is a subgraph of G and G  H is r-equitable, then G is not equitably r-colorable. Kierstead and Kostochka [84] gave a good description of the family of all r-equitable graphs so that all of them can be built up from simple examples in a straightforward way. Their approach to deal with disconnected graphs led them to propose the following conjecture. Conjecture 20 Suppose that .G/ D r > 3 and G is an r-colorable graph. Then G is not equitably r-colorable if and only if the following conditions hold: 1. r is odd. 2. G has a subgraph H D Kr;r . 3. G  H is r-equitable. Chen and Yen [27] have also found sufficient conditions for the nonexistence of equitable -colorings for graphs that are not necessarily connected. Theorem 84 Suppose that .G/ D r > 3 and G is an r-colorable graph. Then G is not equitably r-colorable if the following conditions hold: 1. r is odd. 2. G has exactly one component H D Kr;r . 3. ˛.G  H / 6 jG  H j=r. Suppose that .G/ D r and G is an r-colorable graph such that G has exactly one Kr;r component. If G D Kr;r , then ˛.G Kr;r / D 0 D jG Kr;r j=r. Otherwise, ˛.G  Kr;r / > jG  Kr;r j=.G  Kr;r / > jG  Kr;r j=.G/ > jG  Kr;r j=r. Hence, Condition 3 can be replaced by an equality. Chen and Yen conjectured that those sufficient conditions are also necessary and established some positive evidence. Conjecture 21 Suppose that .G/ D r > 3 and G is an r-colorable graph. Then G is not equitably r-colorable if and only if the following conditions hold: 1. r is odd. 2. G has exactly one component H D Kr;r . 3. ˛.G  H / D jG  H j=r. Theorem 85 A bipartite graph G satisfying .G/ > 2 is equitably .G/colorable if and only if G is different from Kr;r for all odd r > 3. Theorem 86 A graph G that is .G/-colorable and satisfies .G/ > 1 C jGj=3 is equitably .G/-colorable if and only if G is different from Kr;r for all odd r > 3. Theorem 87 Conjecture 21 holds for .G/ D 3. It was shown in Chen et al. [32] that Conjecture 20 and 21 are in fact equivalent. The proof utilizes the following result that was established in Chen et al. [30].

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Theorem 88 Let r > .G/ > .G/. Then there exists a proper coloring of G using r colors such that some color class has size ˛.G/. Further lemmas are needed in the equivalence proof. Lemma 7 If G is r-colorable and ˛.G/ D jGj=r, then G is r-equitable. Lemma 8 If G is r-equitable, then r D .G/. Lemma 9 Let r > .G/. If G is r-equitable, then ˛.G/ D jGj=r. By Lemma 8, .G/ D r > .G/. By Theorem 88, there exists an r-coloring  of G such that a color class of  is of size ˛.G/. Since G is r-equitable,  is an equitable r-coloring and r divides jGj. Hence ˛.G/ D jGj=r. Lemma 10 Let a graph G satisfy .G/ D r > 3. If G is r-equitable, then G does not have Kr;r as a subgraph. Suppose on the contrary that G has a subgraph H D Kr;r . The subgraph H is a component of G since r D .G/. Hence, jGj D jG  H j C jH j D jG  H j C 2r and ˛.G/ D ˛.G  H / C ˛.H / D ˛.G  H / C r. By Lemma 9, ˛.G/ D jGj=r D jG  H j=r C 2. Therefore, ˛.G  H / D jG  H j=r C 2  r 6 jG  H j=r  1. On the other hand, Lemma 8 implies r D .G/ > .G  H /. Then ˛.G H / > djG H j=.G H /e > djG H j=re > jG H j=r. A contradiction is obtained. Lemma 11 Let a graph G satisfy .G/ D r > .G/. If r > 3, G has a subgraph H D Kr;r , and G  H is r-equitable, then G has exactly one Kr;r component. If G D Kr;r or .G  H / < r, then G has exactly one Kr;r component H . Now, suppose that G ¤ Kr;r and .G  H / D r. Since r > 3 and G  H is r-equitable, G  H does not have Kr;r as a subgraph by Lemma 10. Therefore, G has exactly one Kr;r component H . Using Lemmas 7, 9, and 11, the equivalence of two conjectures follows. Theorem 89 Conjecture 20 and 21 are equivalent. In [84], Kierstead and Kostochka described their conjecture in terms of other equivalent conditions. An r-equitable graph G is said to be r-reducible if V .G/ has a partition fV1 ; : : : ; Vt g into at least two parts such that all induced subgraphs GŒVi  are r-equitable. If such a partition fails to exist, then G is called r-irreducible. Obviously, Kr is r-irreducible. It can be identified that there is one other 5-irreducible graph F1 (Fig. 3), three other 4-irreducible graphs F2 ; F3 ; F4 (Fig. 4), and six other 3-irreducible graphs F5 ; : : : ; F10 (Figs. 5 and 6). Together with Kr , the r-irreducible graphs in the list F1 ; : : : ; F10 are called r-basic graphs.

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Fig. 3 The 5-irreducible graph F1

c a

b

F2

F3

F4

Fig. 4 The 4-irreducible graphs F2 , F3 , and F4

a

c

b

F5

F6

F7

Fig. 5 The 3-irreducible graphs F5 , F6 , and F7

An r-decomposition of G is a partition fV1 ; : : : ; Vt g of V .G/ such that every induced subgraph GŒVi  is r-basic. The graph G is called r-decomposable if it has an r-decomposition. A nearly equitable r-coloring of a graph G is an r-coloring of G such that exactly one color class has size s  1, exactly one color class has size s C 1, and all other color classes have size s.

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a

b

F8

F9

c

F10

Fig. 6 The 3-irreducible graphs F8 , F9 , and F10

Let G.r/ be the class of all graphs whose maximum degree and chromatic number are less than or equal to r. Let G.r; n/ denote the subclass of G.r/ consisting of all graphs of order at most n. Theorem 90 ([84]) Let G 2 G.r/ and r divide jGj. Then the following are equivalent. 1. G is r-equitable. 2. G is r-decomposable. 3. G has an equitable r-coloring, but does not have a nearly equitable r-coloring. Conjecture 20 can now be re-stated as follows. Conjecture 22 Suppose that .G/ D r > 3 and G is an r-colorable graph. Then G is not equitably r-colorable if and only if the following conditions hold. 1. r is odd. 2. G has a subgraph H D Kr;r . 3. G  H is r-decomposable. For r > 6, this conjecture means that, if an r-colorable graph G with .G/ D r has no equitable r-coloring, then r is odd and there exists a partition V .G/ D V0 [

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V1 [    [ Vt such that GŒV0  D Kr;r and GŒVi  D Kr for 1 6 i 6 t. Theorem 90 can be used to derive the corollaries below. Corollary 13 For all positive integers r and n > r, the ECC holds for all graphs in G.r; n/ if and only if Conjecture 22 holds for all graphs in G.r; n/. Corollary 14 Let G 2 G.r/ be r-equitable. Then G has a unique r-decomposition. Corollary 15 There exists a polynomial time algorithm for deciding whether a graph G 2 G.r/ is r-equitable. The following conjecture proposed in [83] is a common extension of Conjecture 17 and 22. Conjecture 23 Let r > 3. An r-colorable graph G with .G/ 6 2r has no equitable r-coloring if and only if r divides jGj, G has a subgraph H D Km;2rm for some odd m, and G  H is r-decomposable. This conjecture is proved to be equivalent to Conjecture 17 for graphs with restricted order and Ore-degree. Theorem 91 Let r > 3. Assume that Conjecture 17 holds for all graphs of order at most n and Ore-degree at most 2r. Let G be an r-colorable graph of order n with .G/ 6 2r. Then G has no equitable r-coloring if and only if r divides n, G has a subgraph H D Km;2rm for some odd m, and G  H is r-decomposable. It follows that Conjecture 23 holds for r D 3.

17

´ Theorem More on the Hajnal-Szemeredi

The Hajnal-Szemer´edi Theorem settled a conjecture raised by Erd˝os in 1964. The complete proof given in [64] was long and complicate, and did not produce a polynomial time algorithm. A simplification came when Kierstead and Kostochka [82] used a discharging argument in an approach similar to the original one. An analysis of their proof leads to a complexity results. Theorem 92 There is an algorithm of time complexity O.n5 / that constructs an equitable .r C 1/-coloring of any graph G with jGj D n and .G/ 6 r. An even shorter proof of the Hajnal-Szemer´edi Theorem was included in the survey paper [86]. Independent of Kierstead and Kostochka, Mydlarz and Szemer´edi also found polynomial time algorithm proof for the Hajnal-Szemer´edi Theorem. These two groups finally worked together to refine their old methods and arrived at a faster algorithm [87].

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Theorem 93 There is an algorithm of time complexity O.rn2 / that constructs an equitable .r C 1/-coloring of any graph G with jGj D n and .G/ 6 r. However, the existence of an algorithmic version of Theorem 82 is still open. Conjecture 24 There is a polynomial time algorithm that constructs an equitable .r C 1/-coloring of any graph G with .G/ 6 2r C 1. When one pays attention to the complement of the graph given in the HajnalSzemer´edi Theorem, the theorem can be stated in the following form of which the first non-trivial case r D 3 had previously been solved by Corr´adi and Hajnal [34]. Theorem 94 Let G be a graph of order n with the minimum degree ı.G/ > If r divides n, then G contains n=r vertex-disjoint cliques of size r.

r1 n. r

In order to extend the above to a multipartite version, the next conjecture was proposed in Csaba and Mydlarz [35]. If all parts of an r-partite graph G have the same size, then G is called a balanced r-partite graph. Let V1 ; V2 ; : : : ; Vr be the parts of such a graph G. The proportional minimum degree of G is defined as follows. deg.v; Vj / Q jj ¤i : ı.G/ D min min 16i 6r v2Vi jVj j Conjecture 25 Let G be a balanced r-partite graph of order rn. There exists a Q constant M > 0 such that, if ı.G/n > r1 r n C M , then G contains n vertex-disjoint cliques of size r. The extra additive constant M is necessary for the case of odd r. Partial results have been obtained by Fischer [50] and Johansson [75]. The conjecture has been confirmed for r D 3 [109] and r D 4 [111]. In [35], Csaba and Mydlarz proved a relaxed version of this conjecture. The proofs commonly used Szemer´edi’s Regularity Lemma [136] and the Blow-up Lemma [88] and are complicate. The cases for r D 3 and r D 4 have been reproved by Han and Zhao [65] using their absorbing method. The paper [80] by Keevash and Mycroft contains results about multipartite graphs with sufficiently large girth. Recently, Conjecture 25 has been verified asymptotically by Lo and Markstr¨om [106]. Balogh et al. [8] extended the result of Corr´adi and Hajnal into the setting of sparse random graphs. In order to find a understandable proof of the Hajnal-Szemer´edi Theorem, Seymour [132] was motivated to pose the following conjecture. The r-th power of a graph is obtained by adding new edges joining vertices with distance at most r. Conjecture 26 Let G be a graph of order n with ı.G/ > the r-th power of Cn .

r rC1 n.

Then G contains

Theorem 77 is the case r D 1. The well-known Posa’s conjecture (cf. [43]) is the case r D 2. Seymour’s conjecture implies the Hajnal-Szemer´edi Theorem since

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any r C 1 consecutive vertices of the r-th power of a cycle induce KrC1 . Seymour’s conjecture has been proved by Komol´os et al. [89, 90] when the order of the graph is extremely large using Szemer´edi’s Regularity Lemma and the Blow-up Lemma. Theorem 95 For every positive integer r, there exists an integer N such that for r every n > N every graph G of order n with ı.G/ > rC1 n contains the r-th power of a hamiltonian cycle.

18

Applications

Graph coloring can be regarded as a partition of resources problem. It is convenient in some situations to require color classes be of approximately the same size. Graph coloring is also a natural model for scheduling problems. Suppose that a number of jobs are given to be completed. A conflict graph can be constructed so that vertices represent jobs and two vertices are adjacent if there is a scheduling conflict between the jobs associated with these vertices. A proper coloring of the conflict graph corresponds to a conflict-free schedule. Some other examples are listed below. 1. The mutual exclusion scheduling problem [7, 134]. 2. Scheduling in communication systems [70]. 3. Round-the-clock scheduling [138]. 4. Parallel memory systems [11, 37]. 5. Load balancing in task scheduling [11, 97]. 6. Constructing university timetables [55]. In applications, only algorithms with low time complexity could be utilized. Although checking D .G/ 6 2 can be achieved in polynomial time, the problem of deciding if D .G/ 6 3 is NP-complete even for line graphs of cubic graphs. The majority of known polynomial time results about equitable coloring are listed in [53] and [55]. Two competitive algorithms for approximate equitable coloring were also supplied. In [112, 113], and [114], M´endez-D´ıaz et al. gave a linear integer programming formulation for the equitable coloring problem. They studied its polyhedral structure and developed a cutting plane algorithm. Experiments on randomly generated graphs have been performed to make behavior comparisons with other algorithms of similar nature. Bahiense et al. [6] also gave two integer programming formulations based on representatives for the equitable coloring problem. They proposed a primal constructive heuristic, branching strategies, and branch-and-cut algorithms based on these formulations. Computational experiments were carried out on randomly generated graphs. Applications of equitable colorings to other mathematical problems include the following. Alon and F¨uredi [5] used equitable colorings to the determination of the threshold function for the edge probability that guarantees, almost surely, the existence of various sparse spanning subgraphs in random graphs. R¨odl and Ruci´nski [128] used equitable colorings to give a new proof of the Blow-Up Lemma.

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P Let X D fX1 ; X2 ; : : : ; Xn g be a collection of random variables with S D i 2Œn Xi and D EŒS . The upper tail probability Prob[S > .1 C / ] and the lower tail probability Prob[S 6 .1  / ] for 0 <  6 1 are subjects of interest. A dependency graph for X has vertex set Œn and an edge set such that for each i 2 Œn, Xi is mutually independent of all other Xj with j non-adjacent to i . In Pemmaraju [124], it is shown that a small equitable chromatic number for a dependency graph leads to sharp tail probability bounds that are roughly as good as those would have been obtained had the random variables been mutually independent. An example is an outerplanar dependency graph and Theorem 23 plays an important role in the proof. Pemmaraju also introduced an interesting notion of proportional coloring. For non-negative integer c and integer ˛ > 1, a proper coloring of G is said to be a .c; ˛/-coloring if all except at most c vertices are colored and jVi j 6 ˛jVj j for any pair of color classes Vi and Vj . Theorem 1 can be extended to show that every tree has a .1; 5/-coloring with two colors and every outerplanar graph has a .2; 5/coloring with four colors. Sharp tail probability bounds can be established if the dependency graph can be .c; ˛/-colored [124]. Pemmaraju believed that D .G/ should depend on the average degree rather than on the maximum degree and he proposed the conjecture below. Conjecture 27 There is a positive constant c such that, if a graph G has maximum degree at most jGj=c and average degree d , then D .G/ D O..G/ C d /. The truth of this conjecture will immediately imply an O.1/ equitable chromatic number for most planar graphs and that will translate into extremely sharp tail probability bounds for the sum of random variables that have a planar dependency graph. Janson et al. [71] and Janson and Ruci´nski [72] also used equitable colorings to get new bounds on tails of distributions of sums of random variables.

19

Related Notions of Coloring

In this section, some coloring notions related to equitable coloring will be discussed.

19.1

Equitable Edge-Coloring

An edge-coloring of a graph G is simply an assignment of colors to the edges of G. An edge-k-coloring of G assigns the k colors f1; 2; : : : ; kg to the edges of G. For a vertex v of G, let ci .v/ denote the number of edges incident with v colored i . This edge-k-coloring is said to be equitable if, for each vertex v, jci .v/  cj .v/j 6 1 and nearly equitable if, for each vertex v,

.1 6 i < j 6 k/

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jci .v/  cj .v/j 6 2 .1 6 i < j 6 k/: The following was proved in [68]. Theorem 96 If k > 2 does not divide the degree of any vertex, then the graph has an equitable edge-k-coloring. Hilton and de Werra made the observation at the end of their paper that the colorings can be so constructed that all color classes have equitable sizes. An edge-coloring is said to be proper if any two edges are colored differently whenever they are incident to a common vertex. The smallest number of colors needed in a proper edge-coloring of G is called the chromatic index of G and denoted by 0 .G/. Obviously, .G/ 6 0 .G/. The above theorem reduces to the following well-known theorem of Vizing [142] when k D .G/ C 1. Theorem 97 For any graph G, 0 .G/ 6 .G/ C 1. An edge cover coloring of a graph G is a coloring of the edges of G so that, for every vertex v, each color appears at least once on some edge incident to v. The maximum number of colors needed for such a coloring is called the edge cover chromatic index of G and denoted by 0c .G/. Let k D ı.G/ C 1. Applying Theorem 96 to the graph obtained from G by adjoining a pendant edge to each vertex v whose degree is a multiple of k, the following result of Gupta [62] is deduced. Theorem 98 For any graph G, ı.G/  1 6 0c .G/ 6 ı.G/. Given k > 2, the k-core of a graph G is the subgraph of G induced by all vertices v whose degree is a multiple of k. A stronger form for Theorem 96 also appeared in [68]. Theorem 99 Given k > 2, if the k-core of G contains no edges, then G has an equitable edge-k-coloring. The following recent result of Zhang and Liu [158] was first conjectured by Hilton and de Werra [68]. Theorem 100 Given k > 2, if the k-core of G is a forest, then G has an equitable edge-k-coloring. A graph G is called edge-equitable if G has an equitable edge-k-coloring for any integer k > 2. If a graph is edge-equitable, then its chromatic index is equal to its maximum degree and its edge cover chromatic index is equal to its minimum degree. All bipartite graphs were shown to be edge-equitable in de Werra [38].

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A circuit is a connected graph without any vertex of odd degree. A circuit is said to be odd, or even, according to its number of edges is odd, or even. It was stated in [38] that a connected graph has an equitable edge-2-coloring if and only if it is not an odd circuit. Wu [152] showed that a connected outerplanar graph is edge-equitable if and only if it is not an odd circuit. This can be generalized to series-parallel graphs. The following result was established by Song et al. [135]. Theorem 101 Any connected series-parallel graph is edge-equitable if and only if it is not an odd circuit. Theorem 96 also has the following corollary. Corollary 16 Let k > 2. Then, for any graph G, there exists an edge-k-coloring of G such that, for each vertex v, 1. jci .v/  cj .v/j D 2 for at most one pair fi; j g of colors, and 2. jcs .v/  ct .v/j 6 1 for all pairs fs; tg ¤ fi; j g of colors. This can be proved by adjoining a pendant edge to each vertex of G whose degree is a multiple of k. Then apply Theorem 96 to this extended graph. The corresponding edge-coloring of G satisfies the above conditions. Thus, every graph has a nearly equitable edge-k-coloring for any given k > 2. In fact, Corollary 16 is equivalent to Theorem 96. To see this, suppose that k does not divide the degree of any vertex of G and the conditions of Corollary 16 are satisfied. For each vertex v of G, since deg.v/ is not a multiple of k, it is not possible to have jci .v/cj .v/j D 2 for one pair fi; j g of colors unless some second pair fs; tg ¤ fi; j g of colors satisfies jcs .v/  ct .v/j D 2 also. Thereby, an equitable edge-coloring with k colors can be produced. For efficient algorithms for nearly equitable edge-coloring problem, the reader is referred to Shioura and Yagiura [133], Xie et al. [154], and references therein. These algorithms can produce color classes that have equitable sizes.

19.2

Equitable Total-Coloring

A total-k-coloring of a graph G D .V; E/ is a mapping f W V .G/ [ E.G/ ! f1; 2; : : : ; kg such that any two adjacent or incident elements have distinct images. The total chromatic number 00 .G/ is the smallest integer k such that G has a totalk-coloring. A total-k-coloring is said to be equitable if jjf 1 .i /j  jf 1 .j /jj 6 1 when 1 6 i < j 6 k. Fu [52] first studied equitable total-coloring under the name equalized total coloring and put forward the following tentative conjecture. Conjecture 28 For each k > 00 .G/, there exists an equitable total-k-coloring of G.

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Fu proved this conjecture for complete graphs, complete bipartite graphs, trees, and complete split graph Km _ In . However, the tentative conjecture does not hold in general. Fu gave a family of counterexamples. Theorem 102 For any n > 3, let G D nK2 _ I2n1 . Then 00 .G/ D 2n C 1, yet G has no equitable total-.2n C 1/-coloring. However, G has an equitable total-kcoloring for any k > 2n C 2 D .G/ C 2. This theorem prompted Fu to make the following refined conjecture. Conjecture 29 For each k > maxf00 .G/; .G/ C 2g, G has an equitable total-kcoloring. Fu also proved that G satisfies the above conjecture if .G/ D jGj  2, or G is a complete multipartite graph of odd order. When G is a multipartite graph of even order, the best Fu could prove was that G has an equitable total-k-coloring for any k > .G/ C 3. Wang [145] proposed the following weaker conjecture in which the equitable total chromatic index 00D .G/ of a graph G is defined to be the least integer k such that G has an equitable total-k-coloring. Wang proved the conjecture holds for graphs G with .G/ 6 3. Conjecture 30 For any graph G, 00D .G/ 6 .G/ C 2. This weaker conjecture is powerful enough to imply the outstanding Total Coloring Conjecture, proposed independently by Behzad [9] and Vizing [143], which asserts 00 .G/ 6 .G/ C 2 for any graph G. For some classes of graphs, the upper bound in the above conjecture can be lowered to .G/ C 1. Chungling et al. [33] proved the following for the square product of cycles. Theorem 103 Let G D Cm Cn . Then, for m > 3 and n > 3, 00D .G/ D .G/ C 1 D 5. The wheel Wn is defined to be the join of a cycle Cn with a center vertex u. The Mycielskian M.Wn / of Wn is constructed as follows. For each vertex x of Wn , a new vertex x 0 is added. Join xi and xj0 with an edge whenever xi and xj are adjacent in Wn . Finally, add one more new vertex w and make it adjacent to every new vertices x 0 . The hypo-Mycielskian HM.Wn / of Wn has the same vertex set as M.Wn / and all edges of M.Wn / except those of the form ux 0 , x ¤ u. Note that .HM.Wn // is equal to 6 when n D 3 or 4 and is equal to n C 1 when n > 5. Theorem 104 ([147]) For n > 3, 00D .M.Wn // D .G/ C 1. Theorem 105 ([148]) For n > 3, 00D .HM.Wn // D .G/ C 1.

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One question left open in [52] asks whether the existence of an equitable totalk-coloring implies that of an equitable total-.k C 1/-coloring.

19.3

Equitable Defective Coloring

A reasonable relaxation of scheduling problem allows conflicts to happen to a certain level. The graph coloring model corresponds to this type of relaxation can be formulated as follows. A d -defective coloring is a coloring of the vertices of a graph in which monochromatic subgraphs have maximum degree at most d . An ordinary proper coloring is precisely a 0-defective coloring. A defective coloring simply means a 1-defective coloring, in which a vertex may share a color with at most one neighbor. The smallest number k such that the graph G has a 1-defective coloring is denoted by df1 .G/. A graph G has an equitable defective k-coloring, or an ED-k-coloring, if G has a coloring using k colors that is both equitable and defective. This notion of coloring was introduced by Williams et al. [150]. Let ED .G/ denote the smallest integer k such that G has an ED-k-coloring, and ED .G/ the smallest integer k such that G has an ED-m-coloring for all m > k. It is clear that df1 .G/ 6 ED .G/ 6 ED .G/. These parameters may differ from each other by an arbitrarily large amount. The following example was provided in [150]. Consider X D Kdn=2e and Y D Ibn=2c . Let G be the graph obtained from X _ Y by removing a matching between X and Y of size bn=2c. Note that .G/ D D .G/ D dn=2e. If X is colored with jX j=2 D dn=4e colors and Y is colored with one extra color, then df1 .G/ 6 dn=4e C 1. If a color class in an ED-coloring contains two vertices of X , then it cannot contain any other vertices. This forces every color class has at most three vertices. However, color classes of size three must contain at least two vertices of Y . Therefore, there are at most jY j=2 color classes of size three. It follows that ED .G/ > d3n=8e. It is easy to see that an ED-coloring with k colors exists for any k > d3n=8e, and hence ED .G/ D d3n=8e. Extending results in [108], Williams et al. proved the following. Theorem 106 If G is a planar graph with minimum degree ı.G/ > 2 and girth g.G/ > 10, then ED .G/ 6 3. The condition ı.G/ > 2 is indispensable since K1;n has no ED-k-coloring when n is sufficiently large for any fixed k. The girth condition cannot be lower than 5 since K2;n has girth 4 but ED .K2;n / is not bounded by any constant. Whether 5 is the smallest girth that a planar graph G can have so that ED .G/ can be bounded by a constant is an open question. Recently, Fan et al. [46] have shown that a graph with maximum degree at most r admits an equitable ED-r-coloring and provided a polynomial-time algorithm for constructing such a coloring.

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19.4

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Equitable Coloring of Hypergraphs

A hypergraph H, is an ordered pair .V; F /, where F is a family of subsets of the finite set V . Elements of V and F are called vertices and hyperedges of H, respectively. The number of hyperedges incident with a vertex of H is called its degree. A hypergraph is said to be k-uniform if each hyperedge has precisely k-elements. Let H be a k-uniform hypergraph on n vertices. A strong r-coloring of H is a partititon of the vertices into r parts, called color classes, such that each hyperedge intersects each part. A strong r-coloring is called equitable if the size of each color class is either dn=re of bn=rc. Let c.H/ and ec.H/ denote the maximum possible number of color classes in a strong coloring and an equitable coloring of H, respectively. Clearly, 1 6 ec.H/ 6 c.H/ 6 k. If no upper bounds are imposed on the maximum degree, then ec.H/ D c.H/ D 1 could happen even k is large. An example is the complete k-uniform hypergraph on 2k vertices that satisfies c.H/ D 1 and maximum degree less than 4k . Let a > 1 be any real number, and let  > 0 be small. Let k be sufficiently large such that there exists an integer in the interval Œk=.1 C  2 =4/a ln k; k=.1 C  2 =8/a ln k. For some in the interval Œ 2 =8;  2 =4, the number k=.1 C /a ln k is an integer. Let H be a k-uniform hypergraph with maximum degree at most k a . Yuster [157] ˙ pproved that

there exists an equitable coloring of H with k=.1 C /a ln k  k =a ln k > .1  /k=a ln k colors and the following was established. Theorem 107 If a > 1 and H is a k-uniform hypergraph with maximum degree at most k a , then ec.H/ > a lnk k .1  ok .1//. The lower bound is asymptotically tight. For all a > 1, there exists k-uniform hypergraphs H with maximum degree at most k a and c.H/ 6 a lnk k .1 C ok .1//. Note that results in Alon [3] have already implied that no equitable coloring of a k-uniform hypergraph could have more than .k= ln k/.1 C ok .1// color classes. Yuster made the following remarks at the end of his paper [157]. Using more involved computations, Theorem 107 can be proved when a is not a constant but satisfies a D a.k/ D o.k= ln k/, i.e., the degree of H is allowed to be any subexponential function of k. It is possible to convert the proof of Theorem 107 into an algorithmic one. In terms of the number of vertices of the hypergraph, but not in its uniformity, a polynomial time algorithm can be found to produce an equitable partition with .1  ok .1//ck=.a ln k/ color classes, where c is a fixed small constant depending only on a. A special case of Theorem 107 gives the following interesting result about graphs. Let G be a k-regular graph. If k is sufficiently large, then G has an equitable

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coloring with .1  ok .1//.k= ln k/ colors such that each color class is a so-called total dominating set, that is a subset of the vertices such that each vertex of G has a neighbor in that subset.

20

Conclusion

The subject of graph coloring occupies a central position in graph theory. Historically speaking, much of the early development of graph theory was motivated by the attempts at solving the four-color conjecture. Later on, a large number of variants or generalizations of graph coloring problem have emerged. Graph coloring involves deep mathematical and computational issues. The possibilities for applications are also wide. When resources are allocated, a certain kind of graph coloring model may be lurking around. The requirement for even distribution is rather natural. The equitable coloring goes to the extreme to make color classes differ in size by at most one. This may be too stringent for applications in the real world. Yet it brings up hard questions to be addressed. Although the concept of an equitable coloring of a graph was introduced in early 1970s, substantial research results have only been accumulated in the last 20 years. The importance of the equitable -coloring conjecture has gradually been recognized. Positive evidence has been collected in this chapter. Another fundamental phenomenon in equitable graph coloring is that in many graph classes “most” members admit equitable colorings with colors not extremely larger than their ordinary chromatic numbers. This phenomenon needs further investigated. Equitable coloring of graphs can be formulated in terms of graph packings. This places equitable coloring in a wider context and gives it some previously unexpected connections and it may provide motivations to generate graph packing problems.

Cross-References Advanced Techniques for Dynamic Programming Advances in Scheduling Problems Faster and Space Efficient Exact Exponential Algorithms: Combinatorial and

Algebraic Approaches On Coloring Problems Online and Semi-online Scheduling Resource Allocation Problems

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Faster and Space Efficient Exact Exponential Algorithms: Combinatorial and Algebraic Approaches Dongxiao Yu, Yuexuan Wang, Qiang-Sheng Hua and Francis C.M. Lau

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Combinatorial Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Zeta Transform and M¨obius Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Subset Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Variants of Subset Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Inclusion-Exclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Space Efficient Exact Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Faster Exact Algorithms in Bounded-Degree Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Algebraic Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Algebraic Sieving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Polynomial Circuit-Based Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1250 1250 1251 1252 1257 1259 1262 1268 1271 1271 1282 1286 1287 1289 1289

Abstract

Exponential algorithms, whose time complexity is O.c n / for some constant c > 1, are inevitable when exactly solving NP-complete problems unless P D NP. This chapter presents recently emerged combinatorial and algebraic techniques for designing exact exponential time algorithms. The discussed

D. Yu • F.C.M. Lau () Department of Computer Science, The University of Hong Kong, Hong Kong, People’s Republic of China e-mail: [email protected]; [email protected] Y. Wang • Q.-S. Hua Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, People’s Republic of China e-mail: [email protected]; [email protected]; [email protected] 1249 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 38, © Springer Science+Business Media New York 2013

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techniques can be used either to derive faster exact exponential algorithms or to significantly reduce the space requirements while without increasing the running time. For illustration, exact algorithms arising from the use of these techniques for some optimization and counting problems are given.

1

Introduction

It is a folklore that any NP-complete problem can be exactly solved in exponential time via exhaustive search. Whether there is a faster way than this kind of bruteforce approach to solve any such problem is still unclear. Nonetheless, many researchers have found exact exponential time algorithms that are faster than trivial exhaustive search for quite many NP-hard optimization problems such as the traveling salesman problem (TSP) [39]. The study of exact exponential algorithms for NP-hard problems has received increasing attention since the seminal survey by Woeginger [55] in 2001. As a result, some classical techniques such as inclusionexclusion (e.g., [19, 20, 32]) have been resurrected, and some new techniques, such as fast subset convolution [12] and the algebraic methods (e.g., [9]), have been developed. Unfortunately, for some optimization problems, although it is possible to obtain faster exponential algorithms, exponential space is required, which renders these algorithms not practical for real-life use. This chapter presents some important recently resurrected or emerged combinatorial and algebraic approaches for designing faster exact exponential time algorithms and discusses how these techniques may be used to reduce the space complexity while not sacrificing the time complexity. Notation. Throughout this chapter, a modified big-O notation is used to hide a polylogarithmic factor: for a positive real constant d and a function , O  ./ denotes a time complexity of the form O. logd /. Structure of the chapter. The main theme of this chapter is to introduce two classes of methods for designing either faster or space-efficient exact exponential time algorithms. Section 2 discusses the combinatorial methods, which covers fast subset convolution and inclusion-exclusion based algorithms. Section 3 presents the recently emerged algebraic methods, which covers the algebraic sieving method and the polynomial circuit-based algebraic method. Each of these two sections will first explain how to use these techniques to design faster exact exponential time algorithms and then discuss how they may be used to reduce the space complexity while not increasing the running time. Various examples to illustrate the techniques will also be given. Section 4 concludes the chapter and Sect. 5 highlights the related work on these two classes of methods.

2

Combinatorial Methods

There are two well-developed and commonly used combinatorial techniques – fast subset convolution and inclusion-exclusion. The fast subset convolution can compute the convolution of any two given functions over some subset lattice in

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O  .2n / time, whereas a direct evaluation needs .3n / time. This fast algorithm is based on fast algorithms for the zeta transform and other related transforms which are actually accelerated dynamic programming algorithms. Thus, it needs exponential space since dynamic programming approaches have to store all the useful auxiliary information. Surprisingly, for certain special type of input functions called singletons, this exponential space could be circumvented. Furthermore, several variants of subset convolution, such as the covering product and the packing product [12], also play an important role in the design of faster exponential time exact algorithms. The inclusion-exclusion principle is a classical sieving method: to determine the size of a set, the set is first overcounted, then subtract from this count, add again, subtract again, until finally arriving at the exact number of elements. This allows to count combinatorial objects indirectly, which is better than direct counting which could be inefficient or even impossible. Furthermore, although it needs to go through all subsets, the inclusion-exclusion technique is powerful in providing polynomial space exact algorithms. Combining it with some dynamic programming-based algorithms, such as the fast zeta transform, gives more surprising results. In fact, the inclusion-exclusion technique solves optimization problems via solving the corresponding counting problems, and hence, it is an important alternative for designing exact algorithms for counting problems. Before the detailed descriptions of these two techniques, some essential tools needs to be introduced – the fast zeta transform and the fast M¨obius transform.

2.1

¨ Zeta Transform and Mobius Transform

Denote by R an arbitrary (possibly noncommutative) ring and by N a set of n elements, n  0. Let f be a function that associates with every subset S  N an element f .S / of the ring R. The zeta transform of f is the function f  that associates every S  N with an element in R: X f .S / D f .X /: (1) X S

The M¨obius transform f of f is defined as X .1/jS nX j f .X /: f.S / D

(2)

X S

Given the zeta transform f , the original function f can be obtained by the following formula which is often called M¨obius inversion: f .X / D

X

.1/jX nS j f .S /:

S X

For more details of these two transforms, please refer to [27].

(3)

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2.1.1 Fast Zeta Transform and Fast Mobius ¨ Transform The straightforward way to compute the zeta transform evaluates f .S / at every S  N , using O.3n / ring additions in total. This can be improved to use only O.n2n / ring operations by applying the Yates’s method [37, 58], and the resulting algorithm is commonly called the fast zeta transform. Without loss of generality, assume that N D f1; 2; : : : ; ng. To compute f , let initially f 0 .S / D f .S / for every S  N . Then iterate for j D 1; 2; : : : ; n and S  N with the following recurrence: ( f j .S / D

f j 1 .S /;

if j … X

f j 1 .S n fj g/ C f j 1 .S /; otherwise:

(4)

It can be verified by induction on j that the above recurrence gives f n .S / D f .S / for any S  N in O.n2n / ring operations. The M¨obius transform can be computed in a similar manner. Initially, let f0 .S / D f .S / for every S  N . Then iterate for j D 1; 2; : : : ; n and S  N as follows: ( fj .S / D

fj 1 .S /;

if j … X

fj 1 .S n fj g/ C fj 1 .S /; otherwise:

(5)

Similarly, it can be proved that fn .S / D f.S / for any S  N . Theorem 1 Let N be a set of n elements. Then the zeta transform and the M¨obius transform on the subset lattice of N can be computed in O.n2n / ring operations.

2.2

Subset Convolution

Definition 1 (Subset convolution) Given a universe set N of n elements, f and g are functions defined on subsets of N , which associate every S  N with an element of a ring R, respectively. The convolution f  g for all S  N is defined as follows: X f .X /g.S n X /: (6) f  g.S / D X S

The subset convolution can yield solutions to many hard computational problems. In particular, by embedding the integer max-sum or min-sum semiring into the sum-product ring, the technique can be used to implement many dynamic programming-based algorithms.

2.2.1 Fast Subset Convolution In principle, the fast subset convolution consists of several efficient dynamic programming instances. In the fast subset convolution, the evaluation of (6) can

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be achieved via the product of “ranked” extensions of the classical zeta transforms of f and g on the subset lattice, followed by a “ranked” M¨obius transform. The fast subset convolution is summarized in Algorithm 1 . For every k D 0; 1; : : : ; n and S  N , the ranked zeta transform of f is defined as X f .X /: (7) f .k; S / D X S;jX jDk

Based on the above defined ranked zeta transform, the inversion can be achieved by X f .X / D .1/jS nX j f .jX j; S /: (8) S X

Although the above formula seems redundant, it will provide a key for fast evaluation of the subset convolution. Note that the ranked zeta transform can be computed in O.n2 2n / ring operations by carrying out the fast zeta transform (4) independently for every k D 0; 1; : : : ; n. Similarly, the ranked inversion (8) can be computed in O.n2 2n / ring operations by carrying out the fast M¨obius transform (5) independently for each k D 0; 1; : : : ; n. For the two ranked zeta transforms, f  and g, define a new convolution f  ~g for all k D 0; 1; : : : ; n and S  N by

f  ~ g.k; S / D

k X

f .j; S /g.k  j; S /:

(9)

j D0

Lemma 1 f  g.S / D

P

X S .1/

jS nX j

f  ~ g.jS j; S /

Proof X

.1/jS nX j f  ~ g.jS j; S / D

X S

X

.1/jS nX j

X S

D

X X S

jS j X

f .j; S /g.jS j  j; S /

j D0

.1/jS nX j

jS j X

X

j D0

U;V X jU jDj;jV jDjS jj

f .U /g.V /: (10)

Because X ranges over all subsets of S , for any ordered pair .U; V / of subsets of S satisfying the condition that jU j C jV j D jS j, the term f .U /g.V / occurs exactly once in the sum with sign .1/jS nX j for subsets X of S iff U [V  X . Then putting the terms associated with each pair .U; V / together, by the Binomial Theorem, the coefficient of f .U /g.V / is

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( ! jS j  jU [ V j 1; if jU [ V j D jS j jS jx D .1/ x  jU [ V j 0; otherwise:

(11)

The conditions that jU j C jV j D jS j and jU [ V j D jS j imply that U [ V D S and U \ V D ;; and it follows that X X S

X

.1/jS nX j f  ~ g.jS j; S / D

f .U /g.V /

(12)

U [V DS;U \V D;

D f  g.S /:

t u

Theorem 2 The subset convolution over an arbitrary ring can be evaluated in O.n2 2n / ring operations using the fast subset convolution as shown in Algorithm 1. Proof The correctness is established by Lemma 1. In Algorithm 1 , the time is mainly consumed in lines 2, 6, and 9. By Theorem 1, the time consumed in lines 2 and 9 is O.n2 2n /, and the time needed in line 6 is O.n2 2n / according to Formula (9). Thus, the time complexity of Algorithm 1 is O.n2 2n /.  The following theorem is implied by Theorem 2. Theorem 3 The subset convolution over the integer sum-product ring can be computed in O  .2n log M / time, provided that the range of the input functions is fM; M C 1; : : : ; M g. Proof By Theorem 2, the subset convolution can be computed in O.n2 2n / ring operations. Thus, it suffices to note that any intermediate results, for which ring operations are performed, are O.n log M /-bit integers. This is the case because the ranked zeta transform of an input function can be computed with integers between M 2n and M 2n, and it follows that the convolution of ranked transforms

Algorithm 1 Fast subset convolution Input: A universe set N of n elements and functions f; g defined on 2U Output: The subset convolution f  g for every set S  N 1: For i D 0 to n do 2: Compute the ranked zeta transforms f .k; X/; g.k; X/ for every X  N using the fast zeta transform. 3: End For 4: For k D 0 to n do 5: For every X  N do 6: Compute the convolution f  ~ g.k; X/ 7: End For 8: End For 9: Compute the subset convolution f  g for every set S  N based on Formula (12) using the fast M¨obius transform.

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can be computed with O.n log M /-bit integers. Furthermore, the ranked M¨obius transform is computed by adding and subtracting O.n log M /-bit integers for O.2n / times.  An obvious disadvantage of the fast subset convolution algorithm is its inapplicability to semirings where additive inverses need not exist. For some special semirings usually appearing in combinatorial optimization problems, for example, the integer max-sum and integer min-sum semirings, one can embed these semirings into the integer sum-product ring. Theorem 4 The subset convolution over the integer max-sum (min-sum) semiring can be computed in O  .2n M / time, provided that the range of the input functions is fM; M C 1; : : : ; M g. Proof Here, we only provide the proof for the max-sum semiring, as that for the min-sum semiring is similar. Without loss of generality, assume that the range of the input functions is f0; 1; : : : ; M g; otherwise, the correct output can be obtained by first adding M to each value of both input functions, then computing the convolution, and finally subtracting 2M from the computed values. Let f; g be the two input functions. Let ˇ D 2n C 1 and M 0 D ˇ M . Define new functions f 0 D ˇ f and g 0 D ˇ f which map the subsets of N to f0; 1; : : : ; M 0 g. By Theorem 3, the subset convolution f 0  g 0 over the integer sum-product ring can be computed in O  .2n log M 0 / D O  .2n M / time. It remains to show that for all S  N , the value for maxT S ff .T / C g.S n T /g can be efficiently deduced given the value of f 0  g 0 .S /. Observe that f 0  g 0 .S / can be expressed as f 0  g 0 .S / D

X

ˇ f .T /Cg.S nT /

T S

D ˛0 .S / C ˛1 .S /ˇ C    C ˛2M .S /ˇ

(13) 2M

:

Due to the choice of ˇ, each coefficient ˛r .S / is uniquely determined and equals the number of subsets T of S for which f .T / C g.S n T / D r. Thus, for each S  N , the largest r for which ˛r .S / > 0 corresponds to the value of f  g.S /. Clearly, this can be found in O.M / time.  Example 1 (Partitioning Problem) The generic problem of partitioning is to divide an n-element set N into k disjoint subsets such that each of which satisfies some desired property specified by an indicator function f on the subsets of N . Given N , k, and f as input, the task is to decide whether there exists a partition fS1 ; S2 ; : : : ; Sk g of N such that f .Sc / D 1 for each c D 1; 2; : : : ; k. Many classical graph partitioning problems are of this form. For example, in graph coloring, f .S / D 1 iff S is an independent set in the input graph with the vertices N . Likewise, in domatic partitioning, f is the indicator of dominating sets. Note that the number of valid partitions of N is given by f k .N /, where f k is defined as

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f k D f  : : :  f : „ ƒ‚ … k t i mes

Thus, the valid partitions can be counted by O.log k/ subset convolutions using i the doubling trick – computing only convolutions f 2 . t u Example 2 (Steiner Tree Problem) Given an undirected graph G D .V; E/, a weight w.e/ > 0 for each edge e 2 E, and a set of vertices K  V , the Steiner tree problem is to find a subgraph H of G that connects the vertices in K and has P the minimum total weight e2E.H / w.e/ among all such subgraphs of G. Here, we only consider the case where the weight of each edge is bounded by a constant. Obviously, H is necessarily a tree. A Steiner tree always refers to such an optimal subgraph. The fast subset convolution can be used to accelerate the famous DreyfusWagner algorithm [25]. For a vertex subset Y  V , denote the total weight of a Steiner tree connecting Y in G by W .Y /. The Dreyfus-Wagner algorithm is based on the following dynamic programming called Dreyfus-Wagner recurrence: for jY j  3, and all q 2 Y , X D Y n fqg. W .fqg [ X / D minfW .fp; qg/ C gp .X / W p 2 V g; gp .X / D minfW .fpg [ D/ C W .fpg [ .X n D// W ;  D  X g:

(14) (15)

The Steiner tree problem can be solved by computing the weight W .K/ via the above recursion. For the base case, observe that for jY j D 1 the weight W .Y / D 0 and for jY j D 2 the weight W .Y / can be determined by a shortest-path computation based on the edge weight w.e/, for example, Johnson’s algorithm [35]. A bottom-up evaluation of W .K/ takes O  .3k n C 2k n2 C nm/ time, where n D jV j, m D jEj and k D jKj. Once the values W .fpg [ Y / and gp .Y / for all Y  K and p 2 V are computed and stored, an actual Steiner tree is easy to construct by tracing backward a path of optimal choices in (14) and (15). The fast subset convolution over the min-sum semiring can expedite the evaluation of the Dreyfus-Wagner recursion in (15). Here the fast subset convolution needs to be implemented in a level-wise manner. For each level l D 2; 3; : : : ; k  1, assume the value W .fqg [ X / has been computed and stored for all X  K and q 2 V n X . Define the function fp for all X  K and p 2 V as ( fp .X / D

W .fpg [ X /; if 1  jX j  l  1 1;

otherwise:

(16)

Obviously, gp .X / D fp  fp (X) for all X  K and jX j D l. Then applying the fast subset convolution over the min-sum semiring, by Theorem 4, the Steiner tree problem can be solved in O  .2k n2 C nm/ time. 

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2.3

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Variants of Subset Convolution

2.3.1 Covering Product and Intersecting Covering Product Definition 2 Given two functions f; g defined on 2N which associate each subset of N to an element of a ring R, the covering product f c g.S / for each S  N is defined as X f c g.S / D f .U /g.V /: (17) U;V S;U [V DS

Theorem 5 The covering product over an arbitrary ring can be evaluated in O.n2 2n / ring operations. Proof The proof is mainly based on the following Lemma. Lemma 2 For any S  N , the following holds: .f c g/.S / D f .S /  g.S /: Proof .f c g/.S / D

X

.f c g/.S /

X S

D

X

X

f .U /g.V /

X S U [V DX

D

X

(18)

f .U /

U S

X

(19) g.V /

V V

D f .S /  g.S /: The third equality comes from the fact that for each U; V  X , there exists exactly one X  S such that U [ V D X .  By Lemma 2, f c g can be obtained by computing the M¨obius transform of f .S / g.S /. Then an O  .2n / time algorithm for the covering product is as follows: first, compute f  and g using the fast zeta transform (4), then multiply f  by g for all S  N , and finally, compute f c g using the fast M¨obius transform (5).  Definition 3 Given two functions f; g defined on 2N which associate each subset of N to an element of a ring R, the intersecting covering product f i c g.S / for each S  N is defined as f i c g.S / D

X U;V S U [V DS;U \V ¤;

f .U /g.V /:

(20)

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An O  .2n / time algorithm can be immediately derived from the fact that f i c g D f c g  f  g. Theorem 6 The intersecting covering product over an arbitrary ring can be evaluated in O.n2 2n / ring operations. Furthermore, more precise control over the allowed intersection cardinalities jU \ V j D l besides the l D 0 .f  g/ and l > 0 .f i c g/ cases can be obtained by modifying (9). Example 3 (Minimum Connected Spanning SubHypergraph (MCSH)) A hypergraph is a pair H D .V; E/, where V is a finite set and E is a set consisting of subsets of V . A hypergraph J D .W; F / is a subhypergraph of H if W  V and F  E. A subhypergraph is spanning if V D W . A path in a hypergraph H is a sequence .x1 ; E1 ; x2 ; E2 ; : : : ; El ; xlC1 / such that (a) x1 ; x2 ; : : : ; xlC1 2 V are all distinct, (b) E1 ; E2 ; : : : ; El 2 E are all distinct, and (c) xi ; xi C1 2 Ei for all i D 1; 2; : : : ; l. Such a path joins x1 to xlC1 . A hypergraph is connected if for all distinct x; y 2 V there exists a path joining x to y. Given a hypergraph H D .V; E/ and a weight w.E/ > 0 for each hyperedge E 2 E, the minimum connected spanning subhypergraph problem is to find a connected spanning subhypergraph of H that has the minimum total weight or to assert that none exists. Here, the weight of every edge is assumed to be bounded by a constant. The MCSH problem can be solved in O  .2n / time using the intersecting covering product over the min-sum semiring, where n D jV j. First, define the k-th power of the intersecting covering product for all k D 2; 3; : : : by k k1 f i c D f i c .f i c /: (21) Here, the order in which the products are evaluated matters because the intersecting covering product is not associative. Define the function f for all E  V by ( f .E/ D

w.E/; if E 2 E 1;

otherwise:

(22)

Now, observe that (a) an MCSH can be constructed by augmenting a connected subhypergraph of H one hyperedge at a time, and (b) at most n1 hyperedges occur k in an MCSH of H . Thus, f i c .V / < 1 is the minimum weight of a connected spanning subhypergraph of H consisting of k hyperedges. By storing the function k values f i c for each k D 1; 2; : : : ; n  1, the actual MCSH can be determined by tracing back the computation one edge at a time.

2.3.2 Other Variants Another important variant of the subset convolution is the packing product which is defined for all S  N as

Faster and Space Efficient Exact Exponential Algorithms

X

f p g.S / D

1259

f .U /g.V /:

(23)

U;V S;U \V D;

Given f and g, the packing product can be evaluated in O.n2 2n / ring operations by first computing the subset convolution f  g and then convolving the result with !  vector 1 with all entries being equal to 1 based on the following fact: !  f p g D f  g  1 :

(24)

Some further variations are possible by restricting the domain, for example, to any hereditary family of subsets of N . For more details, please refer to [12].

2.4

Inclusion-Exclusion

The inclusion-exclusion technique is based on a fundamental counting principle – the inclusion-exclusion principle. Clearly, it is a natural approach for solving counting problems. Here, we mainly focus on demonstrating its power in solving decision versions of optimization problems when direct computation is inefficient or even impossible. Another important feature of the inclusion-exclusion technique is that it does not need exponential space in principle. Hence, this technique is also a good choice for deriving polynomial space algorithms. Lemma 3 (Inclusion-exclusion principle) Let B be a finite set with subsets A1 ; : : : ; An  B. With the convention that \i 2; Ai D B, the following holds: 1. The number of elements of B which lie in none of the Ai is j

n \

Ai j D

i D1

X

.1/jX j j

X f1;:::;ng

\

Ai j:

(25)

i 2X

2. Let w W B ! RP be a real-valued weight function extended to the domain 2B by setting w.A/ D e2A w.e/. Then w.j

n \ i D1

Ai j/ D

X X f1;:::;ng

.1/jX j w.j

\

Ai j/:

(26)

i 2X

Proof The Lemma is proved by analyzing the contribution of every element e 2 B to both sides of the expression. If e is not contained by any Ai , it contributes w.e/ to the left hand side. To the right hand side it contributes w.e/ exactly once, namely, in the term corresponding to X D ;. Conversely, assume that e lies in Ai for all i 2 I ¤ ;. Its contribution to the left hand side is 0. On the right hand side, since e lies in the intersection \i 2X Ai for every X  I , by the Binomial Theorem, its total contribution is

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X

.1/

jX j

X I

! jI j X i jI j w.e/ D w.e/ .1/ D 0: i i D0

(27) 

Example 4 (Set Partition) Given a set system .N; F / and an integer k, where F  2N , a k-partition of the given set system is a tuple .S1 ; : : : ; Sk / over F such that [kiD1 Si D N and Si \ Sj D ; for any 1  i ¤ j  k. Denote Pk .F / as the number of such k-partitions. The set partition problem is to determine the minimum k such that there is a k-partition of .N; F /, that is, the minimum k such that Pk .F / > 0. Clearly, this problem can be solved by computing Pk .F / for at most n different k values. The following lemma shows how to compute Pk .F / by making use of the inclusion-exclusion principle. the number of k-tuples .S1 ; : : : ; Sk / over F for which Lemma 4 Let ak .X / denote P Si \ X D ;.1  i  k/ and kiD1 jSi j D n. Then pk .F / D

X

.1/jX j ak .X /:

(28)

X N

Proof This is a direct application P of Lemma 3, where B is the set of k-tuples .S1 ; : : : ; Sk / over F satisfying kiD1 jSi j D n, and Ai  B are those k-tuples that avoid fi g. Then pk .F / is exactly the number of k-tuples that lie in none of the Ai , and j \i 2X Ai j D ak .X /.  Write f  .l/ .Y / for the number of sets S 2 F with jS j D l and S  Y , and recall that it is the zeta transform of the indicator function ( 1; if S 2 F ; jS j D l .l/ f .S / D (29) 0; otherwise: Thus, the fast zeta transform (4) computes a table containing f  .l/ .Y / for all l and Y in O  .2n / time. Once these values have been computed, ak .X / can be obtained for any fixed X  N by dynamic programming in time polynomial in k and n as follows. Define g.j; m/ to be the number of j -tuples .S1 ; : : : ; Sj / for which Pj Sc \ X D ;.1  c  j / and cD1 jSj j D m, formally X

g.j; m/ D

j Y

f  .lc / .N n X /:

(30)

l1 CClj Dm cD1

Then ak .X / D g.k; n/, and it can be computed by the following recursion: g.j; m/ D

m X lD0

g.j  1; m  l/f  .l/ .N n X /;

(31)

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observing that g.1; m/ D f  .m/.N n X /. Finally, summing up ak .X / according to (28) gives an O  .2n / time algorithm for computing Pk .F /. t u xŠ Example 5 (Cover Polynomial) Denote x i D .xi /Š . A Hamiltonian path of a graph is a walk that contains all the nodes exactly once. If the walk is cyclic, it is called a Hamiltonian cycle. The cover polynomial of a directed graph D D .V; A/ is defined as (cf. [13, 22]) X CV .i; j /x i y i ; (32) i;j

where CV .i; j / can be interpreted as the number of ways to partition V into i -directed paths and j -directed cycles of D. The so-called computing cover polynomial is to compute the coefficient CV .i; j /. For X  V , denote by h.X / the number of Hamiltonian paths in DŒX , and denote by c.X / the number of Hamiltonian cycles in DŒX . Define h.;/ D c.;/ D 0. Note that for all x 2 V , h.fxg/ D 1 and c.fxg/ is the number of loops incident on x. Then CV .i; j / can be expressed as: CV .i; j / D

1 i Šj Š X

X

h.X1 /    h.Xi /c.Y1 /    c.Yj /;

(33)

1 ;:::;Xi ;Y1 ;:::;Yj

where the sum is over all .i C j /-tuples .X1 ; : : : ; Xi ; Y1 ; : : : ; Yj / such that it is a partition of V . The expression for CV .i; j / can be reformulated using the principle of inclusionexclusion. Let z be a polynomial indeterminate. Define for every U  V the polynomials X

HU .z/ D

h.X /zjX j ; CU .z/ D

X U

X

c.Y /zjY j :

(34)

Y U

Viewed as set functions, HU .z/ and CU .z/ are zeta transforms of the set functions h.X /zjX j and c.Y /zjY j , respectively. By inclusion-exclusion, CV .i; j / D

1 X .1/jV nU j fzn gHU .z/i CU .z/j : i Šj Š U V

(35)

Based on the above formula, an O  .2n / time algorithm can be obtained. For S  V , let wS .s; t; l/ denote the number of directed walks of length l from vertex s to vertex t in DŒS . Define wS .s; t; l/ D 0 if s … S or t … S . By inclusion-exclusion again, X X h.X / D .1/jX nS j wS .s; t; jX j  1/; s;t 2V S X

c.Y / D

1 XX .1/jY nS j wS .s; s; jX j/: jY j s2V S Y

(36)

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Observing wS .s; t; l/ can be computed in polynomial time, the fast M¨obius transform (5) can compute h.X / and c.Y / for every X; Y  V in O  .2n / time. Then H and C can be evaluated using the fast zeta transform according to (34). Finally, given H and C , (35) can be evaluated in O  .2n / time. t u

2.5

Space Efficient Exact Algorithms

2.5.1 Polynomial Space Algorithm Based on Subset Convolution For some special input functions whose zeta transforms can be determined easily, polynomial space algorithms for some combinatorial optimization problems can be derived via the subset convolution and the covering product. A particular type of input functions called singletons is used here to demonstrate how to efficiently compute the subset convolution and the covering product values for the element set N using only polynomial space. A function f W U ! R is called a singleton if there exists an element e 2 U such that f .x/ D 0 unless x D e. Theorem 7 Given two functions f; g defined on 2N which associate each subset of N to an element of a ring R, if f; g, then (1) f c g.N / can be computed in O  .2n / time and polynomial space, and (2) f  g.N / can be computed in O  .2n / time and polynomial space. Proof (1) Assume that f .Xf / D ef and g.Xg / D eg , where Xf ; Xg  N and ef ; eg 2 R. Since, f; g are singletons, f .X / D 0 for any X  N other than Xf . The same result is also true for g and any subsets other than Xg . The following Lemma is proved in Sect. 2.3.1. Lemma 5 For any S  N , the following holds: .f c g/.S / D f .S /  g.S /:

(37)

Note that for singletons f; g, the zeta transforms for every Y  N can be easily given as ( f .Y / D ( g.Y / D

ef ; if Xf  Y 0;

if otherwise:

eg ;

if Xg  Y

0;

if otherwise:

(38)

(39)

Thus, the zeta transform of f c g for every subset Y  N can be computed in polynomial time after getting the zeta transforms f  and g. Then f c g.N / can be obtained in O  .2n / time by computing the M¨obius inversion (3). The space used is polynomial since it is not necessary to store .f c g/ values for every Y  N .

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(2) Denote h D f  g. Define a relaxation of a function f W 2N ! R as a sequence of functions fi W 2N ! R, for 0  i  jN j, such that for every 0  i  jN j; Y  N : ( fi .Y / D

f .Y /;

if i D jY j

0;

if i < jY j:

(40)

Observe that a function f can have many different relaxations, since there are no restrictions on what fi .X / should be when i > jX j. The basic idea of the proof is that for the input functions f and g, there exists a relaxation fhi g of h such that the zeta transform of hjN j can be computed efficiently. Since h.N / D hjN j .N /, h.N / can be obtained by computing the M¨obius inversion (3) of hjN j . For input functions f and g, define fi D f and gi D g for all i . It is easy to verify that P ffi g and fgi g are relaxations of f and g, respectively. Then define for all i , hi D ij D0 fj c gi j . Lemma 6 As defined above, fhi g is a relaxation of h. Proof By the definition of the covering product, for every X  N

hi .X / D

i X X

fj .Y /gi j .Z/:

(41)

j D0 Y [ZDX

Consider an arbitrary summand fj .Y /gi j .Z/. There are two cases: Case 1 (jX j > i ) Since Y [ Z D X , jY j C jZj  jX j > i . It concludes that either jY j > j and fj .Y / D 0 or jZj > i  j and gi j .Z/ D 0, because ffi g and fgi g both are relaxations of f and g respectively. Thus, fj .Y /gi j .Z/ D 0. Since fj .Y /gi j .Z/ is an arbitrary term in the summation for hi .X /, it follows that hi .X / D 0. Case 2 (jX j D i ) If jY j C jZj > i , either jY j > j or jZj > i  j and fj .Y /gi j .Z/ D 0 which is analogous to the first case. If jY j C jZj D i , then Y and Z are disjoint. Since ffi g and fgi j g are relaxations of f and g respectively, only fjY j .Y /gjZj .Z/ will contribute to the sum. Thus,

hi .X / D

i X X

fj .Y /gi j .Z/

j D0 Y [ZDX

D

X

Y [ZDX;Y \ZD;

(42) fjY j .Y /gjZj .Z/

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X

D

f .Y /g.Z/

Y [ZDX;Y \ZD;

D h.X /:



P By Lemma 5 and the definition of hi in (41), it follows that hi D ij D0 f j gi j . Thus, the zeta transform of hjN j .X / for every X  N can be computed in polynomial time given the zeta transforms for ffi g and fgi g. From (1), the zeta transforms of ffi g and fgi g can be obtained efficiently. Similar to what is done in (1), h.N / D hjN j .N / can be obtained in O  .2n / time and polynomial space by computing the M¨obius inversion X

hjN j .N / D

.1/jN nX j hjN j .X /:

(43)

X N

 Example 6 (Cover Polynomial Revisited) Define CY .i; j / D

1 i Šj Š

X

hl1 c    hli c cli C1    cli Cj .Y /;

(44)

l1 CCli Cj Dni

where hl .Y / and cl .Y / are the number of Hamiltonian paths and Hamiltonian cycles of length l in DŒY , respectively (here, the parameter l is introduced in order to obtain efficient computable zeta transforms). Note that paths and cycles with l edges contain l C 1 and l nodes, respectively, which follows that the sum of the lengths of the paths and cycles in such a partition will be n  i . Moreover, if V is covered, the paths and cycles are disjoint because of the size restriction. The above definition for CV .i; j / is equivalent to that in Example 5. Note that hl .Y / can be replaced by h0l .Y / which denotes the number of walks of length l in DŒY . Similarly, cl .Y / can be replaced by cl0 .Y / which is the number of cyclic walks of length l. Using the technique introduced in Theorem 7, C Y .i; j / can be computed using the following formula given hl0 .Y / and cl0 .Y /: C Y .i; j / D

1 i Šj Š

X

i Y

l1 CCli Cj Dni t D1

hl0t .Y /

j Y

cl0i Cp .Y /:

(45)

pD1

Finally, hl0 .Y / and cl0 .Y / can be computed in polynomial time using standard dynamic programming (cf. [44]). Then CV .i; j / can be obtained by computing the M¨obius inversion as described in Theorem 7 (by a similar argument as used in proving Lemma 1, it can be proved that all extra items introduced in the process of relaxing hl and cl are canceled through computing the M¨obius inversion). Putting everything together will provide an O  .2n / time and polynomial space algorithm for counting cover polynomial. 

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Combining with the algebraic method to be introduced in Sect. 3.2, the technique introduced in Theorem 7 can be used to give polynomial space algorithms for the traveling salesman problem, the Steiner tree problem, the weighted set cover problem, etc., without increasing the running times as have been derived in the fastest solutions to these problems [12, 20, 28, 36].

2.5.2

Polynomial Space Exact Algorithms Based on Inclusion-Exclusion T By the inclusion-exclusion principle (25), if j i 2X Ai j can be computed in polynomial time for each X  f1; : : : ; ng, there exists a polynomial space algorithm evaluating Eq. (25) in O  .2n / time.

Example 7 (Counting Hamiltonian Paths) Given a graph G D .V; E/, a Hamiltonian path is a walk that contains each node exactly once. The Counting Hamiltonian Path problem is to count the number of Hamiltonian paths. The following inclusion-exclusion based algorithm is due to Karp [36]: define the universe B as all walks of length n  1 in G, and define Av as all walks of length n  1 containing v for each v 2 V . By the inclusion-exclusion principle, the number of Hamiltonian paths is X \ hD .1/jX j j Av j: (46) X V

v2X

For X  V and s 2 X , let wk .s; X / be the number of walks from s of length k in GŒX . Then X \ Av j D wn1 .s; V n X /: (47) j v2X

s2V nX

For fixed X  V , wk .s; X / can be computed in polynomial time using the following recurrence: (

1; wk .s; X / D P

if k D 0 t 2N.s/\X wk1 .t; X /; otherwise:

(48)

n Notice T that wk .s; X / is at most O.n / which needs polynomial bits to represent. Thus, j v2X Av j can be computed in polynomial space and time, which provides an O  .2n / time and polynomial space algorithm to evaluate Eq. (46).

Example 8 (Counting Perfect Matchings) Given two undirected graphs F and G, let sub.F; G/ denote the number of distinct copies of a graph F contained in a graph G. Clearly, if F is a matching of n=2 edges, where n is the vertex number of G, then sub.F; G/ is the number of perfect matchings in G. A homomorphism from F to G is a mapping from the vertex set of F to that of G such that the image of every edge of F is an edge of G. Let hom.F; G/ and inj.F; G/ be the numbers of homomorphisms and injective homomorphisms from F to G, respectively. Furthermore, let aut.F; F / denote the number of automorphisms, that

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is, bijective homomorphisms, from F to itself. The following equation switches the focus to computing inj.F; G/, since aut.F; F / for a graph F on nF vertices can be computed in subexponential time [2] sub.F; G/ D

inj.F; G/ : aut.F; F /

(49)

Let S be a given subset of V .G/, then a homomorphism f from F to G is called S -saturating if .a/ S  f .V .F // and .b/ for all v 2 S , jf 1 .v/j D 1. Let S  hom.F; G/ denote the number of S -saturating homomorphisms. By the inclusion-exclusion principle, inj.F; G/ D

X

.1/jW j S  hom.F; GŒV .G/ n W /:

(50)

W 2V .G/nS

Note that if F is a matching of n=2 edges, S  hom.F; GŒV .G/ n W / can be computed in polynomial time and polynomial space using the following equation, which gives rise to an O  .2n / time polynomial space algorithm for counting the number of perfect matchings in G. Let S D fv1 ; : : : ; va g, then S  hom.F; GŒV .G/ n W / D

n 2

a

! aŠ.

a Y .2degV .G/nW .vi // i D1 n

j2E.GŒV .G/ n .W [ S //j 2 a ;

(51)

where degV .G/nW .vi / is the number of vertices adjacent to vi in V .G/ n W .

2.5.3

Linear Space Exact Algorithms Based on Linear Space Fast Zeta Transform As shown before, inclusion-exclusion together with the fast zeta transform can provide efficient exact algorithms for many hard problems, for example, the covering problem, the partitioning problem, and the packing problem (also cf. [19, 20, 40]). However, all these algorithms need O  .2n / space to carry out the fast zeta transform. The following linear space fast zeta transform (Algorithm 2 ) can help these algorithms achieve a linear space bound without increasing the running time. Theorem 8 (Linear Space Zeta Transform) Suppose f vanishes outside a family F of subsets of N and the member of F can be listed in time O  .2n / and space O  .jF j/. Then the values f .X /, for every X  U , can be listed in time O  .2n / and space O  .jF j/ using Algorithm 2 . Proof Partition U into U1 and U2 such that jU1 j D n1 , jU2 j D n2 , where n2 D dlog jF je and n1 D n  n2 . The correctness of Algorithm 2 is established by the following Lemma.

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Algorithm 2 Linear space algorithm for zeta transform Input: A universe set N of n elements and a functions f defined on a family F of subsets of N Output: The zeta transform f  for all sets in 2N 1: For each X1  U1 do 2: For each Y2  U2 do 0 3: set g.Y2 / 4: End For 5: For each Y 2 F do 6: If Y \ U1  X1 , then 7: set g.Y \ U2 / g.Y \ U2 / C f .Y / 8: End For 9: Compute h g using the fast zeta transform on 2U2 10: For each X2  U2 do 11: output the value h.X2 / as the value f .X1 [ X2 / 12: End For 13: End For

Lemma 7 f .X1 [ X2 / D h.X2 / Proof h.X2 / D

X

(52)

g.Y2 /

Y2 X2

D

X X

ŒY \ U1  X1 ŒY \ U2 D Y2 f .Y /

Y2 X2 Y 2F

D

X

Y 2F

D

ŒY \ U1  X1 ŒY \ U2  X2 f .Y / (53)

X

f .Y /

Y 2F ;Y X1 [X2

D

X

f .Y /

Y X1 [X2

D f .X1 [ X2 /:  Observe that the algorithm runs in O  .2n1 .2n2 C jF j// time and O  .2n2 / space. By the assigned values of n1 and n2 , the algorithm thus runs in O  .2n / time and O  .jF j/ space.  In Algorithm 2 , because the computations are independent for each X1  U1 , they can be executed in parallel on O  .2n =jF j/ processors.

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Example 9 (Counting k-Partitions) The problem and the notations are defined in Example 4. It is convenient to express the k-partitions in terms of generating polynomials. Based on the principle of inclusion-exclusion, (28) can be reformulated as follows in which ak .X / is replaced by a polynomial over an intermediate z:

dk .F / D

X

.1/

jN nX j

X N

n X

!k i

ai .X /z

;

(54)

i D0

where the coefficient aj .X / is the number of subsets Y  X that belong to F and are of size j . Now, dk .F / is a polynomial whose coefficient of the monomial zn is the number P of k-partitions. To evaluate this expression, note that the polynomial a.X / D niD0 ai .X /zi is equal to the zeta transform g.X /, where g.Y / D ŒY 2 F zjY j . Now the linear space fast zeta transform operating in a ring of polynomials lists the polynomials a.X / for all X  N in O  .2n / time and O  .jF j/ space. u t Similar ideas to that used in the linear space fast zeta transform can give space efficient algorithms for other elementary transforms and problems, for example, the intersection transform, the disjoint sum, and the chromatic polynomial. An important feature of this type of algorithms is that they admit efficient parallelization as discussed before. In particular, some algorithms [17] of this type can achieve a trade-off between time and space complexities by adjusting the number of processors executing the algorithm in parallel.

2.6

Faster Exact Algorithms in Bounded-Degree Graphs

For bounded-degree graphs, faster exact algorithms can be derived for certain hard problems. The basic idea is to exploit some special properties so that only a special set of subsets of the vertex set, but not all subsets, need to be considered; then, based on a projection theorem presented by Chung et al. [23], which can be used to bound the size of the special set, the running time can be reduced for bounded-degree graphs. In this section, an inclusion-exclusion based algorithm will exemplify how to derive faster exact algorithms for bounded-degree graphs. This method is not only applicable to inclusion-exclusion based algorithms but also to some other types of algorithms. The following lemma is the key to employ this method. Lemma 8 ([23]) Let V be a finite set with subsets A1 ; A2 ; : : : ; Ar such that every v 2 V is contained in at least ı subsets. Let F be a family of subsets of V . For each 1  i  r define the projections Fi D fF \ Ai W F 2 F g. Then, jF jı 

r Y

jFi j:

(55)

i D1

Example 10 (TSP in bounded-degree graphs) Given a graph G with nodes V and distant function d W V V ! N , the traveling salesman problem (TSP) is to find a minimum-weighted Hamiltonian cycle based on the distance function that

Faster and Space Efficient Exact Exponential Algorithms

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minimizes the total distance. Assume that the maximum degree of G is  D O.1/. A set W of vertices is a connected dominating set if every vertex v 2 V is either in W or adjacent to a vertex in W and the induced subgraph GŒW  is connected. Denote by CD the family of connected dominating sets of G. The starting point is the algorithm by Karp [36], and, independently, by Kohn et al. [39]. Assume that the distance function is bounded by a constant, that is, d.u; v/ 2 f0; 1; : : : ; Bg [ f1g for any pair of vertices u; v, where B D O.1/. The algorithm can be conveniently described in terms of generating polynomials. Select an arbitrary reference vertex s 2 V , and let U D V n fsg. For each X  U , denote by q.X / the polynomial over the indeterminate z for which the coefficient of each monomial zw counts the directed closed walks (in the complete directed graph with vertex set V and edge weights given by d ) through s that (i) avoid the vertices in X , (ii) have length n, and (iii) have finite weight w. Then q.X / can be computed in time polynomial in n by the following recurrence and setting q.X / D p.n; s/ for a fixed X  U . Initialize the recurrence for each vertex u 2 V n X with ( p.0; u/ D

1; if u D s 0; otherwise:

(56)

For convenience, define z1 D 0. For each length l D 1; 2; : : : ; n and each vertex u 2 V n X , let X p.l; u/ D p.l  1; v/zd.v;u/ : (57) v2V nX

Here, note that due to the assumption on bounded weights, each p.l; u/ has at most a polynomial number of monomials with nonzero coefficients. By the principle of inclusion-exclusion, the coefficients of the monomials in the polynomial X QD .1/jX j q.X / (58) X U

count, by weight, the number of directed Hamiltonian cycles. It follows immediately that the traveling salesman problem can be solved in time O  .2n /. For bounded-degree graphs, faster algorithms can be derived by analyzing (58) in more details. It will be convenient to work with a complemented form of (58). For each S  U , let r.S / D q.U n S /. Then (58) can be rewritten as Q D .1/n

X

.1/jS j r.S /:

(59)

S U

Observe that the induced subgraph GŒfsg [ S  need not be connected. Associate with each S  U the unique f .S /  U such that GŒfsg [ f .S / is the connected component of GŒfsg [ S  that contains s. It follows that r.S / D r.f .S // for all S  U . This observation enables the following partition of the subsets of U into 9 f -preimages of constant r-value. For each T  U , let f 1 .T / D fS  U W f .S / D T g. Then rewrite (59) in the partition form

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X

Q D .1/n

X

r.T /

T U

.1/jS j :

(60)

S 2f 1 .T /

Lemma 9 X

( .1/

jS j

D

S 2f 1 .T /

.1/jT j ;

if fsg [ T 2 CD

0;

otherwise:

(61)

Proof Consider an arbitrary T  U . The preimage f 1 .T / is clearly empty if GŒfsg [ T  is not connected. Thus, in the following assume that GŒfsg [ T  is connected. For a set W  V , denote N .W / the set of vertices in W or adjacent to at least one vertex in W . Observe that f .S / D T holds for an S  U iff T  S and S \ N .fsg [ T / D T . In particular, if V n N .fsg [ T / is nonempty, then f 1 .T / contains equally many even- and odd-sized subsets. Conversely, if V n N .fsg [ T / is empty (which means that fsg [ T is a dominating set of G), then f 1 .T / D fT g.  Using the above lemma to simplify (60), we have X

Q D .1/n

.1/jT j r.T /:

(62)

T U;fsg[T 2CD

To arrive at an algorithm with running time jCDjnO.1/ , it suffices to list the elements of CD with delay bounded by nO.1/ . Observe that CD is an upclosed family of subsets of V , that is, if a set is in the family, then so are all of its supersets. Furthermore, whether a given W  V is in CD can be decided in polynomial time. These observations enable listing the sets in CD in a top-down, depth-first manner. Thus, the only remaining task is to bound the size of CD. Lemma 10 Let V be a finite set with r elements and with subsets A1 ; A2 ; : : : ; Ar such that every v 2 V is contained in exactly ı subsets. Let F be a family of subsets of V and assume that there is a log-concave function f  1 and 0  s  r such that the projections Fi D fF \ Ai W F 2 F g satisfy jFi j  f .jAi j/ for each s C 1  i  r. Then, s Y jF j  f .ı/r=ı 2jAi j=ı : (63) i D1

Proof Let ai D jAi j and note that ı

jF j 

s Y i D1

2

ai

Pr

i D1 ai

r Y i DsC1

D ır. By Lemma 8,

f .ai / 

s Y i D1

Since f is log-concave, Jensen’s inequality gives

2

ai

r Y i D1

f .ai /:

(64)

Faster and Space Efficient Exact Exponential Algorithms

1X log f .ai /  log f r i D1 r

r X

1271

! ai =r

! D log f .ı/:

(65)

i D1

Taking exponential and combining with (64) gives jF jı  f .ı/r yields the claimed bound.

Qs

i D1

2jAi j , which 

In order to bound the size of CD, we need only to consider the special case where Ai are defined in terms of neighborhoods of the vertices of G. For each v 2 V , define the closed neighborhood N.v/ by N.v/ D fvg[fu 2 V W u and v are adjacent in Gg. Begin by defining the subsets Av for v 2 V as Av D N.v/. Then, for each u 2 V with degree d.u/ < , add u to   d.u/ of the sets Av not already containing it (it does not matter which ones). This ensures that every u 2 V is contained in exactly  C 1 n sets Av . Combining with the following given bound on the size of CD, an O  .ˇ / C1 1=.C1/ time algorithm can be obtained as described above, where ˇ D .2  2/ . n Cn Lemma 11 An n-vertex graphs with maximum vertex degree  has at most ˇ C1 1=.C1/ connected dominating sets, where ˇ D .2  2/ .

Proof Let CD0 D CD n ffvg W v 2 V g. Then for every C 0 2 CD0 and every Av , C 0 \ Av ¤ fvg. Thus, the number of sets in the projection CD0 v D fF \ Av W F 2 CD0 g is at most 2jAv j  2, since F \ Av ¤ ; for any F 2 CD0 . To obtain the bound, apply Lemma 10 with the log-concave function f .x/ D 2x  1 and s D 0. 

3

Algebraic Methods

The application of algebraic methods in designing exact algorithms can be dated back to the famous paper by Kohn et al. [39] on solving the traveling salesman problem using the generating polynomials (for details, please refer to Example 10), P in which by involving an intermediate x, a polynomial G.x/ D ai x ei is produced, where ei denotes the cost of a possible tour and ai denotes the number of tours having this cost. Then the minimal tour cost, that is, the smallest exponent, can be extracted by evaluating G.x/ at a specific value close to 0 and using a logarithmic technique. In this section, two recently developed algebraic methods are introduced – algebraic sieving method and polynomial circuit-based algebraic method.

3.1

Algebraic Sieving

Sieving methods are commonly used in designing exact and parameterized algorithms and have a long history. For example, the inclusion-exclusion principle is a typical sieving method by which all possibilities are first counted, and then, false contributions are canceled out. Here, a new algebraic sieving method which makes use of determinants over a field of characteristic two to achieve sieving is introduced.

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The basic idea is to construct a polynomial in the underlying variables over a finite field in which at least one unique monomial exists for each required structure (e.g., Hamiltonian cycle) and no monomials results from unwanted structures (e.g., nonHamiltonian cycle covers). The existence of structures being looked for ultimately emerges as a polynomial testing problem. Then the old fingerprint idea, often attributed to Freivalds (cf. [43]), is applied in which the polynomial is evaluated in a randomly chosen point in a finite field. The fingerprint result is used to judge whether the constructed polynomial is zero polynomial or not. The SchwartzZippel Lemma (cf. [43]) ensures that such judgment will be correct with high probability. The key point of this method is how to sieve the unwanted monomials to obtain the final required polynomial. So far only determinants over a finite field are taken advantage of to achieve sieving. Whether there exists some other more powerful tools that can be used for sieving under the algebraic framework needs further studying. Before going into the details, some notations need to be defined. The determinant of an n n matrix A over an arbitrary ring R can be defined by the Leibniz formula: det.A/ D

X

sgn./

 WŒn!Œn

n Y

Ai; .i / ;

(66)

i D1

where the summation is over all permutations of n elements, and sgn is a function called the sign of the permutation which assigns either 1 or 1 to a permutation. The permanent of A is defined by

per.A/ D

n X Y

Ai; .i / :

(67)

 WŒn!Œn i D1

Denote GF .2k / as a finite field of characteristic two. It is well known that the determinant of A over GF .2k / coincides with the permanent, since in such a field every element serves as its own additive inverse, in particular the element 1; the sgn function identically maps 1 to every permutation. Moreover, although the determinant is a sum of an exponential number of terms, it can be computed in polynomial time using for instance LU factorization of the matrix which can be found in any textbook on linear algebra. In fact, to compute the determinant is not harder than square matrix multiplication (cf. [21]). Hence, the determinant can be computed in O.n! / field operations where ! is the Coppersmith-Winograd exponent [24]. The following properties and lemma will be used in deriving algebraic sieving algorithms. Property 1 A is a matrix over GF .2k /, then per.A/ D det.A/. Property 2 For a given n n matrix A, det.A/ can be computed in O.n! / time.

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Lemma 12 (Schwartz-Zippel) Let P .x1 ; : : : ; xn / be a non zero n-variate polynomial of total degree d over a field F . Pick r1 ; : : : ; rn 2 F uniformly at random, then d : (68) Pr .P .r1 ; : : : ; rn / D 0/  jF j

3.1.1 k-Dimensional Matching A hypergraph H D .V; E/ consists of a set V of n vertices and a multiset E of hyperedges which are subsets of V . In particular, hyperedges may include only one (or even no) vertex and may appear more than once. In a k-uniform hypergraph each edge e 2 E has size jej D k. Given a vertex subset U of V , the projected hypergraph of H on U , denoted as H ŒU  D .U; EŒU /, is a hypergraph on U where there is one edge eU 2 EŒU  for every e 2 E, defined by eU D e \ U . Definition 4 (k-Dimensional Matching) Given a k-uniform hypergraph H D .V1 [ V2    [ Vk ; E/, with E  V1 V2    Vk , where jVi j D n for 1  i  k, the k-dimensional matching problem asks whether there exists a k-dimensional matching S  E such that [e2S e D V1 [V2   [Vk and 8e1 ¤ e2 2 S : e1 \e2 D ;. In the following, each hyperedge e is associated with a variable xe over GF .2m / for some m  log n C 1. The next lemma shows how to construct a polynomial only consisting of monomials representing k-dimensional matchings by sieving unwanted monomials. Lemma 13 Denote M as the family of all k-dimensional matchings. Let V D [kiD1 Vi and U D V1 [ V2 , then X

X Y

P .H; U; X / D

X V nU

xe ;

(69)

M 2M e2M

where the computation is over a multivariate polynomial ring over GF .2m /, and P .H; U; X / D

X

Y

xe ;

(70)

M 0 2M0 e2M 0

where M0 is the family of all perfect matchings in H ŒU  avoiding X , that is, for every e 2 M 0 , e \ X D ;. Q Proof For every M 2 M, M only avoids ;. Thus, the monomial e2M xe for every k-dimensional matching M emerges in the constructed polynomial precisely once. For a non-k-dimensional matching M 0 , it avoids all subsets of V n .[e2M 0 e/. Q Then the monomial e2M 0 xe for M 0 will be considered in P .H; U; X 0 / for every X 0  V n .[e2M 0 e/. It is Q well known that a nonempty set has even number of subsets. Hence, all of these e2M 0 xe cancel each other since the operations are in a field of characteristic two. 

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P From Lemma 13, X V nU P .H; U; X / consists of P monomials representing k-dimensional matchings of H . Thus, by evaluating X V nU P .H; U; X / on a randomly chosen point r1 ; : : : ; rjEj 2 GF .2m /, the result is nonzero with probability at least 12 if there exists at least one k-dimensional matching in H by Lemma 12. If repeating the evaluation for polynomial times, the error probability can be exponentially small. The remaining work is how to efficiently compute P .H; U; X / on a randomly chosen point r1 ; : : : ; rjEj 2 GF .2m / for every X  V n U . Clearly, H ŒU  is a bipartite multigraph. Define its Edmonds matrix A.H; V1 ; V2 / as X A.H; V1 ; V2 /ij D xe : (71) eD.i;j /2EŒU ;i 2V1 ;j 2V2

Denote M as the family of all perfect matchings in H ŒU . Then the following lemma generalizes Edmonds’ work [26] for the case of bipartite multigraphs. Q P Lemma 14 det.A.H; V1 ; V2 // D O e2M xe , where the computation is M 2M over a multivariate polynomial ring over GF .2m / and the summation is over all O in H ŒV1 [ V2 . perfect matchings M Proof By the definition of determinant, the summation is over all products of n of the matrix elements in which every row or column is used exactly once. In terms of the bipartite graph, this corresponds to a perfect matching in the graph since rows and columns represent the two vertex sets, respectively. Furthermore, the converse is also true. For every perfect matching there is a permutation describing it. Hence, the mapping is one-to-one. The inner product counts all choices of edges producing a matching described by a permutation  since n Y i D1

Ai; .i / D

n Y

X

i D1 eD.i; .i //

xe D

X

Y

xe ;

(72)

M 2M. / e2M

where M./ is the set of all perfect matchings fe1 ; : : : ; en g such that ei D .i; .i //. Let HX ŒU  denote the projected hypergraph of H on U by only projecting edges disjoint to X in H on U . Based on the above Lemma, P .H; U; X / on a randomly chosen point r1 ; : : : ; rjEj 2 GF .2m / for every X  V n U can be computed by first constructing the Edmonds matrix of HX ŒU , with the variables replaced by the random sample points r1 ; : : : ; rjEj , and then computing its determinant. Putting everything together generates a Monte Carlo algorithm with exponentially small error probability for the k-dimensional matching problem. The runtime bound is easily seen to be O  .2.k2/n / since jU j D jV1 j C jV2 j D 2n and P .H; U; X / for every X  V n U can be evaluated in O.n! / time by Lemma 14 and Property 2. Theorem 9 The k-dimensional matching problem on k n vertices can be solved by a Monte Carlo algorithm with an exponentially small failure probability in n and O  .2.k2/n / runtime.

Faster and Space Efficient Exact Exponential Algorithms

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3.1.2 Hamiltonicity An undirected graph G D .V; E/ on n vertices is said to be Hamiltonian if it has a Hamiltonian cycle, a vertex order .v0 ; v1 ; : : : ; vn1 / such that vi vi C1 2 E for all i . The indices are enumerated modulo n, that is, vn1 v0 is also an edge. The problem of detecting if a graph is Hamiltonian is called the Hamiltonicity problem. Clearly, this problem is a special case of the traveling salesman problem. Before the algebraic sieving algorithm for the Hamiltonicity problem was derived, the dynamic programming recurrence that solves the general TSP in O.n2 2n / time introduced by Bellman [6, 7] and independently by Held and karp [30] in the early 1960s was the strongest result for this special problem. The algebraic sieving for the Hamiltonicity problem is obtained by reducing the problem to a variant of the cycle cover counting problem called labeled cycle cover sum, and then making use of the sieving algorithm to solve this variant. The algorithm will be described after introducing some useful notations. In a directed graph D D .V; A/, a cycle cover is a subset C  A such that for every vertex v 2 V , there is exactly one arc av1 2 C starting from v and exactly one arc av2 2 C ending in v. The graphs considered have no loops. Denote cc.D/ as the family of all cycle covers of D, and hc.D/  cc.D/ as the set of Hamiltonian cycle covers. A Hamiltonian cycle cover consists of one big cycle passing through all vertices. The remaining cycle covers (which have more than one cycle in cc.D/ n hc.D/) are called non-Hamiltonian cycle covers. For undirected graphs, hc.G/ includes the Hamiltonian cycles with orientation, that is, traversed in both directions. For a Hamiltonian cycle H 2 hc.G/ for an undirected graph G, an arc uv 2 G infers that the cycle is oriented from u to v along the edge uv. Denote g W A ! B as a surjective function from the domain A to the codomain B and denote the preimage of every b 2 B as g1 .b/, that is, g 1 .b/ D fa 2 A W g.a/ D bg. For a matrix A, denote by Ai;j the element at row i and column j . The labeled cycle cover sum problem is defined as follows: Definition 5 Given a directed graph D D .V; A/, a finite set L of labels, and a function f W A 2L ! R on some codomain ring R, the labeled cycle cover sum problem is to compute the following defined term over R. ƒ.D; L; f / D

X

X Y

f .a; g 1 .a//;

(73)

C 2cc.D/ gWL!C a2C

where the inner sum is over all surjective functions g. In the above definition, f is introduced to make all non-Hamiltonian cycle covers cancel out in the sum. To do this, as before, the computations will be done over GF .2k / denoting a finite field of characteristic two. In what follows, a directed graph is bidirected if for every arc uv it has an arc in the opposite direction, that is, vu. In order to make sure that the Hamiltonian cycle cover will not be canceled as well, an asymmetry around an arbitrarily chosen special vertex s is defined. A function f W A 2L ! GF .2k / is an s-oriented mirror function if

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f .uv; Z/ D f .vu; Z/ for all Z and all u ¤ s; v ¤ s. The following lemma captures how the non-Hamiltonian cycle covers vanish, which also implies the nonexistence of false positives in the resulting algorithms. Lemma 15 Given a bidirected graph D D .V; A/, a finite set L, and a special vertex s 2 V , let f be an s-oriented mirror function with codomain GF .2k /. Then X X Y ƒ.D; L; f / D f .a; g 1 .a//: (74) H 2hc.D/ gWL!H a2H

Proof By the definition of the labeled cycle cover sum, a labeled cycle cover is a tuple .C; g/ with C 2 cc.D/ and g W L ! C . The labeled non-Hamiltonian cycle covers can be partitioned into dual pairs as follows such that both cycle covers in every pair contribute the same term to the sum. For a labeled non-Hamiltonian cycle cover .C; g/, let C be the first cycle of C not passing through s (note that such cycle must exist since the cycle cover consists of at least two cycles and all cycles are vertex disjoint). Here, “first” is w.r.t. any fixed order of the cycles. Then the dual cycle cover of .C; g/ is defined as R.C; g/ D .C 0 ; g 0 /, where C 0 D C except for the cycle C which is reversed in C 0 , that is, every arc uv 2 C is replaced by the arc in the opposite direction vu in C 0 , and g 01 is identical to g 1 on C n C, and is defined by g01 .uv/ D g 1 .vu/ for all arcs uv 2 C. In other words, the reversed arcs preserve their original labeling. Note that such a dual cycle cover always exists since D is bidirected and .C; g/ ¤ R.C; g/,.C; g/ D R.R.C; g//. Hence, the mapping uniquely pairs up the labeled non-Hamiltonian cycle covers. Since f is an s-oriented mirror function and has f .uv; Z/ D f .vu; Z/ for all arcs uv not incident on s, .C; g/ and R.C; g/ contribute the same product term to the sum for computing ƒ.D; L; f /. Finally, since the computations are done in a field of characteristic two, all these terms will be canceled.  When limiting the computations over GF .2k /, the labeled cycle cover sum can be computed efficiently via determinant. Permanent has a natural interpretation as the sum of weighted cycle covers in a directed graph. Formally, let D D .V; A/ be a directed graph with weights w W A ! GF .2k /, and define a jV j jV j matrix with rows and columns representing the vertices V ( Ai;j D

w.ij /;

if ij 2 A

0;

otherwise:

Then, per.A/ D

X Y C 2cc.D/ a2C

w.a/:

(75)

(76)

Faster and Space Efficient Exact Exponential Algorithms

1277

By Property 1 and 2, per.A/ is identical with det.A/ and can be computed in O.n! / time. Define a polynomial in an indeterminate r as follows, with r aiming at controlling the total rank of the subsets used as labels in the labeled cycle covers: ! X X p.f; r/ D det r jZj Mf .Z/ ; (77) Y L

where

ZY

( Mf .Z/i;j D

f .ij; Z/;

if ij 2 A

0;

otherwise:

(78)

Lemma 16 For a directed graph D, a set L of labels, and any function f W A L ! GF .2k /, Œr jLj p.f; r/ D ƒ.D; L; f /;

(79)

where Œr jLj p.f; r/ denotes the coefficient of r jLj in p.f; r/. Proof p.f; r/ can be rewritten as p.f; r/ D

X

X

X

Y

r jq.a/j f .a; q.a//

Y L C 2cc.D/ qWC !2Y nf;g a2C

D

X

X

X

Y

r jq.a/j f .a; q.a//:

(80)

C 2cc.D/ qWC !2L nf;g [a2C q.a/Y L a2C

For functions q W C ! 2L n f;g such that [a2C q.a/  L, that is, whose union over the elements do not cover all of L, the innermost summation is executed an even number of times with the same term (there are 2jL[a2C q.a/j equal terms). Since the computations are over GF .2k /, these cancel. Then, p.f; r/ D

X

X

P

r

a2C

C 2cc.D/ qWC !2L nf;g [a2C q.a/DL

jq.a/j

Y

f .a; q.a//;

(81)

f .a; q.a//;

(82)

a2C

in which the coefficient of r jLj Œr jLj p.f; r/ D

X C 2cc.D/

X

Y

a2C qWC !2L nf;g [a2C q.a/DL 8a¤bWq.a/\q.b/D;

since [a2C q.a/ D L; [a2C jq.a/j D jLj implies 8a ¤ b W q.a/ \ q.b/D;. Inverting the function q will give the labeled cycle cover sum as defined in Definition 5. 

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The above lemma is the base identity that enables a relatively efficient algorithm for computing the labeled cycle cover sum. Lemma 17 The labeled cycle cover sum ƒ.D; L; f / over a field GF .2k / on a directed graph D on n vertices, and with 2k > jLjn, can be computed in O..jLj2 nC jLjn1C! /2jLj / arithmetic operations over GF .2k /, where ! is the CoppersmithWinograd matrix multiplication coefficient. Proof The labeled cycle cover sum is evaluated via the identity in Lemma 16. Observing that p.f; r/ as a polynomial in r has maximum degree jLjn, to recover one of its coefficients (the one for r jLj ), one needs to evaluate the polynomial for jLjn choices of r and to use interpolation to solve for the coefficient being sought. This can be done for instance by using a generator g of the multiplicative group in GF .2k / and evaluating the polynomial in the points r D g 0 ; g 1 ; : : : ; gjLjn1 . The requirement 2k > jLjn assures the distinctness of these points, and hence, the interpolation is P possible. For evjZj ery fixed r, the algorithm begins by tabulating T .Y / D Mf .Z/ ZY r jLj for all Y  L through the fast zeta transform (4) in O.jLj2 / field opP ! jLj erations. Then p.f; r/ D det.T .Y // can be evaluated in O.n 2 / Y L operations by Property 2. Once all values are computed, the polynomial time Lagrange interpolation can be used to compute the coefficient. Summing up the number of field operations required over all jLjn values of r, the lemma follows.  Next, f will be defined to associate the argument arc, and label set with a nonconstant multivariate polynomial such that f .su; X / and f .us; X / will not share variables for any su; us 2 A to ensure that the Hamiltonian cycles oriented in opposite directions will contribute different terms to the sum. Based on the definition of f , the labeled cycle cover sum is represented by a polynomial in the underlying variables with one unique monomial per oriented Hamiltonian cycle and without monomials resulting from non-Hamiltonian cycle covers, which is a consequence of Lemma 15. Then, the fingerprint idea is employed to evaluate the polynomial in a randomly chosen point, and the Schwarz-Zippel Lemma ensures that with high probability, it can be detected whether the polynomial resulting from the labeled cycle cover sum is identically zero (i.e., no Hamiltonian cycles) or not (there is at least one Hamiltonian cycle) by identifying the evaluating result with the polynomial. For simplicity of analysis, assume that n is even. For a uniformly randomly chosen partition V D V1 [ V2 with jV1 j D jV2 j D n=2 and a fixed Hamiltonian cycle H D .v0 ; : : : ; vn1 / in G. An arc vi vi C1 is called unlabeled by V2 if both vi and vi C1 belong to V1 , and the set of all unlabeled arcs is denoted as U.H /. The remaining arcs are referred to as labeled by V2 and denote the set of all labeled arcs as L.H /. Define hc.G; V2 ; m/ as the subset of hc.G/ of Hamiltonian cycles which have precisely m unlabeled arcs by V2 . Assign every edge uv 2 E two variables xuv and xvu , and xuv and xvu are identified except when u D s or v D s.

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1279

Consider a complete bidirected graph D D .V1 ; F / and use V2 as some of the labels. In addition to V2 , a set Lm of size m of extra labels aiming at handling arcs unlabeled by V2 . For each edge uv in GŒV1  and every element d 2 Lm , new variables xuv;d and xvu;d are introduced. And xuv;d coincides with xvu;d except when u D s or v D s. For two vertices u; v 2 V , and a subset X  V , define Pu;v .X / as the family of all simple paths in G from u to v passing through exactly the vertices in X . For uv 2 F and X  V2 , define X

f .uv; X / D

Y

xwz :

(83)

P 2Pu;v .X / wz2P

For every arc uv 2 F such that uv is an edge in GŒV1 , and d 2 Lm , define f .uv; fd g/ D xuv;d . In other points f is set to zero. Lemma 18 With G; D; V2 ; U./; L./; m; L as above, m and f defined Q P Q P x (i) ƒ.D; V2 [ Lm ; f / D H 2hc.G;V2 ;m/ uv2L.H / uv  WU.H /!Lm uv2U.H /  xuv; .uv/ ; (ii) ƒ.D; V2 [ Lm ; f / is a zero polynomial iff hc.G; V2 ; m/ D ;. Proof (i) Since D is bidirected and f is s-oriented mirror function, by Lemma 15, X

ƒ.D; V2 [ Lm ; f / D

X

Y

f .a; g 1 .a//;

(84)

H 2hc.D/ gWV2 [Lm !H a2H

where the inner sum is over all surjective functions g. In order to prove the claim, it needs to expand the Hamiltonian cycles in D into Hamiltonian cycles of G. Note that the arcs of a Hamiltonian cycle in D in the sum above are labeled either by an element of Lm or by a nonempty subset of V2 , since only in such cases f is nonzero. Next, the definition of labeled and unlabeled arcs are extended (which were defined previously for Hamiltonian cycles in G only). For a Hamiltonian cycle H 2 hc.D/ labeled by the surjective function g W V2 [Lm ! H , an arc uv 2 H is called labeled by V2 if g 1 .uv/  V2 and unlabeled by V2 if g 1 .uv/ 2 Lm . Since every arc uv unlabeled by V2 , g 1 .uv/ is an element of Lm , only the Hamiltonian cycles in D with exactly m arcs unlabeled by V2 leave a nonzero contribution. Then, X

ƒ.D; V2 [ Lm ; f / D

0

X

@

H 2hc.D/ HL [HU DH HL \HU D; jHU jDm

0 @

X

Y

 WHU !Lm a2HU

Y

X

1 f .a; g 1 .a//A 

gWV2 !HL a2HL

1

f .a; .a//A ;

(85)

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where the summation is over all surjective functions g and one-to-one functions . Replace f in the above expression, then X

ƒ.D; V2 [ Lm ; f / D

0

X

@

H 2hc.D/ HL [HU DH HL \HU D; jHU jDm 8a2HU Wa2E

0 @

X

Y

Y

X

X

gWV2 !HL a2HL P 2Pu;v

Y

1 xwz A 

.g 1 .uv// wz2P

1

xuv; .uv/ A ;

 WHU !Lm uv2HU

(86) Every vertex in V2 is mapped to precisely one arc in F on a Hamiltonian cycle H in D by g. Moreover, such a cycle H leaves a nonzero result iff there are precisely m arcs in the cycle not being mapped to by g (i.e., unlabeled by V2 ) and they are also edges in G. Rewriting the expression as a sum of Hamiltonian cycles in G,

ƒ.D; V2 [ Lm ; f / D

X H 2hc.G;V2 ;m/

0 @

Y uv2L.H /

10 xuv A @

X

Y

1 xuv; .uv/ A :

 WU.H /!Lm uv2U.H /

(87) (ii) If the graph G has no Hamiltonian cycles, the sum is clearly zero. For the other direction, each Hamiltonian cycle contributes a set of mŠ different monomials per orientation of the cycle (one for each permutation ), in which there is one variable per edge along the cycle. Monomials resulting from different Hamiltonian cycles are different since for each of two different Hamiltonian cycles, one has an edge that the other does not have. Every pair of opposite directed Hamiltonian cycles along the same undirected Hamiltonian cycle also traverse s through opposite directed arcs. Since the variables tied to the opposite directed arcs incident on s are different, these are also unique monomials in the sum.  The above Lemma describes how to transform the input graph G given m, V1 ; V2 into a symbolic labeled cycle cover sum ƒ.D; V2 [ Lm ; f / on a bidirected graph D on n=2 vertices and n=2 C m labels V2 [ Lm . The next lemma shows that only the cases m  n=4 need to be considered. Lemma 19 Let G D .V; E/ be a Hamiltonian undirected graph. For a uniformly randomly chosen equal partition V1 [ V2 D V , jV1 j D jV2 j D n=2, 0 1 n=4 X Pr @ jhc.G; V2 ; i /j > 0A  i D0

1 : nC1

(88)

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Proof Consider one Hamiltonian cycle .v0 ; : : : ; vn1 / in G. Let X denote the random variable representing the number of unlabeled arcs by V2 . Clearly, P r.vi / D 12 for any i , and P r.vi ; vi C1 2 V1 / < 14 since the events vi 2 V1 and vi C1 2 V1 are almost independent. Hence, the expected number of arcs unlabeled 1 by V2 , E.X / < n=4. By Markov’s inequality, P r.X > .1 C 1=n/E.X // < 1C1=n , and the lemma follows. 

Before discussing the algorithm, the only remaining problem is how to calculate f for all subsets of V2 . This can be achieved by running a variant of the BellmanHeld-Karp recursion. Formally, let fO W .V V / 2V ! GF .2k / be defined by fO.uv; X / D

X

Y

xwz :

(89)

P 2Pu;v .X / wz2P

Then f .uv; X / D fO.uv; X / for u; v 2 V1 ; X  V2 , and the recursion (

xuv ; fO.uv; X / D P

if jX j D 0 O w2X;uw2E xuw f .wv; X n fwg/; otherwise:

(90)

can be used to tabulate f .; X / on X  V2 . The running time is O  .2jV2 j /. Next, the algorithm is described in detail. First, pick a partition V1 [ V2 D V uniformly at random with jV1 j D jV2 j D n=2. Then, the algorithm loops over m, the number of edges unlabeled by V2 along a Hamiltonian cycle from 0 to n=4. For each value of m, by Lemma 18, the problem is transformed to determine whether the symbolic labeled cycle cover sum ƒ.D; V2 [ Lm ; f / is a nonzero polynomial. As described in Lemma 17, ƒ.D; V2 [ Lm ; f / will be evaluated in a randomly chosen point over the field GF .2k / with 2k > cn for some c > 1. By Lemma 12, with probability at least 1  1=c it will result in a nonzero answer iff ƒ.D; V2 [ Lm ; f / was a nonzero polynomial (i.e., G has a Hamiltonian cycle with m edges unlabeled by V2 ). Then repeat the algorithm for every m  n=4, and by Lemma 19, with probability at least .1  1=c/=.n C 1/, the presence of a Hamiltonian cycle can be detected if it exists. Running the algorithm a polynomial number of times in n, the probability of failure will be reduced exponentially small. The runtime is bounded by a polynomial factor in m and n of the runtime in Lemma 17. The worst case occurs for m D 4=n in which case the time bound is O  .23n=4 /. Theorem 10 There is a Monte Carlo algorithm detecting whether an undirected graph with n vertices is Hamiltonian or not in O  .23n=4 / time, with no false positives and no false negatives with probability exponentially small in n.

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Polynomial Circuit-Based Algorithms

This method only needs polynomial space. The problems to be solved will be translated into polynomials represented by polynomial circuits, where the translation is done in such a way that the instance considered is a “yes”-instance iff the coefficient of a special monomial in the polynomial is nonzero; this allows us only to compute the coefficient of the special monomial. A discrete Fourier transformbased technique provides an efficient way to achieve the goal.

3.2.1 Coefficient Interpolation Given a set R and a set of variables X , an arithmetic circuit C over .R; C; / is a directed acyclic graph in which each source gate is labeled by a variable x 2 X or an element of R and other gates is either an addition gate or a product gate. The size jC j of a circuit C is the number of gates in C . The depth .C / of a circuit C is the length of a longest directed path in C . Denote by N and C the naturals and the complex numbers, respectively. Computing the coefficient of some special monomial can be abstracted as the following coefficient interpolation problem which can be solved efficiently based on the discrete Fourier transform. Definition 6 (Coefficient Interpolation) Given an arithmetic circuit C over .N; C; / and variables x1 ; x2 ;    ; xˇ over N that computes a polynomial P .x1 ; : : : ; xˇ /, together with ˇ integers t1 ; t2 ; : : : ; tˇ , the problem is to compute t the coefficient of the term x1t1 x2t2    xˇˇ . Denote the coefficients of P as a ˇ-dimensional array fcn1 ;n2 ;:::;nˇ g. Furthermore, assume that the coefficients of any polynomial outputted by a gate of C are bounded by m and any exponent of xi in the terms of P is bounded by Ni  1 for i  ˇ. Note that P can also be interpreted as a polynomial over the complex numbers. The discrete Fourier transform of the ˇ-dimensional array fcn1 ;n2 ;:::;nˇ g is the array fCn1 ;n2 ;:::;nˇ g given by the following formula:

Ck1 ;k2 ;:::;kˇ D

Nˇ 1 ˇ N 1 1 N 2 1 X X X Y 1  cn1 ;n2 ;:::;nˇ !Nkiini N1 N2    Nˇ n D0 n D0 n D0 i D1 1

2

ˇ

(91)

k

D P .!Nk11 ; !Nk22 ; : : : ; !Nˇˇ /;

where !Ni is the i -th complex root of unity, given by e 2 i=Ni . Then the inverse discrete Fourier transform can give the coefficient cn1 ;n2 ;:::;nˇ , as follows:

Faster and Space Efficient Exact Exponential Algorithms

ct1 ;t2 ;:::;tˇ D

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Nˇ 1 ˇ N 1 1 N 2 1 X X X Y 1  Cn1 ;n2 ;:::;nˇ !Nti i ni N1 N2    Nˇ n D0 n D0 n D0 i D1 1

2

ˇ

Nˇ 1 ˇ N 1 1 N 2 1 X X X 1 nˇ Y .Ni ti /ni n1 n2 D  P .!N1 ; !N2 ; : : : ; !Nˇ / !Ni : N1 N2    Nˇ n D0 n D0 n D0 i D1 1

2

ˇ

(92) Thus, by evaluating the polynomial P for N1 N2    Nˇ different arrays of values, which can be done using the arithmetic circuit C , ct1 ;t2 ;:::;tˇ can be computed based on the above formula. Hence, ct1 ;t2 ;:::;tˇ can be computed with O.N1 N2    Nˇ jC j/ real multiplications and additions, and only O.ˇ C jC j/ real numbers need to be stored at any point of the calculation. In order to overcome the problem that the real number cannot be exactly stored and worked with, the binary representations of the numbers with the bits after the l most significant bits truncated will be used instead during the execution of the algorithm. Here, the number l is chosen such that if rounding the resulting estimate c 0 of ct1 ;t2 ;:::;tˇ to the nearest integer, ct1 ;t2 ;:::;tˇ can be correctly obtained. Observe that addition of these estimates can be done in time O.l/, and that multiplication can be carried out in O  .l/ time using a fast algorithm for integer multiplication, such as the F¨urer’s algorithm [29]. Hence, the total time spent by the algorithm is O  ..N1 N2    Nˇ /l/ and the total space is O..ˇ C jC j/l/. The following lemma makes sure the existence of such a choice for l.

Lemma 20 ([42]) Let C be an arithmetic circuit over .C; C; / computing a polynomial P .x1 ; x2 ; : : : ; xˇ / such that (a) all constants of C are positive integers, (b) all coefficients of P are at most m, and (c) P has degree d .d / C has depth ˛. Let x1 ; x2 ; : : : ; xˇ 2 C; x1 0 ; x2 0 ; : : : ; xˇ 0 2 C be such that for every i , kxi k D 1 and kxi xi 0 k  . Let p 0 be the result of applying the circuit C to x1 0 ; x2 0 ; : : : ; xˇ 0 using the floating point arithmetic, truncating intermediate results after l bits. Suppose . C 2  2l /.4md /˛  1. Then kp 0  P .x1 ; : : : ; xˇ /k  . C 2  2l /.4md /˛ . For a fixed value of l, one can compute for every i  ˇ an estimate of !Ni whose distance to !Ni is at most 2  2l . Now, given n1 ; : : : ; nˇ ; t1 ; : : : tˇ ; N1 ; : : : ; Nˇ , based on the circuit C for computing P , a circuit C 0 can be constructed for computing ˇ nˇ Y .Ni ti /ni n1 n2 0 xi : (93) P D P .x1 ; x2 ; : : : ; xˇ / i D1

Clearly, the depth of C 0 is ˛ C log N C log ˇ, where N D maxi Ni . Furthermore, the degree of C 0 is at most d CˇN , and the largest coefficient is at most md . Choose l such that 10  2l .4.d C ˇN /md /˛Clog N Clog ˇ  1. Applying Lemma 20 on C 0 yields an estimation error for

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D. Yu et al. ˇ    Y n n .N t /n P 0 !Nn11 ; !Nn22 ; : : : ; !Nˇˇ D P !Nn11 ; !Nn22 ; : : : ; !Nˇˇ !Ni i i i ;

(94)

i D1

N N N

which is less than 12 . The total estimation error for ct1 ;t2 ;:::;tˇ is less than 2N11 N22 Nˇˇ D 12 . Thus, c 0 rounded to the nearest integer is ct1 ;t2 ;:::;tˇ . Finally, observe that l D ..˛ C log d /.log m C log d //, which yields the following concluding theorem. Theorem 11 The coefficient interpolation problem can be solved in O  .N1    Nˇ ..C /Clog d /.log mClog d // time and O.ˇCjC j..C /Clog d /.log mClog d // space. Remark Here, it is necessary to point out that the indegree of the constructed circuit C is implicitly required to be bounded by a constant. By transforming the problems to polynomials, the above technique can provide pseudo-polynomial time polynomial space algorithms for several hard problems, for example, the Knapsack problem. Example 11 (Knapsack Problem) Given a set S D fs1 ; : : : ; sn g of n items each of which has positive integer weight wi and value vi , and two positive integers w and v, the Knapsack problem asks whether there exists a subset S 0 of items whose total weight is at most w and total value is at most v. It may further assume that all items have weight at most w and value less than v, since otherwise this problem is trivial. The Knapsack problem can be reduced to coefficient interpolation as follows. Pnw Pnv i j Define the polynomial PS .x; y/ D i D0 j D0 cij x y , where cij is the 0 number of item sets S  S whose total weight is exactly i and the total value of the items not in S 0 is exactly j . The Knapsack problem is equivalent to P checking whether there exists a weight w0  w and value v0  . si 2S vi /v such that cw0 v0 is nonzero. Clearly, this can be done by solving coefficient interpolation for each pair of possible values for w0 and v0 , but this would incur an unnecessary overhead ofPvw in thePrunning time. Instead, define the polynomial nv i j 0 P 0 S .x; y/ D pS .x; y/. nw i D0 x /. j D0 y / and set cij to be the coefficient of 0 0  the term Pxi yj in P S . Then it suffices to check whether c wv is nonzero, where0  v D si 2S vi  v. The remaining task is to construct a polynomial circuit C for P 0 S such that faster polynomial space algorithms can be obtained by applying Theorem 11. Q Note that PS .x; y/ can be factorized into PS .x; y/ D niD1 .x wi C y vi / which yields a circuit of size O.n log w C n log v/ and depth O.log w C log v C log n/ for computing PS . BasedP on the following given circuit of size and depth O.log n C wi log w/ for computing and circuit of size and depth O.log n C log v/ for si 2S x P vi computing si 2S y , a circuit C 0 of size O.n log w C n log v/ and depth O.log w C log v C log n/ for computing P 0 S can be constructed.

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Lemma 21 There exists a circuit of size and depth O.log m/ that computes Pm i x . i D0 Proof The following recurrence can be turned into a circuit for computing p X j D0

( j

x D

.x p=2 C 1/ x.x

bp=2c

Pp=2

C

j j D0 x ; Pbp=2c 1/ j D0

if p is even j

x C 1; otherwise:

Pm

i D0 x

i

.

(95)

To make the circuit it needs to construct gates evaluating x bp=2c ; x bp=4c ; x bp=8c , etc. This can be done using an identical trick to the one above by observing that x p D .x p=2 /2 if p is even and x p D x.x bp=2c /2 if p is odd.  Finally, note that the coefficients of pS are at most 2n , and hence, the coefficients of P 0S are at most 2n wvn2 . Also note that the degree of PS 0 is at most 2nv C 2nw. Then applying Theorem 11, one can get an algorithm for the Knapsack problem with time complexity O  .n4 vw.log v C log w/2 / and space complexity O  .n2 .log v C log w/2 /. 

3.2.2 Combination with Subset Convolution By introducing some special gates (e.g., the subset convolution gate and the covering product gate introduced in Sect. 2.2) in the constructed polynomial circuits, some polynomial space algorithms for many other problems can be derived, for example, the TSP problem, the weighted Steiner tree problem, and the weighted set cover problem. Based on the polynomial space result for the subset convolution and the covering product (Theorem 7 in Sect. 2.5.1) and using the embedding technique (introduced in Sect. 2.2), the following result can be obtained by Theorem 11. Whether some other operations and a wider range of input functions can be involved in the polynomial circuit framework to give polynomial space algorithms or faster exponential space algorithms for hard problems, as well as whether this approach can be applied to a wider range of problems, need further studies. Theorem 12 ([42]) Let C be a circuit over ff W 2V ! Mg with operation gates including subset convolution, covering product, addition, and product gates. Let CO be the circuit over fgO W 2V ! Ng obtained by interpreting C over .N; C; /, and let u be such that u  h.Y / where h is the output of any gate of CO and Y  V . If the input function is a singleton and is bounded by W , then it can be decided whether fout .V /  k in O  .2jV j W log u/ time and O  .log u log W / space. Example 12 (Weighted Set Cover) Given a set U , a family F D fS1 ; : : : ; Sl g  2U , a weight function Pw W F ! N, and an integer t.PA set cover of U is a family C  F such that U  S 2C S . The weight w.C/ D S 2C w.S /. The weighted set cover problem is to decide whether there exists a set cover of U with weight at most t.

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Define mi .Y / as the minimum weight of a set cover C of Y such that jCj D i . Then the problem is to determine whether mn .U /  t. mi .Y / can be computed using the following recurrence: 8 ˆ ˆ 0 for uv 2 E and .uv/ D 0 for uv … E, this is equivalent to dG.u/ D uv2E .uv/. Then the minimum degree of G is ı.G/ D fd.v/=v 2 V g, and the maximum degree of G is .G/ D _fd.v/=v 2 V g. Definition 3 The strength of connectedness between two vertices u and v is 1 .u; v/ D supfk .u; v/=k D 1; 2 : : : :g where k .u; v/ D supf.uu1 / ^ .u1 u2 / ^ : : : :: ^ .uk1 v/=u1 : : : : : : uk1 2 V g: Definition 4 An edge uv is a fuzzy bridge of G W .; / if deletion of uv reduces the strength of connectedness between pair of vertices. Definition 5 A vertex u is a fuzzy cut vertex of G W .; / if deletion of u reduces the strength of connectedness between some other pair of vertices. Definition 6 Let G W .; / be a fuzzy graph such that G  .V; E/ is a cycle. Then G is a fuzzy cycle if and only if there does not exist a unique edge xy such that .xy/ D _f.uv/=.uv/ > 0g. Definition 7 The order ofP a fuzzy graph G is O.G/ D fuzzy graph G is S.G/ D uv2E .uv/.

P u2V

.u/. The size of a

Definition 8 Let G W .; / be a fuzzy graph on G  .V; E/. If dG .v/ D k for all v 2 V , that is, if each vertex has some degree k, then G is said to be a regular fuzzy graph of degree k or a k-regular fuzzy graph. This is analogous to the definition of regular graphs in crisp graph theory. Definition 9 Let G W .; / be a fuzzyP graph on G  .V; E/. TheP total degree of a vertex u 2 V is defined by tdG .u/ D u¤v .UV/ C .u/ D uv2E .UV/ C .u/ D dG .u/ C .u/. If each vertex G has the same total degree k, then G is said to be a totally regular fuzzy graph of total degree k or a k-totally regular fuzzy graph. The most important areas for the application of fuzzy graphs and fuzzy relations are logic, topology, pattern recognition, information theory, control theory, artificial intelligence, neural networks, operations research, planning, and systems analysis. Bhattacharya [4] established some connectivity concepts regarding fuzzy cut nodes and fuzzy bridges. The author also introduced the notions of eccentricity and center. Moreover, Bhattacharya and Suraweera [11] introduced an algorithm to find the connectivity of a pair of nodes in a fuzzy graph [4].

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Furthermore, Tong and Zheng [12] presented an algorithm to find the connectedness matrix of a fuzzy graph. While, Xu [13] introduced connectivity parameters of fuzzy graphs to problems in chemical structures, Sunitha and Vijayakumar [7] focused on characterizing fuzzy trees using its unique maximum spanning tree. In addition, a sufficient condition for a node to be a fuzzy cut node is presented in [14]. Also, center problems in fuzzy graphs [15], blocks in fuzzy graphs [16], and properties of self-complementary fuzzy graphs [17] were introduced by the same authors. Therefore, using the concept of the strongest paths, they have obtained a characterization for blocks in fuzzy graphs [15]. Bhutani and Rosenfeld have focused on the concepts of strong arcs [18], fuzzy end nodes [19], and geodesics in fuzzy graphs [20]. In [17], the authors have defined the concepts of strong arcs and strong paths. They have pointed out the existence of a strong path between any two nodes of a fuzzy graph and have studied the strong arcs of a fuzzy tree. In [15], the concepts of fuzzy end nodes and multimin and locamin cycles are studied. The concept of strong arc in maximum spanning trees [21] and its applications in cluster analysis and neural networks were presented by Sameena and Sunitha [21, 22]. According to Mathew and Sunitha, there are different types of arcs in fuzzy graphs and they have obtained an arc identification procedure [14].

2

Fuzzy Linear Programming

Linear programming is one of the most widely used decision-making tools for solving real-world problems. Many researchers focus on the area of fuzzy linear programming because of the fact that the real-world situations are characterized by imprecision rather than exactness. In order to have a clear understanding, few of such researches are discussed/shown below. Gupta and Mehlawat studied a pair of fuzzy primal–dual linear programming problems and calculate duality results using an aspiration level approach. They use an exponential membership function, which is in contrast to the earlier works that relied on a linear membership function. As the fuzzy environment causes a duality gap, they investigate how choosing the exponential membership function impacts the gap [23]. Amiria and Nasseria applied a linear ranking function to order trapezoidal fuzzy numbers. Then, they establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results. In particular, they prove that the auxiliary problem is indeed the dual of the linear programming problems with trapezoidal fuzzy variables (FVLP). Having established the dual problem, the results will then follow as natural extensions of duality results for linear programming problems with crisp data. Finally, using the results, they develop a new dual algorithm for solving the FVLP problem directly, making use of the primal simplex tableau [24]. In his paper, Ramik introduced a broad class of FLP problems firstly and defined the concepts of ˇ-feasible and .˛; ˇ/-maximal and minimal solutions of FLP problems. The class of classical LP problems can be embedded into the class of

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FLP ones. Furthermore, he defines the concept of duality and proves the weak and strong duality theorem generalizations of the classical ones for FLP problems [25]. On the other hand, Inuiguchi treats fuzzy linear programming problems with uncertain parameters whose ranges are specified as fuzzy polytopes in his study. The problem is formulated as a necessity measure optimization model. It is shown that the problem can be reduced to a semi-infinite programming problem and solved by a combination of a bisection method and a relaxation procedure. An algorithm in which the bisection method and the relaxation procedure converge simultaneously is proposed. A simple numerical example is given to illustrate the solution procedure [26]. Wua et al. present an efficient method to optimize such a linear fractional programming problem. First, some theoretical results are developed based on the properties of max-Archimedean t-norm composition. Then, the result is used to reduce the feasible domain. The problem can thus be simplified and converted into a traditional linear fractional programming problem and eventually optimized in a small search space. A numerical example is provided to illustrate the procedure [27]. In addition to above-mentioned studies, a new method to find the fuzzy optimal solution of same type of fuzzy linear programming problems is proposed by Kumar et al. It is easier to apply the proposed method, compared to the existing method in order to solve the fully fuzzy linear programming problems with equality constraints occurring in real-life situations. To illustrate, the proposed method numerical examples are solved, and the obtained results are discussed [28]. Additionally, Lotfi et al. discussed full fuzzy linear programming (FFLP) problems of which all parameters and variables are triangular fuzzy numbers. They use the concept of the symmetric triangular fuzzy number and introduce an approach to defuzzify a general fuzzy quantity [29]. By proposing the fuzzy multiobjective linear programming (FMOLP) model with triangular fuzzy numbers, Zenga et al., transformed the FMOLP model and its corresponding fuzzy goal programming (FGP) problem to crisp ones. These can be solved by the conventional programming methods. The FMOLP model was applied to crop area planning of Liang Zhou region, Gansu province of northwest China, and then the optimal cropping patterns were obtained under different water-saving levels and satisfaction grades for water resources availability of the decision-makers (DM) [30]. Further, Liang developed an interactive fuzzy multiobjective linear programming (i-FMOLP) method for solving the fuzzy multiobjective transportation problems with piecewise linear membership function. Proposed i-FMOLP method aims to simultaneously minimize the total distribution costs and the total delivery time. i-FMOLP method also has reference to fuzzy available supply and total budget at each source and fuzzy forecast demand with maximum warehouse space at each destination. Additionally, the above-mentioned method describes a systematic framework that facilitates the fuzzy decision-making process, enabling a decision-maker (DM) to interactively modify the fuzzy data and related parameters until a set of satisfactory solutions are obtained. An industrial case is presented to demonstrate the feasibility of applying such proposed method to real transportation problems [31].

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Peidro et al. modeled supply chain (SC) uncertainties by fuzzy sets and develop a fuzzy linear programming model for tactical supply chain planning in a multiechelon, multiproduct, multilevel, multi-period supply chain network in their study. In this approach, the demand, process, and supply uncertainties are jointly considered. The aim is to centralize multi-node decisions simultaneously to achieve the best use of the available resources along the time horizon so that customer demands are met at a minimum cost. With this intention, Peidro et al.’s proposal is tested by using data from a real automobile SC. Thus, fuzzy model provides the decision-maker (DM) with alternative decision plans with different degrees of satisfaction [32]. Other solutions are found by Bucley and Feuring to the fully fuzzified linear program where all the parameters and variables are fuzzy numbers in their study. First, they change the problem of maximizing a fuzzy number, the value of the objective function, into a multiobjective fuzzy linear programming problem. Then, they prove that fuzzy flexible programming can be used to explore the whole nondominated set to the multiobjective fuzzy linear program. Hence, an evolutionary algorithm is designed to solve the fuzzy flexible program, and they apply this program to two applications to generate good solutions [33]. Comparatively, Chanas and Zielinski analyzed the linear programming problem with fuzzy coefficients in the objective function. The set of nondominated (ND) solutions with respect to an assumed fuzzy preference relation, according to Orlovsky’s concept, is supposed to be the solution of the problem. Special attention is paid to unfuzzy nondominated (UND) solutions (the solutions which are nondominated to the degree 1). The main results of their paper are sufficient conditions on a fuzzy preference relation which allows reducing the problem of determining UND solutions to that of determining optimal solutions of a classical linear programming problem. These solutions can thus be determined by means of classical linear programming methods [34]. Another major study is by Stanciulescu et al., modeling a multiobjective decision-making process by a multiobjective fuzzy linear programming problem with fuzzy coefficients for the objectives and the constraints. Moreover, the decision variables are linked together because they have to sum up to a constant. Most of the time, the solutions of a multiobjective fuzzy linear programming problem are crisp values. Thus, the fuzzy aspect of the decision is partly lost, and the decision-making process is constrained to crisp decisions. However, they propose a method that uses fuzzy decision variables with a joint membership function instead of crisp decision variables. First, they consider lower-bounded fuzzy decision variables that set up the lower bounds of the decision variables. Second, the method is generalized to lower– upper-bounded fuzzy decision variables that also set up the upper bounds of the decision variables. The results are closely related to the special type of the problem they are coping with, since they embed a sum constraint in the joint membership function of the fuzzy decision variables. Numerical examples are presented in order to illustrate their method [35]. By the same token, as a case study, Sadeghi and Hosseini tried to demonstrate the method of application of FLP for optimization of supply energy system in Iran.

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They used FLP model comprises fuzzy coefficients for investment costs. Following the mentioned purpose, it is realized that FLP is an easy and flexible approach that can be a serious competitor for other confronting uncertainties approaches, that is, stochastic and minimax regret strategies [36]. On the other hand, Liu proposed a new kind of method for solving fuzzy linear programming problems based on the satisfaction (or fulfillment) degree of the constraints. Using a new ranking method of fuzzy numbers, the fulfillment of the constraints can be measured. Then the properties of the ranking index are discussed. With this ranking index, the decision-maker can make the constraints tight or loose based on his optimistic or pessimistic attitude and get the optimal solution from the fuzzy constraint space. The corresponding value of objective distribution function therefore can be obtained. A numerical example illustrates the merits of the approach [37]. Chen and Ko proposed fuzzy linear programming models to determine the fulfillment levels of PCs under the requirement to achieve the determined contribution levels of design requirements (DRs) for customer satisfaction. In addition, by considering the design risk, they incorporate failure modes and effect analysis (FMEA) into quality function deployment (QFD) processes, which are treated as the constraint in the models. In order to cope with the vague nature of product development processes, fuzzy approaches are used for both FMEA and QFD. The illustration of the suggested models is performed with a numerical example to indicate the applicability in practice [38]. Eventually, Katagiri et al. considered multiobjective linear programming problems with fuzzy random variables coefficients. A new decision-making model is proposed to maximize both possibility and probability, which is based on possibilistic programming and stochastic programming. An interactive algorithm is constructed to obtain a satisficing solution satisfying at least weak Pareto optimality [39].

3

Fuzzy Integer Programming

The linear programming models that have been discussed thus far all have been continuous, in the sense that decision variables are allowed to be fractional. Often this is a realistic assumption. For instance, we might easily produce 102 3/4 gallons of a divisible good such as wine. It might also be reasonable to accept a solution giving an hourly production of automobiles at 58 1/2 if the models were based upon average hourly production, and the production had the interpretation of production rates. At other times, however, fractional solutions are not realistic, and we must consider the optimization problem: Xn Maximize cj xj ; (1) j D1

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subject to Xn

aij D bi ; .i D 1; 2; : : : ; m/;

(2)

xj  0.j D 1; 2; : : : ; n; /;

(3)

xj integer .for some or all j D 1; 2; : : : ; n/:

(4)

j D1

This problem is called the (linear) integer programming problem. It is said to be a mixed-integer program when some, but not all, variables are restricted to be integer and is called a pure integer program when all decision variables are integers. The implications of fuzzy set theory in optimal decision-making were first recognized by Bellman and Zadeh [40] and were later extended to linear programming problems with fuzzy constraints and multiple objectives by Zimmermann [41]. The latter formulation was subsequently extended to integer programming problems [42–45]. Fuzzy techniques have recently been applied to the analysis and optimization [46,47]. Also, linear and integer linear programming are known to capture well optimization problems relevant to high-speed networks [48]. The fuzzy integer programming (FIP) methods provide another type of potentially useful approach for integer programming under uncertainty. Major shortcomings with the FIP methods are that, firstly, it may be difficult to obtain membership information for all system components in practical problems; secondly, FIP methods may lead to more complicated submodels that are computationally difficult for practical applications; and thirdly, most of the FIP solution algorithms are indirect approaches containing intermediate control variables or parameters, which are difficult to determine by certain criteria. They are thus unable to communicate uncertainty directly into the optimization processes and resulting specific solutions [49]. Tan et al. present integer programming optimization models for planning the retrofit of power plants at the regional or national level at their study. In addition to the base case (i.e., non-fuzzy or crisp) formulation, two fuzzy extensions are given to account for the inherent conflict between environmental and economic goals, as well as parametric uncertainties pertaining to the emerging carbon capture technologies. Case studies are shown to illustrate the modeling approach [50]. Asratian and KuzjurinP considered covering programs with 0–1 variables and cost function of the form j xj under the assumption that they know a pattern of coefficients in constraints, which are nonzero and only those coefficients can vary in some interval Œ1; M  (zero elements do not change their value). As their main result, they found some sufficient conditions guaranteeing the variation of integral optimum in the average case (over all zero–nonzero patterns) is close to 1 as the number of variables tends to infinity. This means that for typical patterns the values of nonzero elements in A can vary without affecting significantly the value of the optimum of the integer program (i.e., the optimum value depends mostly on the pattern but not on the values of nonzero elements) [48].

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Gharehgozli et al. present a new mixed-integer goal programming (MIGP) model for a parallel-machine scheduling problem with sequence-dependent setup times and release dates. Two objectives are considered in the model to minimize the total weighted flow time and the total weighted tardiness simultaneously. Due to the complexity of the above model and uncertainty involved in real-world scheduling problems, it is sometimes unrealistic or even impossible to acquire exact input data. Hence, they consider the parallel-machine scheduling problem with sequencedependent setup times under the hypothesis of fuzzy processing time’s knowledge and two fuzzy objectives as the MIGP model [51]. Li et al. developed a two-stage fuzzy robust integer programming (TFRIP) method for planning environmental management systems under uncertainty. Their approach integrates techniques of robust programming and two-stage stochastic programming within a mixed-integer linear programming framework. It can facilitate dynamic analysis of capacity-expansion planning for waste-management facilities within a multistage context. The TFRIP method is applied to a case study of long-term waste-management planning under uncertainty. Generated solutions for continuous and binary variables can provide desired waste-flow allocation, capacity-expansion plans with a minimized system cost, and maximized system feasibility [52]. Allahviranloo and Afandizadeh formulated an investment model to find the optimum investment steps by application of operational research science and fuzzy logic concept to model the available uncertainties. Fuzzy integer linear programming models are used to determine the optimum investment and development of a port [53]. Also, Emam studied a bi-level integer nonlinear programming problem [54] with linear or nonlinear constraints, where nonlinear objective function at each level is maximized. The bi-level integer nonlinear programming (BLI-NLP) problem can be thought as a static version of the Stackelberg game, which is used as a leader–follower game. A Stackelberg game is used by the leader, or the higher-level decision-maker (HLDM), given the rational reaction of the follower, or the lowerlevel decision-maker (LLDM). He proposed a two-planner integer model and a solution method for solving this problem. This method uses the concept of tolerance membership function and the branch-and-bound technique to develop a fuzzy max– min decision model for generating Pareto optimal solution for this problem; an illustrative numerical example is given to represent the obtained results [53].

4

Fuzzy Spanning Trees

In classical mathematical programming, the coefficients of objective functions or constraints in problems are assumed to be completely known. However, in real systems, they are uncertain than constant. In order to deal with such uncertainty, stochastic programming [55] and fuzzy programming [56] were considered. Both are useful tools for the decision-making under a stochastic environment or a fuzzy environment, respectively [57].

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In parallel Gao and Lu formulated a fuzzy quadratic minimum spanning tree problem as expected value model, chance-constrained programming, and dependent-chance programming according to different decision criteria in their study. Then the crisp equivalents are derived when the fuzzy costs are characterized by trapezoidal fuzzy numbers. Furthermore, a simulation-based genetic algorithm using Pr¨ufer number representation is designed for solving the proposed fuzzy programming models as well as their crisp equivalents, and a numerical example is displayed to illustrate the effectiveness of the genetic algorithm at the end of the study [57]. Katagiri et al. [58] investigated bottleneck spanning tree problems where each cost attached to the edge in a given graph is represented with a fuzzy random variable. The problem is to find the optimal spanning tree that maximizes a degree of possibility or necessity under some chance constraint. After transforming the problem into the deterministic equivalent one, they introduce the subproblem which has close relations to the deterministic problem. Utilizing fully the relations, they give a polynomial order algorithm for solving the deterministic problem. Additionally, Katagiri et al. [59] studied minimum spanning tree problems where each edge weight is a fuzzy random variable. A fuzzy goal for the objective function is defined to capture the imprecise judgment of the decision-maker. They also proposed a decision-making model based on a possibilistic programming model and the expectation optimization model in stochastic programming.

5

Fuzzy Shortest Path

The shortest path problem (SPP) is one of the most fundamental and well-known combinatorial optimization problem that appears in many applications, including communications, transportation, routing, supply chain management, or models involving agents as a subproblem [60]. The problem of finding the shortest path from a specified source node to the other nodes is a fundamental matter in graph theory and one that is currently being greatly studied [61–68]. The classical problem seeks to select a path with minimum length from a finite set of paths. In real-world problems, arc lengths represent traveling time, cost, distance, or other variables. However, in practice, uncertainty cannot be avoided, and usually, the arc lengths cannot be determined precisely. For instance, on road networks, for several reasons, that is, traffic, accidents, or weather condition, arc lengths representing the vehicle travel time are subject to uncertainty. In such situations, a fuzzy shortest path problem (FSPP) seems to be more realistic and reliable. According to Hernandes et al., an iterative algorithm assumes a generic ranking index for comparing the fuzzy numbers involved in the problem, in such a way that each time in which the decision-maker wants to solve a concrete problem(s) he/she can choose (or propose) the ranking index that best suits that problem. Hernandes et al.’s algorithm is based on the Ford–Moore–Bellman algorithm for classical graphs. In concrete, it can be applied in graphs with negative parameters,

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and it can detect whether there are negative circuits. For the sake of illustrating the performance of the algorithm in the study, it has been developed using only certain order relations. However, it is not restricted at all to use these comparison relations exclusively. As the theoretical base of a decision support system’s concern is to solve this kind of problems, the proposed iterative algorithm is easy to understand [69]. Tajdin et al. concerned with the design of a model and an algorithm for computing a shortest path in a network having various types of fuzzy arc lengths. Firstly, to obtain membership functions for the considered additions, they developed a new technique for the addition of various fuzzy numbers in a path using ˛-cuts by proposing a linear least squares model. Then, using a recently proposed distance function for comparison of fuzzy numbers, they present a dynamic programming method for finding a shortest path in the network [70]. Moazeni discussed the shortest path problem in his study. A positive fuzzy quantity is assigned to each arc as its arc length on a network. He defines an order relation between fuzzy quantities with finite supports. Then by applying Hansen’s multiple labeling method together with Dijkstra’s shortest path algorithm, he proposes a new algorithm for finding the set of nondominated paths with respect to the extension principle. Moreover, he shows that the only existing approach for this problem, Klein’s algorithm, may lead to a dominated path in the sense of extension principle [60]. Keshavarz and Khorram concentrated on a shortest path problem on a network where arc lengths (costs) are not deterministic numbers, but imprecise ones in their study. Here, costs of the shortest path problem are fuzzy intervals with increasing membership functions, whereas the membership functions of the total cost of the shortest path are a fuzzy interval with a decreasing linear membership function. By the max–min criterion suggested in [71], the fuzzy shortest path problem can be treated as a mixed-integer nonlinear programming problem. They pointed out that this problem can be simplified into a bi-level programming problem that is very solvable. In order to solve the bi-level programming problem, they propose an efficient algorithm based on the parametric shortest path problem. An illustrative example is used to denote their algorithm [72]. In a network, the arc lengths may represent time or cost. In practical situations, it is reasonable to assume that each arc length is a discrete fuzzy set. It is called the discrete fuzzy shortest path problem. There are several methods reported to solve this kind of problem in the literature. In these methods, they can obtain either the fuzzy shortest length or the shortest path. In their study, Chuang and Kung claimed a new algorithm which can obtain both of them. The discrete fuzzy shortest length method is proposed to find the fuzzy shortest length, and the fuzzy similarity measure is utilized to get the shortest path. An illustrative example represents their proposed algorithm [62]. Ji et al. considers the shortest path problem with fuzzy arc lengths in their study. According to different decision criteria, the concepts of expected shortest path, ˛-shortest path, and the shortest path in fuzzy environment are originally introduced, and three types of models are formulated. In order to solve these models, a hybrid intelligent algorithm integrating simulation and genetic algorithm are asserted and numerous examples illustrate its effectiveness [73].

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Okada deals with a shortest path problem on a network in which a fuzzy number, instead of a real number, is assigned to each arc length in his study. Such a problem is “ill-posed” because each arc cannot be identified as being either on the shortest path or not. Therefore, based on the possibility theory, he introduces the concept of “degree of possibility” that an arc is on the shortest path. Every pair of distinct paths from the source node to any other node is implicitly assumed to be noninteractive in the conventional approaches. This assumption is unrealistic and also involves inconsistencies. To overcome this drawback, he defines a new comparison index between the sums of fuzzy numbers by considering interactivity among fuzzy numbers. An algorithm is presented to determine the degree of possibility for each arc on a network. This algorithm is evaluated by means of largescale numerical examples. Consequently, this approach is found efficient even for real-world practical networks [66]. The fuzzy shortest path (SP) problem aims at providing decision-makers with the fuzzy shortest path length (FSPL) and the SP in a network with fuzzy arc lengths. In their study, Chuand and Kung represent each arc length a triangular fuzzy set and propose a new algorithm to deal with the fuzzy SP problem. First, they proposed a heuristic procedure to find the FSPL among all possible paths in a network. It is based on the idea that a crisp number is a minimum number if and only if any other number is larger than or equal to it. It owns a firm theoretic base in fuzzy sets theory and can be implemented effectively. Second, they propose a way to measure the similarity degree between the FSPL and each fuzzy path lengths. The path with the highest similarity degree is the SP. An illustrative example is added to display their proposed approach [61].

6

Fuzzy Network Flows

Network flow problems have a wide range of engineering and management applications such as analyses and design of computer networks, cable television networks, transportation systems, communication networks, project schedules, queuing systems, inventory systems, and manpower allocation [74]. In parallel, Liu and Kao [74] studied the network flow problems in that the arc lengths of the network and the objective value are fuzzy numbers. The illustrative example is on the problem of multimedia transmission over Internet. Flows in networks provide very useful models in a number of practical contexts. The major techniques in the area revolve around celebrated max-flow min-cut theorem (MFMCT) and associated algorithms. In this context, Diamond develops analogues of the MFMCT and Karp–Edmonds algorithm for networks with fuzzy capacities and flows in their study. The principal difference between fuzzified and traditional crisp versions is that although the maximum fuzzy flow corresponds to a minimum fuzzy capacity, the latter may incorporate a number of network cuts. Preliminary results are for interval-valued flows and capacities which, in

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themselves, provide robustness and estimates for flows in an uncertain environment. In turn, a fuzzy theorem and algorithm are obtained by regarding these intervals as level sets of fuzzy numbers [75]. Georgiadis considered a general class of optimization problems regarding spanning trees in directed graphs (arborescence). In his study, Georgiadis considered the following optimization problem. He assumed that edge costs take values in a set U endowed with a dyadic relation and a dyadic operation. He sought to find the directed spanning tree whose cost (with respect to the dyadic operation) is minimal with respect to the dyadic relation. Then, he provided an algorithm for solving the problem, which can be considered as generalization of Edmonds’ special cases of this problem provide algorithms for the minimum sum, bottleneck and lexicographically optimal arborescence, as well as the widest-minimum sum spanning arborescence problem [76].

7

Fuzzy Minimum Cost Flow Problems

Minimal cost flow problem (MCFP) is an important problem in combinatorial optimization and network flows and has many applications in practical problems, such as communication, transportation, urban design, and job scheduling models [77]. The aim of the minimal cost flow problem (MCFP) in fuzzy nature, denoted with FMCFP, is to find the least cost of the shipment of a commodity through a capacitated network in order to satisfy imprecise concepts in supply or demand of network nodes and capacity or cost of network links [78]. Ghatee and Hashemi presented a model in which the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy numbers. For easier reference, they refer to this group of problems as fully fuzzified MCFP. Hukuhara’s difference and approximated multiplication are used to represent their model. Thereafter, they sort fuzzy numbers by an order using a ranking function and show that it is a total order, that is, a reflexive, antisymmetric, transitive, and complete binary relation. Utilizing the proposed ranking function, they transform the fully fuzzified MCFP into three crisp problems solvable in polynomial time [79]. Also, Ghatee and Hashemi deal with fuzzy quantities and relations in multiobjective minimum cost flow problem in their study. When t-conorms and t-conorms are available, the goal programming is applied to minimize the deviation among the multiple costs of fuzzy flows and the given targets. To obtain the most optimistic and the most pessimistic satisficing solutions of the problem, two different polynomial time algorithms are introduced by applying some network transformations. To verify the performance of this approach in actual substances, network design under fuzziness is considered, and an efficient scheme is proposed including genetic algorithm including fuzzy minimum cost flow problem [80]. Similarly, Ghatee et al. study the minimal cost flow problem (MCFP) with fuzzy link costs, say fuzzy MCFP, to understand the effect of uncertain factors in applied shipment problems. With respect to the most possible case, the worst case, and the

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best case, the fuzzy MCFP can be converted into a three-objective MCFP. Applying a lexicographical ordering on the objective functions of derived problem, two efficient algorithms are provided to find the preemptive priority-based solution(s), namely, p-successive shortest path algorithm and p-network simplex algorithm. In both of them, lexicographical comparison is used which is a natural ordering in many real problems especially in hazardous material transportation where transporting through some links jeopardizes people’s lives. Presented schemes maintain the network structure of the problem, and so, they are efficiently implementable. A sample network is added to illustrate the procedure and to compare the computational experiences with those of previously established works. Finally, the above-presented approach is applied to find an appropriate plan to transit hazardous materials in Khorasan roads network using annual data of accidents [81].

8

Fuzzy Matching Algorithm

Matching problem is one of the most important optimization problems. The model is widely used in practice such as pattern recognizing, system optimization, and job assignment problem. A matching of a graph is an independent subset M of edge set. In other words, a matching is a set of pairwise disjoint edges. To explain this algorithm, first, we will explain fuzzy maximum matching algorithms. A maximum matching has as many edges in it as possible. Another version of this general problem is called the maximum weighted matching problem, and significantly more difficult will be captured in the second subsection of this section. A maximum weighted matching problem is to choose a maximum matching in a given graph so that the sum of weight of the edges in it is at the maximum level. Nonbipartite weighted matching appears to be one of the “hardest” combinatorial optimization problems that can be solved in polynomial time. Another version of weighted matching algorithms is a well-known problem called the assignment problem (AP). It can also be called as the minimum weight perfect matching problem in bipartite graphs. The fuzzy assignment problem (FAP) is also introduced at the Sect. 8.2.1. There are several other inexact matching methods based on continuous optimization that have been proposed in the recent years. Among them we can cite the fuzzy graph matching (FGM) that is a simplified version of weighted graph matching (WGM) problem and based on fuzzy logic. In FGM, as the cost of matching two nodes does not depend on the matching of the other nodes of the graph, the objective function is considerably simpler than the objective function in WGM problem [82]. Medasani et al. [83] provided a relaxation approach to graph matching based on a fuzzy assignment matrix. Hence, the authors are able to derive that iterative FGM algorithm, which has a matching accurate than the graduated assignment algorithm. Medasani and Krishnapuram [84] introduced a fuzzy approach to content-based retrieval of segmented images using fuzzy attributed relational graphs (FARGs) and an efficient FGM. In sum, the FGM algorithm is of the order O.n2 m2 / and hence suitable for image retrieval.

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1371

Fuzzy Maximum Matching Algorithms

The maximum matching model is also called cardinality matching problem. An independent subset of the edges set is called a matching of a graph, and a matching with as many edges in it as possible is called a maximum matching. Matching problems with fuzzy weights arise when the decision-maker has some vague information about some data of the real-world system. In other words, the parameters of a system can be characterized by fuzzy variables. Essentially, Liu et al. initialized the concepts of expected maximum fuzzy weighted matching, the ˛-maximum fuzzy weighted matching, and the most maximum fuzzy weighted matching. According to various decision criteria, by using the credibility theory, with crisp equivalents are also given, the maximum fuzzy weighted matching problem is formulated as expected value model, chanceconstrained programming, and dependent-chance programming. Furthermore, a hybrid genetic algorithm is designed to solve the proposed fuzzy programming models. Finally, a numerical example is given. The weights of the edges are trapezoidal fuzzy variables in the numerical example [85]. Liu and Gao [86] suggested concepts of the maximum random fuzzy weighted matching problem: (a) the expected maximum random fuzzy weighted matching, (b) the .˛; ˇ/-maximum random fuzzy weighted matching, and (c) the most maximum random fuzzy weighted matching. Later they formulated the maximum random fuzzy weighted matching problem as an expected value model, chanceconstrained programming, and dependent-chance programming. To get the expected maximum random fuzzy weighted matching, the expected maximum random fuzzy weighted matching problem can be formulated as follows: 8 P ˆ max .i;j /2E xij ˆ ˆ ˆ rg; respectively.

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The credibility measure of fuzzy event f  rg is defined by Liu and Liu [95] as Crf  rg D

1 .Posf  rg C Necf  rg/: 2

It can easily verified that Crf  rg C Crf > rg D 1 for any fuzzy event f  rg. Liu [96] also presented two critical values, say ˛-optimistic value which was defined as sup .˛/ D supfrjCrf  rg  ˛g and a pessimistic value which was defined as inf .˛/ D inffrjCrf  rg  ˛g; which serve as roles to rank fuzzy variables, where ˛".0; 1 and  is a fuzzy variable. The expected value of a fuzzy variable  is defined as Z1 EŒ D

Z0 Crf  rgdr 

Crf  rgdr

1

0

provided that at least one of the two integrals is finite. For more detailed properties of credibility measure, the interested reader may consult Liu [97]. For this model, it is thought that all the graphs are undirected and simple. Let G.V; E; / be a graph, where V, E, and  are the vertices set, edges set, and weights vector of the graph G. Sometimes, we employ E(G) and V(G) to denote the edges set and the vertices set of graph G, respectively. The degree and the set of incident edges of the vertex v " V are denoted by dG(v) and Ne[v], respectively. For a subset S of E or V, G[S] denotes the subgraph induced by the subset S. The weight of edge .i; j /"E is denoted by a fuzzy variable ij , where ii D 0 and ij D ji ; i; j D 1; 2; : : : ; ŒV .G/: Let M  E be a matching of G.V; E; /. Let ( xij D

1;

if.i; j / 2 M;

0;

otherwise:

Then the matching M can also be denoted by such a vector x, and the cost function of matching x can be defined as f .x; / D

X .i;j /2E

ij xij0

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where x and  denote the vectors consist of xij and ij ; .i; j /"E, respectively. Definition 10 The ˛-cost of f .x; / is defined as the cost value fN such that Crff .x; /  fNg  ˛; where ˛ is a predetermined confidence level between 0 and 1. It is clear that the cost function f .x; / is also a fuzzy variable when the vector  is a fuzzy vector. In order to rank f .x; /, different ideas are employed in different situations. If the decision-maker wants to find a maximum weighted matching with maximum expected value of the cost function f .x; /, then the following concept is induced. Definition 11 A maximum matching x  is called the expected maximum fuzzy weighted matching if EŒx    EŒf .x; / for any maximum matching x. In other situation, the decision-maker hopes to maximize the a-cost value fN with Crff .x; /  fNg  ˛: For this case, we propose the concept of a-maximum fuzzy weighted matching as follows: Definition 12 A maximum matching x  is called a-maximum fuzzy weighted matching if maxffNjCrff .x  ; /  fNg  ˛g  maxffNjCrff .x; /  fNg  ˛g for any maximum matching x, where a is a predetermined confidence level. Furthermore, the most maximum weighted matching arises when the decisionmaker gives a cost value fN and hopes that the credibility of the cost value exceeding fN will be as maximized as possible Definition 13 A maximum matching x  is called the most maximum fuzzy weighted matching if Crff .x  ; /  fNg  Crff .x; /  fNg for any maximum matching x, where fN is a predetermined cost value.

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Based upon these, we will recast the maximum fuzzy weighted matching problem as multiobjective expected value model, chance-constrained programming, and dependent-chance programming, respectively. Liu and Liu [95] initialized the fuzzy expected value model. Its main idea is to optimize the expected value of the objective function subjected to some expected constraints. Therefore, the expected maximum fuzzy weighted matching problem can be formulated as 8 P ˆ ˆ /2E xij ˆmax .i;j ˆ 0). Also the quality matrix [qij ] that represents the highest quality associated with the worker i performing the job j.0 < qij  1/ is defined. In most real cases, 0 < qij < 1 and then cij is a subnormal fuzzy interval. The membership function of cij is defined as the linear monotone increasing function shown in Fig. 2, which suggests that any expense exceeding ˇij is useless since the quality (worker’s job performance or customer’s satisfaction) can no longer be increased at its upper limit qij condition. xij D 1 is added to (13) because there is no real expense if xij D 0 in any feasible solution x of (12). 8 ˆ ˆ 1, then u is a potential strongly stuck node. In such a case, †u1 uu2 is called a stuck angle. Notice that all the information needed to find a stuck angle at a node is its 1-hop neighborhood, which is completely local.

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Fig. 15 An illustration of Tent Rule and stuck angle

O12

u1

u2 u u3

uk

Theorem 24 ([24]) If u is a strongly stuck node, then there exists a stuck angle at u. Theorem 24 provides a necessary condition to guarantee that all the strongly stuck nodes can be found using Tent Rule. Having found a strongly stuck node u, the strong-hole containing u can be found by the algorithm BoundHole proposed in [24], which can be implemented locally by using only angles between adjacent edges. With the holes detected by BoundHole, geometric routing can be done as follows: When the greedy forwarding fails at a stuck node u, u must be on the boundary of a hole (by Theorem 23 or Definition 4). After that, recovery mode is activated and the packet is routed along the boundary of the hole, say, by the right-hand rule. When the packet gets to a node v which is closer to the destination D than u, the greedy forwarding mode is resumed. When D is outside the hole, such a node v always exists. When D is inside the hole, the packet might go around the boundary of the hole back to u without coming across any node closer to D than u. As long as D is reachable from u, there must be a path from u to D which intersects the boundary of the hole, say, edge x 0 y 0 on the path intersects edge xy on the boundary of the hole. One of x 0 ; y 0 , say x 0 , must be a 1-hop neighbor of x or y in the unit disk graph. In this case, a restricted flooding mode is initiated, in which every node on the boundary of the hole sends the packet to all its 1-hop neighbors. In particular, x 0 can receive the packet and forward it to D.

4.2

Statistical Method

Assuming a uniform random distribution of sensors, Fekete et al. proposed statistical methods to recognize nodes near the boundaries [25, 26].

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The method used in [25] is based on the intuition that nodes on the boundaries have a much lower average degree than those in the interior. Suppose A is the whole region containing n nodes. Use jAj to denote the Lebesgue measure of A. Under the assumption of uniform distribution, a node u in the interior has an estimated neighborhood size of jBR .u/j ;

D .n  1/ jAj where R is the transmission range of the nodes and BR .u/ is the disk with center u and radius R. Given a constant ˛ < 1, nodes with degree less than the threshold ˛

claim themselves to be boundary nodes. Notice that is not known a priori since the region A is exactly what needs to be determined. The performance of the method depends on the estimation of and the choice of ˛. In [25], is estimated based on a node degree histogram, and ˛ is computed by sampling possible values of ˛ and keeping track of the number of connected boundary components. The method proposed in [26] utilizes centrality measures which are commonly used in social networks. The idea is that the centrality of nodes on the boundary is much lower than those in the interior. There are two major classes of centralities: local and global. In fact, degree used in [25] is one type of measures of local centrality. Eigenvalues of a matrix is an example of global centrality. Three centrality measures were used in [26]. The idea is based on the intuition that more shortest paths will pass through interior nodes than boundary nodes. Use uv .w/ to denote the number of shortest .u; v/-paths containing node w. The stress centrality at node w is defined as X st.w/ D uv .w/; u;v2V

where V is the set of nodes in the region. Let uv be the number of shortest .u; v/-paths. Assume that a packet with source u and destination v chooses any shortest .u; v/-path with the same probability. Then the probability that node w is used to transmit the packet is uv .w/=uv . The betweenness centrality at node w is defined as X betw.w/ D uv .v/=uv : u;v2V

To make the estimation localized, one restricts the attention to those nodes which are nearby. The restricted stress centrality at node w is defined as

str .w/ D

X

uv .w/;

u;v2Nı .w/

where Nı .w/ is the set of nodes at most ı-hops away from w. Boundary nodes are distinguished  by a properly set threshold. For example, for ı D 1, any node with str .w/   2 claims itself to be a boundary node, where is the estimated average degree and  is a parameter playing the same threshold role as ˛ in [25]. It was

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proved in [26] that if the density is sufficiently large, nodes are classified correctly with a high probability using the restricted stress centrality with ı D 1. The major limitation of the above two methods is the assumption on the sensor distribution and the high density of the sensors.

4.3

Graph Method

This subsection introduces two kinds of graph-based methods in identifying the boundaries.

4.3.1 Contour-Based Method The methods proposed by Funke and Klein in [30, 31] are based on the observation that contours are broken at boundaries. Unlike the continuous field, contour has no definition in wireless sensor networks since the distances are counted by hops in the communication graph (which is a unit disk graph G in [30, 31]). Funke proposed isoset which resembles contour. For a node u (called beacon in [30] or seed in [31]) and a fixed integer k, the set I.k/ D fv j dhop .u; v/ D kg is called an isoset of level k, where dhop .u; v/ is the hop distance between u and v in G. Suppose the subgraph of G induced by I.k/ has t-connected components C1 .k/; C2 .k/; : : : ; Ct .k/. If t  2, the isoset I.k/ is broken and holes might exist between the connected components. The ends of each Ci .k/ indicate the locations of the holes, and the ends can be identified as the nodes in Ci .k/ having the largest radius in Ci .k/ (the radius of a node w in a connected graph H is the hop distance between w and the node in H which is farthest from w). The ends are marked as near boundaries. The above method is not local. To overcome it, more seeds are chosen, and for each seed s, only the isosets in a bounded neighborhood around s are examined, namely, the isosets I.k/ with hmin  k  hmax . By choosing the seed set to be a maximal independent set, the running time can be bounded. In fact, it was proved in [31] that with hmax being a constant, the running time is O.n C m/, where n and m are the number of vertices and the number of edges of the unit disk graph, respectively. Funke and Klein [31] also provided a theoretical guarantee under the assumption that the holes are all circles which are not too small, the distances between different holes are large enough, and the density of sensors in the non-hole area is high enough. To be more precise, they defined an "-good sensor distribution and proved the following theorem. A set S of sensors is "-good if for every point p 2 R2 which is not in a hole, there is a sensor s 2 S with kspk  ". Theorem 25 ([31]) For circular holes with radius at least rhole D 72, if the minimum distance between holes is at least hole D 18, the sensor distribution is "-good for " D 1=64, and by taking hmin D 4 and hmax D 8, Funke and Klein’s algorithm marks for each boundary point a sensor node within distance at most 4.8, and every marked sensor node has a boundary point within distance 2.8.

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4.3.2 Identifying Inner Nodes The methods used in [52] and [75] aim to recognize inner nodes. All the other nodes are considered to be on the boundaries. The graph model considered in these two papers is quasi unitpdisk graph (QUDG) (see Sect. 3.3 for the definition). It was assumed that d  1= 2 in [52] and [75]. This value is crucial to the geometrical analyses, as can be seen from the following lemma: p Lemma 2 ([52]) Let G D .V; E/ be a d -QUDG with d  1= 2, u; v; x; y be four distinct nodes in V with uv; xy 2 E. If the two straight-line segments p.u/p.v/ and p.x/p.y/ intersect, then at least one of the four edges ux; uy; vx; vy is in E. A cycle C D u0 u1 : : : uk1 u0 is chordless if ui uj 62 E for any i; j with ji  j j > 1, where the operation “” is modulo k. The embedding p.C / of a chordless cycle C divides the plane into an infinite face and at least one finite face (there might be more than one finite faces if C is self-intersecting). For a node set U , N.U / is the set of nodes which are adjacent with at least one node in U , and N ŒU  D N.U / [ U is the closed neighborhood of U . The following result is a corollary of Lemma 2: p Corollary 3 ([52]) Let G D .V; E/ be a d -QUDG with d  1= 2, C be a chordless cycle in G, and U be a connected node set of G. If N ŒC  \ U D ;, then either U is completely contained in the inside of C or completely contained in the outside of C . Denote by f i td .k/ the size of a maximum independent set I which can be placed inside a chordless k-cycle in a d -QUDE such that I \ N ŒC  D ;. Based on the above property, the following result is easy to see, which provides a criterion for a connected set of nodes to lie outside of a chordless cycle. Lemma 3 ([52]) Under the assumption of Corollary 3, if U has an independent node subset I with jI j > f i td .jC j/, then U is completely outside of C . To decide whether there is a node inside of a chordless cycle is much more complicated. For this purpose, [52] proposed a structure called m-flower and proved that for a flower, there is a node inside of its related chordless cycle. The algorithm in [52] looks for a family of node S sets U D fUi g and a family of chordless cycles C D fCb g such that the nodes in i Ui are guaranteed to lie inside of the cycles. Such a pair .C; U/ is called a feasible geometry description (FGD). The goal is to find an FGD with as many inside nodes as possible (recall that the remaining nodes are regarded as boundary nodes; hence, the boundary might be clearer with more inner nodes identified). The algorithm starts from some FGD and then iteratively extends it to another FGD with more inside nodes. The initial FGD is found by detecting a flower. As explained in the previous paragraph, this provides the initial I and C. Then, the FGD is enlarged by the so-called augmenting cycles.

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The success of the above method depends on the existence of a flower. Graph G is called locally "-dense in a region A  R2 , if every "-disk contained in A contains a node of G, that is, for any point x 2 A with B" .x/  A, there is a node u 2 V .G/ with kuxk  ". p Lemma 4 ([52]) For d D 1 and 0 < " < 32  2, if there exists a point x 2 R2 such that G is locally "-dense in the disk B3 .x/, then G contains a 4-flower. Notice that the above lemma insures the existence of a flower only when the network is dense enough, which cannot be guaranteed in a real-world network. To overcome this problem, Saukh et al. [75] proposed a generic classes of patterns to ensure some node inside of the pattern, which makes the algorithm in [75] works much better in a sparse network.

4.4

Topological Method

Homology, being a rigorous mathematical method for detecting and categorizing holes, steps naturally into the way of hole recognition in wireless sensor networks. The task is how to adapt it to suit the discrete settings and the special requirements of wireless networks. The method used in [88] is based on the observation that holes incur a void in a shortest path tree. The rough idea is as follows: The flow of a shortest path tree T forks near a hole and meets again after going along opposite borderlines of the hole. The nodes where the tree flow meets again were called cuts. By tracking their least common ancestor (LCA) on the tree (which is far away because of the hole) will trap the hole. To be more concise, Wang et al. [88] defined a pair of neighboring nodes u; v to be a cut pair, if: (a) The hop distances dhop .u; w/ and dhop .v; w/ are above a threshold ı1 , where w D LCA.u; v/ is the least common ancestor of u; v on T . (b) Maxfdhop .xu ; xv /g is above a threshold ı2 , where the maximum is taken over all node pairs .xu ; xv / with xu being a node on the .u; w/-path on T and xv being a node on the .v; w/-path on T . The first condition indicates that the LCA is far away from the cut pair. The second condition indicates how fat is the hole that one wants to detect. Topologically, cut is a counterpart of cut locus [15] in the continuous setting (roughly speaking, the cut locus of a point x in a manifold is the set of points for which there are multiple minimizing geodesics connecting them from p). It should be noted that the communication graph is not required to be a quasi unit disk graph in [88]. The problem considered in [16, 17, 37] is different from the above boundary recognition problem. The question asked by Ghrist et al. is: assuming disk sensing range of the sensors, can the domain be covered without holes? The problem is ˘ abstracted into a Cech complex, and the detection of sensing holes is equivalent

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to determining whether the domain boundary circle is null-homologous in the ˘ Cech complex. Since this computation requires accurate locations of the nodes and a global coordinate system, which is not suitable for the setting of wireless network, Ghrist et al. continued to find a sufficient but not necessary condition to detect sensing holes using Vietoris-Rips complex, based only on the pairwise communication data.

5

Conclusion

This chapter addresses three classes of geometric optimization problems in wireless sensor networks: localization, routing, and boundary recognition. The survey outlines the main ideas of recent methods for handling the new challenges brought into these problems by large-scale sensor wireless transmission. A number of successful approaches are covered to overcome difficulties in the geometric optimization due to the absence of central control unit, limited transmission range, limited computation and storage capability, etc. Acknowledgements This work is supported by National Natural Science Foundation of China (61222201) and Specialized Research Fund for the Doctoral Program of Higher Education (20126501110001). It is also supported by National Science Foundation of the USA under grants 1020 CNS0831579 and CCF0728851.

Cross-References Efficient Algorithms for Geometric Shortest Path Query Problems.

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Gradient-Constrained Minimum Interconnection Networks Marcus Brazil and Marcus G. Volz

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Motivation and Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Underground Mining Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Network Optimization Applied to Underground Mine Design. . . . . . . . . . . . . . . . . . . . . . 2.3 Minkowski Spaces and the Gradient Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Fermat-Weber Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Steiner Tree Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Gilbert Arborescence Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Two-Dimensional Gradient-Constrained Steiner Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Local Minimality and Exact Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Computational Complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Three-Dimensional Gradient-Constrained Steiner Problem. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Properties of Steiner Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Local Minimality for a Fixed Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Algorithms and Software Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Gilbert Problem in Minkowski Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Two-Dimensional Gradient-Constrained Gilbert Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Edge Vectors and Fundamental Properties of MGAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A Classification of Steiner Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Three-Dimensional Gradient-Constrained Gilbert Problem. . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Classification of Steiner Points of Degree at Most Four. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Maximum Degree of Steiner Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M. Brazil () Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, VIC, Australia e-mail: [email protected] M.G. Volz TSG Consulting, Melbourne, VIC, Australia e-mail: [email protected] 1459 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 73, © Springer Science+Business Media New York 2013

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Abstract

In two- or three-dimensional space, an embedded network is called gradient constrained if the absolute gradient of any differentiable point on the edges in the network is no more than a given value m. This chapter gives an overview of the properties of gradient-constrained networks that interconnect a given set of points and that minimize some property of the network, such as the total length of edges. The focus is particularly on geometric Steiner and Gilbert networks. These networks are of interest both mathematically and for their applications to the design of underground mines.

1

Introduction

In two- or three-dimensional space, an embedded network is called gradient constrained if the absolute gradient (with respect to a vertical axis) of any differentiable point on the edges in the network is no more than a given value m. This chapter is an overview of the properties of interconnection networks that are minimum under such a constraint. The sorts of networks that will be considered here are geometric Steiner and Gilbert networks. Both of these involve constructing a network that interconnects a given set of points (known as terminals) and that minimizes some property of the network, such as the total length of edges. Unlike minimum spanning trees, Steiner and Gilbert networks may contain nodes other than the terminals; these extra nodes are referred to as Steiner points. The challenge in optimizing these networks lies in determining both the locations of the Steiner points and the graph topology of the network. Gradient-constrained minimum interconnection networks are interesting both from a mathematical point of view and because of their potential applications. Mathematically, they lie between Euclidean networks and fixed orientation networks. In practical terms, they have an important role in the efficient design of underground mines. Both of these aspects will be explored in the course of this chapter. The structure of this chapter is as follows. Section 2 outlines the motivation to this problem, particularly in terms of its application to the design of underground mines, and presents some of the required mathematical background such as properties of Minkowski spaces and the Fermat-Weber problem. Sections 3 and 4 focus on the gradient-constrained Steiner problem (in two and three dimensions) where the objective is to minimize the total edge length of the network. In Sects. 5–7, the gradient-constrained Gilbert problem is introduced; this involves designing the network so as to minimize the cost associated with a flow on the edges.

2

Motivation and Background

One of the main motivations for studying gradient-constrained interconnection problems lies in their application to the design of underground mines. Mining is a significant industry worldwide. Reducing the cost of mining operations is an

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important issue for mining companies in an extremely competitive marketplace. For many years, there have been well-developed methods for modeling and optimizing the operation of open-cut mines. Commercial optimization mine planning software packages have been developed based on a technique developed in 1965 by Lerchs and Grossmann [46]. These packages have revolutionized the design of open-pit mines, allowing mining engineers and planners to manipulate and visualize data associated with optimal designs. Until recently, comprehensive optimization tools for the design and planning of underground mining operations have not been available to the mining industry. The design process is typically facilitated by planning CAD software packages (see, e.g., [31]) which require high levels of user intuition and experience. The usefulness of these programs lies in their ability to efficiently generate a range of potential designs, from which the “best” feasible solution can be selected and refined. However, since these programs do not harness the benefits of optimization, there is no guarantee that the final solution is even close to optimal. A number of research groups, however, are now in the process of designing and developing optimization tools for the design of underground mines. An example is the access optimization tool, Decline Optimisation Tool (DOT), which has been developed at The University of Melbourne. This application efficiently determines a near-optimal (where optimal means least cost) underground mining network servicing a given set of points associated with an ore body [19]. Several constraints are imposed on the geometry of the tunnels in the network, to ensure that the tunnels are navigable by haulage trucks. One such constraint is that the slope (steepness or grade) of a tunnel cannot be too large, since the maximum grade at which haulage vehicles can operate is typically 1:7. A network satisfying this constraint is called gradient constrained. The first mathematical study of such networks was conducted by the Network Research Group at The University of Melbourne, who initially studied them in a vertical plane [10] and later in three dimensions [13]. An algorithm for computing least-cost gradient-constrained networks was implemented into a software application [16], called Underground Network Optimiser (UNO). Although this application does not account for other practical constraints, such as truck turning circle navigability, UNO is especially useful for efficiently determining optimal layouts for large-scale mining networks where the effects of constraints other than the gradient constraint on the optimal layout of the mine are not significant. Initially, research on gradient-constrained networks focused on networks which minimize length. While length-minimizing networks optimize the total development (or infrastructure) cost associated with an underground mine, they do not account for the effects of haulage on the optimal design of the mine. Haulage costs can have a significant impact on the structure and layout of an underground mine, and considerable savings can be achieved by including both cost components in the objective function. A network facilitating flows between given pairs of vertices in the network is called a flow-dependent network. In the mining context, these networks can incorporate both development and haulage costs into the objective function.

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This section looks first at the nature of the underground mining problem, and the application of gradient-constrained interconnection networks to this problem, and then introduces some of the main mathematical ideas required for this problem.

2.1

Underground Mining Networks

This background on mining is drawn primarily from [2] and [8]. An underground mine consists of a series of interconnecting tunnels, ore passes (near-vertical chutes down which ore is dropped), and vertical shafts (used to hoist ore up to the surface). Its purpose is to allow extraction of ore containing valuable minerals (such as gold, silver, lead, zinc, and copper) from underground locations to a predetermined surface portal (or breakout point), from where it is transported to a processing mill. Ore zones (or stopes) are identified by geological tests such as surface and infill drilling. From this information, mining engineers can determine suitable draw points, which are the locations on the boundary of each stope from which the ore is accessed. The ore is then excavated using one of a number of possible mining methods (such as stoping, caving, room, and pillar) and is transported to the surface via large haulage trucks. Given that these laden trucks must be able to traverse the ramps, the following important constraints must be imposed on the mine: • Ramps must have absolute gradient not greater than some constant m. This constant is typically in the range 1:9–1:7 depending on equipment specifications. • Ramps must satisfy a minimum turning radius (typically in the range 15–30 m), so that they are navigable by the trucks. • Ramps must avoid certain no-go regions such as the interior of an ore body or other ramps in the mine. A set of tunnels satisfying these constraints and interconnecting the draw points and a point at the surface can be modeled by a mathematical network T , where the draw points correspond to fixed vertices and the tunnels correspond to edges. The cost C associated with a network with a set E of edges is typically modeled by a function of the form X C.T / D .d C hte /le e2E

where d is a development cost rate (typically $4,000/m), h is a haulage cost rate (typically $0.75/t.km), te is the total quantity of ore to be transported P along an edge e over the life of the mine, and le is the length of e. The first term d leP can be viewed as the total development cost of the mine and the second component hte le as the total haulage cost associated with the mine over its life. If h is set to zero, then the cost of the network is proportional to its length, and a network minimizing this objective function is a length-minimizing network.

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2.2

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Network Optimization Applied to Underground Mine Design

This section is a brief survey of the use of network theory in underground mine design. A number of the papers mentioned here will be discussed in more detail later in the chapter. Lizotte and Elbrond [47] reported an early application of network theory to some underground mine layout problems but limited their work to network techniques in a horizontal plane. Brimberg et al. [24] considered a network model to minimize the cost of an underground mine with a vertical hoisting shaft connected to ore deposits by a series of horizontal tunnels. A heuristic method for a more general version of this problem was developed in [36], using fuzzy set theory. Lee first raised the more general problem of setting up and optimizing a threedimensional network modeling the access infrastructure costs of an underground mine layout in a 1989 presentation at the NATO Workshop on Topological Network Design, Denmark. Since then, a significant body of research has been developed in this area, some of which has been implemented into the aforementioned software products, UNO and DOT. Much of this work has been conducted by the Network Research Group at The University of Melbourne. The following is a summary of this research to date: • A network optimization model for minimizing development and haulage costs in underground mines was estsblished. The possibility of modeling the haulage cost of an edge as a function of the slope of the edge was also facilitated in the cost function, and conditions under which the cost function is convex were discussed. Application of the model was demonstrated via a prototype network interconnecting draw points and a vertical hoisting shaft and several case studies in which the model was applied to mines at Pajingo and Kanowna Belle [2, 8, 11, 14, 17]. • The gradient-constrained Steiner problem (in a vertical plane) was introduced. This problem asks for a shortest network interconnecting a set of points, called terminals, in a vertical plane such that no part of the network has absolute gradient greater than a given positive constant m satisfying m  1. Geometric structures of these networks were used to construct a finite algorithm for solving the problem which, when suitably discretized, was shown to be NPcomplete [10]. • The study of the gradient-constrained Steiner problem was extended to three dimensions. The gradient metric was introduced to measure the lengths of edges in gradient-constrained networks, and the terms flat, maximum, and bent were introduced to label edges whose respective gradients are less than, equal to, or greater than m. It was shown that a Steiner point, which is a vertex in the network not among the given set of terminals, has either three or four incident edges, and Steiner points were classified in terms of the labelings of their incident edges. The UNO software product was concurrently developed with this work [13]. • The terms labeled minimal and locally minimal were introduced to describe Steiner points for which a label-preserving perturbation and, respectively, any

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perturbation of the Steiner point cannot shorten the length of the network. Properties of labeled-minimal Steiner points were studied, and necessary and sufficient conditions for Steiner points to be locally minimal were obtained. A formula for computing labeled-minimal degree 3 Steiner points was derived, and an algorithm for computing locally minimal Steiner points was developed [21, 61]. • The Steiner ratio for gradient-constrained networks connecting three points in three dimensions was investigated. The Steiner ratio is the smallest ratio of the length of a Steiner tree to the length of a minimum spanning tree when the two networks interconnect the same set of given points. It was shown that this minimum ratio for gradient-constrained networks tends to 0.75 as the value of m tends to zero [50, 51]. • A method for optimizing declines in underground mines was developed, a decline being a gradient-constrained, turning-circle-constrained path connecting level access points to a surface portal while avoiding certain no-go regions. A procedure for finding an optimal decline was implemented into the DOT software product, which was applied to several case studies including a decline servicing the Jandam gold mine at Pajingo and an access decline at Olympic Dam. The software was further developed to handle networks of declines [15, 16, 19, 20, 62]. • The other area that has been receiving more attention in recent years is that of understanding the effect of weighted edges, associated with flows, on the optimal design of gradient-constrained networks. The principal aim of this research is to obtain a comprehensive understanding of minimum-cost gradient-constrained flow-dependent networks. Of particular interest are the geometric structures possible in such networks. The geometric properties of optimal networks play a crucial role in improving the efficiency of heuristic algorithms for computing gradient-constrained networks [66–68].

2.3

Minkowski Spaces and the Gradient Metric

An important insight into understanding gradient-constrained networks in Euclidean space is that if the distance between two points is measured as the minimum length of a gradient-constrained path between the two points, then this measure forms a metric. For the purpose of determining lengths of edges, the network can be viewed as being embedded in a normed or Minkowski space. This subsection briefly reviews some of the relevant properties of Minkowski spaces, and defines the gradient metric, for measuring the lengths of edges in gradient-constrained networks. See [63] for a more comprehensive introduction to Minkowski geometry. A Minkowski space (or finite-dimensional Banach space) is Rn endowed with a norm kk, which is a function kk W Rn ! R that satisfies: • kxk  0 for all x 2 Rn , kxk D 0 only if x D 0. • k˛xk D j˛jkxk for all ˛ 2 R and x 2 Rn .

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• kx C yk  kxk C kyk. Informally, a norm provides a way of measuring the lengths of vectors in Rn . A common norm is the `p norm

kxkp D

n X

! p1 jxi j

p

i D1

where p  1 is a real number and the xi are the components of x. The value p D 2 gives the familiar Euclidean norm, denoted throughout this chapter by jj, and p D 1 gives the rectilinear norm, also called the Manhattan, taxicab, or `1 norm. The unit ball associated with a given Minkowski space X is the set B D fx 2 X W kxk  1g. It is a centrally symmetric, closed, and convex subset of X with origin o at its center. A unit ball is called smooth if its boundary, bd.B/, has no sharp (nondifferentiable) points and strictly convex if bd.B/ contains no straight line segments. A supporting hyperplane of B is a hyperplane in X touching the boundary of B such that the interior of B lies on one side of the supporting hyperplane. An exposed face of B is the intersection of B with a supporting hyperplane. It is possible to measure the length of edges in a gradient-constrained network in terms of a metric known as the gradient metric, first introduced in [13]. This means that these networks can be thought of as being in a Minkowski space. Formal definitions are given below for the three-dimensional case, but these results also generalize easily to other dimensions. Let xp ; yp ; zp denote the Cartesian coordinates of a point p in three-dimensional space. Assume that the z-axis is vertical. Then, the gradient of an edge pq is defined here to be the absolute value of the slope from p to q and is denoted by g.pq/. That is, jzq  zp j g.pq/ D p : .xq  xp /2 C .yq  yp /2 Suppose pq is an edge of a minimum gradient-constrained network embedded in Euclidean space, where the maximum permitted gradient m is given. If g.pq/  m, then pq is a straight line joining p and q and is referred to as a straight edge. However, if g.pq/ > m, then pq cannot be represented as a straight line without violating the gradient constraint, but it can be represented by a zigzag line joining p and q with each segment having gradient m. Such edges are referred to as bent edges. The lengths of edges in a gradient-constrained tree can be measured in a special metric, called the gradient metric. Suppose o is the origin and p D .xp ; yp ; zp / is a point in space. Define the vertical metric of the line op to be jopjv D czp where c is a given constant. Then the gradient metric can be defined in terms of the Euclidean and vertical metrics. Suppose m is the maximum gradient allowed in gradient-constrained trees. The length of op in the gradient metric is defined to be

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m=1

m=0.58

m=0.14

Fig. 1 Unit ball in gradient-constrained three-space for selected maximum gradients

 jpqjg D

p jpqj D p.xq  xp /2 C .yq  yp /2 C .zq  zp /2 ; if g.pq/  m; if g.pq/  m. jpqjv D 1 C m2 jzq  zp j;

It is easily checked that this defines a metric, as it is the maximum of two metrics. Note that jopj  jopjg and the gradient metric is convex though it is not strictly convex. In gradient-constrained three-space, three examples of the unit ball Bg are shown in Fig. 1 for m D 1; 0:58, and 0:14.

2.4

The Fermat-Weber Problem

The types of interconnection problems considered in this chapter are those that allow for the possibility of adding extra nodes anywhere in the space in order to minimize the length (or, more generally, cost) of the interconnection network. The simplest and most famous example of such a problem is the Fermat problem, which asks one to find a point such that the sum of its distances from a set of given points is a minimum. The more general version of this problem is called the Fermat-Weber problem; this problem asks for a point, called a Fermat-Weber point, minimizing the sum of weighted distances to a set of given points. The weights are specified by given values wi associated with each of the given points pi . A brief exposition of the historical background to this problem is given by Kuhn [42]. A more detailed account is included in a paper of Kupitz and Martini [44], and another good general reference is [27]. Although special instances of the Fermat-Weber problem can be solved with simple geometric constructions, in general for k  4, the problem cannot be solved by exact methods [3]. Thus, significant research has been undertaken to develop numerical procedures for finding Fermat-Weber points. A famous iterative method for the Fermat-Weber problem (outlined below) was posed in 1937 by Andre Weiszfeld [74]. This algorithm was independently discovered by Kuhn and Kuenne [43] in 1962.

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Let N D fp1 ; : : : ; pk g be the set of given points in Rn with respective positive weights w1 ; : : : ; wk . Let ui denote the unit vector from a point x0 2 Rn to pi ; i D 1; : : : ; k. The first step of the algorithm checks whether x0 coincides with a given point. If not, x0 is determined iteratively. Weiszfeld’s Algorithm 1. If there exists a point pj 2 N such that ˇ ˇ ˇ k ˇ ˇ X ˇ ˇ ˇ w u i i ˇ  wj ; ˇ ˇi D1;i ¤j ˇ then pj is a Fermat-Weber point. 2. Otherwise, select an error estimate . 3. Select an initial point x .0/ 2 int.conv.N //. 4. For  D 0; 1; : : : do P x

.C1/

ˇ ˇ 5. Stop when ˇx ./  x .C1/ ˇ  .

WD

wi pi pi 2N jpi x ./ j P wi pi 2N jpi x ./ j

:

The following example demonstrates the application of Weiszfeld’s algorithm and is illustrated in Fig. 2. Let p1 D .0; 1/, p2 D .2; 0/, and p3 D .2; 2/ be given points in the plane with respective weights w1 D 1, w2 D 1, and w3 D 1:65. By testing the collapse condition, it can be shown that x0 ¤ p1 ; p2 ; p3 . Selecting  D 0:001 and x .0/ D .0:25; 1/, the path of the iteration point converges to x0 , as shown in the figure. In 1995, Brimberg [23], building on the work of Kuhn [41] and Chandrasekaren et al. [25], proved that Weiszfeld’s algorithm converges to the unique optimal solution for all but a denumerable set of starting points if and only if the convex hull of the given points is of dimension n. The Fermat-Weber point can be viewed as the central node of a star network with links from the Fermat-Weber point to each of the given nodes. This network is then a minimum cost interconnection network with this star topology. In this context, one can define the gradient-constrained Fermat-Weber problem, introduced by Volz in [66] and [68], which is the Fermat-Weber problem with the added condition that links are measured using the gradient metric. The problem is formulated as follows: • Given: A set N D fp1 ; : : : ; pk g of points in Euclidean space with respective positive weights w1 ; : : : ; wk and a gradient bound m satisfying 0 < m  1 • Find: A point x0 minimizing the sum of weighted distances from x0 to the points in N such that each distance is the minimum length of a piecewise smooth curve whose gradient at each differentiable point is no more than m In the mining context, p1 ; : : : ; pk represent draw points with associated tonnages t1 ; : : : ; tk . Material is hauled from the draw points via independent paths to the

1468 Fig. 2 Iterations of Weiszfeld’s algorithm for a three-point Fermat-Weber problem

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p3

2

1

x (0)

p1

0

p2 0

1

2

Fermat-Weber point, x0 , which might represent, for instance, the base of a vertical hoisting shaft. The total cost of the mine can be minimized by positioning the facility at an optimal location. The weight on the path from pi to x0 is given by wi D d Chti , where d and h are development and haulage cost rates, respectively. Since the ti can take on any value, there are no restrictions on the values of the weights associated with the points in N . This is in contrast to the Gilbert arborescence problem described in the next section, in which edge weights in the network satisfy certain implicit conditions resulting from the flow structure in the network. Numerical tests have shown that the Weiszfeld algorithm, described above, does not lend itself well to the gradient-constrained problem. In [66] and [68], Volz has proposed a new iterative scheme for finding a point x0 minimizing

f .x/ D

k X

jpi  xjg :

i D1

Standard gradient descent procedures, such as those discussed in [6], cannot be easily applied here due to the fact that f .x/ is not everywhere differentiable. The problem for nondifferentiable convex functions, however, can be solved by the subgradient method, described in [57] and [5], which minimizes f .x/ using the iteration x .C1/ D x ./ C s v./

Gradient-Constrained Minimum Interconnection Networks Fig. 3 Iterations of the subgradient method with diminishing step sizes for a three-point gradient-constrained Fermat-Weber problem

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2

p3

x0

x (0)

z

p1

0

p2 0

1 x

2

where x ./ is the th iterate, v./ is any subgradient of f at x ./P , and s is the th step size. For diminishing step sizes satisfying lim!1 s D 0; 1 D1 s D 1, the algorithm is guaranteed to converge to the optimal value. For the gradient-constrained problem, use of the subgradient method may problematic, despite guaranteed convergence. The convergence can be extremely slow due to the gradient being almost perpendicular to the direction toward the minimum. This problem is demonstrated by the following example. Consider three points in a vertical plane: p1 D .0; 1/, p2 D .2; 0/, and p3 D .2; 2/, as shown in Fig. 3. The three points have respective weights w1 D 1, w2 D 1, and w3 D 1:94. The maximum allowable gradient is m D 1. The exact solution, which can in this case be determined exactly by simple calculus, is x0 D .1:9; 1:9/. Starting at x .0/ D .0:1; 1/ and using an initial step size s0 D 0:345, consider the use of the subgradient method where step sizes diminish at each iterate  according p to s D s0 = . The iterative path, shown in the figure, shows that convergence is very slow due to extreme oscillation about the line through p3 with gradient m. At each iteration, the gradient is almost perpendicular to the direction toward the minimum. In fact, after 50,000 iterations, x ./ is still noticeably distant from x0 . If s is chosen to minimize f .x ./ C s v./ / (i.e., an exact line search is adopted), then x ./ converges to the nonstationary point .1:3385; 1:3385/ after one iteration, due to the optimum step size diminishing to zero too early. Volz [66] and Volz et al. [68] proposes a new method that overcomes these problems by exploiting the fact that f is differentiable everywhere except in the set S of points x in R3 for which pj x is an m-edge for any pj 2 N . The method uses exact directional minimization at each iteration and considers a second search direction at maximum gradient when x ./ converges to a (possibly non-minimal) point in some neighborhood of S . This notion of searching in an m-edge direction was first employed in an algorithm for finding gradient-constrained minimum Steiner trees in [16]. The procedure for the gradient-constrained Fermat-Weber

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Fig. 4 Iterations of a new descent method for a three-point gradient-constrained Fermat-Weber problem

2

x (2) p3

z

x (1) p1

0

x (0)

p2 0

1 x

2

problem is outlined in [66, 68], where it is also shown that the method guarantees rapid convergence in dimensions 2 and 3. Applying this method to the previous example results in the sequence converging to the exact solution in two iterations, f.0:1; 1/; .1:3385; 1:3385/; .1:9; 1:9/g (Fig. 4).

2.5

The Steiner Tree Problem

The minimum Steiner tree problem asks for a shortest network spanning a given set of nodes, usually referred to as terminals, in a given metric space. It differs from the minimum spanning tree problem in that additional nodes, referred to as Steiner points, can be included to create an interconnection network that is shorter than would otherwise be possible. This is a fundamental problem in physical network design optimization, and has numerous applications, including the design of telecommunications or transport networks for the problem in the Euclidean plane (the l2 metric), and the design of microchips for the problem in the rectilinear plane (the l1 metric) [40]. The terminology used for the minimum Steiner tree problem in this chapter is largely based on that used in [40]. Let T be a network interconnecting a set N D fp1 ; : : : ; pn g of points, called terminals, in a Minkowski space. A node in T which is not a terminal is called a Steiner point. In a minimum network, it can always be assumed that the degree of a Steiner point is at least three. Let G.T / denote the topology of T , that is, G.T / represents the graph structure of T but not the embedding of the Steiner points. Then G.T / for a shortest network T is necessarily a tree, since if a cycle exists, the length of T can be reduced by deleting an edge in the cycle. A network with a tree topology is called a tree, its links are called

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edges, and its nodes are called vertices. An edge connecting two vertices a; b in T is denoted by ab and its length by ka  bk. The splitting of a vertex is the operation of disconnecting two edges av; bv from a vertex v and connecting a; b; v to a newly created Steiner point. Furthermore, though the positions of terminals are fixed, Steiner points can be subjected to arbitrarily small movements provided the resulting network is still connected. Such movements are called perturbations and are useful for examining whether the length of a network is minimal. A Steiner tree is a tree whose length cannot be shortened by a small perturbation of one or more of its Steiner points, even when splitting is allowed. By convexity, a Steiner tree is a minimum-length tree for its given topology. A minimum Steiner tree is a shortest tree among all Steiner trees. For many Minkowski spaces, bounds are known for the maximum possible degree of a Steiner point in a Steiner tree, giving useful restrictions on the possible topology of an SMT. For example, in Euclidean space of any dimension, every Steiner point in an ST has degree 3. Given a set N of terminals, the Steiner problem (or minimum Steiner tree problem) asks for a minimum Steiner tree spanning N . The gradient-constrained Steiner problem is studied in more detail in Sects. 3 and 4.

2.6

The Gilbert Arborescence Problem

In 1967, Gilbert [34] proposed a generalization of the SMT problem whereby symmetric nonnegative flows are assigned between each pair of terminals. The aim is to find a least cost network interconnecting the terminals, where each edge has an associated total flow such that the flow conditions between terminals are satisfied, and Steiner points satisfy Kirchhoff’s rule (i.e., the net incoming and outgoing flows at each Steiner point are equal). The cost of an edge is its length multiplied by a nonnegative weight. The weight is determined by a given function of the total flow being routed through that edge, where the function satisfies a number of conditions. The Gilbert network problem (GNP) asks for a minimum-cost network spanning a given set of terminals with given flow demands and a given weight function. A variation on this problem that will be shown to be a special case of the GNP occurs when the terminals consist of n sources and a unique sink, and all flows not between a source and the sink are zero. This problem is of intrinsic interest as a natural restriction of the GNP; it is also of interest for its many applications to areas such as drainage networks [45], gas pipelines [4], and underground mining networks [11]. If the weight function is a concave, increasing function of the flow, then the resulting minimum network has a tree topology and provides a directed path from each source to the sink (this is discussed in more detail in Sect. 5). Such a network can be called an arborescence, and this special case of the GNP is referred to as the Gilbert arborescence problem. Traditionally, the term “arborescence” has been used to describe a rooted tree providing directed paths from the unique root (source) to a

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given set of sinks. Here, it is used for the case where the flow directions are reversed, that is, flow is from n sources to a unique sink. It is clear, however, that the resulting weights for the two problems are equivalent; hence, it is reasonable to also use the term “arborescence” for the latter case. Moreover, if one takes the sum of these two cases and rescales the flows (dividing flows in each direction by 2), then again the weights for the total flow on each edge will be the same as in the previous two cases. This justifies the earlier claim that the Gilbert arborescence problem can be treated as a special case of the GNP. It will be convenient, however, for the remainder of this chapter to treat arborescences as networks with a unique sink. A gradient-constrained version of the Gilbert arborescence problem is studied in Sects. 6 and 7. In three-space, the simplest version of this problem can be formulated as follows: • Given: A set of points N D fp1 ; : : : ; pk g in Euclidean three-space, where p1 ; : : : ; pk1 are sources with respective positive flows t1 ; : : : ; tk1 and pk is a unique sink, a gradient bound satisfying 0 < m  1, and positive constants d and h • Find: A minimum-cost network T interconnecting N which provides flow paths from the sources to the sink, such that the cost of an edge e in T is given by .d C hte /le , where te is the total flow through e and le is the length of e under the gradient metric A network satisfying the above conditions is called a minimum Gilbert arborescence (MGA). Vertices in T not in N are called Steiner points, and a Steiner point is a Fermat-Weber point with respect to its adjacent vertices in T . A key difference between the two problems is that a Steiner point s in a Gilbert arborescence has source edges, which are directed into s, and a unique sink edge, directed outward from s, whereas all incident paths connected to a Fermat-Weber point are directed toward that point.

3

The Two-Dimensional Gradient-Constrained Steiner Problem

The simplest version of a gradient-constrained interconnection problem, beyond that of constructing a minimum spanning tree, is the planar Steiner problem. This problem asks for a shortest network interconnecting a given set of points (referred to as terminals) lying on a vertical plane in Euclidean space and operating under a gradient constraint. Unlike a spanning tree network, this network can contain nodes other than the given terminals. It will be shown in Sect. 3.2 that the inclusion of these extra variable nodes makes this an NP-hard problem. In formal terms, the two-dimensional gradient-constrained Steiner problem can be stated as follows: • Given: A set of points N D fp1 ; : : : ; pk g lying on a vertical plane P in Euclidean three-space and a gradient bound satisfying 0 < m  1 • Find: A network T interconnecting N of minimum total length, where the length of each edge e is computed using the gradient metric (with gradient bound m)

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As usual, the nodes of T that are not terminals are referred to as Steiner points and are assumed to have degree at least 3. A network T , solving this problem, is referred to as an m-constrained minimum Steiner tree. From an applications point of view, this problem is interesting, not only as a specific example of the three-dimensional problem but also for potential applications in its own right. For example, consider the following problem. Imagine an exterior wall of a large multistory warehouse, or other building, with entrances at given positions on the wall. Suppose one is required to join the entrances externally by designing an interconnecting system of ramps whose gradients are sufficiently small to allow trolleys and carts to be wheeled along them, or alternatively to allow disabled access. By modeling the entrances as points and the ramps as line segments, the problem of finding the shortest system of ramps is equivalent to solving the planar version of the gradient-constrained Steiner problem. The precise formulation of the above problem deserves a few comments. The reason that the plane P is restricted to being a vertical plane is that it can then be assumed that the network T lies on P. This follows from the observation that the projection of T onto P does not increase the length of any link of T (since the vertical displacement between the endpoints remains the same and the horizontal displacement does not increase). This observation is no longer true in general if P is not restricted to being vertical. For P vertical, the problem, although set in Euclidean three-space, can be assumed to be two-dimensional. The other important restriction on the problem is that the gradient bound m satisfies m  1. Let ˛ D arctan m be the angle of an edge of maximum gradient from the horizontal. The condition is equivalent to saying that ˛  =4. This condition is imposed largely for convenience. In particular, it immediately gives the following local property of T at any node. Lemma 1 Let T be an m-constrained minimal Steiner tree (for 0 < m  1) in a vertical plane P. Then the angle at which any two links of T meet at a vertex is at least 2˛. Proof Suppose, on the contrary, that there exists a node v of T such that two of the links incident with v, say, av and bv, meet at an angle less than 2˛. Clearly, at least one of the two links, say, av, has gradient of absolute value strictly less than m. The length of T can now be reduced by moving the point where this link connects a to bv away from v, toward b to a new point v0 such that either av0 has absolute gradient m or v0 D b (whichever occurs first), as illustrated in Fig. 5. The shorter tree is still gradient constrained, contradicting the minimality of T .  This constraint on the gradient is usually easily met in practice in applications such as ramp design or underground mine access infrastructure where the typical maximum allowable gradient is generally in the range of 1=9–1=7. It follows from Lemma 1 that all nodes of T have degree at most 4 and hence that all Steiner points of T have degree 3 or 4. Indeed, the lemma has the following easy consequence, giving local geometric properties at each Steiner point.

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Fig. 5 Replacing av by av0 decreases the length of T

a

v v⬘

b

Corollary 1 Let s be a Steiner point of an m-constrained minimum Steiner tree T . Let Ls be the vertical line through s. If s is of degree 3, then s has two incident links, one of gradient m and the other of gradient m, lying on one side of Ls , and a third incident link lying on the other side of Ls . If s is of degree 4, then s has two incident links, one of gradient m and the other of gradient m, lying on each side of Ls .

3.1

Local Minimality and Exact Solutions

Like most other planar Steiner problems in normed spaces, the two-dimensional gradient-constrained Steiner problem is NP-hard, meaning that there is almost certainly no polynomial time algorithm for constructing an optimal solution. Despite this, the problem of finding relatively efficient exact algorithms for planar Steiner problems under a range of different norms has attracted considerable interest and success. The most effective and scalable algorithms are those based on the so-called Geosteiner approach [71, 72]. The key to the Geosteiner approach is to view a minimum Steiner tree as a union of full Steiner trees (FST’s), which are minimum Steiner trees where all terminals are leaves and all Steiner points are interior nodes. The algorithm involves two distinct phases: a generation phase which generates a set of FSTs guaranteed to include those used in a minimum Steiner tree; and a concatenation phase which determines how to optimally combine the FSTs in the given set into a minimum Steiner tree. The concatenation phase is independent of the underlying metric and has been efficiently solved, using a branch-and-cut algorithm, by Warme [69] and Warme et al. [70, 71]. The success of the Geosteiner approach depends on keeping the number of FSTs constructed in the generation phase as small as possible. This relies on finding effective pruning techniques. The idea is to build the FSTs in a bottom-up manner and at each step to apply a range of geometric tests which will allow whole families of potential FSTs to be eliminated. An example of such a test is the “lune test,” which basically states that terminals cannot be too close to long edges in a minimum Steiner tree. For the Euclidean metric, a suite of pruning techniques, based on the strong local geometric restrictions on minimum Steiner trees, have been developed which keep the number of generated FSTs relatively small, for

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randomly generated terminal sets. For other metrics, such as the rectilinear metric and the -geometry metrics (where the network is restricted to using  equally spaced orientations, see [7]), the efficiency of the generation phase depends not only on good pruning techniques but also on the characterization of FSTs in terms of canonical forms. These canonical forms (established for general -geometry metrics in [18]) strongly restrict the number of FSTs that need to be considered and give a simple bottom-up method of building FSTs out of “half FSTs”; for more details, see Nielsen et al. [49]. In practice, for any fixed , the resulting number of generated FSTs is almost linear, and the size of the largest generated FST is bounded by a constant. Geosteiner has been used to solve randomly generated problem instances with 1,000 terminals in less than a hour (for   8), and a single problem instance with 10,000 terminals has been solved in less than 48 h for  D 4. The Geosteiner approach has not been developed for the two-dimensional gradient-constrained Steiner problem, probably because the main applied interest is in the three-dimensional version of the problem, where the generation phase of Geosteiner appears to be much less effective. But the two-dimensional problem is amenable to the same general method as there are very strong structural and geometric constraints on the form of locally minimal FSTs. The theory behind this was developed in [10], and the main properties will now be discussed in the remainder of this section. It is useful to divide the study of the structure of two-dimensional gradientconstrained minimal FSTs into two distinct cases, the first where all edges in the full tree have absolute gradient m and the second where there is at least one edge of gradient k where jkj < m. Each of these two cases will be considered in turn.

Case 1: All Edges Have Absolute Gradient  m Recall that any edge e of absolute gradient greater than m with endpoints a and b can be embedded in the Euclidean plane as a path from a to b composed of two straight line segments, each of gradient m or m, such that the Euclidean length of this path is equal to the length of e under the gradient metric. Such edges are referred to as a bent edges and the internal point where a bent edge changes gradient as the corner-point of the edge. Also, a straight line segment in a gradient-constrained minimal FST, each of whose endpoints is either a node or corner-point and which has no nodes in its relative interior, is referred to simply as a segment. Note that the angle between the two segments at a corner-point is 2˛. This viewpoint of the FSTs being embedded in the Euclidean plane allows properties of Euclidean geometry to be employed in the study of these networks. Let T be a two-dimensional gradient-constrained minimal FST such that every segment in T has absolute gradient m. This is very similar to the situation for rectilinear Steiner trees. Indeed, the characterization of the structure of a full rectilinear minimal Steiner tree does not use, in any essential way, the fact that the two available directions are perpendicular. It follows that by defining substructures of T in line with the standard definitions used in rectilinear Steiner trees, it is

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(i)

(ii)

(iii)

(iv)

Fig. 6 The four canonical forms for full components given in Theorem 1

possible to prove a structure theorem giving a canonical form for T identical to that which has been proved for rectilinear Steiner trees. The definitions for substructures of T imitate the corresponding definitions for rectilinear trees used in [40], which are based largely on the geometric characterizations developed in [52]. A complete line of T is defined to be a sequence of one or more adjacent collinear segments of maximal length; it does not have terminals in its relative interior (since T is full), but it may have terminals as endpoints. A corner-point is incident to exactly two complete lines, one of gradient m and the other of gradient m, referred to as the legs of the corner. If the remaining two endpoints of the legs are terminals, the corner is a complete corner. A cross-point is a degree 4 Steiner point of T , and the two complete lines passing through a cross-point are said to from a cross. Finally, given a complete line L of T , let C.L/ denote the set of segments (not in L) which are incident to L at any endpoints of L that are corners. Then a set A of alternating segments incident to L is defined to be the set of segments of T such that: 1. A contains all segments not in L but incident to L at vertices other than the endpoints of L (all of which, it follows, have a gradient which is minus the gradient of L). 2. Each element of A intersects L at a Steiner point of degree 3. 3. No two segments in A [ C.L/ incident to endpoints of a single segment of L lie on the same side of L. By performing a series of local length-preserving transformations on T , each of which either changes the embedding of a bent edge or “slides” an edge whose two endpoints are both Steiner points (see [40] or [52] for the formal definitions of these transformations in the rectilinear setting), one can obtain the following characterization of a canonical form for T , as illustrated in Fig. 6. This theorem, in the rectilinear setting, was originally due to Hwang [37].

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Theorem 1 Let N be a set of n terminals such that n  2. Suppose every m-constrained minimal Steiner tree on N is full and contains only segments of absolute gradient m. Then there exists an m-constrained minimal Steiner tree on N that either: (i) Is a cross; or (ii) Consists of a single complete line with n  2 alternating segments; or (iii) Consists of a complete corner with n  2 alternating segments incident to a single leg; or (iv) Consists of a complete corner with n  3 alternating segments incident to a one leg and a single alternating segment incident to the other leg Note that the first form in Theorem 1, a cross, only occurs if n D 4. The consequence of this theorem is that for any terminal set N , there exists an m-constrained minimal Steiner tree on N such that every full component either has one of the forms listed in the previous theorem or contains an edge with absolute gradient strictly less than m. The generation of all candidate FSTs with all segments having absolute gradient m (as part of a Geosteiner approach) can be done efficiently in practice using the same techniques that have been developed for rectilinear minimum Steiner trees by Zachariasen [77].

Case 2: There Exists an Edge with Absolute Gradient < m For this case, let T be a two-dimensional gradient-constrained minimal FST such that T contains a segment with gradient k where jkj < m. Such a segment is said to have intermediate gradient. Clearly, the segment must be a straight edge since segments at a corner have absolute gradient m. Furthermore, if a is an endpoint of an edge in T of intermediate gradient, then all edges of T incident with a are straight. This is an easy consequence of Lemma 1, since if there is a bent edge at a, then there exists an embedding of that edge such that two of the segments incident with a have an angle between them of less than 2˛, leading to a contradiction to the minimality of T by the argument in the proof of Lemma 1. By a similar argument, it is also clear that a has degree at most three. A key property of T is that every edge in T with intermediate gradient has the same gradient. More specifically, the following structure theorem holds. Theorem 2 Let T be a full m-constrained minimal Steiner tree containing an edge of gradient k where jkj < m. Then every Steiner point of T is adjacent to three straight edges, one of gradient m, one of gradient m, and one of gradient k. The idea of the proof of this theorem, which appears in [10], is to show that if there are two adjacent Steiner points in T that do not satisfy the theorem, then there exists a simultaneous perturbation of both Steiner points that strictly reduces the length of T . This is illustrated in Fig. 7. Suppose T is the m-constrained tree with terminals a; b; c; d as shown in the figure. Here, the points a; b; c; d all lie in a vertical plane, and all edges have gradient m apart from a single edge of intermediate

1478 Fig. 7 A non-minimum tree where each Steiner point is locally minimal

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a

f

s1

b

m

m s2 m d

m c

gradient, labeled “f” in the figure. Each of the two Steiner points satisfies the conditions of Corollary 1, and it is straightforward to confirm that each Steiner point is locally minimal. In other words, no perturbation of s1 or s2 (which fixes the other Steiner point) can reduce the length of T . However, consider the 2-point perturbation v shown in the figure, where s1 is perturbed along the line through s1 b away from b, s2 is perturbed along the line through s2 d toward d , and s1 s2 remains an m-edge. If L represents Euclidean length, then clearly under this perturbation, P 1 b [ s1 s2 [ s2 d /v D 0, but it is easy to see that L.s P 1 a [ s2 c/v < 0 and hence L.s that v is a strictly length-reducing perturbation. A similar argument always applies unless s1 a and s2 c are parallel, that is, have identical gradient. This idea that the directions of edges at a single Steiner point must propagate throughout the full component if there is no length-reducing perturbation for any pair of Steiner points is a concept that has been shown to also apply to metrics other than the gradient metric. It is well known that it is true for all smooth metrics [29]. More recently, it has been shown that it also applies for all metrics where the unit ball is a simple polygon whose vertices lie on the Euclidean unit circle. Such a metric is referred to as a fixed orientation metric; the terminology comes from the observation in [75] and [76] that if D is the set of orientations (or legal directions) corresponding to the vertices of the polygon, then for any pair of points in the Euclidean plane, there exists a path composed of a pair of line segments with adjacent legal directions so that the Euclidean length of the path equals the distance between the points under the fixed orientation metric. In a minimal FST under such a metric containing at least two Steiner points, each Steiner point has degree 3, and hence the incident edges at that Steiner point use a set of up to six legal directions in their representations as Euclidean embeddings. This set of legal directions is known as the direction set of the Steiner point. It has been shown in [18] (for -geometry Steiner trees) and in [9] (for Steiner trees under more general fixed orientation metrics) that for any set of terminals, there exists a minimum Steiner tree such that in each full component all Steiner points use only a single direction set, and hence the full component itself uses at most six legal directions. In a similar manner to the proof of Theorem 2, the proofs of these results are based on showing

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that if two adjacent Steiner points have different direction sets then there exists a length-reducing perturbation on the FST. For fixed orientation metrics, the usefulness of knowing that an FST uses only a single direction set lies in the fact that there are only a relatively small number of direction sets that need to be considered; in fact, if there are  legal orientations (or equivalently  vertices on the polygonal unit ball), then the number of candidate direction sets that need to be considered is ‚./, and these can be identified in ‚./ time [9]. For the gradient metric, however, the “direction set” at each Steiner point includes an unknown intermediate direction k which belongs to the uncountable set .m; m/. This potential difficulty was resolved in [10] where it was shown that for any m-constrained minimal FST, there is a simple formula for computing the gradient of the intermediate edge in terms of the coordinates of the terminals and information about the topology of the tree. The correct statement of this result requires the following definition. A full Steiner topology, each of whose edges has been assigned with the labels m, m, or k with the property that the three edges meeting at a Steiner point all have different labels, is referred to as a labeled Steiner topology. A full m-constrained minimal Steiner tree is said to have a given labeled topology if there exists a real number k with jkj < m such that the gradient of each edge corresponds to the label on that edge. It can be shown, using the threshold technique developed by Hwang and Weng [39], that m-constrained minimum Steiner trees with any labeled topology can occur. Theorem 3 Let T be a full m-constrained minimal Steiner tree containing an edge of gradient k where jkj < m at every Steiner point. Then given the terminal set and labeled Steiner topology of T , the gradient k can be computed in linear time. Hence, the locations of all Steiner points of T can also be computed in linear time. The details of how to compute k and the correctness of the result are given in [10]. The locating of the Steiner points is then a simple recursive procedure based on determining (via the labeled Steiner topology) the position of each point that has two neighboring nodes in known locations.

3.2

Computational Complexity

The properties discussed in the previous section essentially give an efficient way of solving the two-dimensional gradient-constrained Steiner problem for a given Steiner topology. In practice, it should be possible to use these properties as part of a Geosteiner-type algorithm to solve the general problem in a scalable way for most randomly distributed sets of terminals. In this section, however, it will be shown that the general problem is NP-complete, which suggests that there will always be families of instances for which any exact solution algorithm is unable to scale efficiently.

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This result is not surprising, given that both the rectilinear and Euclidean Steiner problems are NP-hard, as shown in [32] and [33]. Indeed, it is possible to prove a somewhat stronger result, namely, that the (discretized) two-dimensional gradientconstrained Steiner problem is NP-complete even when the terminals are restricted to lying on two vertical lines. This restriction is an interesting one as it has been shown by Aho et al. [1] that imposing it on the rectilinear Steiner problem allows one to find a linear time algorithm, whereas Rubinstein et al. [55] have shown that the discretized Euclidean Steiner problem is still NP-complete under this restriction. The formal decision version of the problem is as follows: The Vertical Line Gradient-Constrained Steiner Problem • Instance: A set of points N D fp1 ; : : : ; pk g contained in two vertical lines, a gradient bound m satisfying 0 < m  1 and a real number l. • Question: Is there an m-constrained Steiner tree T with terminals N and length at most l? As discussed in [33], there are difficulties in describing the complexity of problems such as this caused by the necessity of computing with irrational numbers. A method of circumventing these difficulties is to introduce the discretized Euclidean metric d0 , where d0 .a; b/ D ddE .a; b/e. For the discretized vertical line gradient-constrained Steiner problem, the vertices of the tree are restricted to points with integer coordinates, and the lengths of edges in the tree (or the lengths of segments, in the case of bent edges) are replaced by their discretized Euclidean lengths. Theorem 4 The discretized vertical line gradient-constrained Steiner problem is NP-complete. This theorem was first proved in [10] using a similar strategy to that devised for the Euclidean Steiner problem in [55]. A somewhat simpler, more constructive proof of both of these complexity results was given in [12]. The basic strategy of the proof is to show that solving a specific instance of the discretized vertical line gradient-constrained Steiner problem depends on being able to solve the subset sum problem, that is, the problem of deciding for any given subset S DP fd1 ; : : : ; dn g of integers and integer s whether there exists a subset J  S such that J di D s. The subset sum problem is known to be NP-complete, from which the theorem follows. The key to the construction of the instance encoding the subset sum problem is to place closely spaced triples of terminals on one of the vertical lines in such a way that in a gradient-constrained minimal Steiner tree, each triple will have a Steiner point adjacent to two of the terminals, and there will be a bent edge connecting the third terminal to one of that pair. For each triple, there is a choice as to which pair of terminals has an adjacent Steiner point. The locations of all terminals are contrived so that in the minimum Steiner tree, all edges with intermediate gradient are horizontal, and finding a choice of connection to each triple that makes all intermediate gradient edges horizontal is equivalent to solving the subset sum problem. This construction can easily be done in polynomial time from any instance of the subset sum problem, which completes the proof. The details of the construction and proof of correctness can be found in [12].

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The Three-Dimensional Gradient-Constrained Steiner Problem

The three-dimensional gradient-constrained Steiner problem can be stated as follows: • Given: A set of points N D fp1 ; : : : ; pk g in Euclidean three-space and a gradient bound satisfying 0 < m  1 • Find: A network T interconnecting N of minimum total length, where the length of each edge e is computed using the gradient metric (with gradient bound m) A tree solving the three-dimensional gradient-constrained Steiner problem for some given set of terminals will be referred to as a gradient-constrained minimum Steiner tree. Moving from two to three dimensions significantly increases the inherent difficulty in solving Steiner problems. The three-dimensional Euclidean Steiner tree problem, for example, is known to be considerably more difficult than the planar version. Some of the basic properties of such trees have been studied in [35] and [58]. The three-dimensional gradient-constrained Steiner problem is even more complicated. Despite the high levels of complexity involved in solving the Steiner problem, it is nevertheless important to study the properties of exact minimum Steiner trees. In higher-dimensional cases, such properties have proved crucial in the development of approximation algorithms. For example, Smith’s approximation algorithm for d -dimensional Steiner trees in [58] uses the fact that all angles at Steiner points in exact minimum Steiner trees are 2=3. This section is a survey of the most important properties of gradient-constrained minimum Steiner trees, beginning with the fundamental properties of their Steiner points.

4.1

Properties of Steiner Points

The results in this section are mostly from [13]. Let T be a gradient-constrained minimum Steiner tree. It can be assumed that all edges of T are straight lines whose lengths are given by the gradient metric. Let jT jg be the sum of the lengths of all edges in T . An edge pq in T is called an f-edge, m-edge, or b-edge if pq is labeled “f” (g.pq/ < m), “m” (g.pq/ D m), or “b” (g.pq/ > m) respectively. The label of an edge can be thought of as indicating which of the two metrics that compose the gradient metric is “active” for that edge, with an “m” label indicating that both metrics hold simultaneously. A very useful tool for understanding properties of Steiner points in a gradientconstrained minimum Steiner tree is the variational method, which was developed by Rubinstein et al. in [54]. The variational argument in the Steiner tree problem is as follows: for a minimum tree T , the directional derivative of jT jg is greater than or equal to zero when its Steiner points are perturbed in any directions. Note that under an arbitrarily small perturbation, the only edges which can change labeling are

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m-edges. If no m-edges change their labeling under a given perturbation, then it is easily checked that the variation is reversible, and hence the directional derivative is strictly zero. Suppose e D sa is an edge in T and s is a Steiner point which is perturbed to s 0 in direction u. Let ePu (or simply eP if u is known) denote the directional derivative of the length of e. It is easy to show the following lemma and corollary: Lemma 2 (i) If e is an f-edge, then ePu D  cos.†ass 0 /.p (ii) If e is a b-edge, then ePu D  cos.†zss 0 / 1 C m2 where z is a point on the vertical line through s such that †asz  =2. 0 (iii) If / or  cos.†zss 0 / p e is an m-edge, then ePu is equal to0 either  cos.†ass 0 2 1 C m , depending on whether g.s a/  m or g.s a/ > m. Corollary 2 (i) When s moves horizontally, the length of e does not change if e is a b-edge or if e is an m-edge and becomes a b-edge in the move. (ii) When s moves vertically to the same side (or the opposite side) of the horizontal plane through s as a, e becomes shorter (or, respectively, longer) regardless of the gradient of e. From Lemma 2, it is clear that as in the Euclidean metric, the directional derivative of e in the gradient metric is determined only by the direction of u and is independent of the length of sa. Because an f-edge is still an f-edge under a small perturbation, Lemma 2 gives the following easy corollary: Lemma 3 Any f-edge of a Steiner point s meets other edges incident to s at an angle no less than =2. An important application of the variational argument is that of splitting a small angle in a nonoptimal tree. If a Steiner point s of T is perturbed in direction u, denote the directional derivative of jT jg by TPu or just TP . Now, suppose s is a Steiner point joining a; b, and c in a Euclidean Steiner tree T D sa [ sb [ sc. If †acb  2=3, then s collapses into c. On the other hand, if †acb < 2=3 but s D c, then T cannot be minimum by the following variational argument: let s1 be a point on the bisector u of †asb and close to s, then †ass1 D †bss1 < =3 and TP D 1  cos.†ass1 /  cos.†bss1 / < 0: In general, this process is referred to as splitting the angle at s in the subtree sa [ sb along a vector u. An example of the application of this technique to gradient-constrained minimum Steiner trees comes from understanding when a Steiner point in such a tree can have an incident b-edge. For any point p, denote the horizontal plane through p by Hp . For simplicity, a point or an edge will be said to be above (or below) p if it is above (or below) Hp . It is also useful to follow the convention of saying that two edges incident with s lie on the same side of Hs if they lie in the same closed half-space determined by Hs ; that is, this includes the possibility that one or both edges lie on Hs . The two edges are said to be on different sides of Hs only if their interiors lie in different open half-spaces (determined by Hs ).

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Lemma 4 If sa is a b-edge in a gradient-constrained minimum Steiner tree T , then no other edge incident with s lies on the same side of Hs as a. Moreover, all other edges incident with s are m-edges. Proof Let sb be another edge of T incident with s. Let u be a vector with gradient m, perturbing s to s 0 , such that s 0 lies on the same side of Hs as a and the angle  D †s 0 sb is as small as possible. Applying the variational argument to T by P u . If sb is a b-edge lying on splitting the angle †asb at s along u, note that TP D sb the same side of Hs as a, then TP < 0 by Lemma 2 (ii). If, on the other hand, sb is an m-edge or f-edge on the same side of Hs as a, or sb is an f-edge on the opposite side of Hs to a, then  < =2 (since m  1 in the latter case). So again, TP < 0 by Lemma 2. In each case, there is a contradiction to the minimality of T . Finally, if sb is a b-edge lying on the opposite side of Hs to a, then s has only two incident edges by the above argument and s is not a Steiner point.  A close analysis, using a similar approach, for any Steiner point in T , gives the following lemma, which is proved in [13]. Lemma 5 If s is a Steiner point in a gradient-constrained minimum Steiner tree T , then there are at most two incident edges lying strictly above (or below) Hs . Consequently, the degree of any Steiner point is either three or four. Suppose s is a degree 3 Steiner point in T with two incident edges sa; sb lying on one side of Hs and the third sc lying on the other side of Hs . (This includes the possibility that all three edges lie on Hs .) Let ga ; gb ; gc denote the respective labels of these edges. Then the labeling of this degree 3 Steiner point is denoted by (ga gb /gc ). By symmetry, it can be assumed that a and b both lie on or above Hs . The theorem below shows that of the 18 possible distinct labelings for a degree 3 point, only a small number can occur in a gradient-constrained minimum Steiner tree. Theorem 5 If s is a degree 3 Steiner point in a gradient-constrained minimum Steiner tree T , then up to symmetry, there are five feasibly optimal labelings: (ff/f), (ff/m), (fm/m), (mm/m), and (mm/b). Proof By Lemmas 4 and 5, the only possible labelings of s, other than those listed in the statement of the theorem, are (mm/f) and (fm/f). So suppose, contrary to the theorem, there is only one edge, the f-edge sc, lying below s, and there is an m-edge, say, sa, lying above s. Because g.sa/ > g.sc/, sa shrinks strictly faster than sc stretches when s moves vertically upward. P  0 Since the third edge, sb, lies on the same side of Hs as as and sb P under this move, it follows that T < 0, contradicting the minimality of T .  Now, suppose s is a degree 4 Steiner point in T with two edges sa; sb lying on one side of Hs and the other two edges sc; sd lying on the other side of Hs . Let ga ; gb ; gc ; gd be the respective labels of these edges. Then the labeling of s is denoted (ga gb /gc gd ).

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For degree 4 Steiner points, it can be shown, by a somewhat more sophisticated use of the variational approach, that there is only one feasibly optimal labeling. The proof of this result is highly technical and again can be found in [13]. Note that here a stronger bound on the maximum gradient m is required, but as with the previous bound, it is one that is usually easily satisfied in applications. Theorem 6 If s is a degree 4 Steiner point in a gradient-constrained minimum Steiner tree T and if m < 0:38, then the labeling of s is .mm=mm/. By Theorems 5 and 6, there are six possible labelings at a Steiner point when m < 0:38. The two theorems below use the variational argument to show that for each labeling, it is possible to uniquely locate the Steiner point s in the gradientconstrained minimum Steiner tree T in terms of the adjacent vertices of T . Suppose the edges incident with s are sa; sb; sc (or sa; sb; sc; sd if s is of degree 4), where s has labeling (ga gb /gc ) (or (ga gb /gc gd ) if s is of degree 4). For any point p, let Cp denote the cone generated by rotating a line through p with gradient m about the vertical line through p. So Cp is a right vertical cone whose vertex is p. The results for a Steiner point of degree 3 are as follows. Theorem 7 Let s be a degree 3 Steiner point in a gradient-constrained minimum Steiner tree T . (i) If s has labeling (ff/f), then s is the point on the plane through a; b; c such that †asb D †bsc D †csa D 2=3. (ii) If s has labeling (mm/b), then s is a point at the intersection of Ca , Cb and the vertical plane through b; c. (iii) If s has labeling (mm/m), then s is a point at which the three cones Ca , Cb , and Cc all intersect, and †asb  arccos.1 C m2 /=2  =2. (iv) Suppose s has labeling (ff/m), and let v be a horizontal vector tangent to Cc at !     sa/ D †.v; sb/ and cos †.! cs; ! sa/C s. Then s is a point on Cc such that †.v; ! !  !  cos †.cs; sb/ D 1. (v) If s has labeling (fm/m), then s is a point lying on Cb \ Cc , which is an ellipse, such that sa is perpendicular to this ellipse.

Proof If the labeling at s is (ff/f), then it is clear that the location of s is the same as in the unconstrained three-dimensional Steiner problem, proving Case (i). For the remaining labelings, one can employ the variational argument. Since any vector in space can be decomposed into a sum of three noncoplanar components, to prove s can be a Steiner point of a minimum tree T , by the variational argument, it is sufficient to show that TP  0 for perturbations of s in three noncoplanar directions. For Case (ii), it follows from Lemma 2 that TP  0 when s moves either up or down or in any horizontal direction. This proves Case (ii). The other cases can be proved by similar arguments; see [13] for details.  Finally, consider the case where s has degree 4. If, in this case, the edges incident with s can be partitioned into two pairs, such that each pair lies in a vertical plane through s and has the property that the projections of the two edges onto Hs meet

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p4

Fig. 8 A degree 4 Steiner point with bi-vertically coplanar incident edges

p1

m m

s m

m

p2 p3

at an angle of  at s, then the edges of s are said to be bi-vertically coplanar (see Fig. 8). Note that if two m-edges incident with s lie on opposite sides of Hs and lie in the same vertical plane, then those edges are collinear. The other, noncollinear possibility is illustrated in Fig. 8. The following theorem not only shows how to construct a Steiner point of degree 4 but also shows that the existence of such a point requires a very specific configuration for the neighboring nodes. The proof is given in [13]. Theorem 8 Let s be a degree 4 Steiner point (with labeling (mm/mm)) in a gradient-constrained minimum Steiner tree T . Then s is a point at the intersection of the four cones Ca ; Cb ; Cc ; Cd , and furthermore the edges incident with s are bivertically coplanar. This work has been further extended in [21], where it is shown that a degree 3 or degree 4 locally minimal Steiner point in a gradient-constrained tree in 3-space is unique with respect to its adjacent nodes, even though the gradient metric is not strictly convex. More recently, Thomas and Weng [61] have derived explicit formulae for computing minimal Steiner points with given labelings in a gradientconstrained tree.

4.2

Local Minimality for a Fixed Topology

The characterization of Steiner points in the previous section provides a useful first step in solving the three-dimensional gradient-constrained Steiner problem with a given topology. For a fixed topology, the two-dimensional Euclidean Steiner problem has an efficient polynomial-time solution based on the methods of Melzak [48],

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and similarly, it should be possible to efficiently solve the two-dimensional gradientconstrained Steiner problem using the methods in Sect. 3.2. In three-dimensional space, the problem of solving the Euclidean Steiner problem for a fixed topology is much more difficult. It has been shown in [56] that even on four terminals, a minimum Euclidean Steiner tree in 3-space cannot generally be solved by radicals. The same arguments also apply to minimum gradient-constrained Steiner trees in 3-space. This suggests that iterative methods are required to find good approximations of the solution for the general problem. To solve any minimization problem, one can attempt a descent method. This works well for the problem at hand if the gradient constraint is relaxed. In particular, it is well known that for the analogous problem in d -dimensional Euclidean space, the network is minimal (for a given topology) if and only if there is no arbitrarily small perturbation of a single Steiner point that reduces the length of the network (see, e.g., [58]). This will be expressed by saying that if the network is minimal with respect to one-point moves, then it is minimal, or, more briefly, that one-point moves suffice. For the gradient-constrained minimum Steiner tree problem, given terminals and a topology, one can start with an initial nonoptimal tree and then iteratively move toward a global minimum by moving Steiner points. By the convexity of the gradient metric, it follows that such a strategy will work, in theory, if one can determine the appropriate descent directions in which to move the Steiner points. This involves finding length-reducing perturbations of the positions of the Steiner points. One-point moves do not suffice for this problem, essentially due to the nondifferentiability of the metric. This poses a difficulty algorithmically, and it helps to know which simultaneous multi-point moves might be required. One of the key results from [22] shows that, apart from a few easily classified exceptions, one-point moves and two-point moves suffice; that is, if the tree is not minimal for the given topology, then there exists a length reducing perturbation that fixes all but one or two Steiner points. This makes the task of finding suitable length-reducing perturbations much simpler. There are two situations where one-point moves or two-point moves may not suffice, that is, where a consideration of all possible perturbations of one or two Steiner points is not sufficient for finding a descent direction from a non-minimal configuration. The first is where the tree contains an extreme point, which is defined to be a degree 3 Steiner point whose incident edges all have the maximum permitted gradient and all lie in a single vertical plane. In this case, a special length-reducing perturbation involving arbitrarily many points may be required; however, these perturbations are easily described. The other situation is where the tree degenerates to include zero-length edges. This case is very difficult to deal with in theory but does not seem to cause a problem in practical applications. Both of these situations are described in more detail later in this section. The results in this subsection mainly come from [22]. Before discussing these results in a little more detail, it is important to be precise about what is meant here by a perturbation and the change of length under perturbations. A perturbation v on T is a collection of vectors in three-dimensional Euclidean space, one for

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each Steiner point of T . It is assumed that all Steiner points smoothly move with velocities specified by a given perturbation, resulting in a change of the length of T . Perturbations are used to examine arbitrarily small alterations in the positions of the Steiner points of T and the consequential change in the total length of T . Note that the rate of change in length of T depends only on the directions and relative lengths of the vectors at the Steiner points – the length of each individual vector can be assumed to be arbitrarily small. If T consists of a single edge uw, then the length of T with respect to the gradient metric is denoted by L.T / D L.uw/, and the derivative of this length with respect P to a perturbation v is denoted by L.uw/ v . It is also useful to define the active metric for a perturbation. Suppose L.uw/ 6D 0. If uw is a b-edge or an f-edge, then the active metric for v is the same as that of the edge. If uw is an m-edge, then the active metric for v is defined to be b if the derivative of the gradient of uw is 0 and f if the derivative of the gradient is 0 (and hence both b and f if the derivative of the gradient equals 0). This is equivalent to saying that the active metric for the perturbation is the same as the metric active on the projection of the image of uw (after the perturbation has been applied) onto the vertical plane containing uw. Note that although a perturbation can deactivate a metric on an m-edge, it cannot activate an inactive metric, since the perturbations are considered to be arbitrarily small. For a larger tree T with edge set E, the derivative of L.T / with respect to v is defined to be X P /v D P L.T L.uw/ vjfu;wg ; uw2E

where, as usual, vjfu;wg is the perturbation that acts in the same way as v on u and w and acts trivially on all other nodes of T . Since the terminals of a Steiner tree have fixed predetermined positions, it is assumed that all perturbations act trivially on terminals. The convexity of the gradient metric immediately implies the following result. Lemma 6 Let T be a gradient-constrained Steiner tree with topology t. Then T is P /v  0 for every perturbation v. minimum with respect to t if and only if L.T As was illustrated in Fig. 7, Sect. 3.1, the existence of a perturbation v on the Steiner points of T which decreases L.T / does not imply that there exists a 1-point perturbation that decreases the length of T . It is helpful to examine why, in the example in Fig. 7, v cannot be decomposed into 1-point perturbations on the original tree. Let v1 be the action of this perturbation on s1 and v2 the action on s2 . P /v 6D L.T P /v1 C L.T P /v2 . The reason lies in the fact that the The question is why L.T only active metric of s1 s2 is the f-metric for v1 and the b-metric for v2 . If, in the above example, s1 s2 had a common active metric for both v1 and v2 , then there would exist a length-reducing 1-point perturbation. This would have occurred, for example, if the label of the edge s1 s2 had been f or b.

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T1 v1

s1

s2

v2

Fig. 9 The ellipses indicate the two subtrees T1 and T2 used for determining whether or not the edge s1 s2 is decomposing with respect to s2

Hence, it is evident that it is m-edges which cause difficulties when trying to decompose a length-reducing perturbation into a sum of 1-point perturbations because of the possibility of active metrics being changed in the decomposition. However, if 2-point perturbations are included in the decomposition, the problem of m-edges can often be overcome. Let T be a full gradient-constrained Steiner tree with no zero-length edges. Suppose T has an interior node s2 , whose three adjacent nodes are s1 , v1 , and v2 (each of which may or may not be a Steiner point). Let U1 and U2 be the two connected components of T  s1 s2 containing s1 and s2 , respectively. Let T1 D T jV .U1 /[fs2 ;v1 ;v2 g and let T2 D T jV .U2 /[fs1 g . This is illustrated in Fig. 9. Note that s2 is an interior node of both T1 and T2 . It is possible that a perturbation v may decrease L.T / but increase both L.T1 / and L.T2 /. The edge s1 s2 is said to be decomposing with respect to s2 if for any perturbation v acting on the interior nodes of T , there exist perturbations v1 and v2 acting on the interior nodes of T1 and T2 , respectively, such that P /v D L.T P /v1 C L.T P /v2 : L.T Furthermore, s1 s2 is said to be properly decomposing with respect to s2 if s1 s2 is decomposing with respect to s2 and T1 6D T and T2 6D T (i.e., s1 is not a terminal, and at least one of v1 and v2 is not a terminal). The following lemma shows that if all interior edges in T are properly decomposing, then only 1- and 2-point perturbations need to be considered, when attempting to determine whether or not T is minimal. Lemma 7 Suppose every interior edge of full gradient-constrained Steiner tree T is properly decomposing with respect to at least one of its endpoints. If T is minimal with respect to all 1- and 2-point perturbations, then T is minimal. Proof Suppose, on the contrary, that T is not minimal. Then, by Lemma 6, there P /v < 0. The result follows by a exists a perturbation v on T , such that L.T straightforward induction argument on the number of Steiner points acted on (nontrivially) by v, giving the existence of a length reducing 1- or 2-point perturbation. 

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Define a degree 3 Steiner point s in T to be an extreme point if s is labeled (mm/m) and the three edges incident with s all lie in a vertical plane. Theorem 9 Let T be a full gradient-constrained Steiner tree, minimal with respect to 1- and 2-point perturbations, and containing no extreme points or zero-length edges. Then T is minimal (with respect to its topology). Proof A geometric argument, given in [22], shows that if T contains no extreme points, then each interior edge of T is decomposing with respect to both endpoints. If T contains at least three Steiner points, then each interior edge e is incident with a Steiner point se which is adjacent to a third Steiner point (not an endpoint of e); thus e is properly decomposing with respect to se . Hence, by Lemma 7, T is minimal. On the other hand, if T contains less than three Steiner points, then T is minimal by assumption. 

Let Text denote the subnetwork of T induced by extreme points. In other words, Text is a forest whose nodes are the extreme points of T and whose edges are the edges of T whose endpoints are both extreme. Note that each connected component of Text lies in a vertical plane. In order to understand the structure of T if it contains extreme points, one can define a class of easily identified perturbations known as backbone moves and show that T is minimal if and only if none of these moves is length reducing. Definition Let P be a connected component of Text with node set V .P /, containing k  2 nodes (k  3). It is straightforward to show that P is a path in a vertical plane P whose nodes are collinear. Let s1 and sk be the two nodes of T not in P that are neighbors of the two endpoints of P and are collinear with P (or are the collinear neighbors of the single node of P if k  2 D 1). Then T jV .P /[fs1 ;sk g is defined to be the backbone of P (i.e., the backbone is the union of P and the two extra collinear edges of T incident with the endpoints of P ). A k-point backbone move is a perturbation of V .P / [ fs1 ; sk g restricted to P such that the active metric on all edges in or incident with P remains m; each node of V .P / is perturbed an appropriate relative distance directly toward or away from its neighboring terminals. Hence the backbone remains collinear, if the perturbation is applied. Note that a k-point backbone move is determined entirely by its action on s1 and sk . Finding a length-reducing k-point backbone move is no more complex than finding a length-reducing 2-point perturbation. Furthermore, such backbone moves can be shown to exist in some cases where there are no length-reducing 1- or 2-point perturbations. A careful geometric argument, given in [22], shows that for any non-minimal tree containing no zero-length edges, there exists either a 1- or 2-point perturbation or a k-point backbone move that is length reducing.

1490 Fig. 10 A degenerate tree minimal with respect to 1- and 2-point perturbations but not with respect to 3-point perturbations

M. Brazil and M.G. Volz t1

t2

f

s1

m

m t3 m

t4 f

s2

m t5

Theorem 10 Let T be a gradient-constrained Steiner tree, minimal with respect to 1- and 2-point perturbations, and containing no zero-length edges. If T is not minimal then there exists a length-reducing k-point backbone move on T , for some k  3. The remaining case to consider is where T contains zero-length edges. There are a number of reasons why it is sensible to permit zero-length edges for a given topology: it ensures a minimum network exists for each topology, it is necessary to guarantee compactness of the configuration space for a fixed topology, and it means that all topologies can essentially be dealt with by considering only full topologies. For instance, some minimal Steiner trees have a Steiner vertex of degree 4, and this can be considered instead as a degenerate example of a Steiner tree whose topology has all Steiner points of degree 3 but where two such points have an edge of length zero joining them. However, if the restriction on no zero-length edges is relaxed, then Theorem 10 no longer holds. This is illustrated by the following example. Consider a gradient-constrained tree T in the vertical plane with terminals t1 ; t2 ; t3 ; t4 , and t5 , as shown in Fig. 10. The tree T contains two Steiner points s1 and s2 such that s1 t2 is an f-edge sloping downward from s1 , s2 t4 is an f-edge sloping upward from s2 , and the remaining edges are m-edges. Now T can be viewed as a full (but degenerate) tree containing a third Steiner point s adjacent to s1 , s2 , and t3 , such that st3 is a zero-length edge. It is easy to see that T is minimal with respect to 1- and 2-point moves. However, there is a 3-point move, moving s1 along the line through s1 t1 away from t1 , moving s2 along the line through s2 t5 away from t5 , and moving s away from t3 so that s1 s and s2 s remain m-edges that reduces the length of T as the f-edges come closer to being parallel. This move is indicated in the figure in broken lines. This example does not rely on s being extreme. It is possible to move some of the terminals out of the vertical plane, without changing the edge labels, so that this

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problem persists. It is also possible to construct more complicated examples that require the simultaneous perturbation of arbitrarily many Steiner points (not in a vertical plane) in order to reduce the length of the tree.

4.3

Algorithms and Software Solutions

The importance of solving the three-dimensional gradient-constrained Steiner problem for a fixed topology is that it forms a key step toward the solution of the general problem of finding the minimum network joining a given set of nodes, without specifying the network topology. From the example at the end of the previous subsection, it appears that for the case of zero-length edges, characterizing the constraints on Steiner trees that are minimal with respect to the gradient metric, and having a given topology, is a difficult and probably intractible problem. However, this is not necessarily a major setback for constructing an effective approximation algorithm that seeks to find the minimum gradient-constrained Steiner tree. An example of such a software tool is UNO (Underground Network Optimiser), which makes use of the theoretical results in the previous two subsections. A description of an early version of UNO is given in [11], and a computational study using UNO is presented in [62]. The problem of finding the best topology is in general NP-complete, since this is true for the same problem restricted to the horizontal plane. Hence, any practical algorithm requires a method of changing topologies, or at least considering many different topologies, that is unlikely to be highly efficient. UNO incorporates this aspect by including some random steps. By running for some time and repeatedly randomly encountering approximately minimal trees, the algorithm occasionally finds trees that are reasonably close to the minimal tree, even in the case of the example illustrated in Fig. 10. A sensible descent method (using the results in the previous section characterizing minima with no zero-length edges) then brings the algorithm even closer to the true global minimum. The characterization of local minima described in the previous section appears to be enough for the algorithm to perform well in practical cases. In UNO, the problem of finding a “good” topology, for instance, the topology of the minimum tree, is tackled via a heuristic method based on simulated annealing and a genetic algorithm. In this way, the problem is reduced to one of efficiently minimizing the cost for a single topology, using descent methods.

5

The Gilbert Problem in Minkowski Spaces

Gilbert [34] proposed a generalization of the Euclidean Steiner problem incorporating into the network cost a flow between terminals. This section considers the extension of this concept to networks in general Minkowski spaces. Much of the discussion here is based on [67].

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Fig. 11 An example where split routing is cheaper

p1 p2

t=3 t=1

t=3

q

Let T be a network interconnecting a set N D fp1 ; : : : ; pn g of n terminals in a Minkowski space. For each pair pi ; pj ; i ¤ j of terminals, a nonnegative flow tij D tj i is given. The cost of an edge e in T is w.te /le , where le is the length of e, te is the total flow being routed through e, and w./ is a unit cost weight function defined on Œ0; 1/ satisfying w.0/  0

and w.t/ > 0 for all t > 0;

w.t2 /  w.t1 /

for all t2 > t1  0;

w.t1 C t2 /  w.t1 / C w.t2 /

for all t1 ; t2 > 0;

w./ is a concave function.

(1) (2) (3) (4)

That the function w is concave means by definition that w is convex. A network with a cost function satisfying Conditions (1)–(3) is called a Gilbert network. For a given edge e in T , w.te / is called the weight of e and is also denoted simply by we . The total cost of a Gilbert network T is the sum of all edge costs, that is, C.T / D

X

w.te /le

e2E

where E is the set of all edges in T . A Gilbert network T is a minimum Gilbert network , if T has the minimum cost of all Gilbert networks spanning the same point set N , with the same flow demands tij and the same cost function w./. By the arguments of [26], a minimum Gilbert network always exists in a Minkowski space when Conditions (1)–(4) are assumed for the weight function. Conditions (1)–(3) above ensure that the weight function is nonnegative, nondecreasing, and triangular, respectively. These are natural conditions for most applications. Unfortunately, the first three conditions alone do not guarantee that a minimum Gilbert network is a tree. This can be seen by considering the following example of a Gilbert network problem with two sources and one sink in the Euclidean plane, where there exists a split-route flow that has a lower cost than any arborescence. For this example, assume there are two sources p1 ; p2 and a sink q which are the vertices of a triangle 4p1 p2 q with edge lengths kp1  p2 k2 D 1 and kp1  qk2 D kp2  qk2 D 10, as illustrated in Fig. 11. The tonnages at p1 and p2 are 2 and 4, respectively. The weight function is w.t/ D d.3t C 1/=2e, that is, .3t C 1/=2 rounded up to the nearest integer. This function is positive, nondecreasing, and triangular, but not concave. For the example, only the following values need to be considered:

Gradient-Constrained Minimum Interconnection Networks Fig. 12 The minimum Gilbert arborescence

p1 p2

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s

t=6

q

t 1234 6 w.t/ 2 4 5 7 10 Routing 1 unit of the flow from p2 via p1 to q (and all other flow directly to q) gives a Gilbert network (Fig. 11) of total cost w.1/kp1  p2 k2 C w.3/kp1  qk2 C w.3/kp2  qk2 D 102: For a Gilbert arborescence, the flows from p1 and p2 to q are routed via some variable point s as in Fig. 12. A minimum Gilbert arborescence can be calculated by using the weighted Melzak algorithm as described in [34] to correctly locate s. The calculation, which involves a straightforward geometric calculation (see [22]), p p results in a total cost of 9982:5 C 7 3890:25 D 102:074 : : : . This shows that split routing can be necessary when the weight function is not concave. For the remainder of the chapter, it will be assumed that the weight function w is concave in addition to the other three conditions (in fact, the weight function will generally be linear). When all of Conditions (1)–(4) apply and there is a single sink, it has been shown in [34] and [60] that there always exists a minimum Gilbert network that is a Gilbert arborescence. This means that for the remainder of this chapter, it is only necessary, without loss of generality, to consider MGAs. (Note that in [26], Condition (3), the triangular condition, was incorrectly interpreted as concavity of the cost function.) The Gilbert network problem is to find a minimum Gilbert network for a given terminal set N , flows tij , and cost function w./. Since its introduction in [34], various aspects of the Gilbert network problem have been studied, with an emphasis generally on discovering geometric properties of minimum Gilbert networks (see [26, 60, 64, 65] and [67]). As in the Steiner problem, additional vertices can be added to create a Gilbert network whose cost is less than would otherwise be possible, and these additional points are again called Steiner points. A Steiner point s in T is called locally minimal if a perturbation of s does not reduce the cost of T . A Gilbert network is called locally minimal if no perturbation of the Steiner points reduces the cost of T . One of the main results established in [67] is that in a smooth Minkowski plane, that is, a two-dimensional Minkowski space where the given norm is differentiable at any x 6D o, every Steiner point in an MGA has degree 3. In Sect. 6, however, it will be shown that for the gradient-constrained Gilbert problem (where the corresponding Minkowski space is not smooth) degree 4 Steiner points can occur.

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The Two-Dimensional Gradient-Constrained Gilbert Problem

The Gilbert arborescence problem asks for a minimum-cost flow-dependent network interconnecting given sources and a unique sink. The gradient-constrained version of the problem in a vertical plane is called the two-dimensional gradientconstrained Gilbert arborescence problem and is formulated as follows: • Given: A set of points N D fp1 ; : : : ; pk g in a vertical plane in Euclidean space, where p1 ; : : : ; pk1 are sources with respective positive flows t1 ; : : : ; tk1 and pk is a unique sink, a gradient bound satisfying 0 < m  1, and positive constants d and h • Find: A minimum-cost network T interconnecting N which provides flow paths from the sources to the sink, such that the cost of an edge e in T is given by .d C hte /le , where te is the total flow through e and le is the length of e under the gradient metric A network satisfying the above conditions is called a gradient-constrained minimum Gilbert arborescence (gradient-constrained MGA, for short). Vertices in T not in N are, as before, called Steiner points, and as was discussed in Sect.5, a Steiner point is a Fermat-Weber point with respect to its adjacent vertices in T . The quantity d Chte represents the weight associated with the edge e. As in the case of the Steiner problem, there exists a gradient-constrained MGA lying in the vertical plane containing the sources and sink; it will be assumed that the gradient-constrained MGA T lies in this vertical plane.

6.1

Edge Vectors and Fundamental Properties of MGAs

As discussed in Sect. 2.3, Euclidean space equipped with the norm associated with the gradient metric is an example of a Minkowski space. Recall that in a Minkowski space X , the unit ball is defined to be the set B D fx 2 X W kxk  1g. The dual space of X , denoted by X  , is the vector space of all linear functionals on X , a linear functional being a mapping from X into RC . Identifying X and X  with Rn , the dual ball B  is the polar body of B, that is, B  D fy W hx; yi  1;

8x 2 Bg

where h; i is the dot product. In a gradient-constrained vertical plane, the unit ball Bg and dual ball Bg are as shown in Fig. 13a, b. Note that the diagonal lines (shown dashed) have gradient m and that the shapes of the unit ball and dual ball depend on the value of m. For any Minkowski space X , a norming functional of x 2 X is a  2 X  such that kk D 1 and .x/ D kxk. The set of norming functionals of x is denoted by @x. By the Hahn-Banach separation theorem [53], each nonzero x 2 X has an associated norming functional. Each @x corresponds to an exposed face of B 

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b

a

Fig. 13 Edge vectors and outward normal vectors for f-, b- and m-edges in the gradient metric. (a) The unit ball. (b) The dual ball

(i.e., the intersection of B  with a supporting hyperplane). Furthermore, a norming functional is equivalent to a vector uı , where u is an outward unit vector normal to the hyperplane supporting s C Bg at the point where the ray from s through p intersects Bg and ı is the distance from s to this hyperplane. These vectors are referred to as edge vectors. Edge vectors are important for establishing an equilibrium condition at Steiner points. In the gradient metric for a vertical plane, the following lemma characterizes vectors for edges connecting a fixed point and a Steiner point (refer to Fig. 13). Lemma 8 Let ps be an edge with an associated weight w in a gradient-constrained MGA connecting a terminal p and a Steiner point s. Then an edge vector v corresponding to an element of w@jp  sjg D [email protected]  s/ is computed as follows: • If sp is an f-edge, v points from s to p and has length w. • If sp is a b-edge, then v points pvertically up or down, if p is above or below s, respectively, and has length w 1 C m2 . • If sp is an m-edge, then any v can be selected from the continuum of vectors having all possible directions between an m-edge and a vertical edge. The length of v is given by w= cos , where is the angle that v makes with the m-edge. Note that the possible vectors v associated with an m-edge all point to one of the straight line segments on the boundary of B  and the union of all these vectors forms a region bounded by a right-angled triangle. Edge vectors, along with the variational argument, provide tools for analyzing the basic properties of Steiner points in MGAs. An optimal Steiner point s in a

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gradient-constrained MGA is necessarily a Fermat-Weber point with respect to its adjacent points p1 ; : : : ; pk . This leads to the following lemma, which follows from the results of Durier and Michelot [30]. Lemma 9 Let T be a gradient-constrained MGA, and let s be a Steiner point in T with incident edges p1 s; : : : ; pk1 s; spk having respective weights w1 ; : : : ; wk . If s is locally minimal with respect to its adjacent vertices, then there exist edge vectors v1 ; : : : ; vk , computed by Lemma 8, for which k X

vi D 0

i D1

This is referred to as the equilibrium condition. Note that the equilibrium condition is necessary but not sufficient for s to be locally minimal with respect to its adjacent vertices. This is in contrast to Fermat-Weber points in the gradientconstrained problem, for which the equilibrium condition is both necessary and sufficient. The next results require the following definition. Definition Let T be a gradient-constrained MGA in a vertical plane, and let s be a Steiner point in T . Then the horizontal and vertical lines passing through s are denoted by Hs and Vs , respectively. The local properties of any Steiner point s in a gradient-constrained MGA are contained in the following two theorems and their corollary. These results were established and proved in [66], using the variational argument. Note that these properties depend on the gradient of the sink edge. Theorem 11 Let T be a gradient-constrained MGA in a vertical plane, and let s be a Steiner point in T with incident sink edge spk . If spk is a b-edge, then the degree of s is three, and the two source edges are m-edges on the opposite side of Hs to spk . Theorem 12 Let T be a gradient-constrained MGA in a vertical plane, and let s be a Steiner point in T with incident sink edge spk , where spk is not a b-edge. Let L denote the infinite line passing through s; pk , and let H C ; H  denote the two open half-planes into which L divides the vertical plane, such that H C is the upper half-plane if and only if the slope of L is positive. Then: 1. The Steiner point has at most one incident edge in H C , and this edge is either an f-edge or an m-edge on the opposite side of Vs to spk . 2. If spk is an f-edge, s has no incident edges on L and either one or two incident edges in L . In the latter case, one of the two edges must be an m-edge on the same side of Vs as spk , and the other must be either an f-edge or an m-edge on the opposite side of Vs .

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3. If spk is an m-edge, s has at most one incident edge on L, which is necessarily an m-edge, at most one incident edge in L , which is necessarily an m-edge on the same side of Vs as spk , and at most one incident edge in LC , which is either an f-edge or an m-edge. These two theorems imply the following results. Corollary 3 Let T be a gradient-constrained MGA in a vertical plane and let s be a Steiner point in T . Then: 1. The Steiner point s has at most one incident b-edge. 2. If the incident sink edge to s is an m-edge, then s has at most one incident f-edge. 3. The degree of s is either three or four.

6.2

A Classification of Steiner Points

Unlike minimum Steiner trees, gradient-constrained MGAs in a vertical plane do not appear to have a strong geometric structure or a canonical form. However, there are further structural restrictions at Steiner points (beyond those in Corollary 3) that have the potential to be exploited as part of an exact or heuristic algorithm. Steiner points can be classified in terms of the labels of the incident edges, in a similar way as was done in Sect. 4.1. This section outlines a classification, both for degree 3 and 4 Steiner points. The results are proved in [66], using methods of subdifferential calculus and Minkowski sums – see [53] for some background on these topics.

6.2.1 Degree 3 Steiner Points Let T be a gradient-constrained MGA in a vertical plane, and let s be a degree 3 Steiner point in T , with incident edges p1 s; p2 s; sp3 , where p1 s; p2 s are source edges with positive flows t1 ; t2 and sp3 is the sink edge. The incident edges p1 s; p2 s; sp3 have respective weights w1 D d C ht1 , w2 D d C ht2 and w3 D d C h.t1 C t2 /. Let g1 ; g2 ; g3 denote the respective labels of p1 s; p2 s; sp3 . Then, here, the labeling of s is denoted .gi gj gk /, where i; j; k 2 f1; 2; 3g. In Sect. 4.1, a 0 =0 sign was used to distinguish between edges lying above and below the horizontal plane Hs through a Steiner point s in a gradient-constrained SMT. The situation is more complex for gradient-constrained MGAs, and distinguishing between edges above and below Hs becomes less useful. Hence, in this section, there is no 0 =0 sign in the labeling. The convention employed here is that the edge labels are stated in order of increasing gradient, that is, “f” followed by “m” followed by “b.” Since each gi 2 ff; m; bg, there are ten possible labelings of s to examine: .fff/ .ffm/ .ffb/ .fmm/ .fmb/ .fbb/ .mmm/ .mmb/ .mbb/ .bbb/

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Fig. 14 Feasibly optimal labelings for degree 3 Steiner points

A labeling that can occur in a gradient-constrained MGA is called feasibly optimal. Note that for gradient-constrained minimum Steiner trees in a vertical plane, only three of these labelings are feasibly optimal by Corollary 1, namely, (fmm), (mmm), and (mmb). For gradient-constrained MGAs, however, additional labelings are possible. Three of the labelings, those involving more than one “b” label, can immediately be eliminated as not being feasibly optimal by Corollary 3. It is shown in [66] that all the other labelings can occur but with some restrictions on the labeling of the sink edge. Theorem 13 Let T be a gradient-constrained MGA in a vertical plane, and let s be a degree 3 Steiner point in T . Then s has seven feasibly optimal labelings: (fff), (ffm), (ffb), (fmm), (fmb), (mmm), and (mmb). The labeling (fmm) has two feasibly optimal geometries, and hence there are eight possible configurations for s, as shown in Fig. 14. In the figure, edges which can feasibly be sink edges are indicated by arrows. It is worth noting that, of the eight possible geometries for degree 3 Steiner points in gradient-constrained MGAs in a vertical plane, the three configurations (iv), (vii),

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and (viii) are the only labelings in which any edge can feasibly be the sink edge. Also note that these are the only three feasibly optimal labelings for Steiner points in gradient-constrained SMTs in a vertical plane.

6.2.2 Degree 4 Steiner Points Suppose s is a degree 4 Steiner point in a gradient-constrained MGA T in a vertical plane with four edges p1 s; p2 s; p3 s; sp4 , where p1 s; p2 s; p3 s are source edges with respective positive flows t1 ; t2 ; t3 and sp4 is the sink edge. Let g1 ; g2 ; g3 ; g4 be the respective labels of these edges. Then the labeling of s is denoted by .g1 g2 g3 g4 /. There are 15 potential labelings to examine: .ffff/ .fffm/ .fffb/ .ffmm/ .ffmb/ .ffbb/ .fmmm/ .fmmb/ .fmbb/ .fbbb/ .mmmm/ .mmmb/ .mmbb/ .mbbb/ .bbbb/ For gradient-constrained SMTs in a vertical plane, only one of these labelings is feasibly optimal, namely, (mmmm), which forms a cross (therefore, a gradientconstrained vertical plane is an example of an X -plane discussed in [59]). Again, additional labelings are possible for gradient-constrained MGAs in a vertical plane. Nevertheless, most of the labelings can be eliminated as not being feasibly optimal. Theorem 14 Let T be a gradient-constrained MGA in a vertical plane and let s be a Steiner point in T . Then the labeling of s is not (ffff), (fffb), (ffmb), (ffbb), (fmmb), (fmbb), (fbbb), (mmmb), (mmbb), (mbbb), or (bbbb). Theorem 15 Let T be a gradient-constrained MGA in a vertical plane and let s be a degree 4 Steiner point in T . Then the following three labelings are feasibly optimal for s: (ffmm), (fmmm), and (mmmm). Examples have been constructed of Steiner points with each of these labelings, and the optimality of each case has been proved in [66]. Feasible configurations for degree 4 Steiner points are shown in Fig. 15. In the figure, edges which can feasibly be sink edges are indicated by arrows. Whether or not the labeling (fffm) is feasibly optimal remains open to question. If the labeling is feasible, then one of the f-edges is the sink edge sp4 , the m-edge is on the same side of Vs as sp4 (and on the opposite side of Hs to sp4 ), and the two remaining f-edges are on the opposite side of Vs to sp4 . There is a sense in which the feasibly optimal labeling (fmmm) is nongeneric. Definition Let T be a gradient-constrained MGA, and let s be a Steiner point in T . Then the labeling of s is said to be unstable if an arbitrarily small change in one of the edge weights causes the labeling to change.

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Fig. 15 Feasibly optimal labelings for degree 4 Steiner points. Edges which can feasibly be sink edges are indicated by arrows

Fig. 16 Steiner point with labeling (fmmm)

Theorem 16 Let T be a gradient-constrained MGA in a vertical plane and let s be a Steiner point in T . Then the labeling (fmmm) is unstable. The situation is illustrated in Fig. 16. If s is perturbed in direction r1 , then the variation is greater than or equal to zero when

t1 

d cos : h 1  cos

Similarly, if s is perturbed in direction r2 , then the variation is greater than or equal to zero when d cos : t1  h 1  cos Therefore, for the variation to be greater than or equal to zero, it must be exactly zero, and this can only occur when

t1 D

d cos : h 1  cos

An arbitrarily small change in t1 will cause the labeling of s to change, and hence Theorem 16 follows.

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The Three-Dimensional Gradient-Constrained Gilbert Problem

The previous two sections introduced the gradient-constrained Gilbert arborescence problem and gradient-constrained minimum Gilbert arborescences (MGAs) lying in a vertical plane. While the vertical plane problem has a number of potential applications, such as designing the haulage infrastructure for underground mines where the access points to the ore bodies roughly lie in a vertical plane, most mining networks are genuinely three-dimensional. The aim of this section is to establish fundamental properties for gradient-constrained MGAs in three dimensions. Here, a classification of degree 3 and 4 Steiner points in gradient-constrained MGAs in three-space is presented. As before, the classification of a Steiner point is in terms of the labels of its incident edges, where a label indicates whether the gradient of the straight line segment connecting the endpoints of an edge is less than, equal to, or greater than m. However, unlike gradient-constrained Steiner minimum trees or MGAs in the plane, there is no upper bound on the degree of a Steiner point in a gradient-constrained MGA in three dimensions. This issue is addressed in Sect. 7.2. Let Hs denote the horizontal plane passing through a point s. The other notation used in this section is identical to that in Sect.6. The results in this section are mainly from [66].

7.1

Classification of Steiner Points of Degree at Most Four

As before, Steiner points of degree 3 and 4 are considered separately.

7.1.1 Degree Three Steiner Points In [13], it was shown that there are five feasibly optimal labelings for degree 3 Steiner points in gradient-constrained SMTs in R3 . They are (fff), (ffm), (fmm), (mmm), and (mmb). These and additional labelings are possible for gradientconstrained MGAs in three-space. Let T be a gradient-constrained MGA in three-space, and let s be a Steiner point in T . By Theorem 13, the labelings (fff), (ffm), (ffb), (fmm), (fmb), (mmm), and (mmb) are feasibly optimal for gradient-constrained MGAs in a vertical plane. Since the vertical plane case is a special case of the three-dimensional case, these seven labelings are also feasibly optimal in gradient-constrained MGAs in threespace. The remaining labelings are (fbb), (mbb), and (bbb). It can be shown, by a similar argument to that used in proving Lemma 4, that if s has an incident b-edge, then it has no other incident b- or m-edges on the side of Hs containing the b-edge. Hence, labelings (mbb) and (bbb) are not feasibly optimal, since two edges with respective labels m,b or b,b cannot coexist on the same side of Hs . The labeling (fbb) is not feasibly optimal by noting that there is no horizontal vector component to counterbalance that of the f-edge vector (since the two b-edge vectors

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are vertical). Hence the equilibrium condition cannot be satisfied. This implies the following theorem. Theorem 17 Let T be a gradient-constrained MGA in three-space, and let s be a degree 3 Steiner point in T . Then s has seven feasibly optimal labelings: (fff), (ffm), (ffb), (fmm), (fmb), (mmm), and (mmb). Steiner points with these labelings are shown in Fig. 17, where the m-cones associated with the Steiner point are included to distinguish between f-, m-, and b-edges in three-space. Note that in three-space, for the labelings (fff), (ffm), (fmm), (mmm), and (mmb), any of the three edges can potentially be the sink edge. However, it can be shown that for the labelings (ffb) and (fmb), the sink edge must be labeled “f” and “m,” respectively.

7.1.2 Degree 4 Steiner Points In Sect. 4.1 (and [13]), it was shown that the labeling (mmmm) is the only feasibly optimal labeling for degree 4 Steiner points in gradient-constrained SMTs in threespace. For this labeling, two of the m-edges must lie above Hs and the other two below. Moreover, recall that the four edges incident to s are necessarily bi-vertically coplanar, meaning the two m-edges above Hs lie in the same vertical plane and the two m-edges below Hs lie in the same vertical plane (Fig. 8). When the two planes align to form a single vertical plane, then the four m-edges form a cross, a structure that is feasibly optimal for gradient-constrained MGAs in a vertical plane, as discussed in the previous section. Additional labelings are possible for degree 4 Steiner points in gradientconstrained MGAs in three-space and are listed in the theorem below. As before, the potential minimality of these labelings can be proved by direct construction and checking for each construction that certain necessary and sufficient conditions are satisfied. Theorem 18 Let T be a gradient-constrained MGA in three-space, and let s be a Steiner point in T . Then the labelings (fffm), (ffmm), (fmmm), and (mmmm) are feasibly optimal. Specific examples of Steiner points with each of these feasible configurations for degree 4 Steiner points are shown in Fig. 18. For each of the two labelings (ffmm) and (fmmm), two distinct configurations of edges are shown. In all cases, the sink edge is indicated by arrows. The m-cones (generated by taking a line through s with gradient m and rotating it 360ı about the vertical line through s) are shown to help distinguish between f-, m-, and b-edges. Note that in general, the sink edge can have any of the available labels except for the labeling (fffm), in which case the sink edge must be an f-edge. The question now is whether the list in Theorem 18 is complete. As before, a b-edge incident with s cannot lie on the same side of Hs as another incident b- or m-edge, which immediately eliminates the labelings (fmbb), (fbbb), (mmbb), (mbbb),

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Fig. 17 Feasibly optimal labelings for degree 3 Steiner points in three-space

and (bbbb). A similar argument also shows that (ffbb) is never feasibly optimal. In [67], it is shown that all Steiner points in Euclidean MGAs in three-space have degree 3. It follows that the labeling (ffff) is not feasibly optimal. The remaining labelings not accounted for are (fffb), (ffmb), (fmmb), and (mmmb). The following conjecture, if true, would prove that none of these four labelings are feasibly optimal. Conjecture 1 Let T be a gradient-constrained MGA in three-space, and let s be a Steiner point in T . If the degree of s is greater than 3, then s has no incident b-edges. There is strong evidence to suggest that this conjecture is true. The main support for this belief is that, in three-space, a b-edge path from a source to a sink can

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Fig. 18 Feasibly optimal labelings for degree 4 Steiner points in three-space

be embedded in many different ways. Numerical experiments indicate that by embedding such a path using at least three zigzag components, other edges can attach to different corner-points on the zigzag so as to reduce the cost of the network. To demonstrate this, consider the following example. Let p1 D .0:707; 0:707; 0/, p2 D .0:707; 0:707; 0/, and p3 D .1; 0; 0:5/ be sources with respective flows t1 D 1, t2 D 1, and t3 D 100, and let p4 D .1; 0; 0:5/ be the sink. The cost parameters are d D h D 1 and the maximum gradient is m D 0:5. It can be verified that the network shown in Fig. 19, in which the Steiner point s has labeling (ffmm), is a gradient-constrained MGA. If p3 is perturbed by  in the negative x-direction, it can be observed that the topology suddenly changes to the one shown in Fig. 20 (in which  D 0:1). The path from p3 to p4 now has more space such that it has three straight-line segments instead of two. Consequently, the edges incident to p1 and p2 can connect to different corner-points on the path to reduce the cost of the network. This behavior

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Fig. 19 Steiner point with labeling (ffmm)

p4 0.5

z

m 0

f

p1

s m

f

p3

p2

−0.5 1 0.5 0 y

−0.5

−1

−1

−0.5

0.5

0 x

1

p4 0.5

z

m 0

s2

f

p1

m f s1 p2

−0.5 1

Fig. 20 Change in topology resulting from a perturbation of a terminal

m p3

0.5 0 y

−0.5

−1

−1

−0.5

0 x

0.5

1

has been frequently observed when attempting to construct MGAs with the above labelings.

7.2

Maximum Degree of Steiner Points

The degree of a Steiner point in a gradient-constrained Steiner minimum tree (SMT) in three dimensions is either three or four, subject to some constraints on m (see Lemma 5). It has been shown by Cox in [26] that in general there is no upper bound on the degree of a Steiner point in a minimum Gilbert network. The Steiner problem is a special case of the Gilbert arborescence problem, which is itself a special case of the Gilbert network problem. This prompts the question as to whether, for a Steiner point in a gradient-constrained MGA in three dimensions, the degree is limited to three or four as for gradient-constrained SMTs, is unbounded as for general minimum Gilbert networks, or is bounded above by some finite integer value greater than four.

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b

Fig. 21 Large-degree Steiner points, where the sink edge is (a) a b-edge and (b) an m-edge

Resolving this issue is far from trivial. For example, at first it may appear that the configuration illustrated in Fig. 21a, in which the Steiner point s has k  1 incident source m-edges, where k is arbitrarily large, and the sink edge is a b-edge, could potentially be minimum. In this network, each path from pi ; i D 1; : : : ; k  1 to pk is a geodesic under the gradient metric. Therefore, the flow cost component cannot be improved. It is shown in [66], however, that in this case the development cost can be improved and that in fact the following result holds. Theorem 19 Let T be a gradient-constrained MGA in three-space and let s be a Steiner point in T with incident sink edge spk . If spk is a b-edge, then s has degree 3, and the two source edges are vertically coplanar m-edges on the opposite side of Hs to the sink edge. Suppose, however, that the network in Fig. 21a is changed so that the sink edge is now an m-edge instead of a b-edge (Fig.21b). The paths from pi ; i D 1; : : : ; k1 to pk are still geodesics under the gradient metric, so the flow cost cannot be improved. However, it can be shown in this case that if the flow costs are very large compared to the fixed cost, then it is possible for the network to have the shortest length among all networks connecting p1 ; : : : ; pk for which the paths from the sources to the sink are geodesics. This implies that the development costs are minimum and leads to the following result. Theorem 20 There is no upper bound on the degree of a Steiner point in a gradientconstrained MGA in three dimensions.

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Using the UNO software product, many examples of high-degree Steiner points satisfying the properties described above have been found.

8

Conclusion

This chapter has described the theory and applications of minimum gradientconstrained interconnection networks. The focus has been on two problems: the geometric Steiner problem, which asks for a network interconnecting a given set of points of minimum total length, and the Gilbert arborescence problem, which asks for a minimum-cost flow-dependent network interconnecting a set of given sources and a unique sink. These problems have been studied in the setting of gradientconstrained space, which is a type of Minkowski space with several intriguing properties. The aim is to understand the local geometric and topological structure of these networks in order to develop efficient exact and heuristic algorithms. Acknowledgements The research for and writing of this chapter was supported by an ARC Linkage Grant. We gratefully acknowledge the contributions of our collaborators to much of the theory described in this chapter, particularly Professor Doreen Thomas at The University of Melbourne.

Cross-References Network Optimization Steiner Minimal Trees: An Introduction, Parallel Computation, and Future Work Steiner Minimum Trees in E 3 : Theory, Algorithms, and Applications

Recommended Reading 1. A.V. Aho, M.R. Garey, F.K. Hwang, Rectilinear Steiner trees: efficient special-case algorithms. Networks 7, 37–58 (1977) 2. C. Alford, M. Brazil, D.H. Lee, Optimisation in underground mining, in Handbook on Operations Research in Natural Resources, ed. by A. Weintraub et al. (Springer, New York, 2007), pp. 561–577 3. C. Bajaj, The algebraic degree of geometric optimization problems. Discret. Comput. Geom. 3(2), 177–191 (1988) 4. S. Bhaskaran, F.J.M. Salzborn, Optimal design of gas pipeline networks. J. Oper. Res. Soc. 30(12), 1047–1060 (1979) 5. S. Boyd, A. Mutapcic, Subgradient methods, lecture notes of EE364b, Stanford University, Winter Quarter, 2006–2007 (Available at http://www.stanford.edu/class/ee364b/ lectures/subgrad method notes.pdf) 6. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, New York, 2004) 7. M. Brazil, Steiner minimum trees in uniform orientation metrics, in Steiner Trees in Industry, ed. by X. Cheng, D.-Z. Du (Kluwer Academic Publishers, Dordrecht/Boston/London, 2001), pp. 1–28

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8. M. Brazil, D.A. Thomas, Network optimization for the design of underground mines. Networks 49(1), 40–50 (2007) 9. M. Brazil, M. Zachariasen, Steiner trees for fixed orientation metrics. J. Glob. Optim. 43, 141–169 (2009) 10. M. Brazil, D.A. Thomas, J.F. Weng, Gradient-constrained minimal Steiner trees, in Network Design: Connectivity and Facilities Location, ed. by P.M. Pardalos, D.-Z. Du DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 40 (American Mathematical Society, Providence, 1998), pp. 23–38 11. M. Brazil, D.H. Lee, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, Network optimisation of underground mine design. Australas. Inst. Min. Metall. Proc. 305(1), 57–65 (2000) 12. M. Brazil, D.A. Thomas, J.F. Weng, On the complexity of the Steiner problem. J. Comb. Optim. 4, 187–195 (2000) 13. M. Brazil, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, Gradient-constrained minimum networks. I. Fundamentals. J. Glob. Optim. 21(2), 139–155 (2001) 14. M. Brazil, D.H. Lee, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, A network model to optimise cost in underground mine design. Trans. S. Afr. Inst. Electr. Eng. 93(2), 97–103 (2002) 15. M. Brazil, D.H. Lee, M. van Leuven, J.H. Rubinstein, D.A. Thomas, N.C. Wormald, Optimising declines in underground mines. Min. Technol. 112, 164–170 (2003) 16. M. Brazil, D.H. Lee, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, Optimisation in the design of underground mine access, in Orebody Modelling and Strategic Mine Planning: Uncertainty and Risk Management, ed. by R. Dimitrakopoulos. Spectrum Series, vol. 14 (Australasian Institute of Mining and Metallurgy, Carlton, 2005), pp. 121–124 17. M. Brazil, D.A. Thomas, J.F. Weng, J.H. Rubinstein, D.H. Lee, Cost optimisation for underground mining networks. Optim. Eng. 6(2), 241–256 (2005) 18. M. Brazil, D.A. Thomas, J.F. Weng, M. Zachariasen, Canonical forms and algorithms for Steiner trees in uniform orientation metrics. Algorithmica 44, 281–300 (2006) 19. M. Brazil, P.A. Grossman, D.H. Lee, J.H. Rubinstein, D.A. Thomas, N.C. Wormald, Constrained path optimisation for underground mine layout, in International Conference of Applied and Engineering Mathematics, London, July 2007, pp. 856–861 20. M. Brazil, P.A. Grossman, D.H. Lee, J.H. Rubinstein, D.A. Thomas, N.C. Wormald, Decline design in underground mines using constrained path optimisation. Min. Technol. 117, 93–99 (2008) 21. M. Brazil, D.A. Thomas, J.F. Weng, Gradient-constrained minimum networks. II. Labelled or locally minimal Steiner points. J. Glob. Optim. 42, 23–37 (2008) 22. M. Brazil, J.H. Rubinstein, D.A. Thomas, J.F. Weng, N.C. Wormald, Gradient-constrained minimum networks, III. Fixed topology. J. Optim. Theory Appl. 155(1), 336–354 (2012). 23. J. Brimberg, The Fermat-Weber location problem revisited. Math. Program. 71(1), 71–76 (1995) 24. J. Brimberg, H. Juel, A. Sch¨obel, Linear facility location in three dimensions—models and solution methods. Oper. Res. 50(6), 1050–1057 (2002) 25. R. Chandrasekaran, A. Tamir, Open questions concerning Weiszfeld’s algorithm for the Fermat-Weber location problem. Math. Program. 44, 293–295 (1989) 26. C.L. Cox, Flow-dependent networks: existence and behavior at Steiner points. Networks 31(3), 149–156 (1998) 27. Z. Drezner, H.W. Hamacher (eds.), Facility Location: Applications and Theory (Springer, Berlin, 2004) 28. D.-Z. Du, F.K. Hwang, Reducing the Steiner problem in a normed space. SIAM J. Comput. 21(6), 1001–1007 (1992) 29. D.-Z. Du, B. Gao, R.L. Graham, Z.-C. Liu, P.-J. Wan, Minimum Steiner trees in normed planes. Discret. Comput. Geom. 9(1), 351–370 (1993) 30. R. Durier, C. Michelot, Geometrical properties of the Fermat-Weber problem. Eur. J. Oper. Res. 20(3), 332–343 (1985)

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60. D.A. Thomas, J.F. Weng, Minimum cost flow-dependent communication networks. Networks 48(1), 39–46 (2006) 61. D.A. Thomas, J.F. Weng, Gradient-constrained minimum networks: an algorithm for computing Steiner points. J. Discret. Optim. 7, 21–31 (2010) 62. D.A. Thomas, M. Brazil, D.H. Lee, N.C. Wormald, Network modelling of underground mine layout: two case studies. Int. Trans. Oper. Res. 14(2), 143–158 (2007) 63. A.C. Thompson, Minkowski Geometry. Encyclopedia of Mathematics and Its Applications, vol. 63 (Cambridge University Press, Cambridge/New York, 1996) 64. D. Trietsch, Minimal Euclidean networks with flow dependent costs — the generalized Steiner case, May 1985, discussion paper no. 655, Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, IL, 1985 65. D. Trietsch, J.F. Weng, Pseudo-Gilbert-Steiner trees. Networks 33, 175–178 (1999) 66. M.G. Volz, Gradient-constrained flow-dependent networks for underground mine design. PhD thesis, Department of Electrical and Electronic Engineering, The University of Melbourne, 2008 67. M.G. Volz, M. Brazil, C.J. Ras, K.J. Swanepoel, D.A. Thomas, The Gilbert arborescence problem Networks (accepted, May 2012) 68. M.G. Volz, M. Brazil, D.A. Thomas, The gradient-constrained Fermat-Weber problem Networks (submitted 2011) 69. D.M. Warme, Spanning trees in hypergraphs with applications to Steiner trees. PhD thesis, Computer Science Department, The University of Virginia, 1998 70. D.M. Warme, P. Winter, M. Zachariasen, Exact solutions to large-scale plane Steiner tree problems, in Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore (1999), pp. 49–53 71. D.M. Warme, P. Winter, M. Zachariasen, Exact algorithms for plane Steiner tree problems: a computational study, in Advances in Steiner Trees, ed. by D.Z. Du, J.M. Smith, J.H. Rubenstein (Kluwer Academic Publishers, Boston, 2000), pp. 81–116 72. D.M. Warme, P. Winter, M. Zachariasen, GeoSteiner 3.1. Department of Computer Science, University of Copenhagen (DIKU) (2001), http://www.diku.dk/geosteiner/ 73. A. Weber, Uber den Standort der Industrien (Verlag J.C.B. Mohr, Tubingen, 1909), (Translation by C. J. Friedrich Theory of the Location of Industries (University of Chicago Press, Chicago, 1929) 74. E. Weiszfeld, Sur le point pour lequel la somme des distances de n points donn´es est minimum. Tohoku Math. J. 43, 355–386 (1937) 75. P. Widmayer, Y.F. Yu, C.K. Wong, Distance problems in computational geometry with fixed orientations, in Proceedings of the Symposium on Computational Geometry, Baltimore, MD (1985), pp. 186–195 76. P. Widmayer, Y.F. Yu, C.K. Wong, On some distance problems in fixed orientations. SIAM J. Comput. 16(4), 728–746 (1987) 77. M. Zachariasen, Rectilinear full Steiner tree generation. Networks 338, 125–143 (1999)

Graph Searching and Related Problems Anthony Bonato and Boting Yang

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Searching Undirected Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Robber Is Invisible and Active. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Robber Is Visible or Lazy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Searching Digraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Robber Is Visible or Lazy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Robber Is Invisible and Active. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Other Searching Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Mixed Searching with Multiple Robbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nondeterministic Graph Searching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fast Searching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Graph Guarding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Minimum Cost Searching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Pebbling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Open Problems on Searching Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cop-Win Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Higher Cop Number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Upper Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Lower Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Graph Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Algorithmic Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Random Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Constant p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Dense Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Sparse Case and Zigzag Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Bonato () Department of Mathematics, Ryerson University, Toronto, ON, Canada e-mail: [email protected] B. Yang Department of Computer Science, University of Regina, Regina, SK, Canada e-mail: [email protected]

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11 Infinite Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Cop Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Chordal Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Large Families of Cop-Win Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 A Dozen Problems on Cops and Robbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1548 1549 1550 1550 1551 1552 1553

Abstract

Suppose that there is a robber hiding on vertices or along edges of a graph or digraph. Graph searching is concerned with finding the minimum number of searchers required to capture the robber. Major results of graph searching problems are surveyed, focusing on algorithmic, structural, and probabilistic aspects of the field.

1

Introduction

Graph searching is a hot topic in mathematics and computer science now, as it leads to a wealth of beautiful mathematics and since it provides mathematical models for many real-world problems such as eliminating a computer virus in a network, computer games, or even counterterrorism. In all searching problems, there is a notion of searchers (or cops) trying to capture some robber (or intruder or fugitive). A basic optimization question here is: What is the fewest number of searchers required to capture the robber? There are many graph searching problems motivated by applied problems or inspired by some theoretical issues in computer science and discrete mathematics. These problems are defined by (among other things) the class of graphs (e.g., undirected graphs, digraphs, or hypergraphs), the actions of searchers and robbers, conditions of captures, speed, or visibility. Graph searching has a variety of names in the literature, such as Cops and Robbers games and pursuit-evasion problems. The reader is referred to survey papers [4, 21, 60, 62, 74] for an outline of the many graph searching models. A recent book by Bonato and Nowakowski [29] covers all aspects of Cops and Robbers games. In this chapter, a broad overview of graph searching is given, focusing on the most important results. The first part considers the so-called searching games where the robber may occupy an edge or vertex and the robber can usually move at high speeds at any time (but not always, depending on the model). The second part considers so-called Cops and Robbers games where the cops and robber occupy only vertices and they move alternatively to their neighbors. Searching games are often related to various width-type parameters, while there

1 2

Research was supported in part by NSERC, MITACS, and Ryerson University. Research was supported in part by NSERC and MITACS.

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is no such connection in Cops and Robbers. Owing to space constraints, proofs are omitted (however, references are given for all results). The results presented here span probabilistic, algorithmic, and structural results. As such, the chapter is not entirely self-contained; the reader can find most of the relevant background in the books [29, 49, 134]. Note that this chapter is the first reference to give a comprehensive overview of both searching and Cops and Robbers games (although searching is discussed briefly in [29]). The chapter finishes each part with a dozen problems and conjectures which are arguably the most important ones in the field of graph searching. Let G D .V; E/ (D D .V; E/) denote a graph (or digraph) with vertex set V and edge set E. Use uv to denote an edge in a graph with two end vertices u and v and .u; v/ to denote a directed edge in a digraph with tail u and head v. All graphs and digraphs considered in this chapter are finite and simple, unless otherwise stated.

Part 1. Searching Games The moniker searching is used for the case when the robber can move at great speed at any time or at least when a searcher approaches him (not just move to a neighbor in his turn, as is the case in Cops and Robbers). Searching has been linked in the literature to pathwidth and vertex separation of a graph [52, 89], to pebbling (and hence, to computer memory usage) [89], to assuring privacy when using bugged channels [58], to VLSI (i.e., very large-scale integrated) circuit design [56], and to motion planning of multiple robots [127]. A large number of graph searching games have been introduced. We will focus here on some of the basic models. Many graph searching games are closely related to graph width parameters, such as treewidth, pathwidth, and cutwidth. They provide some interpretation of width parameters, and often they can give a deeper insight into the graph structures and produce efficient algorithms. Examples include the proof of a min-max theorem on treewidth [122], the linear time algorithm for computing pathwidth of a tree [52,87], the polynomial time algorithm for computing branchwidth of a planar graph [123], the linear time algorithm for computing cutwidth of a tree [97], and the computation of the topological bandwidth [40, 98]. Part 1 of the chapter surveys searching games on undirected graphs and digraphs. There are four sections in this part. Section 2 deals with the undirected graph searching games in which the robber can move at high speeds along any searcherfree path. Section 3 deals with the digraph searching games in which the robber usually moves at high speeds along a searcher-free directed path (depending on the game setting). Section 4 deals with more recently introduced searching games. As with the end of Part 2, Part 1 closes with a dozen of open problems in the area.

2

Searching Undirected Graphs

The first searching model was introduced by Parsons [109], after being approached by the author of [36] (a paper dealing with finding a spelunker lost in a system of caves). Let G be a connected, undirected graph embedded in R3 with the Euclidean

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distance metric such that no pair of edges intersect at a point that is not a common endpoint. Imagine there is a robber in G who can be located at any point of G, that is, anywhere along an edge or at a vertex. We want to capture the robber using the minimum number of searchers. For each positive integer k, let Ck .G/ be the set of all families F D ff1 ; f2 ; : : : ; fk g of continuous functions fi W Œ0; 1/ ! G. A continuous search strategy for G is a family F 2 Ck .G/ such that for every continuous function h W Œ0; 1/ ! G (corresponding to the robber), there is a th 2 Œ0; 1/ and fi 2 F satisfying h.th / D fi .th /. We say that fi captures the robber at time th . The continuous search number of a graph G is the smallest k such that there exists a search strategy for G in Ck .G/. There are some contexts for which Parsons searching is the model required. For example, in the case of searching for someone lost or a robber hiding in a cave system, the robber and the searchers move continuously. But the notion of Parsons searching presents certain difficulties because the model involves the action of continuous functions defined on the nonnegative real numbers. In this survey, only discrete versions of Parsons searching are considered.

2.1

Robber Is Invisible and Active

When the robber is invisible and active, there are three basic searching games: edge searching, node searching, and mixed searching, which involve placing some restriction on searchers, but placing no restrictions on the robber. In these search games, the discrete time intervals (or time-steps) are introduced. Initially, G contains a robber who is located at a vertex in G, and G does not contain any searchers. Each searcher has no information on the whereabouts of the robber (i.e., robber is invisible), but the robber has complete knowledge of the location of all searchers. The goal of the searchers is to capture the robber, and the goal of the robber is to avoid being captured. The robber always chooses the best strategy so that he evades capture. Suppose the game starts at time t0 and the robber is captured at time tN and the search time is divided into N intervals .t0 ; t1 , .t1 ; t2 , : : :, .tN 1 ; tN  such that in each interval .ti ; ti C1  (also called step), exactly one searcher performs one action: placing, removing, or sliding. The robber can move from a vertex x to a vertex y in G at any time in the interval .t0 ; tN / if there exists a path between x and y which contains no searcher (i.e., robber is active). In the edge search game introduced by Megiddo et al. [100], there are three actions for searchers: placing a searcher on a vertex, removing a searcher from a vertex, and sliding a searcher along an edge from one end to the other. The robber is captured if a searcher and the robber occupy the same vertex on G. In the node search game introduced by Kirousis and Papadimitriou [89], there are two actions for searchers: placing a searcher on a vertex and removing a searcher from a vertex. The robber is captured if a searcher and the robber occupy the same vertex of G or the robber is on an edge whose endpoints are both occupied by searchers.

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In the mixed search game introduced by Bienstock and Seymour [22], searchers have the same actions as those in the edge search game. The robber is captured if a searcher and the robber occupy the same vertex on G or the robber is on an edge whose endpoints are both occupied by searchers. The edge search number of G, denoted by es.G/, is the smallest positive integer k such that k searchers can capture the robber. Analogously, define the node search number of G (written ns.G/) and mixed search number of G (written ms.G/). The following theorem demonstrates the relationships between search numbers. Theorem 1 ([22, 89]) If G is a connected graph, then the following inequalities hold: (1) ns.G/  1  es.G/  ns.G/ C 1: (2) ms.G/  es.G/  ms.G/ C 1. (3) ms.G/  ns.G/  ms.G/ C 1. The following result shows that all three search problems are NP-complete. Theorem 2 ([22, 89, 93, 100]) Given a graph G and an integer k, the problem of determining whether es.G/  k .ns.G/  k or ms.G/  k/ is NP-complete. Megiddo et al. [100] only showed that the edge search problem is NP-hard. This problem belongs to NP owing to the monotonicity result of [93], in which LaPaugh showed that recontamination of edges cannot reduce the number of searchers needed to clear a graph. A search strategy is monotonic if the set of cleared edges before any step is always a subset of the set of cleared edges after the step. Monotonicity is an important issue in graph search problems. Bienstock and Seymour [22] proposed a method that gives a succinct proof for the monotonicity of the mixed search problem, which implies the monotonicity of the edge search problem and the node search problem. Fomin and Thilikos [61] provided a general framework that can unify monotonicity results in a unique min-max theorem. Theorem 3 ([22, 93]) The edge search, node search, and mixed search problems are monotonic. Search numbers have close relationships with pathwidth and treewidth [117,118]. Given a graph G, a tree decomposition of G is a pair .T; W / with a tree T D .I; F /, I D f1; 2; : : : ; mg, and a family of nonempty subsets W D fWi  V W i D 1; 2;S : : : ; mg, such that (1) m i D1 Wi D V . (2) For each edge uv 2 E, there is an i 2 I with fu; vg  Wi . T (3) For all i; j; k 2 I , if j is on the path from i to k in T , then Wi Wk  Wj . The width of a tree decomposition .T; W / is maxfjWi j  1 W 1  i  mg:

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The treewidth of G, denoted by tw.G/, is the minimum width over all tree decompositions of G. A tree decomposition .T; W / is a path decomposition if T is a path; the pathwidth of a graph G, denoted by pw.G/, is the minimum width over all path decompositions of G. One can find more information on treewidth and related problems in the survey papers [23, 115]. Another related graph parameter is the vertex separation introduced by Ellis et al. [52]. A layout of a connected graph G is a one-to-one mapping L: V ! f1; 2; : : : ; jV jg. Let VL .i / Dfx W x 2 V , and there exists y 2 V such that xy 2 E, L.x/  i and L.y/ > i g. The vertex separation of G with respect to L, denoted by vsL .G/, is defined as vsL .G/ D maxfjVL .i /j W 1  i  jV jg: The vertex separation of G is defined as vs.G/ D minfvsL .G/ W L is a layout of Gg: Yang [135] introduced the strong-mixed search game, which is a generalization of the mixed search game. The strong-mixed search game has the same setting as the mixed search game except that searchers have an extra power to clear a neighborhood subgraph; that is, the subgraph induced by N Œv (a set contains v and its neighbors) is cleared if all vertices in N.v/ (a set contains only the neighbors of v) are occupied by searchers. The strong-mixed search number of G, denoted by sms.G/, is the smallest positive integer k such that k searchers can clear G. The following relationships were given in [87, 89, 135]. Theorem 4 ([87, 89, 135]) If G is a connected graph, then sms.G/ D pw.G/ D vs.G/ D ns.G/  1: For an edge search strategy, if the subgraph induced by cleared edges is connected in every step, then call it connected search. We denote by cs.G/ and mcs.G/, respectively, the connected and monotonic connected search numbers of a graph G defined in the natural way. The following theorem from [141, 143] demonstrates that the connected search game is not monotonic. Theorem 5 ([141, 143]) For any positive integer k, there is a graph Wk such that cs.Wk / D 280k C 1 and mcs.Wk / D 290k. Fraigniaud and Nisse [57] investigated visible robber connected node search. They proved that recontamination does help as well to catch a visible robber in a connected way. Barri`ere et al. [15] proved that mcs.T /  2es.T / for any tree T . Dereniowski [48] extended the result to graphs as follows.

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Theorem 6 ([48]) For any graph G, mcs.G/  2es.G/ C 3. Fomin et al. [63] introduced the domination search game, in which the robber hides only on vertices and searchers are placed or are removed from vertices of a graph. In the domination search game, searchers are more powerful than those in the node search game. A searcher “captures” the robber if she can “see” him; that is, a searcher on a vertex v can clear v and all its neighbors. The following theorem gives a relation between the domination search number, written dsn.G/, node search number, and the domination number, written ”.G/. Theorem 7 ([63]) For any graph G, let G 0 be a graph obtained from G by replacing every edge by a path of length three and H be a graph obtained from G by connecting every two nonadjacent vertices by a path of length three. Then, ns.G/ D dsn.G 0 / and ”.G/  dsn.H /  ”.G/ C 1: Kreutzer and Ordyniak [92] extended the domination game to distance d -domination games, in which each searcher can see a certain radius d around her position. That is, a searcher on vertex v can see any other vertex within distance d of v, and if this vertex is occupied by the robber, then the searcher can see the robber and capture him. They showed that all questions concerning monotonicity and complexity about d -domination games can be reduced to the case of d D 1. They gave a class of graphs on which two searchers can win on any graph in this class, but the number of searchers required for monotone winning strategies is unbounded. Due to the following result, the domination search problem is much harder than standard graph search problems. For background on PSPACE and other complexity classes, see Spiser [125]. Theorem 8 ([92]) The domination search problem is PSPACE-complete. Moreover, the problem of deciding whether two searchers have a winning strategy on a graph is PSPACE-complete.

2.2

Robber Is Visible or Lazy

Dendris et al. [47] introduced the lazy-robber game, which has the same setting as the node searching game except that the robber stays only on vertices and moves just before a searcher is going to be placed on the vertex currently occupied by the robber (i.e., robber is lazy). When a searcher is going to be placed on a vertex u currently occupied by the robber, the robber can move from u to another vertex v if there is a searcher-free path from u to v; otherwise, the robber is captured. The following result from Dendris et al. [47] shows that the lazy-robber game is monotonic and the search number of a graph in the lazy-robber game is equal to the treewidth of the graph plus one.

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Theorem 9 ([47]) Let G be a graph. For a lazy robber with unbounded speed, the monotonic search number of G is equal to the search number of G, and moreover, it is equal to tw.G/ C 1. Dendris et al. [47] also considered the speed-limited lazy robber. They showed that if the speed is 1, then the search number minus 1 is equal to a graph parameter, called width, which is polynomial time computable for arbitrary graphs. Given a layout L of a connected graph G, the width of a vertex v 2 V with respect to the layout L is the number of vertices which are adjacent to v and precede v in the layout. The width of the layout L is the maximum width of a vertex of G. The width of G is the minimum width of a layout of G. Theorem 10 ([47]) Let G be a graph. For a lazy robber with speed 1, the monotonic search number of G is equal to the search number of G, and moreover, it is equal to the width of G plus 1. Seymour and Thomas [122] introduced another variant of graph searching, visible-robber game. Its setting differs from node search in that the robber stands only on vertices and is visible to searchers. They showed that the visible-robber game is monotonic and the search number of a graph in the visible-robber game is equal to the treewidth of the graph plus one. They also showed that if there is a non-losing strategy for the robber, then there is a “nice” non-losing strategy with a particularly simple form. In order to show these results, Seymour and Thomas introduced jump-search and haven. For a graph G and X  V , let G  X be the graph obtained from G by deleting X . The vertex set of a component of G  X is called an X-flap. Let ŒV  k if and only if there k is a mapping f W V .G / ! 2V .G/ with the following properties: k (1) For every u 2 V .G /, ; ¤ f .u/  V .G/ n NG Œu: k (2) For every uv 2 E.G /,

f .u/  NG Œf .v/:

Consider Algorithm 1 for determining whether c.G/  k based on Theorem 66. The following theorem gives Theorem 65 as a corollary. Theorem 67 ([33]) Algorithm 1 runs in time O.n3kC3 /.

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If k is not fixed (and hence can be a function of n), then the problem becomes less tractable. Theorem 68 ([67]) The problem k-COP NUMBER is NP-hard. Theorem 68 does not say that k-COP NUMBER is in NP; that is an open problem! See Sect. 12 below. Theorem 68 is proved in Fomin et al. [67] by using a reduction from the following NP-complete problem: Domination: Given a graph G and an integer k  2; is ”.G/  k‹

10

Random Graphs

Random graphs are a central topic in graph theory; however, only recently have researchers considered the cop number of random graphs. Define a probability space on graphs of a given order n  1 as follows. Fix a vertex set V consisting of n distinct elements, usually taken as Œn; and fix p 2 Œ0; 1. Define the space of random graphs of order n with edge probability p, written G.n; p/ with sample n space equaling the set of all 2.2/ (labeled) graphs with vertex set V; and n

P.G/ D p jE.G/j .1  p/.2/jE.G/j : Informally, G.n; p/ may be viewed as the space of graphs with vertex set V , so that two distinct vertices are joined independently with probability p: Even more informally, toss a (biased) coin to determine the edges of your graph. Hence, V does not change, but the number of  edges is not fixed: it varies according to a binomial distribution with expectation n2 p: Despite the fact that G.n; p/ is a space of graphs, it is often called the random graph of order n with edge probability p: Random graphs were introduced in a series of papers by Erd˝os and R´enyi [53–55]. See the book [26] for a modern reference. The cases when p is fixed and when it is a function of n are considered. Graph parameters, such as the cop number, become random variables in G.n; p/: For notational ease, the cop number of G.n; p/ is referred to simply by c.G.n; p//: An event holds asymptotically almost surely (or a.a.s. for short) if it holds with probability tending to 1 as n ! 1. For example, if p is constant, then a.a.s. G.n; p/ is diameter two and not planar.

10.1

Constant p

The cop number of G.n; p/ was studied in Bonato et al. [30] for constant p; where the following result was proved. For p 2 .0; 1/ or p D p.n/ D o.1/; define

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Ln D log

1 1p

n:

Theorem 69 ([30]) Let 0 < p < 1 be fixed. For every real " > 0; a.a.s. .1  "/Ln  c.G.n; p//  .1 C "/Ln:

(6)

In particular, c.G.n; p// D .log n/: The upper bound in Theorem 69 follows by considering the domination number of G.n; p/ [50], while the lower bounds follow by considering an adjacency property. If the case of p D 1=2 is considered, then G.n; p/ corresponds to the case of uniformly choosing a labeled graph of order n from the space of all such graphs. Hence, Theorem 69 may be interpreted as saying “most” finite graphs of order n have cop number approximately log n: Properties of randomly chosen k-cop-win graphs (with the uniform distribution) are described next. For this, it is equivalent to work in the probability space G.n; 1=2/: Let cop-win be the event in G.n; 1=2/ that the graph is cop-win, and let universal be the event that there is a universal vertex. If a graph has a universal vertex w; then it is cop-win; in a certain sense, graphs with universal vertices are the simplest cop-win graphs. The probability that a random graph is cop-win can be estimated as follows: P.cop-win/  P.universal/ D n2nC1  O.n2 22nC3 / D .1 C o.1//n2nC1: A recent surprising result of Bonato et al. [35] showed that this lower bound is in fact the correct asymptotic value for P.cop-win/: Theorem 70 ([35]) In G.n; 1=2/; P.cop-win/ D .1 C o.1//n2nC1: Hence, almost all cop-win graphs contain a universal vertex.

10.2

The Dense Case

Now p consider the cop number of dense random graphs, with average degree pn at least n. The main results in this case were given in Bonato et al. [32]. Theorem 71 ([32]) (1) Suppose that p  p0 where p0 is the smallest p for which

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  log .log2 n/=p p =40  log n 2

holds. Then, a.a.s.     Ln  L .p 1 Ln/.log n/  c.G.n; p//  Ln  L .Ln/.log n/ C 2: p (2) If .2 log n/= n  p D o.1/ and !.n/ is any function tending to infinity, then a.a.s.   Ln  L .p 1 Ln/.log n/  c.G.n; p//  Ln C L.!.n//: By Theorem 71, the following corollary gives a concentration result for the cop number. In particular, for a wide range of p; the cop number of G.n; p/ concentrates on just the one value Ln: Corollary 3 ([32]) If p D no.1/ and p < 1, then a.a.s. c.G.n; p// D .1 C o.1//Ln: From Theorem 71 part (1), it follows that if p is a constant, then there is the concentration result that c.G.n; p// D Ln  2L log n C .1/ D .1 C o.1//Ln:

10.3

The Sparse Case and Zigzag Theorem

Bollob´as, Kun, and Leader [27] established the following bounds on the cop number in the sparse case, when the expected degree is np D O.n1=2 /. Theorem 72 ([27]) If p.n/  2:1 log n=n, then a.a.s. p 1 log log.np/9 1 2 log log.np/ n  c.G.n; p//  160;000 n log n : .np/2

(7)

In particular, Theorem 72 proves Meyniel’s conjecture for random graphs up to a logarithmic factor of n from the upper bound in (7). Recent work by Prałat and Wormald [112] removes the log n factor in the upper bound of (7) and hence proves the Meyniel bound for random graphs. p It would be natural to assume that the cop number of G.n; p/ is close to n also for np D n˛Co.1/ , where 0 < ˛ < 1=2: The so-called zigzag theorem of Łuczak and Prałat [96] demonstrated that the actual behavior of c.G.n; p// is much more complicated.

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Fig. 6 The zigzag-shaped graph of the cop number of G.n; p/

1 2 2 5 1 3

11 1 65 4

1 3

1 2

1

Theorem 73 (Zig-Zag Theorem, [96]) Let 0 < ˛ < 1, and d D d.n/ D np D n˛Co.1/ W (1) If 2j 1C1 < ˛ < 2j1 for some j  1, then a.a.s. c.G.n; p// D .d j / : (2) If

1 2j

0. (3) G is strongly 0-e.c. We note that the strongly 0-e.c. graphs are precisely the spanning subgraphs of the infinite random graph; see Cameron [37].

11.2

Chordal Graphs

As another instance where results from finite graphs do not translate to infinite ones, consider chordal graphs. The graph G is chordal if each cycle of length at least four has a chord. A vertex of G is simplicial if its neighborhood induces a complete graph. Every finite chordal graph contains at least two simplicial vertices, and the deletion of a simplicial vertex leaves a chordal graph. As a simplicial vertex is a corner, a finite chordal graph is dismantlable and so cop-win. However, an infinite tree containing a ray is chordal but not cop-win. Such trees have infinite diameter. Inspired by a question of Martin Farber which asked if infinite chordal graphs (more generally, bridged graphs) of finite diameter are cop-win, it was shown in Hahn et al. [76] that there exist infinite chordal graphs of diameter two that are not cop-win. The difficulty lies in finding examples with finite diameter. Theorem 77 ([76]) For each infinite cardinal , there exist chordal, robber-win graphs of order  with diameter two.

11.3

Large Families of Cop-Win Graphs

From Theorem 77, the cop number of infinite graphs behaves rather differently than in the finite case. Vertex-transitive cop-win finite graphs are cliques; see Nowakowski and Winkler [107]. However, this fails badly in the infinite case, which is the focus of this section. A class of graphs is large if for each infinite cardinal , there are 2 many nonisomorphic graphs of order  in the class. In other words, a large class contains as many as possible non-isomorphic graphs of each infinite cardinality. For example, the classes of all graphs, all trees, and all k-chromatic graphs for k finite are large. Recall that a graph G is vertex-transitive if for each pair of vertices x and y, there is an automorphism of G mapping x to y: The following result of Bonato et al. [34] showed that for any integer k > 0, there are large families of k-cop-win graphs that are vertex-transitive.

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Theorem 78 ([34]) The class of cop-win, vertex-transitive graphs with the property that the cop can win in two moves is large. To describe the large family in Theorem 78, recall some properties of the strong product of a set of graphs over a possibly infinite index set. Let I be an index set. The strong product of a set fGi W i 2 I g of graphs is the graph i 2I Gi defined by V .i 2I Gi / D ff W I !

[

V .Gi / W f .i / 2 V .Gi / for all i 2 I g;

i 2I

E.i 2I Gi / D ffg W f ¤ g and for all i 2 I; f .i / D g.i / or f .i /g.i / 2 E.Gi /g: Products exhibit unusual properties if there are infinitely many factors. Fix a vertex f 2 V .i 2I Gi / and define the weak strong product of fGi W i 2 I g with base f as the subgraph If Gi of i 2I Gi induced by the set of all g 2 V .i 2I Gi / such that fi 2 I W g.i / ¤ f .i /g is finite. The graph If Gi is connected if each factor is, and if jI j   and jV .Gi /j   for each i 2 I , then jV .If Gi /j  . For i 2 I , the projection mapping i W If Gi ! Gi is defined by i .g/ D g.i /. Let fGi W i 2 I g be a set of isomorphic copies of G. Denote by I G the strong product i 2I Gi . If f 2 V .I G/ is fixed, denote by GfI the weak strong product If G with base f . One particular power of a graph is of special interest as it allows us to construct vertex-transitive graphs out of non-transitive ones. Let  be a cardinal, and let H be a graph of order . Let I D  V .H /, and define f W I ! V .H / by f .ˇ; v/ D v. The power HfI of H with base f will be called the canonical power of H and will be denoted by H H . As automorphisms of the factors applied coordinate-wise yield an automorphism of the product, it follows that the weak strong product of vertex-transitive graphs is vertex-transitive. However, the following technical lemma from [34] demonstrates the paradoxical property that if there are infinitely many factors, the weak strong product may be vertex-transitive even if none of the factors are! Lemma 2 ([34]) If H is an infinite graph, then the canonical power of H H is vertex-transitive.

12

A Dozen Problems on Cops and Robbers

To serve as a record and a challenge, 12 open problems on Cops and Robbers are stated. These are arguably the most central (and relevant) problems in the field at the moment. Solutions to problems (1) and (11), in particular, would be exceptional breakthroughs: (1) Most likely the deepest open problem in the area is to settle Meyniel’s conjecture: If G is a graph of order n; then

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p c.G/ D O. n/:

(8)

(2) An easier problem than (1) would to settle the so-called soft Meyniel’s conjecture: For a fixed constant > 0, c.n/ D O.n1 /; (3) Meyniel’s conjecture remains open for familiar graph classes. For example, does it hold in graphs whose chromatic number is bounded by some constant k‹ (4) While the parameters mk are nonincreasing, an open problem is to determine whether the Mk are in fact nonincreasing. A possibly more difficult problem is to settle whether mk D Mk for all k  1: (5) Can the lower bound (Eq. 4) r c.n/ 

n  n0:2625 2

be improved for sufficiently large n‹ (6) For k > 1; it is conjectured in [35] that almost all k-cop-win graphs in fact have a dominating set of cardinality k: If this is the case, then the number labeled k-cop-win graphs of order n satisfies  nk .nk / Fk .n/ D 2o.n/ 2k  1 2 2 : (7) Characterize the planar graphs with cop numbers 1; 2; and 3: (8) Determine the cop number of the giant component of G.n; p p/; where p D .1=n/: In particular, show that the cop number is a.a.s. O. n/: (9) Schroeder’s conjecture: If G is a graph of genus g, then c.G/  g C 3: (10) Is the decision problem k-COP NUMBER in NP? (11) Is COP NUMBER EXPTIME-complete? Goldstein and Reingold [72] proved that the version of the Cops and Robbers game on directed graphs is EXPTIME-complete. They also proved that the version of the game on undirected graphs when the cops and the robber are given their initial positions is also EXPTIME-complete. (12) Are there large classes of infinite cop-win graphs whose members are k-chromatic, where k  2 is an integer?

13

Conclusion

The chapter focused on two major aspects of graph searching: searching games and Cops and Robbers games. A broad overview of searching was given for both graphs and digraphs, focusing on the cases when the robber is invisible and active or is visible or lazy. Related games such as minimum cost searching and pebbling were also considered. Structural characterizations were given in these cases, along with

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complexity results and bounds on various searching parameters. In the game of Cops and Robbers, a summary of known bounds and techniques were presented, focusing on Meyniel’s conjecture as one of the most important problems on the cop number. Results were presented on algorithmic and probabilistic aspects of the cop number, and results were considered on infinite graphs. For both searching and Cops and Robbers, several problems and conjectures were given. Given the broad diversity of techniques, applications, and problems in graph searching, it is evident that the field is now occupying an increasingly central position in graph theory and theoretical computer science. The subject has seen an explosive growth of interest in recent years. The wealth of new research on graph searching points to many more years of fruitful research in the field.

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Graph Theoretic Clique Relaxations and Applications Balabhaskar Balasundaram and Foad Mahdavi Pajouh

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Cliques and Related Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Degree-Based Clique Relaxation: k-Plex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Complexity and Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Polyhedral Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Distance-Based Clique Relaxations: k-Cliques and k-Clubs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Complexity and Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Polyhedral Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Density-Based Models: Densest t -Subgraph and -Quasi-clique. . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Densest t -Subgraph Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Maximum -Quasi-clique Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Selected Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Cliques and graph theoretic clique relaxations are used to model clusters in graph-based data mining, where data is modeled by a graph in which an edge implies some relationship between the entities represented by its endpoints. The need for relaxations of the clique model arises in practice when dealing with massive data sets which are error prone, resulting in false or missing edges. The clique definition which requires complete pairwise

B. Balasundaram () • F. Mahdavi Pajouh School of Industrial Engineering and Management, Oklahoma State University, Stillwater, OK, USA e-mail: [email protected]; [email protected] 1559 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 9, © Springer Science+Business Media New York 2013

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adjacency in the cluster becomes overly restrictive in such situations. Graph theoretic clique relaxations address this need by relaxing structural properties of a clique in a controlled manner via user-specified parameters. This chapter surveys such clique relaxations available in the literature primarily focusing on polyhedral results, complexity studies, approximability, and exact algorithmic approaches.

1

Introduction

A clique in a graph is a set of pairwise adjacent vertices. Given a simple, undirected, and finite graph, finding a clique of maximum cardinality, clustering the graph into minimum number of cliques that cover or partition the vertex set, and enumerating inclusionwise maximal cliques are fundamental combinatorial optimization problems associated with cliques. This basic model and the associated problems have been well studied in graph theory, polyhedral combinatorics, and complexity theory, leading to many deep results in these areas. Cliques and many graph theoretic clique relaxations were proposed in social network analysis (SNA) to model cohesive subgroups in social networks [149]. Such applications of the clique model in SNA can be viewed within the broader scope of graph-based clustering and data mining [18, 41, 55] where data is modeled by a graph in which an edge implies similarity, dissimilarity, or some other relationship between the entities represented by the endpoints of the edge. One could formalize the notion of a graph-theoretic cluster by requiring one or more of the following structural properties: 1. Each vertex has a large number of vertices adjacent to it inside the cluster. 2. The pairwise distances between vertices in the cluster are small. 3. The cluster has a large number of edges with both endpoints inside the cluster. 4. The cluster has a large fraction of the maximum possible number of edges, with both endpoints inside the cluster. 5. The cluster requires a large number of vertices or edges to be deleted in order to be disconnected. Clique is an ideal graph model for this purpose as it represents a cluster in which every vertex has maximum possible number of adjacent vertices, maximum possible edges within a cluster, and minimum possible pairwise distances within a cluster. Furthermore, for a clique with n vertices, at least n  1 edges must be deleted to disconnect the clique or n  1 vertices must be deleted resulting in a trivial graph. The need for relaxations of the clique model arises in practice when dealing with massive data sets which are inevitably prone to errors that manifest in the graph model as false or missing edges. The clique definition which requires complete pairwise adjacency in the cluster becomes overly restrictive in such situations. Graph theoretic clique relaxations address this need by relaxing the aforementioned structural properties in a controlled manner via user-specified parameters.

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This chapter surveys graph theoretic clique relaxations that have witnessed a resurgent interest in recent literature in operations research, computer science, and graph-based data mining. Several comprehensive surveys on the maximum clique problem and its applications are available that also cover application areas other than data mining such as coding theory and fault diagnosis [31, 122]. This chapter adds to recent focused surveys on clique relaxations [27, 103, 125] by providing a more comprehensive overview and surveying some more recent developments. In particular, this chapter focuses on polyhedral results, complexity studies, approximability, and exact algorithmic approaches addressing clique relaxation models. When appropriate, proofs for some of the relevant results are also included from the original works.

1.1

Notations

In this chapter, an arbitrary, simple, finite, undirected graph of order n and size m is denoted by G D .V; E/ where V D f1; : : : ; ng and .i; j / 2 E when vertices i and j are adjacent, with jEj D m. For an introduction to graph theory, readers are referred to texts by West [150] or Diestel [62]. The edge density of a nontrivial graph G is the ratio of number of  edges in G to the number of maximum possible edges in this graph, that is, m= n2 . The edge density of a single vertex graph is taken to be unity. The complement graph of G is denoted by G D .V; E/ where E is the complement of the edge set E. Given S  V , induced subgraph GŒS  is obtained by deleting from G all vertices (and incident edges) in V n S . The graph obtained by the deletion of a vertex i , an edge e, a set of vertices I , or a set of edges J from G is denoted by G  i , G  e, G  I , and G  J , respectively. The complete graph on n vertices is denoted by Kn , and the complete bipartite graph with partitions of sizes p and q is denoted by Kp;q . In particular, the graph K1;n is called a star, and K1;3 is called a claw. Cycle and path on n vertices are denoted by Cn and Pn , respectively. The adjacency matrix AG is a symmetric 0,1matrix of order n  n with ai;j D 1 , .i; j / 2 E. For a vertex i 2 V; NG .i / denotes the set of vertices adjacent to i in G, called the neighborhood of i , and NG Œi  D fi g [ NG .i / denotes the closed neighborhood of i . The set of non-neighbors of i is given by V n NG Œi . Denote by degG .i /, the degree of vertex i in G given by jNG .i /j. The maximum and minimum degrees in a graph are denoted, respectively, by .G/ and ı.G/. For two vertices i; j 2 V , dG .i; j / denotes the length of a shortest path between i and j in G. By convention, when no path exists between two vertices, the shortest distance between them is infinity. The diameter of a graph is defined as d i am.G/ D maxi;j 2V dG .i; j /. For a graph G D .V; E/ and positive integer k, the kth power of G is defined as G k D .V; E k /, where E k D f.i; j / W i; j 2 V; dG .i; j /  kg: The simple graph G k is constructed from the original graph by adding edges corresponding to all pairs of vertices with distance no more than k between them in G.

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The vertex connectivity .G/ and edge connectivity .G/ of a graph are the minimum number of vertices and edges, respectively, whose removal results in a disconnected or trivial graph. A graph is said to be t-connected if .G/  t and t-edge connected if .G/  t. It is known that when G is nontrivial, .G/  .G/  ı.G/ [150]. If the graph G under consideration is obvious, the subscript G in the neighborhood, degree, and distance notations is sometimes dropped for simplicity.

1.2

Cliques and Related Models

Definition 1 A clique is a subset of vertices that are pairwise adjacent in the graph. Definition 2 An independent set (stable set, vertex packing) is a subset of vertices that are pairwise nonadjacent in the graph. The maximum clique problem is to find a clique of maximum cardinality. The clique number !.G/ is the cardinality of a maximum clique in G. The maximum cardinality of an independent set of G is called the independence number of the graph G and is denoted by ˛.G/. The associated problem of finding a largest independent set is the maximum independent set problem. Clearly, I is an independent set in G if and only if I is a clique in G and consequently ˛.G/ D !.G/. In this chapter, maximality and minimality of sets are always defined based on inclusion and exclusion, respectively. For instance, a maximal clique is one that is not a proper subset of another clique. The maximum clique and independent set problems on arbitrary graphs are NP-hard [74] and are hard to approximate within n1 for any  > 0 [79]. Furthermore, finding cliques and independent sets parameterized by solution size are not known to be fixed-parameter tractable problems and, in fact, are basic W Œ1-hard problems [65]. Definition 3 A proper coloring of a graph is one in which every vertex is colored such that no two vertices of the same color are adjacent. A graph is said to be t-colorable if it admits a proper coloring with t colors. Vertices of the same color are referred to as a color class, and they induce an independent set. The chromatic number of the graph, denoted by .G/, is the minimum number of colors required to properly color G. For any graph G, !.G/  .G/, as different colors are required to color the vertices of a clique. Definition 4 A vertex cover is a subset of vertices such that every edge in the graph is incident at some vertex in the subset. Clearly, C is a vertex cover of G if and only if V n C is an independent set in G. The minimum vertex cover problem seeks to find a vertex cover of minimum cardinality.

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Definition 5 A dominating set is a subset of vertices such that every vertex in the graph is either in this set or has a neighbor in this set. The minimum cardinality of a dominating set is called the domination number, denoted by .G/. It should be noted that every maximal independent set is also a minimal dominating set.

2

Degree-Based Clique Relaxation: k-Plex

The degree-based clique relaxation, called a k-plex, was introduced by Seidman and Foster for modeling cohesive subgroups in social network analysis [133]. The parameter k  1 is a positive integer that controls the “degree” of relaxation and includes clique as a special case. A complementary model to k-plex can also be defined along similar lines that relaxes independent sets. Definition 6 A subset S  V is a k-plex if ı.GŒS /  jS j  k. Definition 7 A subset J  V is a co-k-plex if .GŒJ /  k  1. Clearly, 1-plexes and co-1-plexes are simply cliques and independent sets, respectively. For k > 1, a k-plex is a degree-based relaxation of the clique definition which allows for at most k  1 non-neighbors in S , while the co-k-plex is a degreebased relaxation of the independent set definition which allows for at most k  1 neighbors in J . Clearly, S is a k-plex in G if and only if S is a co-k-plex in G. Figure 1 illustrates this concept. The set f1; 2; 3; 4g is a 1-plex (clique), sets f1; 2; 3; 4; 5g and f1; 2; 3; 4; 6g are 2-plexes (maximal and maximum), and the entire vertex set forms a 3-plex. In the same graph f5; 6g forms a co-1-plex (independent set), while f1; 5; 6g forms a co-2-plex. The maximum k-plex problem is to find a largest cardinality k-plex in G, the cardinality of which is the k-plex number of G denoted by !k .G/. The maximum co-k-plex problem is similarly defined with the co-k-plex number denoted by ˛k .G/. Some early results on k-plexes established by Seidman and Foster [133] are summarized in Theorem 1. An alternate characterization of k-plexes, Theorem 2, was also established therein. Theorem 1 ([133]) Let graph G be a k-plex on n vertices. Then, 1. Any vertex-induced subgraph of G is a k-plex. 2. If k < nC2 2 , then d i am.G/  2. 3. .G/  n  2k C 2. Proof 1. Every induced subgraph of G can be obtained by deleting vertices, one at a time. If G is a k-plex, ı.G/  n  k. In G 0 obtained by deleting some vertex from G,

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Fig. 1 Illustration of k-plexes and co-k-plexes

6 4 2

3 5 1

the degree of all the vertices and hence the minimum degree can drop by at most 1. Hence, the new graph continues to be a k-plex as ı.G 0 /  .n  1/  k. 2. For any vertex v in the k-plex G, the number of vertices in the closed neighborhood jN Œvj  n  k C 1. Thus, for two arbitrary vertices u; v we have jN Œuj C jN Œvj  2n  2k C 2 > n by the given condition. Hence, N Œu and N Œv are not disjoint. Thus, for all u; v, dG .u; v/  2 ) d i am.G/  2. 3. Suppose we delete t vertices from G. The resulting graph G 0 is still a k-plex and the above argument still holds. If k < nt2C2 , then d i am.G 0 /  2. Thus, G 0 is still connected if t < n  2k C 2 ) .G/  n  2k C 2. Remark 1 Note that the condition for a diameter-two bound is sharp. Consider the graph Kk1 [ Kk1 which is a k-plex for every k. Thus, a k-plex could be disconnected even for k D nC2 2 . However, no isolated vertex could be present in a k-plex of size k C 1 or more. Remark 2 It is also useful to note that in an arbitrary graph G, any k-element subset of vertices is a k-plex and any k-plex is also a k C 1-plex. Theorem 2 ([133]) G D .V; E/ is a k-plex if and only if every k-element subset of vertices forms a dominating set in G. Proof Suppose there exist k vertices that do not form a dominating set, then there must be some vertex v distinct from these k vertices that is not adjacent to any of these k vertices. This contradicts the fact that G is a k-plex. Now suppose that G is not a k-plex. Hence, jV j  k C 1, and there exists a vertex v 2 V such that jN.v/j  jV j  k  1. Hence, jV n N Œvj  k, and it does not dominate G, specifically the vertex v.  A characterization of 2-plexes via the co-2-plex number is provided in Proposition 1 [105]. This characterization does not extend to general k in a straightforward manner. Proposition 1 ([105]) G D .V; E/ is a 2-plex if and only if ˛2 .G/ D minf2; jV jg.

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Proof Suppose G is a 2-plex and ˛2 .G/  3. Then, V contains a subset S D fu; v; wg that forms a co-2-plex. At least one vertex in S must have zero degree in GŒS , which contradicts the fact that GŒS  is a 2-plex. Conversely, suppose G is graph with jV j  3 and ˛2 .G/ D 2. If G is not a 2-plex, there exists u 2 V such that jN.u/ \ V j  jV j  3. Hence, there exist vertices v; w 2 V n N Œu, and the set fu; v; wg forms a co-2-plex of size 3.  There is a close relationship between the maximum k-plex problem and the minimum bounded-degree-d vertex deletion problem (d -BDD) studied in [114]. The d -BDD problem defined next is a generalization of the minimum vertex cover problem. Definition 8 Given a graph G D .V; E/ and a fixed integer d  0, a bdd-d -set is defined as a subset S  V for which GŒV n S  is a graph with maximum vertex degree at most d [113]. The minimum bounded-degree-d vertex deletion problem is to find a minimum cardinality bdd-d -set in G. Note that for d D 0, a bdd-0-set is precisely a vertex cover of G. Further note that if S is a bdd-d -set, then V n S by definition is a co-k-plex with k D d C 1. Lemma 2 shows the relationship between the maximum k-plex problem and the minimum d -BDD problem. Proposition 2 ([113]) A graph G D .V; E/ contains a k-plex of size c if and only if G contains a bdd-d -set of size jV j  c with d D k  1. Proof It follows from the observation that S  V is a k-plex of size c in G if and only if S is a co-k-plex in G.  A natural degree-based generalization of graph coloring is the following. Definition 9 A proper co-k-plex coloring of a graph G D .V; E/ is an assignment of colors to V such that each vertex has at most k  1 neighbors in the same color class. Co-k-plexes have also been called .k  1/-dependent sets in the literature [64] and co-k-plex coloring has also been studied under other names such as defective coloring [59] and k-improper coloring [80]. A proper co-k-plex coloring of G partitions G into co-k-plexes and is equivalent to a k-plex partitioning of G. Obviously, an independent set cannot contain a clique of size larger than one, while k-plexes of size two or more can be found inside a co-k-plex for k  2. A sharp upper bound on the maximum size of the intersection was established in [20] showing that if .G/  k  1, then !k .G/  2k  2 C .k mod 2/. In light of this observation, two graph invariants can be defined to relax the graph coloring problem as discussed next.

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The minimum number of colors required in a proper co-k-plex coloring of G is called the co-k-plex chromatic number denoted by k .G/ [21]. The weighted cok-plex chromatic number denoted by wk .G/ is defined as the minimum weighted sum of the color classes, where the weight of a color class is the size of a maximum k-plex contained in the color class [104]. These definitions are equivalent when k D 1, but not for k > 1. The latter provides a tighter upper bound on the k-plex number as the following inequality is true for any graph G [20, 104]: !k .G/  wk .G/  Œ2k  2 C .k mod 2/k .G/:

2.1

Complexity and Approximation

For arbitrary k, where the parameter is considered as part of the input for each instance of the problem, the maximum k-plex problem is trivially NP-hard as it includes the maximum clique problem as a special case. However, the complexity of the problems for fixed k (prescribed as part of the problem definition) is a nontrivial question. A graph property … is said to be hereditary on induced subgraphs, if G is a graph with property … and the deletion of any subset of vertices does not produce a graph violating …. Property … is said to be nontrivial if it is true for a single vertex graph and is not satisfied by every graph. A property … is said to be interesting if there are arbitrarily large graphs satisfying …. The maximum … problem is to find the largest order induced subgraph that does not violate property …. Naturally, this optimization problem is meaningful only if … is nontrivial and interesting. Yannakakis [153] obtained a general complexity result for such properties … showing that the maximum … problem for nontrivial, interesting graph properties that are hereditary on induced subgraphs is NP-hard. Inapproximability of such problems was also established by Lund and Yannakakis in [96]. Since k-plexes and co-k-plexes are nontrivial, interesting, and hereditary, the maximum k-plex and co-k-plex problems are NP-hard based on this result and hard to approximate within O.n / for some  > 0 unless P D NP . The NP-hardness of the maximum k-plex problem and the maximum co-kplex problem for every fixed k was also independently established in [20, 64], respectively. In [61], the maximum co-k-plex problem was shown to be NP-hard even on planar and bipartite graphs for fixed k  2. Recently in [81, 88], this problem was shown to be NP-hard for every fixed k on unit-disk graphs (UDGs, intersection graphs of unit-disks in a Euclidean plane [54]). Note that the maximum independent set problem is NP-hard on planar graphs [74] and unit-disk graphs [54], but it is polynomial time solvable on bipartite graphs through graph-matching techniques [56]. Analogous to what is known for cliques and vertex covers, the maximum k-plex problem parameterized by solution size is W Œ1-hard [91], while the d -BDD problem is fixed-parameter tractable when parameterized by solution size [91, 118]. The maximum co-k-plex problem admits a polynomial time approximation scheme (PTAS) on UDGs [81, 88] and a 3k-approximation algorithm on

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UDGs [21]. The PTAS employs a “divide-and-conquer” approach similar to that used for independent sets in UDGs [84], and the constant factor approximation employs approaches similar to those developed in [100] for independent sets. The minimum co-k-plex coloring problem is also NP-hard for every fixed k on UDGs [81, 88]. A polynomial time 6-approximation algorithm for this problem was developed in [81,88], and a 3-approximate algorithm for this problem was presented in [21]. Some bounds on the graph invariants !k .G/; ˛k .G/, and k .G/ were also developed in [21], used to establish the aforementioned approximation guarantees for UDGs and generalized claw-free graphs. These identities summarized in Theorem 3 generalize some well-known results for cliques, independent sets, and coloring (see for instance [83, 100]). Theorem 3 ([21]) Consider a graph G D .V; E/, 1. ˛k .G/  k˛.G/. 2. !k .G/  k!.G/. 3. If G is a .p C 1; k/-claw-free, that is, ˛k .GŒN.v//  p 8 v 2 V , then jJ j 

˛k .G/ p

for any maximal co-k-plex J . 4. If G is a unit-disk graph with an embedding specified in the Euclidean X -Y plane, then ˛k .GŒN.v//  5k 8 v 2 V , and if v is the vertex corresponding to the minimum X -coordinate, then ˛k .GŒN.v//  3k. 5. k .G/  b d.G/ c C 1 where the polynomial time computable invariant d.G/ k denotes the largest d such that G has an induced subgraph of minimum degree at least d . 6. If G is a unit-disk graph, then k .G/  d.G/ . 3k Proof 1. Suppose ˛k .G/  k˛.G/ C 1. Let G .0/ be a co-k-plex induced in G of size k˛.G/ C 1. Apply the following greedy algorithm for a maximal independent set in graph G .0/ , with set I initially empty and i D 0. Pick any vertex v from V .G .i / /, let I I [ fvg and G .i C1/ G .i /  N Œv. Increment i and repeat until .i C1/ .0/ G is null. Since G is a co-k-plex, so are all G .i / , and thus .G .i / /  k 1. So jV .G .i C1/ /j D jV .G .i / /j  1  jN.v/ \ V .G .i //j  jV .G .i / /j  k. After ˛.G/ iterations jI j D ˛.G/ and jV .G .˛.G// /j  k˛.G/ C 1  k˛.G/ D 1. So the algorithm can add at least one more vertex to independent set I , a contradiction. 2. Implied by 1. 3. Suppose J  is a maximum co-k-plex in G. Further assume that p  k, as the only .p C 1; k/-claw-free graphs with p < k are themselves co-k-plexes. Then J n

[ v2J

N Œv D ;I

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otherwise, there exists a v 2 J  such that N Œv \ J D ; contradicting the maximality of J . Hence, we have jJ  j D ˛k .G/ 

X

jN Œv \ J  j  pjJ j:

v2J

4. By Part 1 we have ˛k .N.v//  k˛.N.v//  5k where the last inequality follows from the result of Marathe et al. [100] that for a UDG, ˛.N.v//  5 8 v 2 V . The bound for the minimum X-coordinate can be obtained similarly by extending an analogous result from [100]. 5. Denote by d.G/ the largest d such that G has an induced subgraph of minimum degree at least d . The invariant d.G/ can be found in polynomial time using a simple algorithm [83] along with an associated ordering v1 ; : : : ; vn of vertices of G such that any vi has at most d.G/ neighbors in v1 ; : : : ; vi 1 . In that order, for each vertex, assign the smallest available color. A color is said to be available at a vertex if it has been assigned to at most k  1 neighbors of the current vertex, already processed in the list ordering. Consider any vertex vi , i D 2; : : : ; n. It can have at most d.G/ neighbors already colored in the list so far. There must exist a color from the set f1; : : : ; b d.G/ c C 1g that has been assigned to at most k  1 neighbors of vi . k If not, the number of colored neighbors of vi exceeds d.G/. Since this is true for any vi , the above algorithm uses no more than b d.G/ c C 1 colors. k 6. Let H be the induced subgraph of G such that ı.H / D d.G/. Consider a vertex v 2 V .H / of minimum X -coordinate, and let H 0 denote the graph induced by N.v/. Any proper co-k-plex coloring of G is also proper for H 0 , and it partitions H 0 into co-k-plexes. Hence, we have k .G/  k .H 0 / 

2.2

d.G/ jV .H 0 /j  : ˛k .H 0 / 3k

t u

Polyhedral Results

The maximum k-plex problem can be formulated as the following integer program: !k .G/ D max

X

xi

i 2V

Subject to X

xj  .k  1/xi C jV n N Œi j.1  xi /

j 2V nN Œi 

xi 2 f0; 1g

8 i 2 V:

8i 2V

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Given a graph G D .V; E/, the k-plex polytope of G denoted by Pk .G/ is the convex hull of the incidence vectors of all k-plexes in G. The k-plex polytope is full dimensional and the variable 0,1-bound constraints are facet inducing as stated in Theorem 4. The case k D 1 which corresponds to the well-known clique polytope is excluded from consideration. Theorem 4 ([20]) Given a graph G D .V; E/ and an integer k > 1, 1. d i m.Pk .G// D n, 2. For each i 2 V , inequalities xi  0 and xi  1 are facet inducing for Pk .G/. Several facets and valid inequalities for Pk .G/ for arbitrary and restricted graph classes are available from [20, 105]. The results for the co-k-plex polytope developed in [105] are adapted via complementation for Pk .G/ in this chapter. It should be noted that these results extend several analogous results known for the clique and stable set polytopes [49, 50, 116, 117, 121, 142].

2.2.1 Co-k-plex and Independent Set Inequalities Proposition 3 ([20]) (Co-k-plex inequalities) Given a graph G D .V; E/ and an integer k  1, let rk D 2k  2 C .k mod 2/. If I  V is a maximal co-k-plex of size at least rk C 1 in G, then the following inequality is valid for Pk .G/: X

xi  rk :

i 2I

Proposition 3 can be further strengthened for the case, k D 2 as outlined in Theorem 5. Theorem 5 ([20]) (Co-2-plex Pfacets) Given a graph G D .V; E/ and J  V such that jJ j  3, the inequality xi  2 induces a facet of P2 .G/ if and only if J is a maximal co-2-plex.

i 2J

Proof The validity of this inequality for P2 .G/ follows from Proposition 3. So, it suffices to show that the face of P2 .G/ induced by this inequality, ( x 2 P2 .G/ W

X

) xi D 2 ;

i 2J

is of dimension n  1. Note that if J is a maximal co-k-plex, for every v 2 V n J , at least one of the following conditions must hold: 1. 9 j 2 J \ N.v/ such that jN.j / \ J j D k  1 and including v would cause degree of j in the induced subgraph to be k. 2. jN.v/\J j  k and upon inclusion, v would have degree k or more in the induced subgraph. For every v 2 V n J , the above two conditions for a maximal co-2-plex imply the existence of two vertices u; w 2 J such that fv; u; wg is a 2-plex. Indeed, if the first

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case holds, let u 2 J \ N.v/, then N.u/ \ J D fwg and fv; u; wg is a 2-plex. If the second case holds, fu; wg  J \ N.v/ and again fv; u; wg is a 2-plex. W.l.o.g. assume that J D f1; : : : ; rg and V n J D fr C 1; : : : ; ng, where r  3. Let ei 2 Rn denote the unit vector with i th component one and all others zero. The affinely independent vectors are constructed as follows: x v D ev C er ; 8 v D 1; : : : ; r  1; x r D e1 C e2 (note that x r is distinct from x 1 ; : : : ; x r1 as r  3), x v D ev C eu C ew ; 8 v D r C 1; : : : ; n; where˚for each v 2 V nP J , u; w 2 J are particular vertices described before. Clearly, x v 2 x 2 P2 .G/ W xi D 2 ; 8 v 2 V , and it is easy to verify that they are i 2J

affinely independent. P Conversely, suppose xi  2 induces a facet of P2 .G/ and J is not a co-2i 2J

plex. Then, there exists some v 2 J with 2 neighbors in J which form a 2-plex that violates the facet-inducing inequality, leading to a contradiction. Hence, J must be a co-2-plex. If J is not maximal, then there exists a valid maximal co-2-plex inequality that dominates the given facet-inducing inequality. Hence, J must be a maximal co-2-plex.  Padberg [121] (see also [71]) showed that the maximal independent set inequalities are facet inducing for the clique polytope, and Theorem 5 is a direct analogue for the 2-plex polytope. However, for k  3, co-k-plex inequalities are not necessarily facets of Pk .G/ for arbitrary graphs. It should be noted that the bound rk while sharp is not necessarily achieved in all co-k-plexes and the k-plex number could be strictly smaller than rk . Even in co-k-plexes where the bound is achieved, there may not be enough affinely independent incidence vectors of kplexes achieving that bound. This motivated the study of k-plex rank inequalities of Pk .G/ [105]. For any I  V , the following k-plex rank inequality is valid for Pk .G/: X xi  !k .GŒI /: i 2I

A sufficient condition under which the k-plex rank inequality derived based on the k-plex number of G induces a facet of Pk .G/ was developed in [105], extending a result of Chv´atal [53]. Definition 10 Given a graph G D .V; E/ and an edge e 2 E in the complement graph G, e is called k-plex critical if !k .G C e/ D !k .G/ C 1 where G C e D .V; E [ feg/. Theorem 6 ([105]) Let E   E be the set of all k-plex critical edges P of G. If the graph G  D .V; E  / is connected, then the k-plex rank inequality xi  !k .G/ induces a facet of Pk .G/.

i 2V

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Maximal independent set inequalities play an important role in describing the clique polytope of graphs. Specifically, along with nonnegativity constraints, they are sufficient to completely describe the clique polytope of perfect graphs [58]. Preliminary results in this direction established in [105] show that the co-2-plex facets and the variable bounds are sufficient to describe the 2-plex polytope of paths, mod-3 cycles (see Definition 11), and co-2-plexes. The co-k-plex inequalities for the k-plex polytope generalize the independent set inequalities for the clique polytope. Note that r1 D 1, and hence, the co-1-plex inequalities are precisely the independent set inequalities for the 1-plex (clique) polytope. An alternate approach based on the observation that an independent set can contain at most k vertices from a k-plex for any k leads to the independent set inequalities for the k-plex polytope. Proposition 4 ([20]) (Independent set inequalities) Given a graph G D .V; E/ and an integer Pk  1, if I  V is a maximal independent set of size at least k C 1 in G, then i 2I xi  k is a valid inequality for Pk .G/. Furthermore, if E D ; and n  k C 1, the inequality induces a facet of Pk .G/. When k D 1, co-1-plex and independent set inequalities described here coincide with the well-known inequalities for the clique polytope. For k D 2, independent set inequalities are dominated by the facet-inducing co-2-plex inequalities since rk D k D 2. However, that is not the case for k  3 as maximal independent set inequalities upon lifting can produce facets of Pk .G/ that are not identified as co-k-plex inequalities or k-plex rank inequalities [20]. In addition to independent set inequalities, some other well-known class of inequalities for the clique polytope are hole and antihole inequalities [116,121], wheel inequalities and their generalizations [23, 49], web and antiweb inequalities [142], and many others [24, 33, 75]. Extensions of some of these results to the k-plex polytope is presented next.

2.2.2 Hole Inequalities and Special Cases Definition 11 A hole is an induced chordless cycle of size 4 or more. For a positive integer k, a hole on n vertices with n  0.mod k/ is called a mod-k hole, otherwise it is an odd mod-k hole. The complement of a hole is called an antihole. Valid inequalities based on holes were developed in [20] for the k-plex polytope. These are also known to be facet inducing for Pk .Cn / depending on n and k. Proposition 5 ([20]) (Hole inequalities) Given a graph PG D .V; E/ and a positive integer k, if I  V is a hole of size  k C 3, then i 2I xi  k C 1 is a valid inequality for Pk .G/. Theorem 7 ([20]) Consider a cycle Cn on n vertices, positive integer Pk with n  k C 3 such that n and k C 1 are relatively prime. Then, the inequality xi  k C 1 induces a facet of Pk .Cn /.

i 2V

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Theorem 7 for odd mod-.k C1/ holes is a direct analogue of the result developed by Nemhauser and Trotter [116]. Interestingly, however, mod-.k C 1/ holes can also induce facets of Pk .Cn / [20]. Theorem 8 ([20]) Consider a cycle Cn on n vertices and a positive P integer k  4 such that n  k C 3 and n is a multiple of k C 1. Then, the inequality xi  k C 1 i 2V

induces a facet of Pk .Cn /.

Note that antiholes are k-plexes for k  3. However, when k D 2, antiholes do lead to valid inequalities for the 2-plex polytope [105]. Lifting this inequality leads to the antiwheel inequality, details of which can be found in [105]. Theorem 9 ([105]) Consider the complement of a cyclePon n vertices C n with n  4 such that n 6 0.mod 3/. Then, the inequality xi  b 2n 3 c induces a i 2V

facet of P2 .C n /. Definition 12 Given a graph G D .V; E/ and an integer p, suppose n  2, 1  p  n2 . If E D f.i; j / W j D i C p;    ; i C n  p; 8i 2 V g where sums are written modulo n, then G is called a web, denoted by W .n; p/. The complement of a web is an antiweb. Webs were introduced by Trotter [142] to generalize odd hole and antihole inequalities for the stable set polytope. This result was extended in [105] to show that antiwebs induce facets of the 2-plex polytope as outlined next. It should be noted that W .n; 1/ is a complete graph, for p  2, W .2p C 1; p/ is an odd hole and W .2p C 1; 2/ is an odd antihole. Theorem 10 ([105]) (Web inequalities) Consider the antiwebPW .n; p/ where n and p C 1 are relatively prime and p < b n2 c, then the inequality xi  p C 1 induces i 2V

a facet of P2 .W .n; p//. This section is concluded with valid inequalities produced by generalizing claws. These were originally developed for the co-k-plex polytope in [105]. Definition 13 Given a graph G D .V; E/ and an integer k  1, if there exists a vertex u 2 V such that N.u/ D V n fug, N.u/ is a co-k-plex, and jN.u/j  maxf3; kg, then G is called a k-claw. Vertex u is called the center of the k-claw. Theorem 11 ([105]) Consider a graph G D .V; E/ such thatP G is a k-claw with center u 2 V and integer k  2, the inequality .n  k/xu C xi  n  1 is a i …NG Œu

facet of Pk .G/. Of particular note is that a k-claw can properly contain another k-claw, and both would give rise to distinct facets.

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Algorithms

Exact algorithms for the maximum k-plex problem can be broadly categorized as combinatorial approaches and polyhedral approaches. A branch-and-cut framework is developed in [20] that incorporates maximal independent set and cok-plex inequalities identified via heuristic separation procedures. An iterative peel-branch-and-cut algorithm to solve the maximum k-plex problem to optimality on “scale-free” graphs is also developed in [20] that uses the branchand-cut algorithm in combination with preprocessing and graph decomposition approaches. Scale-free graphs [22, 52] are very large and sparse graphs which exhibit a power-law degree distribution. The large-scale and sparse nature of such graphs present computational challenges that are addressed by exploiting the power-law nature of such graphs. The need for such specialized techniques are motivated by graph-based data mining applications where the graph models exhibit a power-law degree distribution [20]. Numerous exact combinatorial algorithms are known for the maximum clique problem such as those developed by Bron and Kerbosch [38], Balas and Yu [16], Applegate and Johnson [7], Carraghan and Pardalos [43], Babel [13], Balas ¨ and Xue [15], Wood [151], Sewell [134], Osterg˚ ard [120], and Tomita and Kameda [141]. ¨ Osterg˚ ard’s algorithm [120], which extends the approaches proposed by Applegate and Johnson [7] and Carraghan and Pardalos [43], has been extended to the maximum k-plex problem by McClosky and Hicks [102, 106] and by Trukhanov et al. [143, 144]. Design of such algorithms and variations, implementation details, computational results on DIMACS benchmarks [63], as well as comparisons between algorithms can be found in [106, 114, 144]. ¨ Algorithm 1 presents a basic extension of Osterg˚ ard’s clique algorithm [120] to the maximum k-plex problem, which works by first ordering vertices in V based on degree or other criteria as v1 ; v2 ;    ; vn . Let set Si  V be defined as Si D fvi ; vi C1 ;    ; vn g and let c.i / denote the size of the maximum k-plex in GŒSi . Clearly c.n/ D 1 and c1 D !k .G/. Algorithm 1 recursively computes c.i / starting from c.n/ and down to c.1/ for which the following holds:  c.i / D

c.i C 1/ C 1; if the corresponding solution contains vi ; c.i C 1/; otherwise.

A different approach is undertaken in [114] where the close relationship to the minimum bounded-degree-d vertex deletion problem and its fixed-parameter tractability with respect to the solution size is exploited to develop an exact algorithm for the maximum k-plex problem. Another important result in advancing this line of research is the following. A basic result of Nemhauser and Trotter [117] is fundamental to many fixed-parameter tractable and approximation algorithms for the minimum vertex cover problem. In [70], a generalization of this result is

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Algorithm 1 Maximum k-plex algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34:

procedure M AX KPLEX (G) Order vertices in V as v1 ; : : : ; vn max WD 0 for i WD n downto 1 do C WD fv 2 Si n fvi g W fv; vi g is a k-plexg f ound WD f alse FIND KPLEX (C , fvi g) c.i / WD max end for return max end procedure procedure FIND KPLEX (C,P) if C D ; then if jP j > max then max WD jP j f ound WD t rue end if return end if while C ¤ ; do if jC j C jP j  max then return end if i WD minfj W vj 2 C g if c.i / C jP j  max then return end if C WD C n fvi g; P 0 WD P [ fvi g; C 0 WD fv 2 C W P 0 [ fvg is a k-plexg FIND KPLEX (C 0 ,P 0 ) if f ound D t rue then return end if end while end procedure

presented which can further improve the parameterized algorithmics of the d -BDD problem.

3

Distance-Based Clique Relaxations: k-Cliques and k-Clubs

The distance-based clique relaxations were introduced in social network analysis by Luce [94] soon after cliques were used to model cohesive subgroups [95]. These models relaxed the adjacency requirement of cliques to a short path requirement, with the models being refined and redefined by others [3, 112]. A discussion of the different definitions can be found in [19]. Two basic models based on relaxing pairwise distances can be defined as follows.

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Fig. 2 A graph in which set S D f1; 2; 3; 4; 5g is a 2-clique but not a 2-club

6

1

5

4

2

3

Definition 14 A subset of vertices S  V is a k-clique if dG .u; v/  k for all u; v 2 S . Definition 15 A subset of vertices S  V is a k-club if dGŒS  .u; v/  k for all u; v 2 S , that is, d i am.GŒS /  k. By definition, 1-clique and 1-club correspond to the classical clique. Clearly, every k-club is also a k-clique since d i am.GŒS /  k ) dGŒS  .u; v/  k 8 u; v 2 S ) dG .u; v/  k 8 u; v 2 S . The converse, however, is not always true. In fact for k  2, a k-clique S can contain vertices u; v 2 S such that dG .u; v/  k but dGŒS  .u; v/ > k. In Fig. 2 [3], set S D f1; 2; 3; 4; 5g forms a 2-clique which is not a 2-club. It should be noted that dG .1; 5/ D 2 and dGŒS  .1; 5/ D 3. The k-clique model was originally introduced in social network analysis by Luce [94] to model cohesive subgroups. The k-club model [3, 112] was developed to address the drawback of k-cliques where members outside the cohesive subgroup can be used on the short paths required between members inside the group. The k-club number of G denoted by ! k .G/ is the cardinality of a maximum k-club in G, and the maximum kclub problem is to find a k-club of the maximum cardinality in G. The k-clique number of G denoted by e ! k .G/ and the maximum k-clique problem are similarly defined. For a given graph G and a positive integer k, the following inequality is true: !.G/  ! k .G/  e ! k .G/: It should be noted that S  V is a k-clique in G, if and only if S is a clique in the power graph G k . Accordingly, given a graph G and a positive integer k, e ! k .G/ D !.G k /. Hence, the maximum k-clique problem for any k is equivalent to the maximum clique problem on the corresponding power graph. Because of this

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close correspondence between k-cliques in original graph and cliques in the power graph, the k-clique model has not been extensively studied. Two concepts of particular relevance to these problems are distance-k independence [45] and distance-k coloring [107]. Definition 16 Given a graph G D .V; E/, I  V is a distance-k-independent set if for every distinct pair i; j 2 I , dG .i; j /  k C 1. Clearly, distance-1 independence is equivalent to pairwise nonadjacency, which corresponds to a classical independent set. It should be noted that the definition becomes more restrictive as k increases. A distance-.k C 1/ independent set is also distance-k independent, but the converse is not always true. Further, a k-clique or a k-club can intersect a distance-k-independent set in at most one vertex. Definition 17 Given a graph G D .V; E/ and a positive integer k, a proper distance-k coloring is one in which the pairwise distance in G between any two vertices that receive the same color is at least k C 1. A graph is distance-k t-colorable if it has a proper distance-k coloring that uses at most t colors. The minimum number of colors t that admits a proper distance-k t-coloring is the distance-k chromatic number of G denoted by dk .G/. It should be noted that each color class forms a distance-k-independent set. Figure 3 shows a proper distance-2 5-coloring of a graph. By definition, in any proper distance-k coloring of a graph G, a k-clique in G can intersect any color class in at most one vertex. Then the following relationship [97] holds: ! k .G/  e ! k .G/  dk .G/: Some remarks on the nonhereditary nature of k-clubs. For any integer k  1, the k-clique model like the k-plex model is hereditary on induced subgraphs in the sense described in Sect. 2.1. However, this property does not hold for the k-club model for k  2. For example, the graph shown in Fig. 4b is a 2-club which means it is also a 2-clique. Every subset of this set is a 2-clique, but f1; 2; 3; 4g for instance is not a 2-club. Inclusionwise maximality of any polynomially verifiable hereditary property such as k-clique can be tested in polynomial time. For example, to verify maximality of a k-clique S , it suffices to show that there is no single vertex in V nS that could be added to S to form a larger k-clique. However, for the k-club model, nonexistence of a vertex that could increase the size of the k-club by one is a necessary but not sufficient condition for its maximality. For example, in the graph shown in Fig. 4a, set S1 D f1; 2; 3g is a 2-clique and since sets S2 D S1 [ f4g and S3 D S1 [ f5g are not 2-cliques, it can be concluded that S1 is a maximal 2-clique. In Fig. 4b, S1 is not a maximal 2-clique which can be deduced by observing that both S2 and S3 are also

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Fig. 3 A proper distance-2 5-coloring using colors fa; b; c; d; eg

a

b

Fig. 4 Inclusionwise maximality testing of 2-cliques and 2-clubs

2-cliques. Set S1 in the graph shown in Fig. 4b forms a 2-club, while sets S2 and S3 are not 2-clubs. But S1 is not a maximal 2-club since V D S1 [ f4; 5g is a larger 2-club containing S1 . Since the k-club definition is more restrictive than the k-clique, it is possible that a maximal k-club is not a maximal k-clique. For instance, f1; 2; 3; 4g is a maximal 2-club in Fig. 2, but not a maximal 2-clique as it is contained in 2-clique f1; 2; 3; 4; 5g. By contrast, if a maximal k-clique satisfies the diameter requirement, it is also a maximal k-club. However, it is possible in a graph for k  2 that no maximal k-clique is a k-club. Figure 5 shows a graph with exactly two maximal 2-cliques f1; 2; 3; 4; 5; 6; 7g and f1; 2; 3; 5; 6; 7; 8g, neither of which is a 2-club. Hence, even enumerating all maximal k-cliques in the graph to select the ones satisfying the diameter-k condition may not identify a maximum k-club. Furthermore, in the extreme case, it fails to identify a single k-club.

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Fig. 5 A graph in which every maximal 2-clique is not a 2-club

3.1

Complexity and Approximation

The maximum k-clique and k-club problems are NP-hard for any fixed positive integer k [19, 35], and they remain NP-hard even in graphs of fixed diameter with d i am.G/ > k [19]. Hence, the maximum k-club problem is known to be NP-hard for every fixed k even on graphs of diameter k C 1. For given positive integers k and l (k ¤ l), the problem of recognizing whether ! k .G/ D ! l .G/ is also NP-hard. This gap-recognition complexity result was further used to show that for an integer k  2 and in a graph G with ! k .G/ > .G/ C 1, unless PDNP, there does not exist a polynomial time algorithm for finding a k-club of size strictly larger than .G/ C 1 [40]. Recalling that .G/ denotes the maximum vertex degree in G, it is clear that a vertex of maximum degree along with all its neighbors forms a k-club in G for any k  2. The maximum k-club problem for fixed k  2 was shown to be inapproximable 1 within a factor of n 3  for any  > 0, if NP ¤ ZPP [101], which has been 1 strengthened to n 2  recently under the assumption P ¤ NP [12]. Approximation 1 2 algorithms of factor n 2 and n 3 for even and odd k, respectively, have also been proposed recently [12]. In contrast to what is observed for cliques, an encouraging result for kclubs is that it is fixed-parameter tractable when parameterized by solution size [131, 132]. The k-club problem is one of the few parameterized problems for which nonexistence of a polynomial-size many-to-one kernel and existence of a polynomial-size Turing kernel are known [131, 132]. The k-club problem can be solved on trees and interval graphs in O.nk 2 / and O.n2 /, respectively, and it is polynomial time solvable on graphs with bounded treeor cliquewidth [131]. The 2-club problem can be solved on bipartite graphs in O.n5 / [131].

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Fig. 6 An asymmetric partitionable cycle with respect to nodes 1 and 4

Recently, it has been shown that testing inclusionwise maximality of k-clubs is NP-hard, which is a direct consequence of their nonhereditary property [97]. However, maximality of a k-club can be tested in polynomial time in graphs that do not contain an asymmetric partitionable cycle of a certain size [97]. We provide some details on this result next. A k-clique which induces a connected subgraph is called a connected k-clique. Given a graph G D .V; E/, maximality of a connected k-clique D can be checked by testing nodes in V n D that have at least one edge to some node in D, one at a time, to see if they can be added to D to form a larger k-clique. For graphs in which every connected k-clique is also a k-club, maximality of k-clubs can then be checked in polynomial time. By definition, every k-club is a connected k-clique, and in such graphs every connected k-clique is also a k-club, and hence, a maximal connected k-clique is also a maximal k-club. Definition 18 A graph G D .V; E/ is called a partitionable cycle, if it contains a spanning cycle W and a pair of nonadjacent nodes i and j such that every edge in E n E.W / has one endpoint in AW .i; j / and the other endpoint in AW .j; i /, where AW .i; j / and AW .j; i / are the internal nodes on the two paths between i and j in W . A partitionable cycle is said to be asymmetric if jAW .i; j /j ¤ jAW .j; i /j. (See Fig. 6.) Theorem 12 ([97]) Given a graph G D .V; E/ and fixed integer k  2, every connected k-clique is a k-club if G does not contain an asymmetric partitionable cycle on c vertices as an induced subgraph, where 5  c  2k C 1. Corollary 1 ([97]) In a bipartite graph, every connected 2-clique is a 2-club.

3.1.1 Formulations The maximum k-club problem admits the following integer programming formulation [19, 35]: X !N k .G/ D max xi i 2V

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Subject to X

xi C xj  1 C

yijl

8 .i; j / … E

lWPijl 2Pij

xp  yijl xi 2 f0; 1g yijl 2 f0; 1g

8 p 2 V .Pijl /; Pijl 2 Pij ; .i; j / … E 8i 2V 8 Pijl 2 Pij ; .i; j / … E;

where Pij is an indexed collection of all paths of length at most k between vertices i; j in G and Pijl is the path with index l between vertices i; j . The formulation ensures that if two vertices are in a k-club, then all the vertices in at least one path between them with length less than or equal to k are also included in the k-club. It should be noted that the size of Pij could be very large, making this formulation difficult to handle. But the formulation for the maximum 2-club problem stated next is much more compact. X !N 2 .G/ D max xi i 2V

Subject to xi C xj 

X

xk  1 8 .i; j / … E

k2N.i /\N.j /

xi 2 f0; 1g 8i 2 V: It should also be noted that while the exponential formulation for the maximum k-club problem discussed here is unsuitable for explicit use with integer programming solvers for large k, a more compact formulation has been recently developed in [145] that permits such use. We present this formulation next. Recall that AG D Œai;j nn is the symmetric adjacency matrix of G. We use subscripts of the form .u; v/ where u refers to the position of the term and v refers to the position of the summation. For example, .3; 2/ is the index over which the second summation of the third term runs. The formulation is such that 1  v  u  k  1. Specifically, the term using indices .u; :/ models paths of length u C 1. The formulation requires that whenever vertices i and j are included in a k-club, at least one of these terms is 1, thereby requiring a path between i and j of length at most k as well as the requirement that the vertices on that path are included. The compact 2-club formulation presented above can be viewed as a special case of the formulation presented next. !N k .G/ D max

X i 2V

xi

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subject to n X

ai;j C

ai; .1;1/ a .1;1/;j x .1;1/

.1;1/D1 n X

C

n X

ai; .2;1/ a .2;1/; .2;2/ a .2;2/;j x .2;1/ x .2;2/

.2;1/D1 .2;2/D1

C

n X

n X

n X

ai; .3;1/ a .3;1/; .3;2/ a .3;2/; .3;3/ a .3;3/;j x .3;1/ x .3;2/ x .3;3/

.3;1/D1 .3;2/D1 .3;3/D1

CC n X .k1;1/D1

:::

n X

ai; .k1;1/ a .k1;1/; .k1;2/    a .k1;k1/;j x .k1;1/ x .k1;2/

.k1;k1/D1

   x .k1;k1/  xi C xj  1; 8 i; j 2 V W i < j xi 2 f0; 1g

8i 2 V:

To see the correctness of this formulation, consider x 2 f0; 1gn which is an incidence vector of a k-club in G. Then for a pair i; j such that xi D xj D 1 and i < j , the corresponding constraint requires the left-hand side of the constraint to be at least 1. Since x corresponds to a k-club, if .i; j / 2 E then ai;j D 1 and the constraint is satisfied. Otherwise, there exists a path i D i0  i1      iu D j where 2  u  k such that xis D 1; s D 0; : : : ; u and the term corresponding to u has the product corresponding to .u; s/ D is ; s D 1; : : : ; u  1 equal to 1. Hence, every constraint is satisfied by incidence vectors of k-clubs. Conversely, let x 2 f0; 1gn be a feasible solution to the formulation. Then, it suffices to show that S D fi 2 V W xi D 1g is a k-club. Consider i; j 2 S , i < j . Since x satisfies the corresponding constraint, at least one term in the left-hand side is 1. Suppose ai;j D 0, then for some u between 1 and k  1, a product in term u equals 1. Suppose ai; .u;1/ a .u;1/; .u;2/  : : :  a .u;u1/; .u;u/ a .u;u/;j x .u;1/ x .u;2/ : : : x .u;u/ D 1: Hence, .u; s/ 2 S for s D 1; : : : ; u, and i  .u; 1/  .u; 2/      .u; u  1/  .u; u/  j is a path of length u C 1  k in G. This shows S is a k-club. The approach taken here is to formulate the problem as a nonlinear (binary) integer program. Note that the polynomial products of binary variables can be linearized using standard techniques. The authors [145] show that this formulation can be linearized into a linear integer program which uses O.k n2 / variables and constraints by exploiting the properties of a k-club.

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The authors of [145] also introduce the notion of an r-robust k-club which is a subset of vertices S such that GŒS  has at least r internally vertex disjoint, edge disjoint, or simply distinct paths (differing in at least one edge) of length at most k (in GŒS ) between every pair of vertices. Clearly when k D 2, distinct paths are also edge/vertex disjoint, and the three ideas coincide. Note that a 2-club which is 2-connected is not necessarily a 2-robust 2-club although the converse is true. A cycle on five vertices is a 2-connected 2-club, but it is not a 2-robust 2-club since the pair of paths between any two vertices are not both of length at most two. The formulation for the maximum k-club problem presented above can be extended for the case where r distinct paths of length at most k are required between pairs of vertices. This can be accomplished by replacing the right-hand side with r.xi C xj  1/ instead. Specifically, the r-robust 2-club model allows the detection of diameter-two clusters that are robust, in the sense that they preserve the diametertwo bound under vertex or edge deletions (up to r  1).

3.2

Polyhedral Results

Given a graph G D .V; E/, the convex hull of the incidence vectors of k-clubs in G is called the k-club polytope of G denoted by Ck .G/. Some preliminary results on these polytopes, especially the 2-club polytope, are available from [19, 44]. Theorem 13 [19] presents the basic results on C2 .G/ and the first family of facets developed. It should also be noted that the maximal 2-independent set inequalities are analogous to Padberg’s maximal independent inequalities for the clique polytope [121]. Proposition 6 ([17]) (Distance-k-independent set inequalities) Given a graph G D .V; E/, let I  V be a maximal k-independent set. Then the following inequality is valid for Ck .G/: X xi  1: i 2I

Theorem 13 ([19]) Consider the 2-club polytope C2 .G/ of a graph G D .V; E/. 1. d i m.C2 .G// D n. 2. xi  0 induces a facet of C2 .G/ for every i 2 V . 3. For i 2 V , xi  P1 induces a facet of C2 .G/ if and only if dG .i; j /  2 8 j 2 V . 4. The inequality i 2I xi  1 induces a facet of C2 .G/ if and only if I is a maximal distance-2-independent set in G. Proof The full dimension of the 2-club polytope follows from the fact that ;; fi g are 2-clubs for each i 2 V . The nonnegativity facet also follows from the same observation. The unit upper-bound facet is a special case of a 2-independent set facet that is proved next. Suppose I is amaximal distance-2-independent set. We establish the maximality  P of the face FI D x 2 C2 .G/ W xi D 1 , thereby showing it is a facet. Suppose i 2I

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there exists a valid inequality ˛ T x  ˇ such that, F D fx 2 C2 .G/ W ˛ T x D ˇg FI . Since ei 2 FI  F 8 i 2 I , we have ˛i D ˇ 8 i 2 I . Now for every j 2 V n I , there exists a vertex i 2 I such that at least one of the following two conditions are satisfied: 1. .i; j / 2 E and so ei C ej 2 FI  F . 2. N.i / \ N.j / ¤ ;, that is, they have a common neighbor k 2 V n I in which case ei C ej C ek ; ei C ek 2 FI  F . Now for every j 2 V n I , in the first case we obtain, ˛i C ˛j D ˇ ) ˛j D 0 and in the second case we obtain ˛i C ˛k C ˛j D ˇ and ˛i C ˛k D ˇ ) ˛j D ˛k D 0. Thus, FI D F is a facet. Suppose the inequality induces a facet of C2 .G/. If I is not a 2-independent set, then there exist i; j 2 I such that either .i; j / 2 E or if I is independent, there exists k 2 N.i / \ N.j /. In the first case, ei C ej which is a feasible point violates the inequality. In the second case, ei C ej C ek which is feasible violates the inequality. Either situation contradicts the validity of this inequality. Hence, I must be a 2-independent Pset. If I is not maximal, there exists I [ fug which is 2-independent and xu C xi  1 is valid and dominates a facet-defining inequality, i 2I



which is a contradiction. Hence, I is maximal.

This line of research has been recently furthered in [44]. Theorem 14 provides necessary and sufficient conditions under which the common neighborhood constraint in the formulation induces a facet. Theorem 15 introduces another facet-defining inequality for C2 .G/. Theorem 14 ([44]) Given a graph G D .V; E/, let a; b 2 V , and I  V n fa; bg be such that I [ fag and I [ fbg are distance-2-independent sets in G and d i stG .a; b/ D 2. The following inequality is valid for C2 .G/ and induces a facet if and only if I is a maximal set: X

X

xi 

i 2I [fa;bg

xj  1:

j 2N.a/\N.b/

Theorem 15 ([44]) Given a graph G D .V; E/, let a; b; c 2 V , and I  V n fa; b; cg be such that set fa; b; cg is independent and sets I [fag, I [fbg and I [fcg are distance-2-independent in G. The following inequality is valid for C2 .G/ and induces a facet if and only if I is a maximal set: X i 2I [fa;b;cg

xi 

X

˛j xj  1;

j 2V

where 8 < 2; if j 2 N.a/ \ N.b/ \ N.c/, ˛j D 1; if j 2 ŒN.a/ \ N.b/ [ ŒN.a/ \ N.c/ [ ŒN.b/ \ N.c/ n ŒN.a/ \ N.b/ \ N.c/, : 0; Otherwise:

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Recently in [98], a new family of facets called the independent distance-2dominating set inequalities for the 2-club polytope have been introduced that include all known facets other than nonnegativity constraints as special cases. The k-club polytope for k  3 has fewer associated results, and there is ongoing research on polyhedral approaches to these cases. Alternate formulations and valid inequalities for the maximum 3-club problem were introduced along with computational experiments in [4, 5].

3.3

Algorithms

Combinatorial exact algorithms appear to be more prevalent presently for solving the maximum k-club problem for arbitrary k. The results on the 2-club polytope have also led to preliminary branch-and-cut approaches incorporating maximal distance-2-independent set facets discussed in [17, 19]. A sequential cutting-plane method is also proposed in [44]. Heuristic algorithms for finding large k-cliques or k-clubs in a given graph have been proposed in [34, 44, 66] that can be used to produce lower bounds in some exact approaches like the combinatorial branch-andbound algorithms. Due to the intractability of testing inclusionwise maximality of k-clubs and its nonhereditary nature, well-known exact combinatorial algorithms for cliques ¨ like Carraghan-Pardalos and Osterg˚ ard’s algorithms [43, 120] are unlikely to be applicable to k-clubs although they can naturally be applied to the power graph to find k-cliques. The existing branch-and-bound algorithms from [35,97] are classical partitioning approaches based on variable dichotomy but differ in the approach taken to obtain upper bounds. The approach taken in [35] is to obtain upper bounds at each node of the search tree via the k-clique number. In [97], the authors employ the distance-k coloring problem to obtain upper bounds in the search tree. From the inequality presented in Sect. 3, the former has the potential to produce tighter bounds than the latter approach. However, in the first approach, the maximum k-clique problem (which is also NP-hard) must be solved to optimality to produce a valid bound. Whereas, a heuristic coloring algorithm is sufficient with the second approach to produce potentially weaker upper bounds. Details on these algorithms and available benchmark instances can be found in [35, 97]. A fixed-parameter tractable algorithm is presented in [131] to find a k-club of size c if one exists in G in O..c  2/c  cŠ  c 3 n C nm/ time.

4

Density-Based Models: Densest t-Subgraph and -Quasi-clique

Two types of optimization problems can be developed to identify dense clusters in graphs. In the first type, given a graph G D .V; E/ and a fixed positive integer t  n, the problem is to find a subgraph on t vertices of maximum edge density.

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Since the order of the subgraph is fixed (t), this is equivalent to finding a t-vertex subgraph with maximum number of edges or average degree. In the second type, given a graph G D .V; E/ and a fixed real number  2 Œ0; 1, the problem is to find a subgraph of maximum order with edge density at least  . The first type is analogous to finding a k-plex, k-clique, or a k-club of fixed order t with minimum k. In other words, the most cohesive or tightly knit cluster of a given order t where cluster cohesion is quantified by the parameter k. The second type generalizes the maximum clique problem which is a special case when  D 1 and hence is a clique relaxation in a manner similar to the foregoing models in this chapter.

4.1

The Densest t-Subgraph Problem

The problem of the first type is referred to in the literature by various names such as the t-cluster problem [57], the heaviest (unweighted) t-subgraph problem [92], the densest t-subgraph problem [68], and the maximum edge subgraph problem [10]. The edge-weighted version of this problem has also been studied in [10, 11, 78, 129]. A different but closely related densest subgraph problem is to choose a nonempty subgraph G 0 D .V 0 ; E 0 / of an edge-weighted graph G such that the ratio w.E 0 /=jV 0 j is maximized. The densest subgraph problem can be solved in polynomial time using maximum-flow algorithms [73, 92] (see also [46]).

4.1.1 Complexity and Approximation The densest t-subgraph problem is NP-hard even on bipartite graphs, perfect graphs, chordal graphs [57], and planar graphs [89]. No PTAS exists under the assumption that NP does not have subexponential time algorithms [90]. It is shown to have a PTAS on dense instances (jEj D .n2 /) if t D .n/ in [8]. Other types of approximation algorithms for the problem are also available in the literature. A O.n0:3885 / approximation algorithm is available in [92], and an improved 1 approximation guarantee of O.n 3  / for some  > 0 is presented in [68, 69]. A 1 more recent algorithm [25] provides an approximation guarantee of O.n 4 C / for any  > 0. The approximation guarantee of O.n=t/ from the greedy algorithm where the vertex of minimum degree is recursively deleted until a set of size t remains is also established in [10]. A 3-approximate algorithm is available for the problem on chordal graphs [93] and a 1.5-approximate algorithm is known for proper interval graphs and bipartite permutation graphs (the complexity on these graph classes is unknown) [69, 138]. The complexity threshold of this problem was studied in [11], designated as the t-f .t/-dense subgraph problem which asks if G contains a t-vertex subgraph with at least f .t/ edges.  This is the decision version of the maximum clique problem when f .t/ D 2t which is known to be NP-complete. It is established

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that the problem remains NP-complete when f .t/ D ‚.t 1C / and when f .t/ D mt 2 =n2 .1 C O.n1 // for any constant  such that 0 <  < 1. Semidefinite programming-based approximation algorithms are also available for this problem in [69, 138]. A deterministic approximation algorithm for the problem is also presented in [130] that applies rounding techniques to the optimal solution of a linear programming relaxation of the following binary quadratic program: X max xi xj .ij /2E

Subject to X

xi D t

i 2V

xi 2 f0; 1g

8i 2 V:

4.1.2 Polyhedral Results and Algorithms Different integer programming formulations are presented for this problem in [26], obtained by different linearizations and tightening of the aforementioned binary quadratic program. Polyhedral study of a particular linearization of this formulation undertaken in [32] is presented next. Summarized are some families of valid inequalities, sufficient conditions under which they are facet inducing, and the associated separation problems investigated in [32]. max

X

zij

.ij /2E

Subject to X

xi D t

i 2V

zij  xi

8.ij / 2 E

zij  xj

8.ij / 2 E

xi 2 f0; 1g 8i 2 V zij 2 f0; 1g 8.ij / 2 E: Theorem 16 ([32]) Given G D .V; E/ and a positive integer t  n, let Dt .G/ denote the convex hull of feasible solutions to the above integer program. Then, d i m.Dt .G// D n C m  1. Theorem 17 ([32]) (Generalized neighborhood inequalities) Given G D .V; E/, i 2 V , and a set Ai  V n fi g with jAi j  t  2, the following inequality is valid for Dt .G/:

Graph Theoretic Clique Relaxations and Applications

X j 2Ai

xj C

X

1587

zij  jAi j C .t  jAi j  1/xi :

j 2N.i /nAi

Furthermore, if jN.i / \ Ai j  t and t  n  2, the generalized neighborhood inequality induces a facet of Dt .G/. In [32], the authors also show that the generalized neighborhood inequalities can be separated in polynomial time. Another family of inequalities, similar to the odd-set inequalities for the matching polyhedron [67], were also developed and the associated separation problem was shown to be polynomial time solvable in [32]. Theorem 18 ([32]) (Matching inequalities) Given G D .V; E/, T  V and a maximal matching M of GŒV n T , the following inequality is valid for Dt .G/, if jT j C t is odd: X X jT j C t  1 : xj C zij  2 j 2T .ij /2M

In addition, if t  2jM j C 1  jT j  t  1 and t  n  3, then the matching inequality induces a facet of Dt .G/. Similar results on validity and facet-inducing conditions are developed in [32] for more families such as forest inequalities, tree inequalities, and disjoint matching inequalities. Separating these inequalities is shown to be NP-hard. Computational experiments to study the effectiveness of these inequalities in a branch-and-cut framework are also conducted in [32]. This work appears to be the only polyhedral approach to this problem in the literature presently. In addition to the approximation algorithms discussed in Sect. 4.1.1, a variable neighborhood search [111] metaheuristic is also available for the problem in [36].

4.2

The Maximum -Quasi-clique Problem

The second type of problem designed to identify dense clusters in a graph is called the maximum  -quasi-clique problem studied in [1, 2, 124]. Definition 19 Given a graph G D .V; E/ and a real number 0    1, S  V is a  -quasi-clique1 if the edge density of GŒS  is at least  . The maximum  -quasi-clique problem is to find a maximum cardinality  -quasiclique in G, and this cardinality is the  -quasi-clique number of G, denoted by q ! .G/. This problem includes the maximum clique problem as a special case when  D 1 and is shown to be NP-hard for fixed  2 .0; 1/ in [124]. This result is also extended to show a variant of the model studied in [39] is also NP-hard. It should be noted that a subset S  V such that ı.GŒS/  .jSj  1/ has also been called a -quasi-clique in some literature (see for instance [39, 87, 154]).

1

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Similar to k-clubs, the  -quasi-cliques are nonhereditary. But they are said to be quasi-hereditary, a notion introduced in [124]. A graph property … is said to be quasi-hereditary; if G has property …, then there exists a vertex v 2 V such that G  v also satisfies property …. If G has edge density at least  , deleting a vertex with degree no larger than the average degree (specifically a vertex of minimum degree) will result in a graph with edge density at least  [124]. q Summarized next are bounds on ! .G/ extending results known for !.G/ [6, 115] and its relationship to the clique number developed in [124]. Proposition 7 ([124]) Given a graph G D .V; E/, and 0 <   1 !q .G/



C

p  2 C 8m : 2

Moreover if G is connected, !q .G/ If  > 1 



 C2C

p . C 2/2 C 8.m  n/ : 2

1 !.G/ ,

!q .G/ 

!.G/ : 1  !.G/.1   /

The maximum  -quasi-clique problem can be formulated as the following quadratically constrained binary integer program [124]. Recall that AG D Œai;j nn is the symmetric adjacency matrix of G. Denote by M , a zero-diagonal n  n matrix in which every entry except the diagonal entries are ones. !q .G/ D max

X

xi

i 2V

Subject to x T AG x  x T M x xi 2 f0; 1g8i 2 V: Some preliminary computational experiments are performed in [124] by linearizing this formulation into two different linear mixed-integer programs, solved using a commercial solver. These linearizations are presented next. The following formulation is obtained by a simple linearization that introduces a new variable (and associated constraints) for each binary quadratic term. Note that this formulation has O.n2 / variables and constraints.

Graph Theoretic Clique Relaxations and Applications

!q .G/ D max

n X

1589

xi

i D1

Subject to n n X X

.  aij /zij  0

i D1 j Di C1

zij  xi ; 8j > i D 1; : : : ; n zij  xj ; 8j > i D 1; : : : ; n zij  xi C xj  1; 8j > i D 1; : : : ; n zij  0; 8j > i D 1; : : : ; n xi 2 f0; 1g; 8j > i D 1; : : : ; n: The following formulation is also a binary mixed-integer program based on an alternate linearization scheme which results in a more compact formulation with O.n/ variables and constraints. !q .G/ D max

n X

xi

i D1

Subject to n X

yi  0

i D1

li xi  yi  ui xi ; 8i D 1; : : : ; n xi C

n X

.aij   /xj  ui .1  xi /  yi ; 8i D 1; : : : ; n

j D1

xi C

n X

.aij   /xj  li .1  xi /  yi ; 8i D 1; : : : ; n

j D1

xi 2 f0; 1g; 8i D 1; : : : ; n yi 2 R; 8i D 1; : : : ; n where ui D .1   /

n P j D1

aij and li D  n  1 

n P j D1

! aij

.

The asymptotic behavior of the maximum  -quasi-clique size in uniform random graphs has been studied in [146]. A greedy randomized adaptive search procedure (GRASP) was introduced in [1] to find large  -quasi-cliques in a graph. In [2], the

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authors defined a potential function by introducing the notion of the potential of a vertex or an edge set with respect to a given  -quasi-clique. This potential function was used to develop another GRASP to find maximal  -quasi-cliques in a given graph. A combinatorial branch-and-bound algorithm for the maximum  -quasiclique problem was recently developed in [99]. Clearly, theoretical developments on this model are presently limited even though numerous heuristic approaches have been employed to find quasi-cliques in various applications.

5

Selected Applications

Recent advances in high-throughput data collection in different fields such as Internet research, social network analysis, text analytics, bioinformatics, computational finance, and telecommunication among others have led to an increased demand for effective models and tools for data mining. Data mining is the process of summarizing, visualizing, and processing large data sets in order to extract useful knowledge from the data by using advanced mathematical techniques [139]. In a variety of data mining applications, the elements of a system and the relationships among them are modeled as a graph, wherein graph algorithms and optimization techniques are used to uncover the specific patterns in such data sets [55, 148]. Graph-based data mining is particularly advantageous for studying unstructured or partially structured data compared to methods that almost exclusively deal with numerical data. Cliques and their relaxations have been used to identify “tightly knit” clusters which often provide valuable insights in a variety of different application settings. An Internet graph has vertices representing IP addresses, and edges in such graphs are determined based on information from routing protocols or using traceroute probes [37]. In web mining and Internet research, the web can be modeled by a graph in which each webpage is represented by a vertex and two vertices are adjacent if the corresponding webpages are linked together [140]. The 2-club model has been used for topical clustering of hyperlinked documents [110] and web sites [140] to expedite search procedures. A k-plex-based approach to mine databases is proposed in [154]. In call graphs, vertices represent telephone numbers and an edge represents a call placed from one vertex to another in a specified time interval [1]. A call graph representing telecommunications traffic data over one 20-day period with 290 million vertices and 4 billion edges was mined using quasi-cliques in [1]. Stock-market graphs have vertices representing stocks, and two stocks are connected by an edge if they are positively correlated over some threshold value based on calculations over a period of time in history [28–30]. In [29, 30], the authors studied the US stock market and used clique relaxation models in order to classify this market and find robust diversified portfolios. Social networks are graph models representing sociological information such as friendship or acquaintance among people. Vertices usually represent people, and an

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edge indicates a “tie” between two people. A tie could mean that they know each other, they visited the same place, or any other sociological connection. Scientific collaboration networks with vertices representing authors (in a particular field or with publications in a particular journal) and edges indicating coauthorship fall under this category [42, 77, 82, 85, 135]. In social network analysis, finding cohesive subgroups in contact networks, friendship networks, and collaboration networks are traditional applications that employ cliques and clique relaxations. A special case is criminal network analysis [9, 47, 48, 60, 136], used to detect organized criminal activities such as money laundering [119] and terrorism [108, 109]. Detecting cohesive subgroups in contact networks has also been used for disease contagiousness studies and what-if analysis by healthcare providers [76, 135]. Mining friendship networks and finding dense clusters can provide useful information for advertising and marketing purposes [82, 85, 152]. Biological networks such as protein interaction networks and gene co-expression networks are used to model biological information. A protein interaction network is represented by a graph with the proteins of an organism as vertices, and an edge exists between two vertices if the proteins are known to interact based on twohybrid assays and other biological experiments [72, 86, 137]. In gene co-expression networks, vertices represent genes, and an edge exists between two vertices if the corresponding genes are co-expressed with correlation higher than a specified threshold in microarray experiments [127]. Detecting clusters in protein interaction networks helps identify protein complexes and functional modules that influence cellular functions [137]. Clustering of gene co-expression networks has been used to detect network motifs using cliques, quasi-cliques, and k-plexes in [51, 87, 127, 147]. Protein interaction networks of organisms like Helicobacter pylori and Saccharomyces cerevisiae have been mined using k-clubs [19, 123], quasi-cliques [14, 137], and k-plexes [126]. Additionally, finding cliques in graph models of electroencephalogram time series data has been used to help identify epileptic seizures [128].

6

Conclusion

Graph theoretic clique relaxations provide practical alternatives to cliques for modeling clusters in graph-based data mining applications. The variety of models available make this an attractive approach to consider in many such applications where the best clustering model can only be identified after it has proved itself to be useful in the application context. In massive data sets, the clique relaxations discussed in this chapter allow a level of tolerance to edges missed due to data errors. At the same time, these models provide specific structural guarantees controlled by the user, so that the significance of the cluster under edges included in error is also satisfactory. Hence, clique relaxations, for appropriate values of their respective parameters, provide a hierarchy of models that enable the user to derive valuable insights by detecting large clusters in massive data sets.

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The results presented in this chapter are focused on polyhedral and combinatorial approaches, as well as complexity and approximation results that indicate the limits of solvability. More is needed with regard to results of this nature, as well as practical metaheuristic approaches designed for massive data sets. Studying these models under uncertainty with respect to the edges in the graph is also important. With ongoing active research in this area driven by practical needs and theoretical challenges, interesting developments can be expected in the future. Acknowledgements This work was partially supported by the US Department of Energy Grant DE-SC0002051.

Cross-References Combinatorial Optimization in Data Mining Combinatorial Optimization Techniques for Network-Based Data Mining Optimization Problems in Online Social Networks

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133. S.B. Seidman, B.L. Foster, A graph theoretic generalization of the clique concept. J. Math. Sociol. 6, 139–154 (1978) 134. E.C. Sewell, A branch and bound algorithm for the stability number of a sparse graph. INFORMS J. Comput. 10(4), 438–447 (1998) 135. T. Smieszek, L. Fiebig, R.W. Scholz, Models of epidemics: when contact repetition and clustering should be included. Theor. Biol. Med. Model. 6(1), 11 (2009) 136. M.K. Sparrow, The application of network analysis to criminal intelligence: an assessment of the prospects. Soc. Netw. 13, 251–274 (1991) 137. V. Spirin, L.A. Mirny, Protein complexes and functional modules in molecular networks. Proc. Natl. Acad. Sci. 100(21), 12123–12128 (2003) 138. A. Srivastav, K. Wolf, Finding dense subgraphs with semidefinite programming, in Approximation Algorithms for Combinatiorial Optimization, ed. by K. Jansen, J. Rolim. Lecture Notes in Computer Science, vol. 1444 (Springer, Berlin/Heidelberg, 1998), pp. 181–191. doi: 10.1007/BFb0053974 139. P.-N. Tan, M. Steingach, V. Kumar, Introduction to Data Mining (Addison-Wesley, Boston, 2006) 140. L. Terveen, W. Hill, B. Amento, Constructing, organizing, visualizing collections of topically related, web resources. ACM Trans. Comput. Human Int. 6, 67–94 (1999) 141. E. Tomita, T. Kameda, An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Global Opt. 37(1), 95–111 (2007) 142. L.E. Trotter, A class of facet producing graphs for vertex packing polyhedra. Discret. Math. 12, 373–388 (1975) 143. S. Trukhanov, Novel approaches for solving large-scale optimization problems on graphs. Ph.D. thesis, Texas A&M University, 2008 144. S. Trukhanov, B. Balasundaram, S. Butenko, Exact algorithms for hard node deletion problems arising in network-based data mining (2011, submitted) 145. A. Veremyev, V. Boginski, Identifying large robust network clusters via new compact formulations of maximum k-club problems. Eur. J. Oper. Res. 218(2), 316–326 (2012) 146. A. Veremyev, V. Boginski, P. Krokhmal, D.E. Jeffcoat, Asymptotic behavior and phase transitions for clique relaxations in random graphs, working paper, 2010 147. B.H. Voy, J.A. Scharff, A.D. Perkins, A.M. Saxton, B. Borate, E.J. Chesler, L.K. Branstetter, M.A. Langston, Extracting gene networks for low-dose radiation using graph theoretical algorithms. PLoS Comput Biol. 2(7), e89 (2006) 148. T. Washio, H. Motoda, State of the art of graph-based data mining. SIGKDD Explor. Newsl. 5(1), 59–68 (2003) 149. S. Wasserman, K. Faust, Social Network Analysis (Cambridge University Press, New York, 1994) 150. D. West, Introduction to Graph Theory (Prentice-Hall, Upper Saddle River, 2001) 151. D.R. Wood, An algorithm for finding a maximum clique in a graph. Oper. Res. Lett. 21(5), 211–217 (1997) 152. A.G. Woodside, M.W. DeLozier, Effects of word of mouth advertising on consumer risk taking. J. Advert. 5(4), 12–19 (1976) 153. M. Yannakakis, Node-and edge-deletion NP-complete problems, in STOC ’78: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (ACM, New York, 1978), pp. 253–264 154. Z. Zeng, J. Wang, L. Zhou, G. Karypis, Coherent closed quasi-clique discovery from large dense graph databases, in Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’06 (ACM, New York, 2006), pp. 797–802

Greedy Approximation Algorithms Peng-Jun Wan

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Independence Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Maximum Independent Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Maximum-Weight Independent Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Dependence Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Minimum-Cost Submodular Cover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Minimum CDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Minimum-Weight CDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Graph Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1600 1601 1601 1603 1608 1609 1616 1621 1624 1627 1627 1628

Abstract

Greedy strategy is a simple and natural method in the design of approximation algorithms. This chapter presents greedy approximation algorithms for very broad classes of maximization problems and minimization problems and analyzes their approximation bounds. A number of applications of these greedy approximation algorithms are also described in this chapter. For many applications, the greedy approximation algorithm can achieve the best known approximation bound in a straightforward manner. For some applications, they have different formulations, and the corresponding greedy approximation algorithms have different approximation bounds. In addition, some applications can have better approximation algorithm than the greedy approximation algorithm.

P.-J. Wan Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA e-mail: [email protected] 1599 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 48, © Springer Science+Business Media New York 2013

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Introduction

Greedy strategy is one of the major techniques in the design of approximation algorithms for NP-hard optimization problems. A greedy approximation algorithm is an iterative algorithm which produces a partial solution incrementally. Each iteration makes a locally optimal or suboptimal augmentation to the current partial solution, so that a globally suboptimal solution is reached at the end of the algorithm. This chapter presents a number of classes of optimization problems to which the greedy strategy can be applied to produce good approximate solutions. The following standard terms and notations are adopted throughout this chapter. Consider a ground set E and a real function f defined on 2E . f is increasing if, for any two subsets A and B of E, A  B implies f .A/  f .B/. f is proper if, for any X  E, X f .X /  f .e/ : e2X

f is submodular if, for any two subsets A and B of E, f .A/ C f .B/  f .A [ B/ C f .A \ B/: f is a polymatroid function if f .;/ D 0 and f is submodular and increasing. The marginal value of B  E with respect to A  E is defined by B f .A/ D f .A [ B/  f .A/: Similarly, the marginal value of an element e 2 E with respect to A  E is defined by e f .A/ D f .A [ feg/  f .A/: Then, f is increasing if and only if x f .A/  0 for any A  E and x 2 EnA. f is submodular if and only if for any A  E and different x; y 2 E n A, the inequality x f .A/  x f .A [ fyg/ holds (see, e.g., [28]). In addition, the following are equivalent: • f is increasing and submodular. • For any A; B  E, f .B/  f .A/ 

X

x f .A/ :

x2BnA

• For any A  E and x; y 2 E n A, x f .A/  x f .A [ fyg/ :

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1601

Independence Systems

Consider a finite ground set E and a collection I of subsets of E. .E; I/ is called an independence system if I is hereditary (i.e., for any two subsets I and J of E, I  J 2 I implies I 2 I). Each member of I is called an independent set or a stable set. A subset I of E is said to be a maximal independent set (or base) if I 2 I, but for any e 2 E n I , I C e … I. A subset I of E is said to be a circuit if I … I, but for any e 2 I , I  e 2 I. This section presents greedy approximation algorithms for the following two general maximization problems: • Maximum Independent Set (MIS): given an independent system .E; I/, compute a set I 2 I maximizing jI j. • Maximum-Weight Independent Set (MWIS): given an independent system .E; I/ and an increasing nonnegative weight (or profit) function f on 2E , compute a set I 2 I maximizing f .I /.

2.1

Maximum Independent Set

Consider an independent system .E; I/ and an ordering he1 ; e2 ; : : : ; en i of E; an independent set I can be computed by a first-fit greedy algorithm GreedyMIS described in Table 1. We derive an upper bound on the approximation ratio of the algorithm GreedyMIS in terms of the exchangeability of an ordering of E. Let  be an ordering of E. For a subset X  E and an element e 2 X , X  e (respectively, e  X ) means for each x 2 X , x  e (respectively, e  x). The ordering  is said to be q-exchangeable if the following holds: If X  Y 2 I, X  e  Y n X , and X C e 2 I, then there is a set Z  Y n X such that jZj  q and .Y n Z/ C e 2 I. Theorem 1 The approximation ratio of the algorithm GreedyMIS in a q-exchangeable ordering is at most q. of elements selected by the greedy Proof Let x1 ; x2 ; : : : ; xk be the sequence  ˚ algorithm. Denote I0 D ;, and Ij D x1 ; : : : ; xj for each 1  j  k. Let O be an optimal solution. We construct k subsets C1 ; C2 ; : : : ; Ck of O asˇ described in ˇ Table 2. We will show that fC1 ; C2 ; : : : ; Ck g is a partition of O and ˇCj ˇ  q for each 1  j  k, from which the theorem follows. We first claim that for each 1  j  k, Ij [ Bj 2 I and Ij \ Bj D ;. The claim is proved by induction on j . The claim holds trivially for j D 0. Assume that the claim holds for some j  1 with 1  j < k. We consider two cases: ˚  ˚  Case 1: xj 2 Bj 1 . Then, Cj D xj and Bj D Bj 1 n xj . So     Ij [ Bj D Ij 1 C xj [ Bj 1  xj D Ij 1 [ Bj 1 2 I;     Ij \ Bj D Ij 1 C xj \ Bj 1  xj  Ij 1 \ Bj 1 D ;:

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Table 1 The first-fit selection of a maximal independent set

Greedy MIS I ;I For i D 1 to n do if I C ei 2 I then I Output I .

Table 2 A partition of O

I C ei I

B0 OI for j D 1 to k do ˚  if xj 2 Bj 1 then Cj xj I a smallest subset C else Cj   of Bj 1 such that Ij [ Bj 1 n C 2 I I Bj Bj 1 n Cj ; output fC1 ; C2 ; : : : ; Ck g.

Case 2: xj … Bj 1 . Then,   Ij [ Bj D Ij [ Bj 1 n Cj 2 I;     Ij \ Bj D Ij 1 C xj \ Bj  Ij 1 C xj \ Bj 1 D Ij 1 \ Bj 1 D ;: Therefore, the claim holds for j as well. ˇ ˇ Next, we show that ˇCj ˇ  q for each 1  j  k in two cases: ˇ ˇ Case 1: xj 2 Bj 1 . Then, ˇCj ˇ D 1  q. Case 2: xj … Bj 1 . Since Ij 1 [ Bj 1 2 I, we have Ij 1  xj  Bj 1 by the first-fit greedy principle. Since Ij1 C xj D Ij 2 I and Ij 1 \ Bj 1 D  ;, there is a set C  I [ Bj 1 j 1     n Ij 1 D Bj 1 such thatˇ jCˇ j  q and Ij 1 [ Bj 1 n C C xj D Ij [ Bj 1 n C 2 I. Therefore, ˇCj ˇ  q. Finally, we show that fC1 ; C2 ; : : : ; Ck g is a partition of O. By the first-fit greedy principle, Ik is a maximal independent set. Since Ik [ Bk 2 I and Ik \ Bk D ;, we have Bk D ;. As Bk D O n .C1 [ C2 [ : : : [ Ck / ; fC1 ; C2 ; : : : ; Ck g is a partition of O.



For the application of Theorem 1 to specific problems, the major task is to find an ordering of the ground set with small exchangeability. Two prominent examples are seeking the maximum number of interference-free node transmissions [37, 38] or link transmissions [32, 36] in wireless networks. Certain natural node orderings or link orderings have been shown to have constant-bounded exchangeability.

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2.2

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Maximum-Weight Independent Set

Consider an independence system .E; I/ and a nonnegative weight (or profit) function f on 2E . In the general greedy approximation strategy for this instance of MWIS, each iteration optimally or suboptimally solves the augmentation problem: For any I 2 I, find one e 2 E n I with I C e 2 I that maximizes e f .I /. Let A be a polynomial -approximation algorithm for this augmentation (the case  D 1 means the exact algorithm). Table 3 describes the greedy algorithm GreedyMWIS which utilizes the algorithm A in each iteration. To derive the approximation bound of the algorithm GreedyMWIS, we introduce some relevant terms. Consider a subset X of E. A subset I of X is said to be a maximal independent set (or base) of X if I 2 I, but for any x 2 X n I , I C x … I. Let B .X / denote the set of maximal independent subsets of X . The (upper) rank (or independence number) of X is defined by r C .X / D max fjI j W I 2 B .X /g ; and the lower rank of X is defined by r  .X / D min fjI j W I 2 B .X /g : The rank quotient of .E; I/ is defined by  max

 r C .X / W X  E; ` .X / > 0 : r  .X /

Note that an independence system whose rank quotient is one is termed as a matroid. Theorem 2 Let q be the rank quotient of .E; I/. The approximation ratio of the algorithm GreedyMWIS is at most q C 1 if f is polymatroid and at most q if f is linear. The proof of Theorem 2 makes use of the following exchange property of independent systems. Theorem 3 Suppose that .E; I/ has rank quotient q. Let X and Y be two bases and X n Y D fx1 ; x2 ; : : : ; xk g. Denote X0 D X \ Y and Xi D .X \ Y / [ fx1 ; x2 ; : : : ; xi g for 1  i  k. Then, there is a partition fC1 ; C2 ; : : : ; Ck g of Y n X such that for each 1  i  k, (1) jCi j  q, and (2) for each e 2 Ci , Xi 1 C e 2 I.

Proof We construct k subsets C1 ; C2 ; : : : ; Ck of Y n X as described in Table 4. Clearly, for each 1  i  k, (1) jCi j  q, and (2) for each e 2 Ci , Xi 1 [ feg 2 I. So, we only have to show that fCi W 1  i  kg is a partition of Y nX or equivalently Y0 D ;. We prove a stronger claim that for each 0  i  k, jYi j  i q by induction downward on i . As

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Table 3 A partition of Y nX

Table 4 A partition of Y nX

GreedyMWIS I ;; A EI While A ¤ ; do x the augmentation to I output by AI I I C xI A fe 2 A n I W I C e 2 I g; Output I .

Yk Y n X; for each i D k down to 1 do Bi fe 2 Yi W Xi1 [ feg 2 I g; Ci a subset of Bi of cardinality min fjBi j ; qg; Yi1 Yi n Ci ; output fC1 ; C2 ; : : : ; Ck g.

jYk j D jY j  jX \ Y j  q jX j  jX \ Y j  q .jX j  jX \ Y j/ D kq; the claim holds for i D k. Now, suppose that jYi j  i q for some 1 < i  k. We show that jYi 1 j  .i  1/ q in two cases: Case 1: jBi j  q. Then jCi j D q and jYi 1 j D jYi j  jCi j  .i  1/ q. Case 2: jBi j < q. Then, Ci D Bi and Yi 1 D Yi n Bi . By the definition of Bi , Xi 1 is a base of Xi 1 [ Yi 1 D Xi 1 [ .Yi n Bi / : Hence, jYi 1 j  q jXi 1 j D .i  1/ q.



From Theorem 3, we obtain the following weak exchange property of independent systems. Corollary 1 Suppose that .E; I/ has rank quotient q. Let X D fx1 ; x2 ; : : : ; xk g be a base of E and denote X0 D ; and Xj D fx1 ; : : : ; xi g for 1  i  k. Then, for any base Y of E, there is a partition fC1 ; C2 ; : : : ; Ck g of Y such that for each 1  i  k, (1) jCi j  q (2) Xi 1 \ Ci D ;, and (3) for each e 2 Ci , Xi 1 C e 2 I. Now, we prove Theorem 2. Let x1 ; x2 ; : : : ; xk be the sequence ˚ of elements  selected by the greedy algorithm. Denote I0 D ;, and Ij D x1 ; : : : ; xj for each 1  j  k. Let O be an optimal solution. By Corollary 1,ˇ there ˇ is a partition fC1 ; C2 ; : : : ; Ck g of O such that for each 1  j  k, (1) ˇCj ˇ  q, (2) Ij 1 \ Cj D ;, and (3) for each e 2 Cj , Ij 1 C e 2 I. We first consider the case that f is polymatroid. By the greedy rule, for each e 2 Cj ,     e f Ij 1  xj f Ij 1 ;

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which implies that  ˇ ˇ      Cj f Ij 1  ˇCj ˇ  xj f Ij 1  qxj f Ij 1 : Therefore,

f .O/  f .I / 

k X

Cj f .I / 

j D1

k X

  Cj f Ij 1

j D1

k X    q xj f Ij 1 D qf .I / : j D1

So, f .O/  .q C 1/ f .I / : Next,  we  consider the case that f  is linear. In thiscase, for each e 2 Cj , f .e/  f xj , which implies that f Cj  qf xj . So, f .O/  qf .I / : This completes the proof of Theorem 2. Remark For linear f and q D  D 1, Theorem 2 is the classic result on greedy algorithm for matroid due to Rado [29] and Edmonds [9, 10]. For linear f and  D 1, Theorem 2 was proved by Jenkyns [19] and Korte and Hausmann [22]. For polymatroid f and  D 1, Theorem 2 was proved by Fisher et al. [12]. For polymatroid f and   1, the theorem was proved by Calinescu et al. [4]. A matroid .E; I/ is called a uniform matroid if I D fI  E W jI j  kg for some positive integer k. For uniform matroid .E; I/, the algorithm GreedyMWIS achieves better approximation bound. Theorem 4 If .E; I/ is a uniform matroid and f is a polymatroid function, then the approximation ratio of the algorithm GreedyMWIS is at most 1e11= . The proof of the above theorem utilizes the following algebraic inequality. Lemma 1 Suppose that c1 ; c2 ; : : : ; ck are k positive numbers less than a. Then, for any k nonnegative numbers x1 ; x2 ; : : : ; xk ; 0 k

min @ i D1

i 1 X j D1

1 cj xj C axi A 

Pk 1

i D1 ci xi k Y



i D1

1

cj a



:

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Proof For each 1  i  k, let ci a

i D

k Y  1 j Di C1

1

k Y 

1

i D1

Then

Pk

i D1 i

cj a

cj a

 

:

D 1, and hence, 0 k

min @ i D1

i 1 X

1 cj xj C axi A 

k X

j D1

0 i @

i D1

1 k X a @ i C D j A ci xi D c i i D1 j Di C1 k X

0

i 1 X

1 cj xj C axi A

j D1

Pk 1

i D1 ci xi k Y

 1

i D1

cj a



:



So, the lemma follows.

Now, we prove Theorem 4. Let x1 ; x2 ; : : : ; xk be the sequence of elements selected by the greedy algorithm. Then, jOj D k. By the greedy rule, for each 1  i  k and each y 2 O n Ii 1 , yf .Ii 1 /  xi f .Ii 1 / ; and consequently,

f .O/  f .Ii 1 / C

X

yf .Ii 1 /

y2OnIi 1

 f .Ii 1 / C jO n Ii 1 j xi f .Ii 1 / 

i 1 X

  xj f Ij 1 C kxi f .Ii 1 / :

j D1

By Lemma 1, we have 0 k

f .O/  min @ i D1

i 1 X j D1





1

xj f Ij 1 C kxi f .Ii 1 /A

Greedy Approximation Algorithms

Pk 



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j D1 xj f

 1 1



 Ij 1 D k

1 k

f .Ik /  k 1 1  1  k

f .Ik / : 1  e 1=

This completes the proof of Theorem 4. Remark Maximizing submodular functions subject to a uniform matroid constraint appears to have come from an application to a facility location problem in the work of Cornuejols et al. [6]. In [26], Fisher et al. formally proved the special case of Theorem 4 in which  D 1. For this special case, Nemhauser and Wolsey [27] further showed that any algorithm with better approximation bound requires an exponential number of queries of the values of f , and Feige [11] showed that 1 is the best approximation ratio unless P = NP. For   1, Wan and Liu [35] 1e1 obtained the same approximation bound as the one in Theorem 4 of the greedy approximation algorithm for throughput maximization in wavelength-routed optical networks, and Hochbaum and Pathria [18] also derived the same approximation bound of the greedy approximation algorithm for maximum k-coverage. For the application of Theorem 2 to specific problems, the main task is to estimate the rank quotient of the underlying independence system. Theorem 1 can often be applied for this purpose. An independence system .E; I/ is said to be qexchangeable if every ordering of E is q-exchangeable. By Theorem 1, the rank quotient of any q-exchangeable system is at most q. It is easy to verify that: • .E; I/ is q-exchangeable if and only if for any X  Y 2 I and e 2 E with X C e 2 I, there is a set Z  Y n X such that jZj  q and .Y n Z/ C e 2 I. • .E; I/ is q-exchangeable if for any I 2 I and e 2 E, I C e contains at most q circuits. • .E; I/ is q-exchangeable if it is the intersection of q matroids. Below is the list of instances of MWIS with linear weight function and non-matroid independent systems. Maximum-Weight Hamiltonian Cycle: Given an edge-weighted complete graph, find a Hamiltonian cycle with maximum total weight. • Ground set: the set of edges • Independent sets: Hamiltonian cycles or vertex-disjoint paths • Rank quotient: 2 Maximum-Weight Directed Hamiltonian Cycle: Given an edge-weighted complete digraph, find a Hamiltonian cycle with maximum total weight. • Ground set: the set of arcs • Independent sets: directed Hamiltonian cycles or vertex-disjoint paths. • Rank quotient: 3

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Maximum-Weight Stable Set of Unit-Disk Graphs: Given a vertex-weighted unit-disk graph, find a stable set with maximum total weight. • Ground set: the set of vertices. • Independent sets: stable sets. An independent set or stable set of a graph is a set of vertices in the graph, no two of which are adjacent. • Rank quotient: 5. A number of applications of Theorem 4 were surveyed by Goundan and Schulz [15].

3

Dependence Systems

Consider a finite ground set E and a collection C of subsets of E. .E; D/ is called a dependence system if for any two subsets X and Y of E, X  Y and X 2 D implies Y 2 D. Each member of D is called a dependent set. A subset X of E is said to be a minimal dependent set if X 2 D, but for any e 2 X , X  e … D. Given an dependent system .E; I/ and an increasing and proper nonnegative cost function c on 2E , the problem minX 2D c .X / is referred to as Minimum-Cost Dependent Set. For many instances of Minimum-Cost Dependent Set, the dependence system .E; D/ is implicitly defined by an increasing coverage function f on 2E : For each X  E, X 2 D if and only if f .X / D f .E/, or it is implicitly defined by a decreasing coverage deficiency function (also called potential function) f on 2E ; for each X  E, X 2 D if and only if f .X / D 0. If the coverage function is a polymatroid function, the problem Minimum-Cost Dependent Set is commonly known as the Minimum-Cost Submodular Cover, and each dependent set is also called as a submodular cover. Minimum-Cost Submodular Cover is well behaved. In Sect. 3.1, we present a general greedy approximation algorithm for Minimum-Cost Submodular Cover and its approximation bound. For other instances of Minimum-Cost Dependent Set which cannot be cast as Minimum-Cost Submodular Cover, they lack such unified framework for both design and analysis of greedy approximation algorithms. In Sects. 3.2 and 3.3, we present greedy approximation algorithms for Minimum Connected Dominating Set (MCDS) and Minimum-Weight Connected Dominating Set (MWCDS), respectively. A connected dominating set (CDS) of a connected graph G D .V; E/ is a subset of vertices U such that U is a dominating set (i.e., each vertex v 2 V n U is adjacent to at least one vertex in U ) and the induced subgraph G ŒU  is connected. The problem MCDS seeks a CDS of the smallest size in a given graph G, and the problem MWCDS seeks a CDS of the smallest weight in a given vertex-weighted graph G. They are selected as representative examples for the following reasons: • MCDS is associated with a coverage function which is not submodular but has a special shifted submodularity. Such shifted submodularity can still be exploited to design and analyze the greedy approximation algorithm in the manner similar to those for Minimum-Cost Submodular Cover.

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• The naive extension of the greedy approximation algorithm for MCDS to MWCDS has a very poor approximation ratio. • The design and analysis of the greedy approximation algorithm for MWCDS can be slightly modified to some other well-known minimization problems, which are listed in Sect. 3.3.

3.1

Minimum-Cost Submodular Cover

Consider an instance of Minimum-Cost Submodular Cover specified by the ground set E, the polymatroid coverage function f; and the increasing and proper cost function c. In the general greedy approximation strategy for this instance, each iteration optimally or suboptimally solves the augmentation problem: For any given X  E, find one e 2 E nX that maximizes e f .X / =c .e/. Let A be a polynomial -approximation algorithm for this augmentation (the case  D 1 means the exact algorithm). Table 5 describes the greedy algorithm GreedyMCSC which utilizes the algorithm A in each iteration. Let x1 ; x2 ; : : : ; xk be the sequence of elements selected by the algorithm GreedyMCSC. Denote X0 D ;, and Xi D fx1 ; x2 ; : : : ; xi g for each 1  i  k. i/ , which is referred to the price of xi . Let opt For each 1  i  k, let pi D x c.x i f .Xi 1 / be the cost of an optimal solution. When f is integer-valued, c is linear, and A is optimal, it is well known that c .Xk / =opt  H .q/, where q D maxe2E f .e/ and H.q/ D 1 C

1 1 C ::: C 2 q

is the kth Harmonic number. For the general case, we define the curvature to be  ˚P min e2S c .e/ W S is a cover : D opt Note that  D 1 if c is linear. We have the following two general performance bounds. Theorem 5 If f maxe2E f .e/.

is integer-valued, then c .Xk / =opt  H .q/, where q D

Theorem 6 If f .E/  opt and pi  1 for each 1  i  k, then c .Xk / =opt  .E/ 1 C  ln fopt . The proof of both theorems utilizes the following algebraic inequalities. Suppose that `0 ; `1 ; : : : ; `k1 are decreasing positive numbers. Then, k1 X `i 1  `i i D1

`i 1

 ln

`0 : `k1

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Table 5 Greedy algorithm for minimum submodular cover

GreedyMCSC X ;I While f .X/ < f .E/ do x the augmentation to X output by AI X X C x; Output X.

If all of them are integers, then k1 X `i 1  `i i D1

`i 1

 H .`0 /  H .`k1 /  ln

`0 : `k1

We prove Theorem 5 by a charging argument. Let O D fy1 ; y2 ; : : : ; ym g be a cover satisfying that m X   c yj    opt: j D1

 ˚ Denote O0 D ;, and Oj D y1 ; y2 ; : : : ; yj for each 1  j  m. Each xi charges on yj with      pi yj f Xi 1 [ Oj 1  yj f Xi [ Oj 1 : Then, the total charge by xi on O is exactly c .xi / as pi

m X      yj f Xi 1 [ Oj 1  yj f Xi [ Oj 1 j D1

D pi Œ.f .Xi 1 [ O/  f .Xi 1 //  .f .Xi [ O/  f .Xi // D pi Œf .Xi /  f .Xi 1 / D pi xi f .Xi 1 / D c .xi / : P Thus, the total charge on O is kiD1 c .xi /. On  the other hand, we claim that the . Indeed, let l be the first i such that total charge on each y is at most H .q/ c y j j   yj f Xi [ Oj 1 D 0. Then, the charge on yj by each xi with i  l is 0. For   each 1  i  l, yj f Xi 1 [ Oj 1 > 0 which implies that yj … Xi 1 [ Oj 1 , and hence by the greedy rule, we have     c yj c yj c .xi / :    pi D xi f .Xi 1 / yj f .Xi 1 / yj f Xi 1 [ Oj 1

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Thus, l X

     pi yj f Xi 1 [ Oj 1  yj f Xi [ Oj 1

i D1

    l X yj f Xi 1 [ Oj 1  yj f Xi [ Oj 1    c yj yj f Xi 1 [ Oj 1 i D1           c yj H yj f Oj 1  c yj H yj f .;/    H .q/ c yj : 

So, our claim holds. As m X

m X     H .q/ c yj D H .q/ c yj

j D1

j D1

 H .q/   c .O/ D H .q/  opt: The total charge on O is at most H .q/  opt. Therefore,

c .X / 

k X

c .xi /  H .q/  opt:

i D1

This completes the proof for Theorem 5. Next, we prove Theorem 6. Let O be a cover satisfying that X

c .y/    opt:

y2O

For each 0  i  k, let `i D f .E/  f .Xi / be the “coverage deficiency” at the end of iteration i . Let t be the subscript satisfying that `t  opt > `t C1 . We claim that for each 1  i  k,   opt pi  min  ;1 : `i 1 Indeed, the inequality pi  1 holds by assumption. By the greedy rule, we have y f .Xi 1 / 1 1 xi f .Xi 1 /  max D pi c .xi /  y2OŸXi 1 c .y/ P 1 y2OŸXi 1 y f .Xi 1 / P   y2OŸXi 1 c .y/

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1 f .O/  f .Xi 1 / P  y2O c .y/



1 `i 1 `i 1 D :    opt   opt

So, our claim holds. Thus, for each 1  i  k, c .xi / D pi xi f .Xi 1 / D pi .`i 1  `i /   opt ; 1  .`i 1  `i /  min  `i 1   `i 1  `i D min   opt ; `i 1  `i : `i 1 Hence, t X

`t  opt c .xt C1 / `t  `t C1

c .xi / C

i D1

   opt

t X `i 1  `i i D1

`i 1

   opt ln

`0 opt

D opt   ln

f .E/ ; opt

`t  opt C `t

!

and k k X X opt  `t C1 c .xt C1 / C c .xi /  .opt  `t C1 / C .`i 1  `i / D opt: `t  `t C1 i Dt C2 i Dt C2

Summing up the above two inequalities, we have

c .X / 

f .E/ opt: c .xi /  1 C  ln opt i D1

k X

This completes the proof for Theorem 6. Remark For linear c and  D 1, Theorem 5 was proved by Wolsey [39]. For submodular c and  D 1, Theorems 5 and 6 were proved by Wan et al. [34].

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For many specific Minimum-Cost Submodular Cover problems, the coverage function f is not given explicitly. For these problems, the main task is to find the proper submodular coverage function. For some Minimum-Cost Submodular Cover problems, such as set cover, the submodular potential functions can be easily defined. But for some others, it is far from trivial to find the submodular potential function. A prominent example is Weighted Connected Vertex Cover (WCVC). An instance of WCVC is a connected vertex-weighted graph G D .V; E/. A solution to WCVC is a connected vertex cover, which is a subset of vertices inducing a connected subgraph and covering all edges. The objective is to find a connected vertex-cover with the minimum total weight. While the unweighted variant of WCVC has a simple 2-approximation algorithm [13], WCVC has the same approximation hardness as weighted set cover [13]. It can be formulated as a special case of Minimum-Cost Submodular Cover in which the ground set is V and the coverage of a subset U  V is the number of edges in G incident to U minus the number of connected components of the subgraph of G induced by U [7]. By Theorem 5, the approximation ratio of greedy approximation algorithm is at most H .  1/. For some specific Minimum-Cost Submodular Cover problems, they may have different formulations which lead to different approximation bounds. A prominent example is Minimum-Power Spanning Tree. Let G D .V; EI c/ be an edge-weighted graph with c .e/ > 0 for each e 2 E. Any subgraph H induces a power assignment pH W V .H / ! RC defined by pH .u/ D max c .e/ ; 8u 2 V .H / I e2ıH .u/

P the power of H is defined by u2V .H / pH .u/. The problem of finding a spanning tree of minimum power is referred to as Minimum-Power Spanning Tree and may have two different formulations into Minimum-Cost Submodular Cover. In the first formulation [20], the ground set is the set E of edges; for any subset any A  E, its coverage is the rank of the graphic matroid on .V; A/, and its cost is the power of .V; F /. For this formulation, q D  D 1 and the curvature is at most 2. So the approximation ratio of the greedy algorithm is at most 2. In the second formulation [1], the ground set consists of all k-restricted trees (i.e., subtrees of G containing at most k vertices) for some fixed integer k  2. Let mst .G/ denote the weight of a minimum-weight spanning tree of G. For any subgraph H of G, let mst .G W H / denote the weight of the minimum-weight spanning forest F such that H [ F is connected. For any subset of k-restricted trees whose union is H , its coverage is defined to be 2 Œmst .G/  mst .G W H /, and its cost is the power of H . It was proved in [24] that the curvature is at most k  1 C

1 : blog kc  1 C k=2blog kc

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By Theorem 6, the approximation ratio of the greedy algorithm is at most 1C 1C

1 blog kc  1 C k=2blog kc

ln 2:

When k tends to infinity, the above approximation bound approaches to 1 C ln 2, which is better than the approximation bound 2 obtained in the first formulation. The remaining of the subsection provides the formulations of some well-known optimization problems as instances of Minimum-Cost Submodular Cover. For all of them,  D 1. Below is the list of problems with linear cost. Generalized Set Multi-cover [14]: Given a matrix A 2 Nmn and two vectors b 2 Nm1 , compute 8 9 n 1. For each 0  i < k, let `i D n1f .Bi /c , which is referred to as the shifted coverage deficiency. Then n  1  c D `0 > `1 > : : : > `k1  c C 2: By Lemma 3, for each 0  i < k, `i 1  `i 

n  1  f .Bi 1 / `i 1 1 D ; c c

and hence, 1 `i 1  `i  : `i 1 c Therefore, k  1 X `i 1  `i `0 n  1  c :   ln  ln c ` ` c C 2 i 1 k1 i D1 k1

Since c 

n2 , 1

we have n  2  .  1/ c , and hence, n  1  c  .  1/ c C 1  c D .  2/ c C 1 D .  2/ .c C 2/  .2  5/  .  2/ .c C 2/  1 < .  2/ .c C 2/ :

Thus, k1 < ln .  2/ c which implies k  1  c ln .  2/ : This completes the proof of the theorem.



The approximation bound 2 C ln .  2/ in Theorem 7 is slightly better than the approximation bound 2 C ln  derived in [30]. Du et al. [7] presented a generalization of GreedyCDS which chooses at each iteration a set of up to 2k  1 vertices with the maximum gain-cost ratio for some fixed positive parameter k.   integer The approximation ratio of this generalization is at most 1 C k1 .1 C ln .  1//. This implies that for any " > 0, there is a .1 C "/ .1 C ln .  1//-approximation

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algorithm for MCDS. An interesting observation is that for greedy approximation algorithms with submodular coverage functions, the above generalization cannot lead to better performance ratio.

3.3

Minimum-Weight CDS

The algorithm GreedyCDS for MCDS augments the partial solution by a single vertex in each iteration. While it can be modified for MWCDS, the proof of Theorem 7 cannot be extended to prove a logarithmic approximation bound in the weighted case. This subsection presents a different greedy approximation algorithm [16] for MWCDS, which can achieve the logarithmic approximation bound. The main difference is that a group of nodes are selected to augment the partial solution in each iteration. This technique is adapted from the greedy approximation algorithm [21] for Minimum Node-Weighted Steiner Tree. We begin with the definition of the potential (or coverage deficiency) function. Let B be a set of nodes in V . A piece with respect to B is either a vertex not in N ŒB or a connected component of G ŒB. Let f .B/ denote the number of pieces with respect to B. Then, B is a CDS if and only if f .B/ D 1. Let D be the arc-weighted digraph obtained from G by replacing each edge uv in G with two arcs .u; v/ and .v; u/ of weights c .u/ and c .v/, respectively. The greedy algorithm maintains a set B of black nodes, which is initially empty. Repeat the following iteration while f .B/ > 1. Choose an arborescence T in D on some greedy manner, add all internal nodes of T to B, and then update f .B/. When f .B/ D 1, B is output as the CDS. The key ingredient of this algorithm is the greedy strategy used by each iteration to select a proper arborescence T . The detail of this greedy strategy is presented below. For any arborescence T in D, I .T / denotes the set of all internal nodes in T . Fix a subset B of V with f .B/ > 1. The head of a piece with respect to B is defined to be the vertex in this piece with the smallest ID. The price of an arborescence T in D with respect to B is defined to be the ratio of c .I .T // to the number of heads with respect to B contained in T . Ideally, one would greedily wish to use a cheapest (i.e., least priced) arborescence in each iteration. While a cheapest arborescence can be computed easily when f .B/ is small, it is hard to be computed in general. In general, a cheapest arborescence is selected among a polynomial number of specially defined candidates. Let T .B/ denote the set of candidates. The construction of T .B/ is described below. For each vertex u, let Au be the shortest-path spanning arborescence in D rooted at u. Suppose that B is a set of black nodes with f .B/ > 1. Sort all the f .B/ heads in the increasing distance from u in Au . For each 2  j  f .B/, Tj .B; u/ is the minimal arborescence in Au spanning u and the first j heads. Then, T .B/ consists of .f .B/  1/ n candidates: ˚  T .B/ D Tj .B; u/ W u 2 V; 2  j  f .B/ : The greedy algorithm GreedyWCDS is described in Table 7. Next, we derive an approximation bound of GreedyWCDS.

1622 Table 7 Outline of the algorithm GreedyWCDS

P.-J. Wan

GreedyWCDS B ;; while f .B/ > 1, construct T .B/; find a cheapest T 2 T .B/ I B B [ I .T /; output B:

Theorem 8 The approximation ratio of GreedyWCDS is at most 2 .H .n/  1/. To prove the above theorem, we first give a lower bound on the “gain” of each candidate arborescence in T .B/ in terms of the reduction on the number of pieces. Lemma 4 Suppose that f .B/ > 1. Then for any T 2 T .B/ containing h heads with respect to B, f .B/  f .B [ I .T //  h=2: Proof After the addition of I .T /, all pieces with respect to B containing a head in T (and possibly some other pieces with respect to B) get merged into some new (and larger) piece with respect to B [ I .T /. Thus, f .B/  f .B [ I .T //  h  1  h=2: So, the lemma holds.



The next lemma presents an upper bound on the price of the cheapest candidate in T .B/. Lemma 5 Suppose that f .B/ > 1. Then the price of the cheapest tree in T .B/ is at most fopt .B/ . Proof We prove the lemma by a decomposition argument. A spider is an arborescence in D in which all nodes except the root have out-degree at most 2. Initialize T  to be a spanning arborescence of D whose internal nodes form an optimal solution and j to be 0. Repeat the following iteration while T  contains at least two heads. Increment j by one. Repeatedly prune the sinks of T  which are not heads until all sinks are heads. Choose a deepest vertex u in T  such that the maximal u-arborescence contained in T  (which is the subgraph of T  induced by u and all its descendants in T  ) contains at least two heads. Set Sj to be the maximal u-arborescence contained in T  . By the maximum depth of u, every child arborescence of Sj has exactly one head, and hence, Sj is a spider. Delete Sj from T  . If T  still contains one head, then augment the last Sj by the path from the root of Sj to this head.

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Let S1 ; S2 ; : : : ; Sq be the sequence of spiders obtained by this construction. Then, they are vertex disjoint. In addition, their union contains all heads. For each 1  j  q, let hj denote the number of heads contained in Sj . Then, q X

hj D f .B/ ;

j D1

and

q X    c I Sj  opt: j D1

Thus,

   Pq    c I Sj opt j D1 c I Sj Pq min :   1j q hj f .B/ j D1 hj

Let S be the cheapest one among S1 ; S2 ; : : : ; Sq . Then, the price of S is at most opt . f .B/ Let u be the root of S . We show that T` .B; u/ is at least as cheaper as S , from which the lemma follows. Let v1 ; v2 ; : : : ; v` be the heads in S and v01 ; v02 ; : : : ; v0` be the heads in T` .B; u/. By treating S as a subgraph of D, we have c .I .T` .B; u/// C .`  1/ c .u/ 

` X

` ` X   X d i stD u; v0i  d i stD .u; vi /  d i stS .u; vi /

i D1

i D1

i D1

D c .I .S // C .`  1/ c .u/ : So, c .I .T` .B; u///  c .I .S //. As T` .B; u/ and S have the same number of heads, T` .B; u/ is at least as cheaper as S .  Finally, we prove Theorem 8 by applying Lemmas 4 and 5. Suppose that the algorithm runs in k iterations. Let h0 D n which is the number of initial pieces. For any 1  i  k, let Ti be the arborescence selected at iteration i and let `i be the number of pieces just after iteration i . Let bi be the cost of Ti and hi be the number of heads in Ti . Then by Lemma 4, for each 1  i  k, `i 1  `i  hi =2; and hence, hi  2 .`i 1  `i / : By Lemma 5, for each 1  i  k,

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P.-J. Wan

bi opt  hi `i 1 and consequently bi 

hi `i 1  `i opt  2opt : `i 1 `i 1

So, k X

bi  2opt

i D1

k X `i 1  `i

`i 1

i D1

 2opt

`i 1 k X X 1 j i D1 j D`i C1

D 2opt

`0 X j D`k C1

1 j

D 2 .H .`0 /  H .`k // opt D 2 .H .n/  1/ opt: This completes the proof of Theorem 8. By choosing more sophisticated pool T .B/ of candidates as introduced in [16], better approximation bounds 1:5 ln n and .1:35 C "/ ln n for arbitrary constant " > 0 can be achieved. The same algorithmic techniques have been applied to other problems including Minimum Node-Weighted Steiner Tree [16], Minimum-Power Spanning Arborescence [5], and Minimum Node-Weighted Strongly Connected Dominating Set [23].

4

Graph Coloring

Let G D .V; E/ be an undirected graph. A (vertex) coloring of G is an assignment of colors to V satisfying that adjacent vertices are assigned with distinct colors. Equivalently, a vertex coloring corresponds to a partition of V into stable sets. The chromatic number of G, denoted by  .G/, is the smallest number of colors required by any vertex coloring of G. The problem Minimum Vertex Coloring seeks a vertex coloring of a given graph with the smallest colors. It is NP-hard in general. It can be cast as Minimum-Cost Submodular Cover with the following formulation: The ground set is the collection I of all stable sets of G. For a subcollection of k stable sets whose union is U , its coverage is jU j and its cost is k. For this formulation, the curvature is one, and the augmentation problem is Maximum Stable Set in graphs. Suppose that there is a -approximation algorithm A for such Maximum Independent Set problem. The adaptation of the general greedy algorithm

Greedy Approximation Algorithms Table 8 Greedy link scheduling

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GreedyColoring U V; i 0; While U ¤ ; do i i C1 I the stable set of G ŒU  output by AI assign the i th color to all vertices in I ; U U n I.

GreedyMCSC for Minimum Vertex Coloring, called GreedyColoring, is described in Table 8. By Theorem 5, the approximation ratio of GreedyColoring is at most H .˛/, where ˛ is the independence number of G. This section presents yet another greedy algorithm of first-fit type which can outperform GreedyColoring in many graph classes. We introduce some standard notations and terms. Let G D .V; E/ be an undirected graph. For any vertex v 2 V , degG .v/ denotes the degree of v in G, and NG .v/ denotes the set of neighbors of v in G. The minimum degree of G is denoted by ı .G/. Consider a vertex ordering  of V . For each v 2 V , NG .v/ denotes the set of neighbors of v preceding v in the ordering , and NG Œv denotes fvg [ NG .v/. For any subset U of V , G ŒU  denotes the subgraph of G induced by U . Suppose that D D .V; A/ is digraph. For out each v 2 V , degin D .v/ (respectively, degD .v/) denotes the in-degree (respectively, in out-degree) of v in D, and ND .v/ (respectively, NDout .v/) denotes the set of inneighbors (respectively, out-neighbors) of v in D. For any subset U of V , D ŒU  denotes the subgraph of D induced by U . Given a vertex ordering  of V given by hv1 ; v2 ; : : : ; vn i, a coloring of V with colors represented by natural numbers can be produced in the following first-fit manner: Assign the color 1 to v1 . For i D 2 up to n, assign to vi with the smallest color which is not used by any vertex in N  .vi / : Such coloring of V is referred to as the first-fit coloring in the ordering . It is easy to verify that the number of colors used by the first-fit coloring in the ordering hv1 ; v2 ; : : : ; vn i is at most 1 C maxv2V jN  .v/j. The value maxv2V jN  .v/j is referred to as the inductivity of the ordering . The smallest inductivity of all vertex orderings is referred to as the inductivity of G and is denoted by ı  .G/. It was shown in [25] that ı  .G/ D max ı .G ŒU / ; U V

and it can be achieved by a special vertex ordering known as smallest-degree-last ordering, which can be produced iteratively as follows: Initialize H to G. For i D n down to 1, let vi be a vertex of the smallest degree in H and delete vi from H . Then the ordering hv1 ; v2 ; : : : ; vn i is a smallest-degree-last ordering. In the next, we derive the approximation bound of the first-fit coloring in the smallest-degree-last ordering. For any vertex ordering  of V , its backward local independence number (BLIN) is defined to be the parameter

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ˇ ˇ max max ˇI \ NG .v/ˇ : v2V I 2I

The backward local independence number (BLIN) of G, denoted by ˛  .G/, is the smallest BLIN of all vertex orderings of G. An orientation of G is a digraph obtained from G by imposing an orientation on each edge of G. For any orientation D of G, its inward local independence number (ILIN) is defined to be the parameter ˇ ˇ max max ˇI \ NDin .v/ˇ : v2V I 2I

The inward local independence number (ILIN) of G, denoted by ˇ  .G/, is the smallest ILIN of all orientations of G. The following theorem was proved in [32,37]. Theorem 9 The first-fit coloring of G in the smallest-degree-last ordering has approximation bound min f˛  .G/ ; 2ˇ  .G/g. Proof Since the first-fit coloring of G in the smallest-degree-last ordering uses at most 1 C ı  .G/ colors, it is sufficient to show that 1 C ı  .G/   .G/  min f˛  .G/ ; 2ˇ  .G/g : Consider an optimal ˚  coloring which partitions V into  .G/ stable sets Ij W 1  j   .G/ . Let  be a vertex ordering of G whose BLIN is ˛  .G/. Then, for any vertex v 2 V, .G/ Xˇ ˇ ˇN  .v/ \ Ij ˇ  . .G/  1/ ˛  .G/ : jN  .v/j D j D1

Hence, the of the ordering  is at most ˛  .G/ . .G/  1/. This implies that ı  .G/  . .G/  1/ ˛  .G/ : Therefore, 1 C ı  .G/  1 C . .G/  1/ ˛  .G/   .G/  ˛  .G/ : Let D be an orientation of G whose ILIN is ˇ  .G/ and U be the subset of V such that ı  .G/ D ı .G ŒU /. Then, there exists at least one vertex u 2 U such that out degin DŒU  .u/  degDŒU  .u/ :

Thus, out ı  .G/ D ı .G ŒU /  degGŒU  .u/ D degin DŒU  .u/ C degDŒU  .u/

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Xˇ ˇ ˇN in .u/ \ Ij ˇ

.G/ in  2 degin DŒU  .u/  2 degD .u/ D 2

D

j D1

 2 . .G/  1/ ˇ  .G/ : So,

1 C ı  .G/  1 C 2 . .G/  1/ ˇ  .G/  2 .G/  ˇ  .G/ :

This completes the proof of the theorem.



Theorem 9 has found many applications in the wireless schedulings of node transmissions [37, 38] and link transmissions [32, 36]. These wireless scheduling problems can be formulated as the minimum graph coloring of some conflict graphs, which have constant-bounded BLINs and/or constant-bounded ILINs.

5

Conclusion

This chapter presents a number of general greedy approximation algorithms for a broad class of optimization problems. For many specific problems, a straightforward greedy approximation algorithm can achieve the best known approximation bound. But for some problems, they may have different formulations, and the corresponding greedy approximation algorithms have different approximation bounds as illustrated in Sect. 3.1. Furthermore, some problems have approximation algorithms with better approximation bounds than that of the greedy approximation algorithm. Below are some examples of these problems: • The problem of maximizing the sum of weighted rank   functions of matroids over an arbitrary matroid constraint has a 1= 1  e 1 -approximation algorithm [4], while the greedy approximation algorithm has an approximation ratio 2 by Theorem 2. • The problem Maximum-Weight Stable Set of Unit-Disk Graphs described in Sect. 2.2 admits a polynomial-time approximation scheme [17, 31], while the greedy approximation algorithm has an approximation ratio 5. • The two problems Minimum-Weight Vertex Cover and Minimum-Weight Feedback Vertex Set both have 2-approximation algorithms [2], (Bafna et al., 1994, Approximating feedback vertex set for undirected graphs within ratio 2, unpublished), while the greedy approximation algorithms are logarithmic approximation ratios.

Cross-References A Unified Approach for Domination Problems on Different Network Topologies Algorithmic Aspects of Domination in Graphs Connected Dominating Set in Wireless Networks

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Energy Efficiency in Wireless Networks On Coloring Problems Variations of Dominating Set Problem

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24. M. Li, S. Huang, X. Sun, X. Huang, Performance evaluation for energy efficient topologic control in ad hoc wireless networks. Theor. Comput. Sci. 326(1–3), 399–408 (2004) 25. D.W. Matula, L.L. Beck, Smallest-last ordering and clustering and graph coloring algorithms. J. ACM 30(3), 417–427 (1983) 26. G.L. Nemhauser, L.A. Wolsey, M.L. Fisher, An analysis of approximations for maximizing submodular set functions - I. Math. Prog. 14, 265–294 (1978) 27. G.L. Nemhauser, L.A. Wolsey, Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res. 3(3), 177–188 (1978) 28. G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization (Wiley, New York, 1988) 29. R. Rado, A theorem on independence relations. Q. J. Math. 13, 83–89 (1942) 30. L. Ruan, H. Du, X. Jia, W. Wu, Y. Li, K. Ko, A Greedy approximation for minimum connected dominating sets. Theor. Comput. Sci. 329, 325–330 (2004) 31. E.J. van Leeuwen, Better Approximation schemes for disk graphs. Proc. SWAT Springer Lect. Note Comput. Sci. 4059, 316–327 (2006) 32. P.-J. Wan, Multiflows in multihop wireless networks. in Proceedings of the tenth ACM international symposium on Mobile ad hoc networking and computing, New Orleans, LA, USA, pp. 85–94 (2009) 33. Y. Wan, G. Chert, Y. Xu, A note on the minimum label spanning tree. Inform. Process. Lett. 84(2), 99–101 (2002) 34. P.-J. Wan, D.-Z. Du, P. Pardalos, W. Wu, Greedy approximations for minimum submodular cover with submodular cost. Comput. Optim. Appl. 45(2), 463–474 (2010) 35. P.-J. Wan, L.-W. Liu, Maximal throughput in wavelength-routed optical networks, in DIMACS Workshop on Optical Networks, 1998 36. P.-J. Wan, C. Ma, Z. Wang, B. Xu, M. Li, X. Jia, Weighted wireless link scheduling without information of positions and interference/communication radii, in IEEE INFOCOM, 2011 37. P.-J. Wan, Z. Wang, H.W. Du, S.C.-H. Huang, Z.Y. Wan, First-fit scheduling for beaconing in multihop wireless networks, in IEEE INFOCOM, 2010 38. P.-J. Wan, C.-W. Yi, X. Jia, D. Kim, Approximation algorithms for conflict-free channel assignment in wireless ad hoc Networks. Wiley J. Wireless Comm. Mobile Comput. 6(2), 201–211 (2006) 39. L.A. Wolsey, An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 4(2), 385–393 (1982) 40. X. Xu, Y. Wang, H. Du, P.-J. Wan, F. Zou, X. Li, W. Wu, Approximations for node-weighted Steiner tree in unit disk graphs. J. Optim. Lett. 4(3), 405–416 (2010)

Hardness and Approximation of Network Vulnerability My T. Thai, Thang N. Dinh and Yilin Shen

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Hardness Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 NP-Completeness of Critical Edge Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 NP-Completeness of Critical Vertex Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 NP-Completeness of ˇ-Edge Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hardness of ˇ-Vertex Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Approximation Algorithm for ˇ-Edge Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Balanced Tree Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Pseudo-approximation Algorithm and Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Approximation Algorithm for ˇ-Vertex Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Algorithm Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Theoretical Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A Second pLook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 O.plog n/ Pseudo-approximation for ˇ-Edge Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 O. log n/ Pseudo-approximation for ˇ-Vertex Disruptor. . . . . . . . . . . . . . . . . . . . . . . . . . 6 Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Assessing network vulnerability is a central research topic to understand networks structures, thus providing an efficient way to protect them from attacks and other disruptive events. Existing vulnerability assessments mainly focus on investigating the inhomogeneous properties of graph elements, node degree, for example; however, these measures and the corresponding heuristic solutions cannot either provide an accurate evaluation over general network topologies

M.T. Thai () • T.N. Dinh • Y. Shen Department of Computer Science, Information Science and Engineering, University of Florida, Gainesville, FL, USA e-mail: [email protected]; [email protected]; [email protected] 1631 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 23, © Springer Science+Business Media New York 2013

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or performance guarantees to large-scale networks. To this end, this chapter introduces two new optimization models to quantify the network vulnerability, which aim to discover the set of key node/edge disruptors, whose removal results in the maximum decline of the global pairwise connectivity. Results presented in this chapter consist of the NP-completeness and inapproximability proofs of these problems along with pseudo-approximation algorithms.

1

Introduction

Complex network systems such as the Internet, WWW, MANETs, and the power grids are often greatly affected by several uncertain factors, including external natural or man-made interferences (e.g., severe weather, enemy attacks, malicious attacks). Additionally, they are extremely vulnerable to attacks, that is, the failures of a few critical nodes (links) which play a vital role in maintaining the network’s functionality can break down its operation [1]. In order to assess the network vulnerability, we need to address several fundamental questions such as the following: How do we quantitatively measure the vulnerability degree of a network? What are the key nodes (links) that play a vital role in maintaining the network’s functionality? What is the minimum number of nodes (links) do we need to take down in order to break down an adversarial network? Despite numerous methods have been investigated on the network vulnerability, they neither reveal the global damage done on the network when multiple nodes (links) fail simultaneously nor correctly identify the key nodes who play a vital role in maintaining the network’s functionality [2–8]. (More details of related work are discussed later in Sect. 6.) In this chapter, two optimization models along with their four related problems to assess the network vulnerability are discussed as follows: Model 1: Critical Vertex (Edge) Disruptor [CVD (CED)] : Given a graph G D .V; E/ representing a network (where V is a set of nodes and E is a set of links in the network) and a positive integer k < jV j (k < jEj), determine a set of vertices (edges) S  V (S  E) with jS j D k so as to minimize the total pairwise connectivity in the induced graph GŒV n S  (GŒE n S ), obtained by removing S from G, where the total pairwise connectivity in G is defined as the total number of connected pairs of vertices in G. A pair of nodes is connected if there is a path between them in graph G. Note that E is a set of links where links can be physical links or logical links between two endpoints. For example, if there is a communication between nodes u and v in the network, there will be an edge .u; v/ in the representing graph G. If G is used to represent the functional dependency between each node in a network, then there is an edge .u; v/ in G iff there is a functional relationship between u and v in the network. Therefore, the connectivity discussed in this chapter is not simply a physical connectivity in a given network.

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Clearly, this model aims to discover vertices (edges) whose removal maximizes the network fragmented and maximumly destroys the network. The ratio between the size of S and the number of pairwise disconnectivity in GŒV n S  determines the degree of vulnerability of G, that is, the smaller this ratio is, the more vulnerable the network is. Thus, this model represents a new paradigm for quantitatively characterizing the vulnerability of an underlying network. Model 2. ˇVertex (Edge) Disruptor [ˇVD (ˇED)] : Given a graph G D .V; E/ and 0  ˇ < 1, find a subset of vertices (edges) S  V (S  E) with the minimum cardinality so that the total pairwise connectivity of GŒV n S  (GŒE n S )  is at most ˇ n2 . This model also reveals the vulnerability degree of the networks. That is, the more key nodes (edges) there are (i.e., the more nodes (edges) whose deletion is required to meet the requirement of pairwise connectivity), the less vulnerability the network is. Conversely, the fewer the key nodes (edges), the more vulnerability the network will be to the attacks. Model 1 can be used to maximally disrupt the network with a given cost, whereas Model 2 can be used in the study of breaking down an adversarial social network to a certain degree with the minimum cost. For example, this concept can be applied to destroy, that is, arrest, a small number of key individuals in an adversarial social network (e.g., terrorist networks) in order to maximally disrupt the networks, ability to deploy a coordinated attack. The set of nodes S obtained from the two models present critical nodes in networks, such as a key person, a commander of military networks, or a key computer components in computer networks. Moreover, the two models present a deeper meaning and greater potentials on the study of multiple disruption levels (different values of k and ˇ). Several recent studies in the context of wireless networks have aimed to discover the nodes/edges whose removal disconnects the network, regardless of how disconnected it is [9–11]. However, it is not reasonable to limit the possible disruption to only disconnecting the graph, ignoring how fragmented it is since the giant connected component still exists and the network may function well. For example, a scale-free network can tolerate high random failure rates [1], since the destructions to boundary vertices may not significantly decline the network connectivity even though the whole graph becomes disconnected. In addition, different disruption levels may require different sets of disruptors on which these two models can differentiate, whereas existing methods cannot. For example, the node centrality method always returns a set of nodes with nonincreasing degrees regardless of the disruption level. The study of these models also finds applications in several other fields, such as network immunization, offering control of epidemic outbreaks, and containment of viruses. For example, instead of an expensive mass vaccination, we can vaccinate a small number of individuals (corresponding to key nodes of a network). The immunized nodes cannot propagate the virus, thus minimizing the overall transmissibility of the virus. This chapter presents the hardness complexity of the four problems, CED, CVD, ˇED, and ˇVD, along with their pseudo-approximation solutions. The results are discussed in the following sections.

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2

Hardness Results

2.1

NP-Completeness of Critical Edge Disruptor

In this section, the NP-completeness of CED in general graphs and in unit disk graphs (UDGs) is shown. Furthermore, this section presents an NP-completeness of CED in power-law graphs (PLGs), strengthening its intractability. UDGs are often used in the networking research field to model the homogeneous wireless networks where all nodes have the same transmission range where PLGs represent a class of complex networks whose nodes follow the power-law distribution such as the Internet and social and biological networks [12]. That is, the fraction of nodes in the network having degree k is proportional to k ˇ , where ˇ is a parameter whose value is typically in the range 2 < ˇ < 3. The hardness proof of CED in general graphs uses a reduction from the pmultiway cut problem, which is defined as follows: Definition 1 (p-Multiway Cut) Given an undirected graph G D .V; E/ and a subset of terminals T  V with jT j D p, find a minimum set of edges of G whose removal disconnects all vertices in T . That is, each vertex in T belongs to each connected component. Note that p-multiway cut is NP-complete for any p  3 according to E. Dahlhaus et al. [13]. Fact 1 Given a graph G D .V; E/ with p-connected   components, the total pairwise   connectivity of G is lower bounded by p bn=pc and upper bounded by npC1 2 2 where n D jV j. Theorem 1 The critical edge disruptor (CED) problem is NP-complete.

Proof Consider the decision version of CED that asks whether G D .V; E/ contains a set of edges S  E of size k such that the total pairwise connectivity of GŒE n S  is at most c for a given positive integer c. It is easy to see that CED 2 NP since a nondeterministic algorithm needs only guess a subset S of edges and checks in polynomial time whether S has the appropriate size k and the total pairwise connectivity of GŒE n S  is at most c by using the depth-first search. The reduction from the 3-multiway cut (3-MC) problem to CED is used to prove its NP-hardness. Let an undirected graph G D .V; E/ where jV j D n, a set T  V of three terminals and a positive integer k  jEj be any instance of 3-MC. The graph G 0 D .V 0 ; E 0 / is constructed as follows: For each terminal v 2 T , l D n2 , more vertices are added onto v to construct a clique of size n2 C 1, as illustrated in Fig. 1. We now show that there is a 3-MC cut of size k in G iff G 0 has a CED

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Fig. 1 The reduction from 3-multiway cut instance to a CED instance. (a) Graph G with T D fv2 ; v5 ; v6 g. (b) Reduction from 3-multiway cut to CED

S of size k such that the pairwise connectivity of GŒE n S  is at most c where 2  2   2  2 n 2C1 C n Cn2 < c < 2n 2C2 . 2 First, suppose S  E is a 3-MC for G with jS j  k. By our construction and according to Fact 1, the total pairwise connectivity of G 0 ŒE 0 nS  is upper bounded by ! ! n2 C 1 n2 C n  2 2 C (1) 2 2 which is less than c. Thus, S is also a CED of G 0 . Conversely, suppose that S  E 0 with jS j D k  jEj is a CED of G 0 , that is, the total pairwise connectivity of G 0 ŒE 0 nS   c. We show that S indeed is also a 3-MC of G. Note that S \ .E 0 n E/ D ;, that is, for any e 2 S , e cannot be an edge in the constructed cliques since removing k edges cannot disconnect the cliques; thus,  2 the total pairwise connectivity is nC3n , which is larger than c. Therefore, S  E. 2 Now, we further state that set S also disconnects all the three terminals. Assume that it is not, then the two cliques of these two terminals are connected; thus, the  2  total pairwise connectivity is at least 2n 2C2 > c, contradicting to the fact the total pairwise connectivity of G 0 ŒE 0 n S   c.  We now show that CED still remains NP-complete in UDGs by reducing from the planar independent set (PIS) with maximum degree 3. Given a planar graph G D .V; E/, the PIS problem asks us to find a subset S  V with the maximum size such that no two vertices in S are adjacent. Lemma 1 (L.G. Valiant [14]) A planar graph G with maximum degree 4 can be embedded in the plane using O.jV j/ area in such a way that its vertices are at integer coordinates and its edges are drawn so that they are made up of line segments of the form x D i or y D j , for integers i and j .

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Theorem 2 The critical edge disruptor (CED) problem is NP-complete in UDGs. Proof Consider the decision version of CED that asks whether a UDG G D .V; E/ contains a set of edges S  E of size k 0 such that the pairwise connectivity in the induced graph GŒE n S  is at most c for a given positive integer c. Clearly, CED 2 NP in UDGs. To prove that CED on UDGs is NP-hard, we reduce the planar independent set (PIS) with maximum degree 3 to it. Let a planar graph G D .V; E/ where jV j D n with maximum degree 3 and a positive integer k  n be an arbitrary instance of PIS. We must construct in polynomial time a UDG G 0 D .V 0 ; E 0 / and positive integers k 0 and c such that G has an independent set of size k iff G 0 D .V 0 ; E 0 / has a CED of size k 0 where the pairwise connectivity of G 0 after removing this CED is at most c. The construction of G 0 is as follows: Given a planar graph G, we first embed it into the plane according to Lemma 1. For each node vi 2 V , convert it to a triangle Tvi D fvi1 ; vp i 2 ; vi 3 g (dark gray triangles in Fig.2b). Each edge in Tvi is set to be a unit length r D 33 . Since each node vi in G has at most three neighbors, say x, y, and z, we attach these neighbors to vi1 , vi 2 , and vi 3 . That is, three edges .vi ; x/, .vi ; y/, and .vi ; z/ in G become .vi1 ; x/, .vi 2 ; y/, and .vi 3 ; z/. If the degree is less than 3, we only need to arbitrarily choose one or two from them to attach. Afterwards, for each embedded edge euv D .u; v/, we divide it into jeuv j “pieces” according to its length and replace each piece by Pa concatenation of two triangles as shown in Fig. 2b. Thus, G 0 is composed of 2 u;v2V jeuv j K3 cliques where jeuv j is the length of edges between node u andpv after embedded onto the plane. Note that G 0 is a unit P 3 0 disk graph P with radius r D 3 . Finally, we set k D 2 u;v2V jeuv j C 3.n  k/ and c D 3. u;v2V jeuv j C k/. First, suppose S  V is an independent set of G with jS j D k. For each piece in all edges of G, we remove two edges from two concatenation triangles such that each piece has one triangle left. After doing this, all triangles Tvi corresponding to the node vi 2 S will be disconnected from other nodes in G 0 . For each node vi 62 S , we remove all edges in the corresponding triangles Tvi in G 0 . Consider a set S 0 consisting of all the removed edges (2 edges per piece and 3 edges for each Tvi where vi 62 S ). Then clearly jS 0 j D k 0 j and the pairwise connectivity in G 0 after removing S 0 is c. Therefore, S 0 is a CED of G 0 . Conversely, suppose that S 0  V 0 with jS 0 j D k 0 is a CED of G 0 , that is, the pairwise connectivity of G 0 ŒE 0 n S 0  P is at most c. First, it is easy to see that the pairwise connectivity will increase by u;v2V jeuv j=2 if two triangles corresponding to every other pieces are connected since there is one more connected node pair. Thus, either one of the two triangles for each piece in G 0 will be disconnected. It implies that two edges will be removed from each piece. Apart from this, in order to reduce the pairwise connectivity to c, we remove 3.n  k/ more edges to avoid the connectivity between the nodes in different triangles. Note that for each triangle Tvi , either all three edges are removed or none since the removal of edges in a

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b a

Fig. 2 The reduction from a planar independent set instance to a CED instance. (a) An instance G. (b) A reduced graph G 0

connected component does not help to reduce the pairwise connectivity if it cannot disconnect the component. Some edges belonging to K2 are exchangeable with the nodes in other triangles within polynomial time. This implies the nodes vi (in V ) corresponding to Tvi which has all their edges remained is an independent set of G. Since the number of Tvi is k, the IS also has size k.  Now our attention is turned to the study of CED’s NP-completeness in powerlaw graphs. The P .˛; ˇ/ model mentioned in [15] is used to describe a power-law ˛ graph. In this model, the number of vertices of degree d is b de ˇ c, where e ˛ is the normalization factor. To show CED’s hardness, we reduce from the clique separator (CS) problem, which is defined as follows: Definition 2 (Clique Separator (CS)) Given a graph G D .V; E/, find a set of edges S such that the induced graph GŒE n S  has the connected components to be cliques and each clique has size at least 3. A subgraph of G is called a clique iff all its vertices are pairwise adjacent. The CS problem asks us to find such a clique separator with the minimum size. Before proceeding, let us first present the following fundamental lemma, which states that a PLG can be constructed from any given simple graph by choosing an appropriate value ˛: Lemma 2 (Ferrante et al. [16]) Let G1 D .V1 ; E1 / be a simple graph with n nodes and ˇ  1. For ˛  maxf4ˇ; ˇ log n C log.n C 1/g, we can construct a power-law

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graph G D G1 [ G2 with exponential factor ˇ and the number of nodesP e ˛ .ˇ/ by 1 constructing a bipartite G2 as a maximal component in G. Here . / D 1 i D1 i  is the Riemann zeta function. Since the hardness proof of CED is based on the reduction from CS, the hardness of CS must be investigated first as follows: Lemma 3 The CS problem is NP-hard. Proof To prove that CS is NP-hard, we reduce the planar independent set (PIS) problem with maximum degree 3 to it. By using the same reduction in Theorem 2, 0 it can be Pshown that G has an independent set of size k iff G has a CS of size 0 k D 2 u;v2V jeuv j C 3.n  k/. First, suppose S  V is an independent set of G with jS j D k. To disconnect the triangles from other nodes, two edges from two concatenation triangles for each piece are removed. For each Tvi where vi 2 S , since all three nodes will not overlap with other cliques, the triangle Tvi will be selected as a connected component of size 3. For the other Tvi where vi 62 S , Tvi cannot be selected as a clique since at least one of its endpoints overlaps with other triangle. Thus, the removed edges have size k 0 . And after removing these edges, the obtained connected components are cliques of size 3. That said, the removed edges set is a CS of G 0 . Conversely, suppose that S 0  V 0 is a CS of G 0 with S 0 D k 0 . For each piece, at least two edges have to be removed to disconnect the triangle from other nodes. For 0 each Tvi , its edges P belongs to S if at least one has been used 0by the other triangles. 0 Since k D 2 u;v2V jeuv j C 3.n  k/, it is easily to modify S to be an independent set of G.  Theorem 3 The critical edge disruptor (CED) problem is NP-complete in powerlaw graphs. Proof Again, consider the decision version of CED that asks whether a power-law graph G D .V; E/ contains a set of links S  V of size k 0 such that the pairwise connectivity in GŒE n S  is at most c for a given positive integer c. To prove that CED in PLGs is NP-hard, we reduce the clique separator (CS) to it. Given G, a power-law graph G 0 D G [ Gb where the bipartite graph Gb D .Ub ; Vb I Eb / is a maximal component in G 0 is constructed according to Lemma 2. We show that there is a CS of size k in G iff G 0 has a CED S 0 of size k 0 such that the pairwise connectivity of G 0 ŒE 0 nS 0  is at most c, where k 0 D kCjEb jjMb j and c D jEj  k C jMb j. Here Mb denotes the set of edges in the maximum matching of Gb . First, suppose S  V is a clique separator of G with jS j D k. The total pairwise connectivity of GŒE n S  is equal to jEj  k since all components in this graph are cliques. Since the maximum matching on Gb can be found in polynomial time using Hopcroft-Karp algorithm [17], the pairwise connectivity of G 0 is c after removing additional Eb n Mb edges. Therefore, the set S 0 D S [ .Eb n Mb / with size k 0 D k C jEb j  jMb j is a CED of G 0 .

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Conversely, suppose that S 0  V 0 is a CED of G 0 with size k 0 . Let S 0 D A [ Sb , where A and Sb are the removed edges (in S 0 ) in G and Gb , respectively. We show that jSb j = jEb j  jMb j. If jSb j < jEb j  jMb j, the pairwise connectivity of Gb is increased at least two when adding one more edge into the maximum matching. On the other hand, the removal of l edges on G can reduce the pairwise connectivity at most l after removing the CS k. Therefore, we have jSb  jEb j  jMb j in order to maximally reduce the number of pairwise connectivity. If jSb j > jEb j  jMb j, the pairwise connectivity of Gb is reduced by one when removing one more edge from the maximum matching. Meanwhile, an edge added into the residual graph of G will increase the pairwise connectivity at least one if it connects to two independent nodes and at least 3 if it has one endpoint belonging to some component in the residual graph of G. Therefore, maximally reducing the number of pairwise connectivity requires jSb j  jEb j  jMb j. Combining both conditions, we have Sb D jEb j  jMb j. Thus, jAj D k, which should be a CS of G in order for GŒE n A having the pairwise connectivity of size jEj  k. 

2.2

NP-Completeness of Critical Vertex Disruptor

In this section, the NP-completeness of CVD on both UDGs and PLGs are shown. Definition 3 (Minimum Vertex Cover) Given a graph G D .V; E/, find a subset C  V with the minimum size such that for each edge in E at least one end vertex belongs to C . Lemma 4 (Garey et al. [18]) Minimum vertex cover remains NP-complete on planar graphs with maximum degree 3. Lemma 5 For any planar graph G D .V; E/ with maximum degree 3, it can be embedded into a UDG with the radius 1=2 such that there exists an independentspace .v/ for each vertex v 2 V satisfying .v/ ¤ ; and any new vertices in .v/ will be incident to v and independent from all other vertices. Proof After mapping G D .V; E/ into an integer coordinates based on Lemma 1, a Gu D .Vu ; Eu / UDG can be constructed as follows: Each edge e D .vi ; vj / 2 E is replaced by a path .vi ; w1 ; w2 ; : : : ; w2pe ; vj / where pe is the length of e and the radius r is set to 1=2. Now, we prove that such independent-space .v/ exists for each v 2 V . Since G has the maximum degree 3, that is, for each vertex v 2 V , there are at most 3 neighbors (in the form of wi ) in Vu . Thus, there exists a nonempty space .v/ for each v such that the maximum distance between the vertices in this area to vertex v is at most 1/2 and the minimum distance between such vertices to all other vertices is larger than 1/2. As shown in Fig. 3, the shadow area corresponds to .v1 /. 

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W'1

W''1

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V2

Fig. 3 An independent-space .v1 /

Theorem 4 The critical vertex disruptor (CVD) problem is NP-complete in UDGs. Proof Consider the decision version of CVD that asks whether a UDG G D .V; E/ contains a set of vertices S  V of size k 0 such that the total pairwise connectivity of GŒV n S  is at most c for a given positive integer c. Again, it is easy to see that CVD 2 NP . Now, to prove that CVD is in NP-hard, we reduce the planar vertex cover (PVC) with maximum degree 3 to it. Let a planar graph with maximum degree 3 G D .V; E/ where jV j D n and a positive integer k  n be an arbitrary instance of PVC. We must construct in polynomial time a UDG G 0 D .V 0 ; E 0 / and positive integers k 0 and c such that G has a vertex cover of size at most k iff G 0 D .V 0 ; E 0 / has a CVD of size k 0 and the pairwise connectivity of G 0 after removing this CVD is at most c. Such a construction is described as follows: Given G, construct a Gu according to Lemma 5. On Gu , for each independent-space .vi /, a new gray vertex ui is added 0 such that there is an edge P only between vi and ui to obtain G as shown in Fig. 4. 0 Finally, set k D k C e2E.G/ pe and c D n  k. First, suppose S  V is a vertex cover of G with jS j  k. It is easy to verify that G has a vertex P cover S with size at most k iff Gu has a vertex cover Su with size at most k C e2E.G/ pe . That is, Su consists of S and a half of nodes wi on each path .vi ; w1 ; : : : ; w2p.vi ;vj / ; vj /. We show that Su is also a CVD of G 0 . Removing Su from G 0 returns all disjoint edges .vi ; ui / left where vi … S . Thus, the pairwise connectivity is n  k, which is equal to c. Conversely, suppose that S 0  V 0 with jS 0 j D k 0 is a CVD of G 0 , that is, the total pairwise connectivity of G 0 ŒV 0 nS 0  is at most c. If ui 2 S 0 , that is, S 0 contain a gray node, replacing ui with any vi or wi (black or white nodes) will further decrease the 0 pairwise connectivity. Thus, any gray P it can be assumed that S does not contain 0 node. Since jS j D k C e2E.G/ pe , it is easily to modify the set S 0 to be a vertex cover of Gu . In this case, the total pairwise connectivity is at most c D n  k. Thus, S 0 \ V is a VC of G. 

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b

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Fig. 4 The reduction for a planar vertex cover instance to a CVD instance. (a) A planar graph G (b) Embed G onto a UDG Gu . (c) Create G 0 by adding gray modes at independent-spaces

Theorem 5 The critical vertex disruptor (CVD) problem is NP-complete in powerlaw graphs. Proof Again, consider the decision version of CVD that asks whether a power-law graph G D .V; E/ contains a set of nodes S  V of size k 0 such that the pairwise connectivity in GŒV n S  is at most c for a given positive integer c. Now, to prove that CVD in power-law graphs is NP-hard, we reduce the vertex cover (VC) to it. Let a graph G D .V; E/ where jV j D n and a positive integer k  n be any instance of VC. A power-law graph G 0 D .V 0 ; E 0 / is constructed

1642 Fig. 5 The reduction from a vertex cover instance to CVD instance in power-law graphs. For simplicity, all edges in G are omitted. (a) An instance G. (b) Reduced graph G 0 D G1 [ G2

M.T. Thai et al.

b

a

as follows. First, for each node vi 2 V , one additional node ui is added onto it, denoted by G1 where V1 D V [ U and U D fui g. Then according to Lemma 2, a power-law graph G 0 D .V 0 ; E 0 / can be constructed by embedding G1 and a bipartite graph G2 D .V21 ; V22 I E2 / where V21 and V22 are two sets of disjoint nodes in G2 and ˛  maxf4ˇ; ˇ log.2n/ C log.2n C 1/g with some specific ˇ. Note that V21 and V22 are marked gray and white separately as shown in Fig. 5. We show that there is a VC of size k in G iff G 0 has a CVD S 0 of size k 0 such that the pairwise connectivity of G 0 ŒV 0 n S 0  is at most c, where k 0 D k C minfjV21 j; jV22 jg and c D n  k. First, suppose S  V is a vertex cover of G with jS j D k. Therefore, G has a vertex cover S of size k iff G [ G2 has a vertex cover S 0 of size k C minfjV21 j; jV22 jg since VC is polynomially solvable in any bipartite graphs according to K˜onig’s theorem [19]. Then after removing S 0 from G 0 , only all disjoint links .vi ; ui / are left where vi 62 S . Therefore, the pairwise connectivity in G 0 is n  k, which is equal to c. That said, S 0 is a CVD of G 0 . Conversely, suppose that S 0  V 0 with jS 0 j D k 0 is a CVD of G 0 , that is, the total pairwise connectivity of G 0 ŒV 0 n S 0  is at most c. First, if ui 2 S 0 , this ui can be replaced with any vi without increasing the pairwise connectivity. Since jS 0 j D k 0 D kCminfjV21 j; jV22 jg, it is easily to modify the S 0 to be a vertex cover of G [G2 , where the total pairwise connectivity on G 0 is at most c D n  k. Thus, S 0 \ V is a VC of G. 

2.2.1 Inapproximability As the CVD is NP-complete, one will question how tightly one can approximate the solution, leading to the theory of inapproximability. The inapproximability factor gives us the lower bound of near-optimal solution with theoretical performance guarantee. That said, no one can design an approximation algorithm

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with a better ratio than the inapproximability factor. In this section, the inapproximability of CVD is further investigated in general graphs by showing the gap-preserving reduction from maximum clique problem, which is defined as follows: Definition 4 (Maximum Clique) Given a graph G D .V; E/, find a clique of maximum size. Theorem 6 The CVD problem is NP-hard to be approximated within ‚ any  < 1  logn 2 in general graphs unless P D NP .

k n

for

Proof In the proof, a gap-preserving reduction from maximum clique problem to CVD is shown. According to H˚astad [20], maximum clique is proven NP-hard to be approximated within n1 . Let a graph G D .V; E/ and a positive integer k  jV j D n be an instance of maximum clique problem (MC). Now we construct the graph G 0 D .V 0 ; E 0 / as follows: Firstly, a complement graph G D .V; E/ where E D f.vi ; vj /j.vi ; vj / 62 Eg is constructed. Next, for each vertex v in G, l  n1  1, more nodes are added to construct a clique of size l C 1 and obtain G 0 , as shown in Fig. 6. Let  be a feasible solution of MC on G and   ' be a feasible ; the solution of CVD on G 0 where j'j D n  k. Define c D .n  k/ 2l C k lC1 2 completeness and soundness are shown respectively as follows: Completeness: In this step, we need to show that if OP T ./ D k then OP T .'/ D c Let OP T ./ be the maximum clique in graph G, then OP T ./ is the maximum independent set in G. Since the minimum vertex cover and the maximum independent set are complementary on the same graph, removing the set of nodes V n OP T ./ in G will result into a set of only isolated nodes. Thus, the  pairwise   , connectivity in G 0 after removing such n  k critical nodes is .n  k/ 2l C k lC1 2 which is equal to c. It implies that OP T .'/ D c. 1 Soundness: In this step, we need to show that if OP T ./ < n1 k, then k OP T .'/ > ‚ n c.

a

Fig. 6 The gap-preserving reduction from a maximum clique instance to a CVD instance. (a) A graph instance (b) Reduced graph G 0 from MC

b

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One Connected Component

1-

1-

Fig. 7 The lower bound instance of CVD in G 0 given that j'j D n  k and OP T ./
n1 . In this case, 1 0 the lower bound of the pairwise connectivity in G after removing n  n1 k critical k nodes is shown to be ‚ n c. Note that the pairwise connectivity is minimized when the number of connected components are maximized and each connected 1 1 component has the same size as shown in Fig. 7. That is, since 1  n1 > n1 , the lower bound is achieved when the same number of cliques C G1 connects to one C G2 , where C G1 and C G2 denote the group of cliques of size l and cliques of size 1 l C 1, respectively, after removing n  n1 k nodes. Therefore, the following lower bound is obtained:

0 @



1 n1 1 n1

1

.n1



1

C 1 .l C 1/A 1 k n1 2 !  1/.l C 1/ 1 k 2 n1

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.n1 1/.lC1/ D

2

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l  2

C

1 k n1 lC1 c k 2

ln1  

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ln1  2

1

1

 n n lC1 2

>

 D‚

k n

k

c



c

Step 2 holds since bzc  z  1 for any real number z; steps 4 and 5 hold since l  n1  1 and k  n. 

2.3

NP-Completeness of ˇ-Edge Disruptor

In this section, the NP-completeness of ˇ-ED is shown by reducing from the balanced cut problem. This section begins with several needed definitions as follows: Definition 5 A cut hS; V nS i corresponding to a subset S 2 V in G is the set of edges with exactly one endpoint in S . Usually V nS is denoted by SN . The cost of a cut is the sum of its edges’ costs (or simply its cardinality in the case all edges have unit costs) and denoted by c.S; SN /. Finding a min cut in the graph is polynomial solvable [21]. However, if one asks for a somewhat “balanced” cut of minimum size, the problem becomes intractable. A balanced cut is defined as following: Definition 6 (Balanced cut) Let f be a function from the positive integers ˝to the˛ positive reals. An f -balanced cut of a graph G D .V; E/ asks us to find a cut S; SN N  f .jV j/. with the minimum size such that minfjS j; jSjg ˝ ˛ Abusing notations, for 0 < c  12 , c-balanced cut is used to find the cut S; SN N  cjV j. The following results on with the minimum size such that minfjS j; jSjg balanced cut shown in [22] will be used in the proofs: Corollary 1 (Monotony) Let g be a function with 0  g.n/  g.n  1/  1

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Then f .n/  g.n/ for all n, implies f -balanced cut is polynomially reducible to g-balanced cut. Corollary 2 (Upper bound) ˛n -balanced cut is NP-complete for ˛;  > 0. It follows from Corollaries 1 and 2 that for every f D O.˛n /; f -balanced cut is NP-complete. It is now ready to prove the NP-completeness of ˇ-edge disruptor: Theorem 7 ˇ-edge disruptor is NP-complete. Proof Theorem 7 is proven for a special case when ˇ D 12 . For other values of ˇ, the proof can go through with a slight modification of the reduction. In the proof, n is considered to be a large enough number, say n > 103 . Consider the decision version of the problem that asks whether an undirected graph G D .V; E/ contains a 12 -edge disruptor of a specified size: 1 1 -ED D fhG; Ki j G has a -edge disruptor of size Kg 2 2 To show that 12 -ED is in NP-complete, we must show that it is in NP and that all NPproblems are polynomial time reducible to it. The first part is easy; given a candidate subset of edges, it is easily check in polynomial time if it is a 12 -edge disruptor of size K. To prove the second part, we show that f -balanced cut is polynomial time q 2

n 2b n cCn

3 reducible to 12 -ED where f D b c. 2 Let G D .V; E/ be a graph in which one seeks to find an f -balanced cut of size k. Now construct the graph H.VH ; EH / as follows: VH D V 0 [ C1 [ C2 where 2 V 0 D f vi j i 2 V g and C1 ; C2 are two cliques of size b n3 c. The total number of 2 nodes in H is, hence, N D 2b n3 c C n. Beside edges inside two cliques, an edge 2 .vi ; vj / is added for each edge .i; j / 2 E. Next each vertex vi connects to b n4 c C 1 2 vertices in C1 and b n4 c C 1 vertices in C2 so that the degree difference of nodes in the cliques are at most one. The construction of H.VH ; EH / is illustrated in Fig. 8. Now we need to show that there cut of size k in G iff H has an  is2 a f -balanced 2 1 n -edge disruptor of size K D n b 4 c C 1 C k where 0  k  b n4 c. Note that the 2 ˝ ˛ 2 N 2 N  b .jS jCjSj/ cost of any cut S; SN in G is at most jS jjSj c D b n4 c. ˝ ˛ 4 On the one hand, an f -balanced cut S; S N of size k in G induces a cut ˝ ˛ 2 C1 [ S; C2 [ SN with size exactly n b n4 c C 1 C k. If the cut is selected as the   disruptor, the pairwise connectivity will be at most 12 N2 . hand, assume that H has a 12 -edge disruptor of size K D On the other 2

n b n4 c C 1 C k. Because separating n nodes in a clique from the other nodes

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Fig. 8 Construction of H D .VH ; EH / from G D .V; E/

 2 2 in that clique requires cutting at least n.b n3 c  n/ > n b n4 c C 1 C k edges, there are at most n nodes separated from both the cliques. A direct consequence is that 2 at least .b n3 c  n/ nodes remain connected in each clique after removing edges in the disruptor. Denote the sets of those nodes by C10 and C20 , respectively. C10 and C20   cannot be connected otherwise the pairwise connectivity will exceed 12 N2 . Denote by X1 and X2 the set of nodes in V 0 that are connected to C10 and C20 respectively. Since C10 and C20 are disconnected we must have X1 \ X2 D . The disruptor can be modified without increasing its size and the pairwise connectivity such that no nodes in the cliques are split. For each u 2 C1 nC10 , we remove from the disruptor all edges connecting u to C10 and add to the disruptor all edges connecting u to X2 . This will move u back to the connected component 2 that contains C10 while reducing the size of the disruptor at least .b n3 c  n/  n. At the same time, an arbitrary node v 2 X1 is selected and added to the disruptor all remaining v’s adjacent edges. This increases the size of the disruptor at most 2 .b n4 c C 1/ C n while making v isolated. By doing so, the size of the disruptor 2 2 decreases by .b n3 c  n/  n  ..b n4 c C 1/ C n/ > 0. In addition, the pairwise connectivity will not increase when connecting u to C10 and disconnecting v from C10 . If no nodes are left in X1 , a node v 2 X2 can be selected (as in that case jC20 [ X2 j > jC10 [ X1 j) to make sure the pairwise connectivity will not be increased. The same process is repeated for every node in C2 nC20 and at the end of the process, C10 D C1 and C20 D C2 . We will prove that X1 [X2 D V 0 , that is, hX1 ; X2 i induces a cut in V .G/. Assume 2 not, the cost to separate C1 [X1 from C2 [X2 will be at least (b n4 cC1/.jV 0 X1 jC  2 2 2 jV 0 X2 j/ D .b n4 cC1/.2njX1jjX2 j/  .b n4 cC1/.nC1/ > n b n4 c C 1 Ck that is a contradiction. Since X1 [ X2 D V 0 , the  disruptor induces a cut in G. To have the pairwise connectivity at most 12 N2 , both .C1 [ X1 / and .C2 [ X2 / must have size

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. If follows that X1 and X2 must have size at least f .n/ D  2 the cut induced by hX1 ; X2 i in G will be n b n4 c C 1

q 2 n 2b n3 cCn b c. The cost of 2 2 Ck  n.b n4 c C 1/ D k.

2.4



Hardness of ˇ-Vertex Disruptor

Theorem 8 ˇ-vertex disruptor is NP-complete. Proof The details of the proof will be ignored. Instead, only the sketch is presented by showing that the vertex cover is polynomial time reducible to ˇ-vertex disruptor. Let G D .V; E/ be a graph in which one seeks to find a vertex cover of size k. Note that if we remove nodes in a vertex cover from the graph, the pairwise connectivity in the graph will be zero. Hence, by setting ˇ D 0, G has a vertex cover of size k iff G has an ˇ-vertex disruptor of size k. One can also avoid using ˇ D 0 by replacing each vertex in G by a clique of large enough sizes, say O.n/; that ensures no vertices in cliques will be selected in the ˇ-vertex disruptor.  Given the NP-hardness of the ˇ-vertex disruptor, the best possible result is a polynomial-time approximation scheme (PTAS) that given a parameter  > 0, produces a .1 C /-approximation solution in polynomial time. Unfortunately, this is impossible unless P = NP. Theorem 9 Unless P = NP, ˇ-vertex disruptor has no polynomial-time approximation scheme. Proof In the case ˇ D 0, the problem is equivalent to finding the minimum vertex cover in the graph. In [23], Dinur and Safra showed that approximating vertex cover within constant factor less than 1.36 is NP-hard. Hence, a PTAS scheme for ˇ-vertex disruptor does not exist unless P D NP. 

3

Approximation Algorithm for ˇ-Edge Disruptor 3

This section presents an O.log 2 n/ pseudo-approximation algorithm for the ˇ-edge disruptor problem in directed graphs. (Note that in directed graphs, a pair .u; v/ is said to be connected if there exists a directed path from u to v and vice versa, from v to u.) Formally, the algorithm finds in a directed graph G a ˇ 0 -edge disruptor whose 0 3 cost is at most O.log 2 n/OPTˇED , where ˇ4 < ˇ < ˇ 0 and OPTˇED is the cost of an optimal ˇ-edge disruptor. As shown in Algorithm 1 , the approximation solution consists of two main steps: (1) constructing a decomposition tree of G by recursively partitioning the

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graph into two halves with a directed c-balanced cut and (2) solving the problem on the obtained tree using a dynamic programming algorithm and transform this solution to the original graph. These two main steps are explained in the next two subsections.

3.1

Balanced Tree Decomposition

Let us first introduce some definitions before describing the balanced tree decomposition algorithm. Definition 7 Given a directed graph G D .V; E/ and a subset of vertices S  V . The set of edges outgoing from S and ˝the set ˛ of edges incoming to S are denoted by ı C .S / and ı  .S /, respectively. A cut S; SN in G is defined as ı C .S /. A c-balanced ˝ ˛ N  cjV j. The directed c-balanced cut problem is cut is a cut S; SN s.t. minfjS j; jSjg to find the minimum c-balanced cut. And finally, P.G/ denotes the total pairwise connectivity in G. ˝ ˛ Note that a cut S; SN separates pairs .u; v/ 2 S  .SN / so that there is no path u to v, that is, there is no strongly connected component (SCC) containing vertices both in S and SN . The tree decomposition procedure is as follows: As shown in Algorithm 1 (lines 3–13), the procedure starts with a tree T D .VT ; ET / containing only one root node t0 . t0 is associated with the vertex set V of G, that is, V .t0 / D V .G/. For each node ti 2 VT whose V .ti / contains more than one vertex and V .ti / has not been partitioned, the subgraph GŒV .ti / induced by V .ti /q in G is partitioned using a

c-balanced cut algorithm [24] where constant c D 1  ˇˇ0 . For each c-balanced cut on GŒV .ti /, two children nodes ti1 and ti 2 of ti in VT are created. These two children nodes correspond to two sets of vertices returning by the cut. Finally, each node ti is assigned a cost cost.ti / which is equal to the cost of the cut performed on GŒV .ti /. This procedure continues until V .ti / contains only a single vertex. That is, the leaves of T represent nodes in G. Therefore, T has n leaves and n  1 internal nodes where each internal node in T represents a subset of nodes in V . Note that the root node t0 is at level 1. If a node is at level l, all its children are defined to be at level l C 1. Note that all collections of vertices corresponding to nodes in a same level form a partition in V . Lemma 6 The height of T obtained in the above balanced tree decomposition procedure is at most O. log1c 0 n/ Proof Note that the directed c-balanced cutpalgorithm [24] finds in polynomial time a c 0 -balanced cut within a factor of O. log n/ from the optimal c-balanced cut for c 0 D ˛c and fixed constant ˛. Thus, when separating GŒV .ti / using cbalanced cut, the size of the larger part is at most .1  c 0 /jV .ti /j. By induction

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method, it can be shown that if a node ti is on level l, the size of the corresponding collection V .ti / is at most jV j.1c 0 /l1 . It follows that the tree’s height is at most O. log1c 0 n/. 

3.2

Pseudo-approximation Algorithm and Analysis

This subsection presents the second main step which uses the dynamic programming to search for the right set of nodes in T such that the cuts to partition those corresponding sets of vertices in G have the minimum cost and the obtained pairwise connectivity is at most ˇ 0 n2 where ˇ < ˇ 0 < 1. The details of this step are shown in Algorithm 1 (lines 14–23). Denote a set F D ft1 ; t2 ; : : : ; tk g  VT such that V .t1 /; V .t2 /; : : : ; V .tk / is a k ] partition of V .G/ , that is, V .G/ D V .ti / and for any pair ti and tj 2 F , i D1

V .ti / \ V .tj / D ;. Such a subset F is called G-partition. Denote by A.ti / the set

Algorithm 1 ˇ-edge disruptor 1: Input: A directed graph G D .V; E/ and 0  ˇ < ˇ 0 < 1 2: Output: A ˇ 0 -edge disruptor of G fConstructqthe decomposition treeg 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24:

ˇ

c 1  ˇ0 T .VT ; ET / .ft0 g; /, V .t0 / V .G/; l.t0 / D 1 while 9 unvisited ti with jV .ti /j  2 do Mark ti visited, create new child nodes ti1 ; ti2 of ti l.ti1 /; l.ti2 / l.ti / C 1 VT [ fti1 ; ti2 g VT ET [ f.ti ; ti1 /; .ti ; ti2 /g ET Separate GŒV .ti / into two using directed c-balanced cut Assign two obtained partitions to V .ti1 / and V .ti2 / The cost of the balanced cut cost .ti / end while fFind the minimum cost G-partitiong for ti 2 T in reversed BFS  order from root node t0 do for p 0 to ˇ 0 n2 do if P .GŒV .ti //  p then 0 cost .ti ; p/ else minfcost .ti1 ; / C cost .ti2 ; p  / C cost .ti / j   pg cost .ti ; p/ end if end for end for  Find F with P .F / D minfcost .t0 ; p/ j p  ˇ 0 n2 g Return union of cuts used at A.F / during tree construction

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Fig. 9 A part of a decomposition tree. F D ft2 ; t3 ; t5 ; t6 g is a G-partition. The corresponding partition fV .t2 /; V .t3 /; V .t5 /; V .t6 /g in G can be obtained by using cuts at ancestors of nodes in F , that is, t0 ; t1 ; t4

of nodes corresponding to ancestor of ti in T and A.F / D

[ ti 2F

A.ti /. It is clear that

an F is G-partition iff F satisfies: 1. 8ti ; tj 2 F W ti … A.tj / and tj … A.ti / 2. 8ti 2 VT ; ti is a leaf: A.ti / \ F ¤  In case F D ft1 ; t2 ; : : : ; tk g is G-partition, V .t1 /, V .t2 /, : : :, V .tk / can be separated by performing the cuts corresponding to ancestors of nodes in F during the tree construction. For example, a decomposition tree with a G-partition set ft2 ; t3 ; t5 ; t6 g in Fig. 9. The corresponding partition fV .t2 /; V .t3 /; V .t5 /; and V .t6 /g in G can be obtained by cutting GŒV .t0 /, GŒV .t1 /, and GŒV .t4 / successively using c-balanced cuts in the tree construction. The cut cost, hence, will be cost.t0 / C cost.t1 / C cost.t4 /. In general, the total cost of all the cuts to separate V .t1 /; V .t2 /; : : : ; V .tk / is equal to X cost.F / D cost.ti / ti 2A.F /

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And the pairwise connectivity in G is equal to P.F / D

X

P.GŒV .ti //

ti 2F

 Our goal now is to find a G-partition F 2 VT so that P.F /  ˇ 0 n2 with a minimum cost.F / since the cut associated to F on T is the ˇ 0 -edge disruptor of G. Clearly, finding such a set F has an optimal substructure; thus, it can be found in O.n3 / using dynamic programming as described next. Let cost.ti ; p/ denote the minimum cut cost to make the pairwise connectivity in GŒV .ti / be less than or equal to p using only c-balanced cuts corresponding to nodes in the subtree rooted at ti . The minimum cost for a G-partition  subset F that induces a ˇ 0 -edge disruptor of G is then minfcost.t0 ; p/ j p  ˇ 0 n2 g where t0 is the root node in T . The value of cost.ti ; p/ can be calculated using following recursive formula:

cost.ti ; p/ D

8 < 0

if P.GŒV .ti //  p

:min cost.ti1 ; / C cost.ti 2 ; p  / C cost.ti / otherwise p

where ti1 ; ti 2 are children of ti . In the first case, when P.GŒV .ti //  p, no cut is required; thus, cost.ti ; p/ D 0. Otherwise, all possible combinations of pairwise connectivity  in V .ti1 / and p   in V .ti 2 / for all   p can be investigated. The combination with the smallest cut cost is then selected. Based on the above recursive formula, a dynamic programming (shown in lines 14–23 of Algorithm 1 ) can find a minimum cost G-partition set F that induces a ˇ 0 -edge disruptor in G. The only part left is to prove that such a set F exists in T and the cost of the ˇ 0 -edge disruptor induced from F found in the dynamic programming algorithm is 3 no more than O.log 2 n/Optˇ-ED . The proof is shown as follows: Lemma 7 There exists a G-partition subset F in T that induces a ˇ 0 -edge 3 disruptor whose cost is no more than O log 2 n Optˇ-ED . Proof We first show that such a set F  exists. That is, we are able to find a set G-partition set F such that P.F /  ˇ 0 n2 . Denote Dˇ an optimal ˇ-edge disruptor in G. Removing Dˇ from G, we obtain a set of strongly connected components (SCCs), denote as Cˇ D fC1 ; C2 ; : : : ; Ck g. We construct a G-partition subset F based on Cˇ as shown in the Algorithm 2 . Nodes in T are visited in a top-down manner, that is, every parent must be visited before its children. This can be done by visiting nodes in breadth first search (BFS) order from the root node t0 . For each node ti , if there exists some component Cj 2 Cˇ such that V .ti / contains more than .1  c/jV .ti /j nodes in Cj (all leaves in T

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satisfies this condition) and no ancestors of ti has ever been selected into F , then we select ti as a member of F . It is clear to see that F is a G-partition as it satisfies two conditions mentioned earlier. The total pairwise connectivity of G induced by F is bounded as ! X jV .ti /j P.F /  2 t 2F i

D

X

1 X 2 C 2C j

ˇ

jV .ti /j2 

jV .ti /\Cj j.1c/jV .ti /j

0 

1 XB B B 2 C 2C @ j



12 C n C jV .ti /jC  A 2

X r jV .ti /\Cj j

1 X 2 C 2C

s

j

ˇ ˇ0

ˇ0 jCj j ˇ

jV .ti /j

!2 

1 ˇ0 @ X jCj j2  nA  < 2ˇ C 2C 0

n 2

j

n 2 ! n ˇ 2 0

Thus, such a set F exists. 3 Next, we show that cost.F /  O.log 2 n/Optˇ-ED . Let h.T / and LuT denote the height of T and the set of nodes at level u of T , respectively. We have X

h.T /

cost.F / D

X

cost.ti /

(2)

uD1 ti 2.Lu \A.F // T

If ti 2 A.F /, then ti is not selected to F . Hence, there exists Cj 2 Cˇ so that jV .ti / \ Cj j < .1  c/jV .ti /j (otherwise ti was selected into F as it satisfies the conditions in the line 3, Algorithm 2 ). To guarantee c < 1c we constrain c < 1=2, 0 that is, ˇ > ˇ4 . The edges in the optimal ˇ-edge disruptor Dˇ separate Cj from the other SCCs. Hence, Dˇ also separates Cj \ V .ti / from the V .ti /nCj in GŒV .ti /. Denote sep.ti ; Dˇ / the set of edges in Dˇ separating Cj \ V .ti / from the rest in GŒV .ti /. Clearly, sep.ti ; Dˇ / is a directed c-balanced cut of GŒV .ti /. Since, the cut algorithm p we used in the tree construction p has a pseudo-approximation ratio of only O. log n/, we have cost.tu /  O. log n/jsep.ti ; Dˇ /j.

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Algorithm 2 Find a good G-partition set F of T that induces a ˇ 0 -edge disruptor in G F  for ti 2 T in BFS order from t0 do if .9Cj 2 Cˇ W jV .ti / \ Cj j  .1  c/jV .ti /j/ and (A.ti / \ F D ;) then F F [ fti g end if end for

Recall that if two nodes ti and tj are on the same level, then V .ti / and V .tj / are two disjoint subsets. It follows that sep.ti ; Dˇ / and sep.tj ; Dˇ / are also disjoint sets. Therefore, we have X

cost .ti /

ti 2.LuT \A.F //

p  O. log n/

X

jsep.ti ; Dˇ /j

ti 2.LuT \A.F //

p  O. log n/j

[

sep.ti ; Dˇ /j

ti 2.LuT \A.F //

p D O. log n/Optˇ -ED

Since h.T / is at most O.log n/ (Lemma 6), it follows from Eq. 2 that cost.F /  3 O.log 2 n/Optˇ-ED . It completes the proof.  0 Since there exists a G-partition of T that induces a ˇ -edge disruptor  3subset whose cost is no more than O log 2 n Optˇ-ED as shown in Lemma 7 and the dynamic programming is always able to find such a set F , the following theorem follows immediately: 3

Theorem 10 Algorithm 1 achieves a pseudo-approximation ratio of O.log 2 n/ for the ˇ-edges disruptor problem.

4

Approximation Algorithm for ˇ-Vertex Disruptor

This section presents a polynomial-time algorithm (shown in Algorithm 3 ) that finds in a directed graph G D .V; E/ a ˇ 0 -vertex disruptor whose the size is at most O.log n log log n/ times the optimal ˇ-vertex disruptor where 0 < ˇ < ˇ 02 . At the high level, Algorithm 3 consists of two main phases: (1) vertex conversion and (2) size constraint cut. In the first phase, a given graph G is converted into G 0 in a way that removing v 2 G has the same effects as removing edge in G 0 . In the second phase, G 0 is cut into SCCs capping the sizes of the largest component

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while minimizing the number of removed edges. The constraint on the size of each component is kept relaxing until the set of cut edges induces a ˇ 0 -vertex disruptor of G.

4.1

Algorithm Description

Phase 1: Vertex Conversion. Given a directed graph G D .V; E/ for which we want to find a small ˇ 0 -vertex disruptor, a new directed graph G 0 D .V 0 ; E 0 / is constructed as follows: 1. V 0 construction: For each vertex v 2 V , create two vertices v and vC in V 0 . Thus, V 0 D f v ; vC j v 2 V g. 2. E 0 construction: For each vertex v 2 V , add a directed edge .v ! vC / to E 0 . And for each directed edge .u ! v/ 2 E, add a directed edge .uC ! v / 2 E 0 . That is, E 0 D f.v ! vC / j v 2 V g [ f.uC ! v / j .u ! v/ 2 Eg. 3. Edge cost assignment: Assign 1 for all edges .v ! vC / and C1 to other edges in E 0 . An example of this construction is shown in Fig. 10. Based on this conversion, a careful edge cut on G 0 will return a vertex disruptor on G. Indeed, let EV0 D f.v ! vC / j v 2 V g and consider a cut set De0  E 0 that contains only edge in EV0 , we have a one-to-one correspondence between De0 and Dv D fv j .v ! vC / 2 De0 g which is a vertex disruptor set of G. However, G and G 0 have different maximum pairwise connectivity, .n1/n for G and 2 .2n1/2n 0 for G , the fractions of pairwise connectivity remaining in G and G 0 after 2 removing Dv and De0 are, therefore, not simply related to each other. Thus, we next describe what a “careful edge cut” is and how good a vertex disruptor of G can be obtained. Phase 2: Size Constraint Cut. Observe that if edges are removed in order to separate a graph into SCCs, then there is a relation between the pairwise connectivity in the remaining graph and the maximum size of SCC. The smaller the maximum size of SCC, the smaller pairwise connectivity in the graph. However, the smaller the maximum size of each SCC, the more edges are needed to be cut. Therefore, a good range of the maximum size of SCC in G 0 needs to be found in order to return a good vertex disruptor of G with the minimum cut. Along this direction, a binary search can be performed to find a right upper bound ˇjV 0 j and lower bound ˇjV 0 j of the size of each SCC in G 0 . As shown in Algorithm 3 , at each step, we find in G 0 D .V 0 ; E 0 / a minimum cost edge set whose removal partitions Q 0 j, where the graph into strongly connected components, each has size at most ˇjV ˇCˇ ˇQ D b c  . The value of ˇQ is rounded to the nearest multiple of  so that the 2

number of steps for the binary search is bounded by log 1 . The problem of finding a minimum cost edge set to decompose a graph of size n into strongly connected components of size at most n is known as -separator problem. At this step, the

1656 Fig. 10 Phase 1: Vertex conversion which converts a graph G into G 0 such that removing a vertex v 2 G has the same effects as removing an edge in G 0

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a

b

Algorithm 3 ˇ 0 -vertex disruptor Input: Directed graph G D .V; E/ and fixed 0 < ˇ 0 < 1 Output: A ˇ 0 -vertex disruptor of G .; / G 0 .V 0 ; E 0 / V 0 [ fvC ; v g 8v 2 V W V 0 E 0 [ f.v ! vC /g; c.v ; vC / 1 8v 2 V W E 0 E 0 [ fuC ! v g; c.uC ; v / 1 8.u ! v/ 2 E W E 0 0; ˇ 1 ˇ V .G/ DV while .ˇ  ˇ > / do ˇC ˇ ˇQ b 2 c   Q 0j Find De  E 0 to separate G 0 into strongly connected components of sizes at most ˇjV using algorithm in [25] fv 2 V .G/ j .vC ! v / 2 De g Dv  if P .GŒV nDv /  ˇ n2 then ˇ D ˇQ  Remove nodes from Dv as long as P .GŒV nDv /  ˇ n2 if jDV j > jDv j then DV D Dv end if else ˇ D ˇQ end if end while Return DV

algorithm presented in [25] which finds a -separatorin a directed graph G with  a pseudo-approximation ratio of O 12 : log n log log n for a fixed  > 0 is used. Q 0 j. By the cost assignment step in the vertex construction In our context, D ˇjV phase, the edge cut obtained in this step must be a subset of EV0 as other edges e 0 not in EV0 have a cost of C1; hence, e 0 never got selected. Finally, the cut edges in G 0 is converted to vertices in G to obtain the ˇ 0 -vertex disruptor. Simply, for each edge .v ! vC / in a cut set, we have a corresponding vertex v 2 V .

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Theoretical Analysis

Lemma 8 Algorithm 3 always terminates with a ˇ 0 -vertex disruptor. Proof We show that whenever ˇQ  ˇ 0 then the corresponding Dv found in Algorithm 3 is a ˇ 0 -vertex disruptor of G. Consider the edge separator De0 of G 0 induced by Dv . We first show the mapping between SCCs in GŒV nDv  and SCCs in G 0 ŒE 0 nDe0 , the graph obtained by removing De0 from G 0 . Partition the vertex set V of G into (1) Dv : the set of removed nodes; (2) Vsingleton : the set of SCCs whose sizes are one, that is, each SCC has only one node; and (3) Vconnected : the set of remaining U SCCs whose sizes are at least two, denote Vconnected D li D1 Ci ; jCi j  2. Vertices in Vconnected belong to at least one cycle in G, whereas vertices in Vsingle are all singleton. We have the following corresponding SCCs in G 0 ŒE 0 nDe0 : 1. v 2 Dv $ SCCs fvC g and fv g in G 0 ŒE 0 nDe0 . Because after removing .v ! vC /, vC does not have incoming edges and v does not have outgoing edges. 2. v 2 Vsingleton $ SCCs fvC g and fv g. Assume vC belong to some SCC of size at least 2 , that is, vC lies on some cycle in G 0 . Because the only incoming edge to vC is from v , it follows that v is preceding vC on that cycle. Let u and uC be the successive vertices of vC on that cycle. Then u and v should belong to the same SCC in G which yields a contradiction as v 2 Vsingleton . Similarly, v cannot lie on any cycle in G 0 . 3. SCC Ci  Vconnected $ SCC Ci0 D fv ; vC j v 2 Ci g. This can be shown using a similar argument as above. Q Since De0 is a ˇ-separator, the sizes of SCCs in G 0 ŒE 0 nDe0  are at most ˇQ 2n. It Q follows that the sizes of SCCs in GŒV nDv  are bounded by ˇn. Denote the set of SCCs in GŒV nDv  by C with the convention that vertices in Dv become singleton SCC in GŒV nDv . Therefore, we have P.GŒV nDv / D

X Ci 2C



1 0 ! jCi j 1 @X D jCi j2  jV jA 2 C 2C 2

0

i

X

1

1@ Q j/jCi j  jV jA ˇjV 2 C 2C i

! ! 1Q 2 jV j jV j 0 D ˇ > 0. First, it follows from the Lemma 8 that Algorithm 3 returns a ˇ 0 -vertex disruptor Q Dv . At some step, the size of Dv equals to the cost of ˇ-separator De0 in G 0 where ˇQ 0 is at least ˇ   according to Lemma 8 and the binary search scheme. The cost of the separator is at most O .log n log log n/ times the Opt.ˇ/ Q -separator using the algorithm in [25]. Consider an optimal .ˇ 02  9/-vertex disruptor Dv0 of G and its corresponding edge disruptor De0 in G 0 . Denote the cost of that optimal vertex disruptor by Opt.ˇ02 9/-VD . If there exists in GŒV nDv  a SCC Ci so that jCi j > .ˇ 0  2/n, then ! 1 0 n 0 02 P.GŒV nDv / > ..ˇ  2/n  2/..ˇ  2/n  1/ > .ˇ  9/ 2 2 0

when n > 20.ˇ C1/ . Hence, every SCC in G 0 ŒV nDv0  has size at most .ˇ 0  2/.2n/, that is, De0 is an .ˇ 0  2/-separator in G 0 . It follows that Opt.ˇ02 9/-VD  Opt.ˇ0 2/-separator in G 0 . Because ˇQ    ˇ 0  2, we have Opt.ˇ/ Q -separator  Opt.ˇ0 2/-separator  Opt.ˇ02 9/-VD The size of the vertex disruptor jDv j D jDe0 j is at most O .log n log log n/ 0 times Opt.ˇ/ Q -separator . Thus, the size of found ˇ -vertex disruptor Dv is at most O.log n log log n/ times the optimal .ˇ 02  9/-vertex disruptor. As an arbitrary small  can be chosen, setting ˇ D ˇ 02  9 completes the proof. 

5

A Second Look

Another look at the solutions to ˇ-ED and ˇ-VD in order to obtain a better algorithm is discussed in this section. As can be seen earlier, a c-balance cut p for a tree decomposition is used to solve the ˇ-ED problem. This results into a log n ratio for a c-balance cut and another log n for the height of the decomposition tree, leading to a log1:5 n ratio. If a better cut without a need of p decomposing G into a tree is devised, a log n factor can be saved, thus obtaining a log n ratio. In order to do so, let us first describe the following definitions which are the key concepts to a better algorithm by approximating the best “local” cut at each step: ˝ ˛ Cut Ratio. Given a graph G D .V; E/, for any cut S; SN where S  V and N denoted by ˛.S /, is defined as SN D V n S , the cut ratio of hS; Si,

Hardness and Approximation of Network Vulnerability

˛.S / D

c.S; SN / ; N jS jjSj

1659

(3)

where c.S; SN / is the total cost of edges in the cut. The cut ratio metric intuitively allows to find “natural” partitions: The numerator captures the minimum cost criterion, while the denominator favors a balanced partition. Assume that G is connected, the cut disrupts at least jS jjV n S j pairs, those with exact one endpoint in S ; hence, ˛.S / can also be seen as the average cost to disrupt pairs. Sparsest Cut. The sparsest cut is a cut with the smallest cut ratio. Let ˛.G/ denote the minS V ˛.S /, the minimum cut ratio of all cuts in G. Thus, the sparsest cut obtains a cut with the cut ratio ˛.G/. As expected, finding the sparsest cut is also an NP-hard problem [26].

5.1

p O. log n/ Pseudo-approximation for ˇ-Edge Disruptor

As the cut ratio can be seen as the average cost to disrupt the pairwise connectivity, iteratively finding and applying the sparsest cut until obtaining a ˇ-edge separator may result in a small cost ˇ-edge disruptor. We describe a new algorithm using the p sparsest cut in Algorithm 4 and show this algorithm is indeed a log n pseudoapproximation algorithm for the ˇ-ED problem. The recursive bisection algorithm (RBA) iteratively selects a connected component in the graph and calls a subroutine SPARSEST CUT to cut the component into two  smaller ones, until the pairwise connectivity in the graph is no more than ˇ n2 . The subroutine SPARSEST CUT function works as follows: Assume that it takes a connected graph G 0 D .V 0 ; E 0 / as the input it will find a small ratio cut S 0 in G 0 , and returns the pair of hS 0 ; SN 0 i; ˛.S 0 / , that is, the edges in the cut and the cut ratio. At each iteration, a component C  in G that has the minimum cut ratio is selected, and the edges in the sparsest cut of C  are removed from G. After that, C  is no longer a component of G, and new components are introduced into G. The SPARSEST CUT is used again

Algorithm 4 Recursive bisection algorithm (RBA) Eˇ ; for each connected component C in G .CE ; C˛ /  SPARSEST CUT.C / [27] while P .G/ > ˇ n2 Find a component C  of G with minimum cut ratio C˛ Eˇ Eˇ [ CE Remove edges in CE from G for each C 0 in G   0new0component SPARSE CUT.C 0 / CE ; C˛ return Eˇ

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to find cuts and the corresponding cut ratios for the newly created components. This procedure is repeated until the pairwise connectivity in G is no more   than ˇ n2 . Clearly, the performance of the SPARSEST CUT subroutine directly impacts on the performance of RBA. Here the SPARSEST CUT is selected as the algorithm that guarantees the best theoretical result in literature for the min ratio cut problem [27]. N given a The result in [27] gives us a polynomial-time algorithm to find a cut hS; Si p graph G D .V; E/ with the cut ratio at most O. log n/˛.G/. This selection yields p an O. log n/ pseudo-approximation algorithm for the ˇ-edge disruptor problem as analyzed below. Lemma 9 Given a graph G D .V; E/ and a subset of edges M!  E, if ! D !/ P.G/  P.GŒE n M! / > 0, then c.M  ˛min .G/, where ! ˛min .G/ D minf˛.C / j C is a connected component of Gg: Proof We first  prove for the case G is connected, that is, ˛min .G/ D ˛.G/ and P.G/ D n2 . Let C1 ; C2 ; : : : ; Ck be connected components in GŒE n M!  and let Ci .V / denote the set of vertices in component Ci . By definition of ˛.G/, we have ˛.G/  ˛.Ci .V //. Thus,  c Ci .V /; Ci .V /  ˛.G/jCi .V /jjV n Ci .V /j Take the sum over all components Ci , we have X X  c Ci .V /; Ci .V /  ˛.G/ jCi .V /j.n  jCi .V /j/ i

i

Simplify both sides and note that

X

jC.Vi /j D n, we have

i

  X X 2 2c.M! /  ˛.G/ n jC.Vi /j  jC.Vi /j i

 2˛.G/

 n 2

i



X

 P.Ci /  2˛.G/ !

i

Hence, the lemma holds when G is connected. Now, if G is not connected, let T1 ; T2 ; : : : ; Tl be connected components of G, .j / M! be the intersection of M! and the edges in Tj , and Tj0 be the subgraphs .j /

obtained from Tj after removing M! . Applying the above results to the case that the graph is connected on each connected component, we have

Hardness and Approximation of Network Vulnerability

c.M! / D

X

c.M!.j / / 

j

X

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 ˛.Tj / P.Tj /  P.Tj0 /

j

X P.Tj /  P.Tj0 /  ˛min .G/ j

D ˛min .G/ .P.G/  P.GŒE n M! // Thus, the lemma holds for every graph G.



Theorem 12 If SPARSEST CUT is an approximation algorithm with a factor f .n/ for the min cut ratio problem, then RBA returns a ˇ-edge disruptor of cost at most f .n/ OPTeˇ0 , for any 0 < ˇ 0 < ˇ, where OPTeˇ0 denotes the cost of a minimum ˇ 0 -edge disruptor.

Proof By Lemma 9, if removing a set of edges M!  E from G disrupts ! pairs !/ of vertices, then c.M  ˛min .G/. Thus, at any round in the while loop of RBA, ! since a set of edges Eˇ0 in a minimum ˇ 0 -edge disruptor, for some 0 < ˇ 0 < ˇ, can  disrupt at least .ˇ  ˇ 0 / n2 more pairs in G, we have OPTeˇ0    ˛min .G/: .ˇ  ˇ 0 / n2

(4)

From the hypothesis that SPARSE CUT is an f .n/ factor approximation algorithm for the min cut ratio problem, the average cost to disrupt a pair by removing CE is upper bounded by f .n/˛min .G/. By (4), the average cost to disrupt pairs in the graph at any step is at most OPTeˇ0  f .n/ .ˇ  ˇ 0 / n2  Therefore, even when Eˇ disrupt all n2 pairs in G, the total cost is no more than ! OPTeˇ0 n f .n/ n OPTeˇ0 : f .n/  0 ˇ  ˇ0 .ˇ  ˇ / 2 2 Hence, the theorem follows.



Remarks If the underlying graph is directed, the directed sparsest cut algorithm in [24] can be used in the place of the sparsest cut algorithm in [27] to obtain a pseudoapproximation algorithm with the same ratio. Note that for directed graphs, a pair of vertices .u; v/ is said to be connected if there exists a path from u to v and from v to u (i.e., connected in both directions).

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p O. log n/ Pseudo-approximation for ˇ-Vertex Disruptor

5.2

For the ˇ-vertex disruptor problem, the first phase – vertex conversion p – as discussed in Sect. 4 still can be used. However, in order to obtain an O. log n/ pseudoapproximation, a relationship between ˇ-VD and ˇ-ED needs to be defined in order to take advantage of Algorithm 4 instead of using the phase two – size constraint cut. Recall that given a directed graph G D .V; E/ for which we want to find a small ˇ-VD, after the vertex conversion, we should obtain a directed graph G 0 D .V 0 ; E 0 / which will have twice the number of vertices in G, that is, jV 0 j D 2jV j D 2n. Consider a directed edge disruptor set De0  E 0 that contains only edge in EV0 . There is a one-to-one correspondence between De0 to a set Dv D fv j .v ! vC / 2 De0 g in G.V; E/ which is a vertex disruptor set in G. Since G and G 0 have different maximum pairwise connectivity, .n1/n for G and .2n1/2n for G 0 , the fractions 2 2 0 of pairwise connectivity remaining in G and G after removing Dv and De0 are, however, not exactly equal to each other. Lemma 10 A ˇ-edge disruptor set in the directed graph G 0 induces the same cost ˇ-vertex disruptor set in G. 0 Proof Let us denote Dv and De0 forvertex disruptor in G and edge disruptor in G n. 2n 0 0 0 Given P.G ŒE n De /  ˇ 2 we need to prove that: P.GŒV nDv  /  ˇ 2 where n D jV j. Assume GŒV nDv  has l SCCs of size at least 2, say Ci ; i D 1 : : : l. The corresponding SCCs in G 0 ŒE 0 n De0  will be Ci0 ; i D 1 : : : l where jCi0 j D 2jCi j. .k / .2k/ k.nk/  0; for all 0  k  n. We have Since 2n2  n2 D .n1/n.2n1/ . 2/ .2/ l jCi j l jCi0 j X X P.GŒV nDv  / 2 2 n n  D 2n  ˇ 2

i D1

2

i D1



2

Lemma 11 A ˇ-vertex disruptor set in G induces the same cost .ˇ C /-edge disruptor set in G 0 for any  > 0. Proof The same Given   notations in the proof of Lemma 10 are used here. P.GŒV nDv  /  ˇ n2 we need to prove that P.G 0 ŒE 0 n De0 /  .ˇ C / 2n 2 . We have P.G 0 ŒE 0 nDe0 / 2n 2

D

l X i D1

P.GŒV nDv  / jCi j.n  jCi j/ n C .n  1/n.2n  1/ 2

Hardness and Approximation of Network Vulnerability

D

 P.GŒV nDv  / n 1 2

0 is an arbitrary small constant. Proof Let G be a directed graph with uniform vertex costs in which we wish to find a ˇ-vertex disruptor. Construct G 0 as described at the beginning of this section. For ˇ 0 < ˇ, apply the given approximation algorithm to find in G 0 a ˇedge disruptor, denoted by De0 , with the cost at most f .n/  Optˇ0 ED .G 0 /, where Optˇ0 ED .G 0 / is the cost of a minimum ˇ-edge disruptor in G 0 . From Lemma 10, De0 induces in G a ˇ-vertex disruptor Dv of the same cost. We shall prove that Optˇ0 ED .G 0 /  Optˇ0 VD .G/ C 0 ; where Optˇ0 VD .G/ is the cost of a minimum ˇ 0 -vertex disruptor in G and 0 is some positive constant. It follows that the cost of Dv will be at most f .n/  .Optˇ0 VD .G/ C 0 /  .1 C /f .n/Optˇ0 VD .G/ Here, assume that Optˇ0 VD .G/ > 0 

C2

0  ; otherwise, Optˇ 0 VD .G/ can be found in

time

O.n / (polynomial time). From an optimal ˇ-vertex disruptor of edge  construct its corresponding  G, 0 disruptor De in G 0 . If P.G 0 ŒE n De   ˇ 2n /  OptˇVD .G/, 2 , then OptˇED .G   and we yield the proof. Thus, only the case P.G 0 ŒE n De  > ˇ 2n is considered. 2 Among SCCs of G 0 ŒE n De , there must be a SCC of size at leastlˇ2n m or else   2n 1 0  1 ˇ2n G ŒE n De   ˇ  ˇ 2 (contradiction). Remove 0 D ˇ vertices 2 from that SCC. The pairwise connectivity in G 0 ŒE n De  will decrease at least  ˇ2n  ˇ1 ˇ1 D 2n  ˇ12  n for sufficient large n. From Lemma 11, the pairwise connectivity after removing vertices will be less than

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 ˇC

1 2n  1



! ! 2n 2n n ˇ 2 2

Therefore, after removing at most 0 vertices from De , a ˇ-edge disruptor is obtained. Hence, OptˇED .G 0 /  OptˇVD .G/ C 0 : p From the above theorem and the O. log n/ pseudo-approximation algorithm for directed ˇ-edge disruptor (shown in Algorithm 4 ), we have the following result:  Theorem 14 For any 0 pˇ 0 < ˇ, there is an algorithm that finds a ˇ-vertex disruptor of cost at most O. log n/.OPTvˇ0 /, where OPTvˇ0 is the optimal ˇ-vertex disruptor.

6

Literature

Several existing works on network vulnerability and survivability have been investigated, roughly categorized into two periods. Researches during the early period mainly focused on investigating the node’s centrality and prestige, using functions of node degree [3, 5, 28, 29] as the only measure. The main measures of centrality can be classified as degree centrality, betweenness, closeness, and eigenvector centrality. Readers are referred to [30] to find more details about this centrality measurement. As wireless ad hoc networks came forth, several recent studies have aimed to discover the critical nodes whose removal causes the network disconnected, regardless of how fragmented it is! Several centralized, distributed, and localized heuristics have been proposed. Some representatives are centralized DFS-based search [7–9, 31], distributed disjoint path [10, 32], and localized k-critical [11]. Most of these algorithms suffer from high communication overhead to discover the network partition and could not efficiently locate the critical nodes. In addition, none of these methods have been proven theoretically. Unfortunately, none of the above work reveals the global damage done on the network in the case of multiple nodes/links failing simultaneously, thus inaccurately assessing the network vulnerability. None of these methods prove the performance bound of their solutions either. This chapter presents a novel paradigm via two new optimization models for quantitatively characterizing the vulnerability of networks. The proposed approaches aim to identify the key nodes who play a vital role in the network overall performance, such as whose removal maximizes the network fragmented. This study defines the objective function differently in terms of the connected components rather than a traditional direct measure of algebraic connectivity, thereby providing an excellent way to assess the network vulnerability.

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Conclusion

This chapter presents two models along with their complexity hardness and solutions to the network vulnerability. There are a couple of notes that we would like to discuss further as follows: It is easy to see that when k is large enough, the CVD problem cannot be approximated within a factor of ˛.n/ for any polynomial-time computable function ˛.n/ unless P D NP by reducing from the vertex cover problem. To avoid the case of the optimal total pairwise connectivity after deleting S equal to 0, k can be set as k < jEj= where  is the maximum degree of a given graph G. Approximation solutions to this problem are still open. The two optimization models can be modified to further assess the network vulnerability under other objective functions. For example, in the context of networks, QoS (quality of services) is an important concept. Therefore, we can consider maximizing the QoS instead of total pairwise connectivity. Other objective functions include the following: minimize the number of pairwise nodes such as their shortest paths in GŒV n S   d for some distance d minimize the connected vertex disruptor. All these problems can be studied on a probabilistic graph G D .V; E/ where each edge e 2 E has a failure probability pe 2 Œ0; 1, which models a dynamic network. As many complex networks such as the Internet, WWW, biological networks, and social networks can specifically be modeled as power-law graphs, where the number of nodes of degree i is be ˛ = i ˇ c for some constants ˛ and ˇ, it would be interesting to investigate the proposed problems (ˇ-ED, ˇ-VD, CVD, and CED) in power-law networks. As shown early, these problems still remain NP-complete on power-law graphs. Even though, does the power-law distribution play any key factor on the inapproximability? Are we able to obtain a smaller such as a constant approximation algorithm for these problems on power-law graphs?

Recommended Reading 1. R. Albert, H. Jeong, A.-L. Barabasi, Error and attack tolerance of complex networks. Nature 406, 378–482 (2000) 2. R. Ravi, D. Williamson, An approximation algorithm for minimum cost vertex connectivity problems, in Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (ACM, New York, 1995), pp. 332–341 3. S.P. Borgatti, Identifying sets of key players in a social network. Comput. Math. Org. Theory 12(1), 21–34 (2006) 4. E. Zio, C.M. Rocco, D.E. Salazar, G. Muller, Complex Networks Vulnerability: a multipleobjective optimization approach, in Proceedings of Reliability and Maintainability Symposium, Orlando, 2007, pp. 196–201 5. L.C. Freeman, Centrality in social networks i: conceptual clarification. Soc. Netw. 1(3), 215– 239 (1979) 6. P.K. Srimani, Generalized fault tolerance properties of star graphs source, in Proceedings of the International Conference on Computing and Information, Ottawa, Orlando, 1991, pp. 510–519 7. M. Duque-Anton, F. Bruyaux, P. Semal, Measuring the survivability of a network: connectivity and rest-connectivity. Europ. Trans. Telecommun. 11(2), 149–158 (2000)

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8. R. Tarjan, Depth first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972) 9. D. Goyal, J. Caffery, Partitioning avoidance in mobile ad hoc networks using network survivability concepts, in Seventh IEEE Symposium on Computers and Communications (ISCC’02), Taormina-Giardini Naxos, Orlando, 2002 10. M. Hauspie, J. Carle, D. Simplot, Partition detection in mobile ad hoc networks using multiple disjoint paths set, in Objects, Models and Multimedia Technology Workshop, Switzerland, 2003 11. M. Jorgic, I. Stojmenovic, M. Hauspie, D. Simplot-Ryl, Localized algorithms for detection of critical nodes and links for connectivity in ad hoc networks, in Proceedings of 3rd IFIP MED-HOC-NET Workshop, Bodrum, 2004 12. A.L. Barabasi, H. Jeong, Z. Nada, E. Ravasz, A. Schubert, T. Vicsek, Evolution of the social network of scientific collaborations. Phys. A 311, 590 (2002) 13. E. Dahlhaus, D.S. Johnson, C.H. Papadimitriou, P.D. Seymour, M. Yannakakis, The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994) 14. L.G. Valiant, Universality considerations in vlsi circuits. IEEE Trans. Comput. 30(2), 135–140 (1981) 15. W. Aiello, F. Chung, L. Lu, A random graph model for massive graphs. in STOC ’00 (ACM, New York, 2000) 16. A. Ferrante, G. Pandurangan, K. Park, On the hardness of optimization in power-law graphs. Theory. Comput. Sci. 393(1–3), 220–230 (2008) 17. http://en.wikipedia.org/wiki/hopcroft-karp algorithm 18. M.R. Garey, D.S. Johnson, The rectilinear steiner tree problem in np complete. SIAM J. Appl. Math. 32, 826–834 (1977) 19. D´enes K˜onig, Gr´afok e´ s m´atrixok. Matematikai e´ s Fizikai Lapok 38, 116–119 (1931) 20. J. H˚astad, Clique is hard to approximate within n1-, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Washington, DC, 1996), p. 627 21. M. Stoer, F. Wagner, A simple min-cut algorithm. J. ACM 44(4), 585–591 (1997) 22. D. Wagner, F. Wagner, Between min cut and graph bisection, in MFCS ’93: Proceedings of the 18th International Symposium on Mathematical Foundations of Computer Science (Springer, London, 1993), pp. 744–750 23. I. Dinur, S. Safra, On the hardness of approximating minimum vertex cover. Ann. Math. 162, 2005 (2004) 24. A. Agarwal, M. Charikar, K. Makarychev, Y. Makarychev, O(log n) approximation algorithms for min uncut, min 2cnf deletion, and directed cut problems, in STOC ’05: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing (ACM, New York 2005), pp. 573–581 25. G. Even, J.S. Naor, S. Rao, B. Schieber, Divide-and-conquer approximation algorithms via spreading metrics. J. ACM 47(4), 585–616 (2000) 26. S. Arora, S. Rao, U. Vazirani, Expander flows, geometric embeddings and graph partitioning, in STOC ’04: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing (ACM, New York, 2004), pp. 222–231 p Q 2 / time. SIAM J. 27. S. Arora, E. Hazan, S. Kale, O. log n/ approximation to sparsest cut in O.n Comput. 39(5), 1748–1771 (2010) 28. A. Bavelas, A mathematical model for group structure. Human Org. 7, 16–30 (1948) 29. S.P. Borgatti, M.G. Everett, A graph-theoretic perspective on centrality. Soc. Netw. 28(4), 466– 484 (2006) 30. D. Koschitzki, K.A. Lehmann, S. Peters, L. Richter, D. Tenfelde-Podehl, O. Zlotowski, in Centrality Indices. Lecture Notes in Computer Science, vol. 3418 (Springer, Berlin/New York, 2005) pp. 16–61 31. A. Karygiannis, E. Antonakakis, A. Apostolopoulos, Detecting critical nodes for manet intrusion detection systems, in Proceedings of Security, Privacy and Trust in Pervasive and Ubiquitous Computing, Lyon, Orlando, 2006 32. M. Sheng, J. Li, Y. Shi, Critical nodes detection in mobile ad hoc network, in Proceedings of the 20th International Conference on Advanced Information Networking and Applications (AINA’06), Vienna, Orlando, 2006

Job Shop Scheduling with Petri Nets Hejiao Huang, Hongwei Du and Farooq Ahmad

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Deadlock-Free Design of JSSP with Petri Nets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Terminology About Petri Nets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Petri Net Modeling of JSSP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Deadlock-Free Design for JSSP with Multi-resources Sharing. . . . . . . . . . . . . . . . . . . . . 2.4 A Case Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Modular-Based JSSP Design and Makespan Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Operators and Makespan Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modular-Based Design and Optimizatioan: A Case Study. . . . . . . . . . . . . . . . . . . . . . . . . . 4 Genetic Algorithm for JSSP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Petri Net-Based Modeling and Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Genetic Algorithm for FJSSP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Simulation Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In job shop scheduling problem (JSSP), deadlock-free design and scheduling optimization are important issues, which are difficult to tackle in the JSSP having multiple shared resources. To deal with these issues, this chapter introduces a modular-based system design method for modeling and optimizing JSSP. Several operators are presented for handling the combination of JSSP modules and

H. Huang () • H. Du Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, People’s Republic of China e-mail: [email protected]; [email protected] F. Ahmad Information Technology, University of Central Punjab, Lahore, Pakistan e-mail: [email protected] 1667 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 51, © Springer Science+Business Media New York 2013

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resource sharing. For handling deadlock issues with multi-resources sharing, the conditions on “dead transitions” and “circular waiting” are considered. Based on a transitive matrix, deadlock detection and deadlock recovery algorithms are developed in order to get a deadlock-free system. For each operator, the partial schedule and makespan of each module are obtained. The optimal scheduling of JSSP can be derived from these theoretical results step by step. With the modularbased system design method, one cannot only design a correct system model which is live, bounded, reversible, and deadlock-free and terminate properly but also find the optimal scheduling of JSSP in a formal way. For handling large-scale JSSP, the traditional optimization technologies such as genetic algorithm, branch and bound method, hybrid methods, and rule-based methods can be applied together with the modular-based approach in this chapter. Based on the Petri net model, a genetic algorithm is also introduced here.

1

Introduction

The job shop scheduling problem (JSSP) considered in this chapter can be stated as follows: There are a finite set of n jobs, each of which consists of a chain of operations, and a finite set of m machines, which are able to process at most one operation at a time. Each operation must be processed during an uninterrupted time on some given machine, and the purpose is to find a schedule to allocate the resources rationally and finish the task in time. The allocation of resources, such as machines, robots, workers, and AGVs, has to be considered for the modeling and functioning of JSSP, which is a challenge for a system designer. Among others, a JSSP design has the following characterizations and challenges: • Heavy reliance on resource sharing and deadlock avoidance. In a JSSP, resources such as robots, machines, assembly lines, and buffers are sometimes shared by several processes. Therefore, handling resource sharing becomes an important aspect during JSSP design. It is noteworthy that “sharing” does not imply simultaneous usage while simultaneous usage can be handled by re-enterable codes in software systems, which is not allowed in JSSP, and “sharing” requires a resource to be occupied by some part(s) exclusively during utilization and to be released afterward. In single resource-sharing scheduling, if a resource is occupied by others, the job has no other choice but waits until the resource is released. For multiple resources systems, a wrong order of resource allocation, when the resources are limited or distributed unreasonably, deadlocks, or overflows may occur. In a JSSP, deadlock may be the state of “circular waiting” which occurs unexpectedly. Thus, deadlocks must be detected and prevented during system design. In this chapter, handling deadlock issues will be discussed in Sect. 2. • Strong tendency toward component-based architecture. The processes and resources of a JSSP are modeled, built, or owned by different unrelated parties, at different times, and under different environments. Therefore, in order not to have severe interference in their individual developments, it is better for a JSSP to adopt a modular-based architecture. The architecture enlarges the reusability

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of the system modules and increases the system running speed. However, it faces many difficulties during integration, including operation system, interfaces, and communication. In the literature, net-based approaches concerning integration include place merging, transition merging, path merging, and modular synthesis. The related references can be found in [1, 2]. This chapter handles the integration problem by applying property-preserving composition operators [3–6] such as enable, choice, interleaving, and iteration. For each operator, the partial schedule and makespan of each module are obtained. The optimal scheduling of JSSP can be derived from these theoretical results step by step. With the modularbased system design method, one cannot only design a correct system model which is live, bounded, and reversible and terminate properly but also find the optimal scheduling of JSSP in a formal way. The technical details concerning modular-based design of JSSP with Petri net will be introduced in Sect. 3. In the literature, much work involved the research on the above features. The description of the related approaches and a brief review can be found in Deniz K. Terry’s PhD thesis [7].

2

Deadlock-Free Design of JSSP with Petri Nets

Petri nets are suitable for modeling sequence, concurrency, choice, and mutual exclusion in discrete event systems, and they have been extensively used to model and analyze the JSSP [8–12]. Further, they are useful for allocating resources and detecting or preventing unexpected system behaviors, such as deadlock and capacity overflow. Motivated by the features of Petri nets, this chapter uses timed Petri nets (TPNs) to model the JSSP, while deadlock detection and recovery are performed by considering the transitive matrix of Petri net model [13]. In the literature, Petri net-based deadlock-free design for JSSP considers the following three types of approaches. The first one is relying on the techniques concerning siphons and traps. Elementary siphon invariants in Petri net structure are useful for analyzing deadlock in flexible manufacturing system (FMS) [14–17]. The deadlock-free method is addressed by adding a monitor or controller to avoid deadlock structure [16–18]. Recently siphons, traps, and invariants have been extensively studied to prevent deadlock and analyze the properties of Petri nets. However, deadlock analysis for compositional system that shares common resources could not grab much attention of the researchers because the number of siphons grows rapidly with the size of final composite net. The second is about heuristic scheduling algorithm, such as genetic algorithm, used to get an optimum and deadlock-free scheduling of FMS, for example, in [19, 20]. This method is limited to some special problems in JSSP with the given system model; it is not suitable for general cases. The last one is based on the transitive matrix, which is driven from the directed graph in [13]. The place transitive matrix describes the transferring relation from one place to another place through transitions. The analysis of the cyclic scheduling for the determination of the optimal cycle time is studied, and transitive matrix

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has efficiently been used to slice off some subnets from the original net in [21]. Deadlock-free conditions are given in [22, 23] based on transitive matrix, and an algorithm for finding deadlock is given in [24] by using the place transitive matrix. In this chapter, JSSP is studied by using the TPNs. Further, modular-based architecture is adopted and transitive matrix-based deadlock detection and recovery algorithms are presented. Firstly, each job is modeled by a “small” Petri net, and then these “small” Petri nets, representing different jobs to be completed, are merged together by fusing the common resource places. In multi-resources sharing, an efficient algorithm based on place transitive matrix is presented to detect deadlock for each two jobs of JSSP. In addition, a deadlock recovery method is introduced to give the deadlock-free schedule. Finally, a deadlock-free method for system design is obtained based on these two algorithms. A case study is presented to illustrate the efficiency of the given methods.

2.1

Terminology About Petri Nets

In this subsection, the basic definitions about Petri net and its structural classification are presented for completeness. The related terminology is mostly taken from [13, 25–29]. Definition 1 (Petri Net) A Petri net (N , M0 / is a net N = (P , T , F , W / with an initial marking M0 where, P is a finite set of places; T is a finite set of transitions such that P \ T D Ø and P [ T ¤ Ø; F  (P  T /[ (T  P / is the flow relation; and W is a weight function such that W .x, y/ 2 N C if (x, y/ 2 F and W .x, y/ = 0 if (x, y/ … F M0 is a function M0 : P ! N such that M0 .p/ represents the number of tokens in place p 2 P . Definition 2 (Pre-set and Post-set) For x 2 P [ T ,  x = fyj (y; x/ 2 F g and x  = fyj (x; y/ 2 F g are called the pre-set (input set) and post-set (output set) of x, respectively. For a set X  P [ T ,  X D [x2X x and X  D [x2X x  . Definition 3 (Marking) A marking M.k/ =[p1 .k/, p2 .k/, . . . , pm .k/]T is an m-vector of nonnegative integer, where pi .k/ is the number of tokens in place pi immediately after firing the kth transition in a firing sequence. Especially, M.0/ denotes the initial marking; it is the same as M0 . Definition 4 (Incidence Matrix) B  is an m  n matrix of input function, where the entry of row i column j is B  [i , j ]= W(pi , tj /; B C is an mn matrix of output function, where the entry in row i column j is B C [i , j ]= W(tj , pi /. B D B C  B  is called incidence matrix. Definition 5 (Transitive Matrix) Let LP be place transitive matrix, and let LV be transition transitive matrix with m rows and m columns, where LP D B  .B C /T

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and LV D .B C /T B  . The labeled place transitive matrix LVP is an m  m matrix satisfying LVP D B  diag(t1 , t2 , . . . , tn / (B C /T , where the elements of LVP describe the direct transferring relation from one place to another through one or more transitions. Definition 6 (Weighted Place Transitive Matrix) The Weighted place transitive matrix is denoted as LVP  which is a transformation from LVP : If a transition tk appears s times in the same column of LVP , then replace tk in LVP by tk =s in LVP  ; otherwise the elements in LVP  are the same as LVP . Definition 7 (Path) For a Petri net (N , M0 /=(P , T , F , M0 /, a path  = x1 x2 . . . xn is a sequence of places and transitions such that (xi , xi C1 / 2 F , 1  i  n;  = x1 x2 . . . xn is said to be elementary if 8 xi , xj 2 , i ¤ j ) xi ¤ xj . Definition 8 (Transition (Resp., Place) Path) For an elementary path  = x1 x2 . . . x2n in a Petri net N D .P , T , F , M0 /, ’ = x1 x3 . . . x2n1 is said to be a transition (resp., place) path of  if xi 2T (resp., xi 2P), i =1, 3, . . . , 2n  1. Definition 9 (Timed Petri Net) A TPN (N , M0 / is a Petri net with a time parameter. That is N D .P , T , F , ), where P , T , F , and M0 are the same as given in Definition 1 and  is a time function : T ! N (set of positive integers) associated to transitions such that for t 2 T ,  (t/ represents the time duration of firing t. Definition 10 (State Machine, Augmented State Machine) A Petri net (PN) is said to be a state machine (SM) if and only if 8t 2 T : jt  j D j tj = 1. An augmented state machine (ASM) is a Petri net (N , M0 / = (P[R, T , I , O, M0 / whose places consist of two disjoint subsets P and R such that the following conditions hold: 1. The net N obtained by removing all the resource places in R is a state machine. 2. Each resource place r 2 R is associated with k pairs of transitions (ti , tj /, i , j = 1, 2, . . . , k, where r  = f ti ji =1, 2, . . . , kg,  r = f tj jj =1, 2, . . . , kg. 3. Initially, the place in P without input transitions are marked by M0 and other places are not marked. 4. M0 (p) = 1, 8p 2 R: Definition 11 (Place Fusion) Consider two Petri nets (N1 , M10 / = (P1 , T1 , F1 , M10 / and (N2 , M20 / = (P2 , T2 , F2 , M20 /, where P1 \ P2 D R ¤ Ø. Let (N , M0 / = (P , T , F , M0 / be composed from (N1 , M10 / and (N2 , M20 / by operation place fusion. Then the elements in (N , M0 / are defined as follows: P D P1 [ P2 , T D T1 [ T2 , F D F1 [ F2 , and 8 < M10 .p/ M0 .p/ D M20 .p/ : maxfM10 .p/; M20 .p/

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Definition 12 (Place Invariant and Minimal Place Invariant) For a Petri net (N , M0 /, a place invariant is an m-vector y  0 such that B T  y D0, where B is m  n incidence matrix. It is represented by nonzero entries of vector y  0, which is called the support of place invariant. Place invariant is said to be minimal if no proper subset of the support is also a support. Definition 13 (Siphon and Minimal Siphon) A set of places S  P is called a siphon if  S  S  , and a siphon S is called minimal if there does not exist S 0 such that S 0  S .

2.2

Petri Net Modeling of JSSP

In this subsection, JSSP is described, and its modeling through TPN is explained. Problem Description. JSSP discussed in this chapter can be stated as follows: • J D fJ1 ; J2 , . . . , Jn g is a finite set of n jobs. • O D fO1 , O2 , . . . , On g is a set of operations for the jobs, and Oi D fOi1 , Oi 2 , . . . , Oi k g is a chain of operations for job Ji , 1  i  n. • M D fM1 , M2 , . . . , Mm g is a finite set of m machines. • Each machine can handle at most one operation at a time. • f i1 ,  i 2 , . . . ,  i k g is the uninterrupted period of a given length for each operation processed in one or some given machines. • The resources in JSSP may be workers, who are responsible for operating the machines, robots, or other equipments. • The purpose of JSSP is to find a schedule, that is, a reasonable allocation of the operations to time intervals and machines or other resources such that the makespan is minimized. In this chapter, TPN is used to model the JSSP, and there are two types of places, that is, operation places and resource places. First of all, take no account of resources in JSSP, the predefined sequence of operations in each job is formulated as the transition sequence ti1 ti 2 . . . ti k . Then add a set of places P to connect these transitions together, such that pi1 is the input place of ti1 and initially marked with one token. Generally, for each transition tij , its output place pi.j C1/ is the input place of ti.j C1/, j = 1, 2, . . . , k, and i = 1, 2, . . . , n. As a result, each job in the problem is modeled as n state machines with the unique path pi1 ti1 pi 2 ti 2 . . . pij tij . . . pi k ti k pi.kC1/ (i = 1, 2, . . . , n), and each net has an initial marking Mi 0 and a final marking Mif . Now, consider the set of resources R. Each resource here is denoted as a resource place ri initially marked with a token. When an operation tij requires some resource ri , the output of ri is tij . Moreover, resources’ release follows the “hold while waiting” property: Suppose the resource rj is needed by operation tij of job Ji , after operation tij is finished, if the next operation ti.j C1/ of job Ji also needs rj ,

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then rj continue to be held by operation ti.j C1/ of job Ji ; otherwise the resource rj is released. After adding resource place, the model of each job is an ASM. When some operations are ready to execute, there will be one token in the operation places, which is called operation enabled, and if the resources are available to use, there will be one token in the resource places, which is called resource enabled. A transition ti can be fired only when its operation place and resource place are both enabled. Property 1 Every ASM for an individual job is executable. Proof Every ASM without resource places is a sequence of operations represented by the transitions in the form of a state machine. Such type of state machine is executable in sequence if the input place of first operation in sequence is marked by a token. Since marked resource places are added to the transitions in such a way that they become output and/or input places of transitions in the form of ASM, furthermore, resources are used exclusively and follow the “hold while waiting” property, every ASM is executable in sequence. Property 2 For every ASM of an individual job, the final marking is reachable.

Proof The Property 2 is the consequence of the Property 1. Property 3 For each r 2 R, which is not on self-loop in the ASM of an individual job, there exists a minimal place invariant with only marked place r 2 R: Proof Each resource place, which is not on a self-loop, represents a resource utilized by the sequence of operations. Further, each such type of resource place is added to the sequence of transitions representing the chain of operations in such a way that it become input place for the first transition and output place for last transition in that sequence. Therefore, allocation of each resource place to the chain of operations is represented by the directed cycle with only marked place r 2 R representing the availability of a resource. Hence, allocation of resources imposes the existence of minimal place invariant for each r 2 R which is not on a self-loop. Property 4 Every place invariant in the ASM of an individual job is a minimal siphon with only one marked place r 2 R. Proof From Property 3, since place invariants in ASM are the only directed cycles, each place invariant stands for a minimal siphon with only marked place r 2 R. The cost of the implementation of operation is described as time function (t/ in TPN for transition t. As described above, each job is modeled as an ASM. Since there are identical shared resources in each ASM, these ASM can be merged

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together by applying place fusion operator, and a compositional net is built for modeling the whole JSSP. Thus, the scheduling purpose is to find a firing sequence in the resultant Petri net model from the initial marking M0 to the final marking Mf and to get a minimum makespan. If there is no common resource in two jobs, the makespan is the maximal one of these two sub-models since the two jobs can be processed in parallel. Property 5 The compositional net, obtained by merging the ASMs of individual jobs, is an ASM. Proof Since the compositional net fulfills the conditions of ASM described in Definition 10, which are given below: 1. The net N obtained by removing all the resource places in R is a state machine. 2. Each resource place r 2 R is associated with k pairs of transitions (ti , tj /, i , j = 1, 2, . . . , k, where rP = f ti ji =1, 2, . . . , kg, rP = f tj jj =1, 2, . . . , kg. Therefore, the compositional net, which is obtained by merging the independent ASMs modeling different jobs to be completed in JSSP, is again an ASM.

Property 6 Every cycle in the compositional ASM is marked. Proof R is the only set of marked places in ASM. From Property 3, by removing every place r 2 R; there will be no place invariant and consequently no directed cycle in ASM. Thus, every cycle in compositional ASM contains a place from R, which is marked.

Property 7 For a compositional ASM, for every r 2 R, there exists a minimal siphon which contains only one marked place r. Proof According to Property 6, every cycle contains a place from R. From Property 4, every minimal siphon in ASM contains a cycle. Therefore, every siphon in the compositional ASM, which contains at least one minimal siphon, is marked with r 2 R: Figure 1 is an example of two jobs of JSSP sharing two machines. Each job contains two operations which are associated with time functions (t1 /, (t2 /, (t3 /, and (t4 / respectively. The operations t1 , t2 in Job1 are both processed on machine r1 , and operations t3 , t4 in Job2 are processed on machines r1 and r2 , respectively. The original states of Job1 and Job2 are M10 = (1, 0, 0, 1)T and M20 = (1, 0, 0, 1, 1)T , respectively, and the final states are M1f = (0, 0, 1, 1)T and M2f = (0, 0, 1, 1, 1)T , respectively. Since machine r1 is occupied by both jobs, these two sub-models are merged together by place fusion, and the resultant Petri net model is shown in Fig. 2. Then the scheduling problem is to find a firing sequence in the resultant Petri net,

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from the initial marking M0 = (1, 0, 0, 1, 0, 0, 1, 1)T to the final marking Mf = (0, 0, 1, 0, 0, 1, 1, 1)T such that the makespan is minimized.

2.3

Deadlock-Free Design for JSSP with Multi-resources Sharing

Deadlock is an unexpected state. In a multi-resource-sharing system, deadlock may occur when the resources are limited or distributed unreasonably. In this section, an algorithm is presented to detect deadlock and describe a method to recover deadlock for multi-resource-sharing systems.

2.3.1 Basics of Deadlock • Dead Marking In JSSP model, a marking M is said to be a dead marking if M is not the final marking Mf , and there is no transition which can be fired at this marking M . Formally, the dead marking can be expressed as 8 M 2 R.N , M0 /, if M ¤ M f and 8 t 2 T :  M [t >, then M is a dead marking.

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Fig. 3 An example to illustrate dead transition

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• Dead Transition In JSSP model, a transition t is said to be a dead transition if t is operation enabled but not resource enabled. In this chapter, dead transition always appears in the post-set of the operation place with tokens in dead marking. Figure 3 illustrates the dead transition in a Petri net model. The initial marking is M0 = (1, 0, 0, 0, 0)T and the final marking is Mf = (0, 0, 0, 1, 0)T . In this model, after t1 fires, the marking M1 = (0, 1, 0, 0, 0)T ¤ Mf . Since there are no tokens in p5 , t2 cannot be fired and there is no reachable marking for M1 . Then marking M1 is a dead marking and transition t2 is a dead transition. • Circular Waiting In manufacturing system, “circular waiting” is the state that for two or more tasks, each of them holds a machine, but all of them are waiting for each other to release the machine. In this chapter, the state of two tasks is considered. It is one of the inducements of deadlock in manufacturing system. In Petri net, “circular waiting” can be demonstrated as follows: for Petri nets Ni (i =1, 2), r1 and r2 are resource places in Ni , and ti a ti b ti c tid (i =1, 2) is a transition path associated with resource place. In N1 , r1 is associated with the pairs of transitions (t1a , t1c /, and r2 is associated with the pairs of transitions (t1b , t1d /; in addition, rP1 D t1a , rP1 D t1c and rP2 D t1b , rP2 D t1d . in N2 , r1 is associated with the pairs of transitions (t2a , t2c /, and r2 is associated with the pairs of transitions (t2b , t2d /; in addition, rP1 D t2b , rP1 D t2d and rP2 D t2a , rP2 D t2c . Figure 4 is a model of “circular waiting” which satisfies rP1 Dft1a , t2b g, rP1 Dft1c , t2d g and rP2 Dft1b , t2a g, rP2 Dft1d , t2c g. • Deadlock Based on the place fusion in JSSP, deadlock occurs when there are dead transitions that cause “circular waiting” in the system. • Deadlock-Free A Petri net system is said to be deadlock-free if there is at least one firing sequence from the initial marking M0 to the final marking Mf ; otherwise it is said that the system contains deadlock. • Marking Transformation The transformation of marking is defined as MR .k C 1/T D M.k/T LVP  , where MR .kC1/ is an m-vector of nonnegative integer and is a marking calculated from reachable marking M.k/=[p1 .k/, p2 .k/, . . . , pm .k/]T and LVP  is the weighted place transitive matrix (Definition 6) [13].

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The transformation equation shows the flow relation of a token in pi .k/ from M.k/ to MR .k C 1/ by the weighted place transitive matrix. In this P Pequation, if the coefficient of ti in the function pi LVPij  is an integer, then let pi LVPij  D 1, i

i

which P means tokens in pi .k/ can be transferred from M.k/ to MR .kC1/; otherwise, let pi LVPij  D 0, which means tokens in pi .k/ cannot be transferred from M.k/ i

to MR .k C 1/. However, MR .k C 1/ only presents the flow relation of tokens; it may not correspond to a reachable marking M.k C 1/. In order to improve this situation, some constrained conditions are added to this equation: P When the coefficient of ti in the function pi LVPij  is an integer, i P for example, pi LVPij  D 1 in MR .k C 1/, and the tokens in these places

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appearing in this function are consumed, then these tokens can be transferred from M.k/ to M.k C 1/. P Condition (2) When the coefficient of ti in the function pi LVPij  is not i

an integer and the tokens in the corresponding places of this entry are not previously consumed, then the tokens in these places are preserved; otherwise P let pi LVPij  D 0 in M.k C 1/. i

From this calculation, a series of reachable marking can be found from M0 . In addition, if there is no marking reachable from Mi , then there are dead transitions at Mi . Thus, the system contains deadlock. Figure 5 is an

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Fig. 5 An example to explain these conditions

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example to illustrate these conditions. The initial marking of Fig. 5 is M0 = (1, 2 0, 0, 0, 0, 1, 1)T , and the weighted place transitive matrix is LVP  D 3 p1 0 t1 =2 0 0 0 0 0 6 p2 6 0 0 0 t2 0 7 t2 0 7 6 p3 6 0 0 0 t3 =2 0 0 0 7 7 6 7 p4 6 0 0 0 0 t4 0 t4 7. 6 7 p5 6 0 0 0 0 0 0 07 6 7 r1 4 0 t1 =2 0 0 0 0 05 r2 0 0 0 t3 =2 0 0 0 MR .k C 1/T D M.k/T LVP  = [0, t1 .p1 .k/ C r1 .k///2, t2 .p2 .k//, t3 .p3 .k/ C r2 .k///2, t4 .p4 .k//, t2 .p2 .k//, t4 .p4 .k//]T . Since p1 .0/ = 1, r1 .0/ = 1 and r2 .0/ = 1 in the initial marking M0 , MR .1/ D M.0/LVP  D M0 LVP  = [0, t1 .1 C 1//2, t2  0, t3 .0 C 1//2, t4  0, t2  0, t4  0]T D [0, t1 , 0, t3 /2 0, 0, 0]T . The second element in MR .1/ is t1 and the coefficient of t1 is an integer; at the same time, the tokens in p1 .0/ and r1 .0/ are consumed, and by Condition (1), the corresponding value in reachable marking M2 is 1. On the other side, the coefficient of t3 is 1/2 and the corresponding place is r2 .0/ whose tokens are not consumed previously; by Condition (2), the tokens in r2 .0/ are preserved. Consequently, the reachable marking of M0 is M1 = (0, 1, 0, 0, 0, 0, 1)T . Based on the two constraints, a reachable marking can be obtained by calculating the transformation of marking.

2.3.2 Deadlock Detection In this subsection, the transformation equation is applied to identify the dead transitions in two Petri nets. Based on the dead transition and the conditions of “circular waiting,” a deadlock detection algorithm is proposed to find out whether there is deadlock or not in the compositional system. 2.3.3 Deadlock Recovery When there is a deadlock in the net, the system falls into the local deadlock state and even stops running. Therefore, deadlock recovery is important for system design. In this subsection, an efficient method is presented to recover a deadlock for the composition of two Petri nets. Since the deadlock is caused by “circular waiting” and by removing it, deadlock can be removed. In the following, the recovery method is proposed based on the definition of “circular waiting” and “self-loop” in Petri net.

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Algorithm 1: Deadlock detection Input: N1 D .P1 , T1 , F1 , M10 ,  1 /, N2 D .P2 , T2 , F2 , M20 ,  2 / (where P1 \ P2 D R D fr1 , r2 g), the initial marking M10 , M20 , and the final marking M1f , M2f Output: Deadlock or deadlock-free Step 1: Do place fusion on N1 and N2 , and then generate a net N with the initial marking M0 and the finial marking Mf . Step 2: Construct the weighted place transitive matrix-LVP  for compositional net N: Step 3: For k = 0, k < n, k++: Calculate the next marking MR .k C 1/ from the marking M.k/ by MR .kC1)T D M.k/T LVP  . If MR (k+1) = Mf , stop and output “there is no deadlock in the net.” Else if there are dead transitions in the net and they form a “circular waiting,” then output “there is a deadlock.” Else output is “there is flaw in the system model.”

Algorithm 2: Deadlock recovery Input: Dead transitions which cause deadlock (suppose they are t1b and t1b 0 / Output: A deadlock-free model Step 1: Compare # (t10 t11 . . . t1.a1/ ) and # (t20 t21 . . . t2.a1/ ), and find out the larger quantity, without loss of generality, suppose it is # (t10 t11 . . . t1.a1/ ). Step 2: Add two new arcs (t1.b1/ , r1 / and (r1 , t1b / into the model, and then the pair (t1a , t1c / is decomposed as (t1a , t1.b1/ / and (t1b , t1c / which are both associated with r1 . Step 3: Detect deadlock by Algorithm 1. If output is “there is a deadlock,” do the recovery again by repeating Steps 1 and 2. Else, output is “there is no deadlock in the net.”

Let N1 and N2 be two Petri net sub-models, with two kinds of common resource places r1 and r2 . They satisfy the “circular waiting” conditions mentioned in Sect. 2.3.1. Suppose  i D ti 0 ti1 . . . ti m is a transition path in Ni , i0 D ti a ti b ti c tid is a sub-path of i (i = f1, 2g), and # (ti 0 ti1 . . . ti.a1/ / is the makespan of ti 0 ti1 . . . ti.a1/ . After the recovery given above, the subsystem becomes deadlock-free. The method proposed above only changes the input and output of one resource place in one Petri net sub-model. If the complexity of the model is not considered, the similar change is done in the second Petri net sub-model. After this change, the Petri net model can be live and the makespan can also be minimized.

2.3.4 Deadlock-Free Design Based on the algorithms given above, a compositional deadlock-free model for a multitasking JSSP can be constructed. The detailed process is as shown in Algorithm 3.

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Algorithm 3: Deadlock-free design Input: n Petri nets Output: A compositional deadlock-free model for these n Petri nets Step 1: For each pair of Petri nets which have common resource places, run Algorithm 1 to detect deadlock. Step 2: If there is a deadlock in any two Petri nets, run Algorithm 2 to recover deadlock, and go to Step 3. Else, if all of these pairs are deadlock-free, go to Step 3. Step 3: Merge these n nets together by place fusion to get a deadlock-free model for this JSSP.

2.4

A Case Study

A JSSP example with four jobs and two kinds of resources (e.g., machines and a worker) is shown in Table 1. J1 , J2 , J3 , and J4 are four jobs; O1 , O2 , O3 , and O4 are operations of each job; M1 , and M3 are machines and M2 is the worker; and the integers following Mi represent the processing time of each operation. The second and the third operations of each job need both machine and worker to process, but the first and the fourth operations of each job need only one resource. In order to obtain a schedule, this problem is illustrated step by step: Step 1: Construct the model for this JSSP: • According to Sect. 2.3, the Petri nets (ASMs) for four jobs are depicted in Fig. 6, and ti1 , ti 2 , ti 3 , ti 4 are corresponding to the four operations in job i (i = f1, 2, 3, 4g); r1 , r2 , and r3 are corresponding to three resources. • When there are more than two shared resources in two jobs, deadlock may occur. Therefore, after the system model is built, deadlock detection should be done to make sure the system is deadlock-free. Deadlock detection is performed for each compositional ASM obtained by merging the pair of ASMs of individual jobs. Since there are four Petri nets (ASMs for individual jobs), totally there are six detections. Step 2: Choose two Petri nets with shared resources and detect deadlock; if there exists a deadlock, recover it: • Consider Petri nets N1 and N2 . Figure 7 is the compositional system for N1 and N2 . In the model N12 , M.k/ =[p11 .k/, p12 .k/, p13 .k/, p14 .k/, p15 .k/, p21 .k/, p22 .k/, p23 .k/, p24 .k/, p25 .k/, r1 .k/, r2 .k/]T , the initial marking M (0) = M0 = (1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1)T and the final marking Mf = (0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1)T . • Construct the weighted place transitive matrix-LVP for Fig. 7: Table 2 is the weighted place transitive matrix, M.k/=[p11 .k/, p12 .k/, p13 .k/, p14 .k/, p15 .k/, p21 .k/, p22 .k/, p23 .k/, p24 .k/, p25 .k/, r1 .k/, r2 .k/,]T , and the initial marking is M0 = (1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1)T . • Calculate the reachable marking by MR .kC1)T D M.k/T LVP  , and get the next reachable marking: MR .kC1)T D M.k/T LVP  = [0, t11 .p11 .k/Cr1 .k///2, t12 .p12 .k/Cr2 .k///2, p13 .k/t13 , p14 .k/t14 , 0, t21 .p21 .k/Cr2 .k///2, t22 .p22 .k/Cr1 .k///2, p23 .k/t23 , p24 .k/t24 , p13 .k/t13 C p24 .k/t24 , p14 .k/t14 C p23 .k/t23 ]T .

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O1 O2 O3 O4

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Fig. 6 The Petri net models of the JSSP

MR (1)D M (0)LVP  = [0, t11 .p11 .k/ C r1 .k///2, t12 r2 .k//2, 0, 0, 0, t21 .p21 .k/ C r2 .k///2, t22 r1 .k//2, 0, 0, 0, 0]T , based on the two conditions presented in Sect. 2.3.1, M (1) =[0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]T . • Analyze if there exists a deadlock or not: in M (2), transitions t12 and t22 are dead transitions. Since r1 and r2 are associated with transitions t12 and t22 , which involve in “circular waiting, the system is confronted a deadlock. • Recover the deadlock for N1 and N2 : – Since (# t11 = 7) is greater than (# t21 = 3), then the first sub-model should be modified. – Add two arcs t11 , r1 and (r1 , t12 /, and then (t11 , t11 / and (t12 , t13 / are the new transition pairs associated with resource r1 (Fig. 8 is the model after deadlock recovery). – Detect deadlock again; since there is no deadlock, stop.

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Fig. 7 The model after place fusion

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Table 2 Place transitive matrix of Petri net in Fig. 7

LVP 

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0 60 6 60 6 60 6 60 6 6 60 6 60 6 60 6 60 6 60 6 40 0

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0 0 t12 =2 0 0 t13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t12 =2 0

0 0 0 t14 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 t21 =2 0 0 0 0 0 t21 =2

0 0 0 0 0 0 t22 =2 0 0 0 t22 =2 0

0 0 0 0 0 0 0 t23 0 0 0 0

0 0 0 0 0 0 0 0 t24 0 0 0

0 0 t13 0 0 0 0 0 t24 0 0 0

3 0 0 7 7 0 7 7 t14 7 7 0 7 7 7 0 7 7 0 7 7 t23 7 7 0 7 7 0 7 7 0 5 0

Step 3: Choose other pairs of Petri nets and do the same work like Step 2: • The Petri net pairs are (N1 , N3 /, (N1 , N4 /, (N2 , N3 /, (N2 , N4 /, and (N3 , N4 /. • For the pairs (N1 , N3 /, (N1 , N4 /, (N2 , N3 /, and (N2 , N4 /, there is no deadlock in each of them. However, there is deadlock in integrating N3 and N4 . • Through analyzing the transformation equation MR .kC1)T D M.k/T LVP  , the dead transitions t32 and t42 are found, which cause deadlock. • Recover deadlock on transition t32 and t42 , and the resultant model is similar to Fig. 8.

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Fig. 8 The recovery model for N1 and N2

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Fig. 9 The deadlock-free model of the given JSSP

Step 4: Deadlock-free design for JSSP After deadlock detection and recovery, there is no deadlock in each two submodels. By Algorithm 3, do place fusion on these four Petri nets and a deadlockfree model is obtained (Fig. 9).

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σ = t41 t21 t22 t23 t24 t42 t11 t43 t44 t31 t32 t33 t34 t12 t13 t14 0

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Idle work

Fig. 10 The Gantt chart of the given JSSP

Step 5: Scheduling solution Since it is deadlock-free, a scheduling solution for this JSSP can be obtained. Here CPN Tool [30] is applied to simulate the resolutions and choose a best one. The minimum makespan in the report is 48, and one optimal schedule is given in Fig. 10.

3

Modular-Based JSSP Design and Makespan Calculation

In this section, several operators for the design specification of JSSP are presented. For each operator, the property about the makespan calculation is also provided. Based on these operators and properties, a larger and more complex module, even for the whole system, can be deduced from the smaller ones step by step, and the schedule of larger modules can also be calculated. As a component of a variety of methodologies developed to generate JSSP schedules, Petri net is very powerful for modeling the concurrent processing, resource sharing, routing flexibility, sequential steps, concurrent use of multiple resources, machine setup times, job batch sizes, and so on in JSSP [31–34]. As for schedule optimization, there are two types of Petri net-based methodologies for JSSP: (1) heuristic searching algorithms based on reachability graph of Petri net models [35–37] and (2) GA algorithms for optimizing scheduling [38–40]. Petri nets are attracting the attention of many researchers for the design of JSSP and the optimization of the scheduling of the system [41–43]. However, current Petri net-based methodologies have the following shortages: (1) the Petri net models are put forward by researchers for solving their specified problems. Therefore, the effect of Petri nets cannot be explored efficiently. Although it is good for obtaining better optimization results, they cannot be applied for general problems. (2) the real-time JSSP is usually quite complex and large; consequently,

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the size of the Petri net model will be quite large. The current Petri net analysis methods are not applicable because of the state exploding problem of Petri nets. In order to overcome the hardship mentioned, this chapter handles JSSP problem from two aspects: system design and scheduling optimization. On system design aspect, this chapter applies a modular-based approach for JSSP design specification and verification. The basic module model is an extended Petri net process (EPNP) similar to the Petri net process (PNP) discussed in [2–4]. The difference is that EPNP is an extended Petri net while PNP is a general Petri net. Hence, all the results on property-preserving Petri net process algebra (PPPA) [3–6] will also be true for the JSSP design in this chapter. With modular-based approach, the JSSP is correct from the viewpoint that the system contains some desired properties such as liveness, boundedness, reversibility, and proper termination. As for system scheduling optimization, based on EPNP, this chapter provides some information on finding the possible schedule (the firing sequences of the Petri net model) and makespan of the composite modules from the constituent modules for the integration operators in PPPA. In order to handle resource-sharing problem, the place-merging operator and the integration operators are combined for system modeling. The firing sequences and the upper bound of the makespan for these combined operators are concluded. The algorithm for finding the makespan and its corresponding scheduling is also presented for the combined operators. With the modular-based method for system design and scheduling optimization, the schedule and makespan of the composite modules can be calculated based on their constituent modules. In order to reduce the calculation time, traditional algorithms such as GAs and branch-and-bound methods can be applied together with the algorithms presented in this chapter for optimizing the schedule of JSSP. For completeness, a GA algorithm based on Petri net model will be introduced in Sect. 4. The readers are supposed to have some Petri net-related foundations. Most of the terminologies can be found in [3, 4, 45]. The results in this section are mainly derived from [44].

3.1

Operators and Makespan Optimization

Definition 14 An EPNP B= (N , pe , px /, where N = (P, T, F, M c ; C; W , D/ is a colored TPN (called extended Petri net), pe is the entry place, and px is the exit place of EPNP. The performance and characteristics, for example, proper initialization, proper termination, and handling resource sharing, of EPNP similar to PNP defined in [3,4]; the difference is that in EPNP, N is an extended Petri net, while in PNP, N is a general Petri net. Hence, all the theoretical results in [3–6] are also true for EPNP and can be applied to design a correct job shop scheduling system. Note that since the design specification and verification are similar to [2–4], this chapter will pay much more attention to the design specification and optimization of JSSP.

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b a

t

t

In job shop scheduling systems, operations either do not need any resources or need resources such as machines, robots, and workers. Hence, the corresponding models for the operations in a JSSP have two different forms. For the first case, a transition t represents an operation; t has a single input place representing the beginning of the operation and a single output place representing the end of the operation (Fig. 11a). For the second case, one more place representing the resources should be associated to t and a self-loop is formed. Initially, there are some tokens assigned to the resource place. Tokens with different colors are used to represent different resources (Fig. 11b). Based on the relationship of the operations and the control flow of JSSP, some operators are defined to combine these operations in JSSP. In this chapter, the basic model is the EPNP. Although EPNP and the general PNP [3, 4] have different structural and behavioral properties and firing rules, the pictorial representation of these operators will be the same as that in [3]. Hence, in this chapter, the same definition and the same pictorial representation as that in Mak’s thesis [3] and Huang’s thesis [4] are used for the definition of all the operators, that is, action prefix, iteration, enable, choice, interleave, parallel, disable, disable-resume, place merging, and the mix operators. In the following definition, notation B = (N , pe , px / (resp., Bi = (Ni , pi e , pix // represents an EPNP, where N = (P, T, F, M c ; C; W , D/ (resp., Ni = (Pi ; Ti ; Fi ; Mi c ; Ci ; Wi , Di // is an extended Petri net with an initial static marking Mc , pe , and px (resp., Mi c , pi e , and pix / are the entry and exit places of the process, respectively. The initial marking of B (resp., Bi / is Me D pe C Mc (resp., Mi e D pi e C Mi c ), and the exit marking of B (resp., Bi / is Mx D px C Mc (resp., Mix D pix C Mi c /. For each operator defined in this chapter, the firing sequence and the potential partial schedule in the composite process will be concluded based on the possible firing sequences of the constituent processes. These conclusions will be presented as properties. Since the results are trivial, the details of the proof will be omitted. The conclusions can be used to find the optimal scheduling solutions. Some propertypreserving results can also be obtained for these operators. Since most property-preserving results can refer to the related results in [3, 4], the behavioral property-preserving results will be omitted in this chapter. While the property-preserving results can be used to specify and verify the JSSP, the work

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related to the application of property-preserving technology for system specification and verification can be found in [2, 4]. In JSSP, before processing some operations, a preparation work is usually needed. For example, after a batch of pieces has finished processing, the machine may need repairing before handling the next batch of pieces. In the system modeling, the following formally defined action prefix operator is used to combine the preparation action and the operation process. It is a special case of operator enable which will be introduced later. Definition 15 (Action prefix) For a process B = (N , pe , px / and a transition b … T representing an action, the operator action prefix applied to B, in notation bI B, is defined as the process B0 = (N0 , pe 0 , px 0 ), where pe 0 is the new entry place, px 0 = px , P0 = P[ fpe 0 g, T0 = T [ fbg, F0 = F [ f(pe 0 , b/, (b, pe /g, Mc 0 = Mc , C0 = C , W0 = W [fw(pe 0 , b/, w(b, pe /g, D0 = D [ fd.b/g. Property 8 Suppose (B0 , Me 0 ) is induced from (B, Me / by applying the operator action prefix. If Me [ > Mx in (B, Me /, then Me 0 [b > Mx 0 in (B0 , Me 0 ). If  has the minimum makespan in (B, Me /, then b  has the minimum makespan in (B0 , Me 0 ). Definition 16 (Iteration) For a process B= (N , pe , px /, an operator iteration applied to B, in notation B0 , is defined as the process B0 = (N0 , pe 0 , px 0 ), where P0 = P [ fpe 0 , px 0 g; T0 = T [ f"1 , "2 , "3 g; F0 = F [ f.pe 0 , "1 /, ("1 , pe /, ("2 , pe /, (px , "2 /, (px , "3 /, ("3 , px 0 )g, Mc 0 = Mc , C0 = C , W0 = W [ fw.pe 0 , "1 /, w("1 , pe /, w("2 , pe /, w(px , "2 /, w(px , "3 /, w("3 , px 0 )g, D0 = D [ fd."1 /, d ("2 /, d ("3 /g. Iteration modifies a process in such a way that it can be executed repetitively while preserving the option of exiting after any iteration. For example, in JSSP, iteration operator can be used to model the case of processing a batch of pieces in a machine. Property 9 Suppose (B0 , Me 0 ) is induced from (B, Me / by applying iteration operator. If Me [ > Mx in (B, Me /, then Me 0 ["1  "2  "2 : : :  "2  "3 > Mx 0 in (B0 , Me 0 ). If  has the minimum makespan in (B, Me /, then "1  "3 has the minimum makespan in (B0 , Me 0 ). Definition 17 (Enable) For two processes Bi = (Ni , pi e , pix /, i D 1; 2, their composition by enable, in notation B1 >> B2 , is defined as the process B = (N , pe , px /, where P D P1 [ P2 , with pe D p1e and px D p2x ; T D T1 [ T2 [ f"g; F D F1 [ F2 [ f.p1x ; "/, (", p2e /g, Mc D M1c [ M2c , C D C1 [ C2 , W D W1 [ W2 [ fw.p1x , "), w(", p2e /, D D D1 [ D2 [ fd."/g. Enable B1 B2 models the execution of two processes B1 and B2 in sequence. That is, B1 is firstly executed and B2 is executed after the successful termination of B1 . However, if B1 does not exit successfully, B2 will never be activated.

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Property 10 Suppose (B, Me / is induced from (B1 , M1e / and (B2 , M2e / by applying the operator enable. If M1e [1 > M1x in (B1 , M1e / and M2e [2 > M2x in (B2 , M2e /, then Me [1 " 2 > Mx in (B, Me /. If 1 has the minimum makespan in (B1 , M1e / and 2 has the minimum makespan in (B2 , M2e /, then 1 " 2 has the minimum makespan in (B, Me /. Definition 18 (Choice) For two processes Bi = (Ni , pi e , pix /, i = 1, 2, their composition by choice, in notation B1 [ ] B2 , is defined as the process B= (N , pe , px /, where P = (P1 [ P2 – fp1x , p2x g) [ fpe , px g, where px is the place merging p1x and p2x ; T D T1 [ T2 [ f"1 , "2 g; F D F1 0 [F2 0 [ f(pe , "1 /, (pe , "2 /, ("1 , p1e /, ("2 , p2e /g, where Fi 0 = Fi – f(t, pix /jt 2  pix g [ f(t, px /jt 2  pix g, i = 1, 2, Mc D M1c [ M2c , C D C1 [ C2 , W D W1 .F1 0 ) [W2 (F2 0 )[ fw(pe , "1 /, w(pe , "2 /, w("1 , p1e /, w("2 , p2e /, D= D1 [ D2 [fd ("1/, d ("2 /g. Choice B1 [ ] B2 models the selection for execution between two processes B1 and B2 . Property 11 Suppose (B, Me / is induced from (B1 , M1e / and (B2 , M2e / by applying the operator choice. If M1e [1 > M1x in (B1 , M1e / or M2e [2 > M2x in (B2 , M2e /, then Me ["1 1 > Mx and Me ["2 2 > Mx in (B, Me /. If 1 has the minimum makespan in (B1 , M1e / and 2 has the minimum makespan in (B2 , M2e /, then the firing sequence with the minimum makespan in (B, Me / is minf"1 1 , "2 2 g. Definition 19 (Interleave) For two processes Bi = (Ni , pi e , pix /, i = 1, 2, their composition by interleave, in notation B1 jjjB2 , is defined as the process B = (N , pe , px /, where P = P1 [ P2 [ fpe , px g; T D T1 [ T2 [ f"1 , "2 g; F D F1 [ F2 [ f(pe , "1 /, ("1 , p1e /, ("1 , p2e /, (p1x , "2 /, (p2x , "2 /, ("2 , px /g, Mc D M1c [ M2c , C D C1 [ C2 , W D W1 [ W2 [ fw(pe , "1 /, w("1 , p1e /, w("1 , p2e /, w(p1x , "2 /, w(p2x , "2 /, w("2 , px /g, D D D1 [ D2 [fd ("1/, d ("2 /g. Interleave B1 jjjB2 models the concurrent but independent execution of two processes B1 and B2 with synchronized exit. Property 12 Suppose (B, Me / is induced from (B1 , M1e / and (B2 , M2e / by applying the operator interleave. If M1e [1 > M1x in (B1 , M1e / and M2e [2 > M2x in (B2 , M2e /, then Me ["1  "2 > Mx in (B, Me /, where  is a transition sequence consisting of all of the transitions in 1 and 2 such that the appearance order of the transitions conforms to the order in 1 and in 2 (i.e., the appearance order is preserved) . If 1 has the minimum makespan in (B1 , M1e / and 2 has the minimum makespan in (B2 , M2e /, then  has the minimum makespan in (B, Me / and # = max f#1 , #2 g, where the notation # is the makespan value of the firing sequence .

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For example, suppose 1 = abc and 2 = de. Then  can be any one of the following ten sequences: deabc, daebc, dabec, dabce, adebc, adbec, adbce, abdec, abdce, and abcde. Definition 20 (Composition by the Operator Parallel) For two processes Bi = (Ni , pi e , pix /, i = 1, 2 with a “common” set of transitions G D T1 \ T2 to be synchronized, their composition by Parallel, in notation B1 jGjB2 , is defined as the process B = (N , pe , px / where P = P1 [ P2 [ fpe , px g; T D T1 [ T2 [ f"1 , "2 g, where T1 \ T2 ¤ Ø; F D F1 [ F2 [ f.pe , "1 /, ("1 , p1e /, ("1 , p2e /, (p1x , "2 /, (p2x , "2 /, ("2 , px /g, Mc D M1c [ M2c , C D C1 [ C2 , W D W1 [ W2 [ fw..pe , "1 /, w("1 , p1e /, w("1 , p2e /, w(p1x , "2 /, w(p2x , "2 /, w("2 , px /g, D D D1 [ D2 [fd ("1 /, d ("2 /g. Parallel B1 jGjB2 models the concurrent execution of two processes B1 and B2 while they have to be synchronized at some points and at their exit points. Synchronization has been one of the most complicated problems in the research of concurrent systems. Property 13 Suppose (B, Me / is induced from (B1 , M1e / and (B2 , M2e / by applying the operator parallel. If M1e [˛1 ˇ1 > M1x in (B1 , M1e / and M2e [˛2 ˇ2 > M2x in (B2 , M2e /, then Me ["1 ˛ˇ"2 > Mx in (B, Me /, where ˛ is a transition sequence consisting of all of the transitions in ˛1 and ˛2 such that the appearance order of the transitions in ˛ conforms to the order in ˛1 and in ˛2 , and ˇ is a transition sequence consisting of all of the transitions in ˇ1 and ˇ2 such that the appearance order of the transitions in ˇ conforms to the order in ˇ1 and in ˇ2 . If 1 D ˛1 ˇ1 has the minimum makespan in (B1 , M1e / and 2 D ˛2 ˇ2 has the minimum makespan in (B2 , M2e /, then ="1 ˛ˇ"2 has the minimum makespan in (B, Me / and # = max f#˛1 , #˛2 g+ #+ max f#ˇ1 , #ˇ2 g. Definition 21 (Disable) For two processes Bi = (Ni , pi e , pix /, i = 1, 2, their composition by disable, in notation B1 [i B2 , is defined as the process B = (N , pe , px /, where P = (P1 – fp1x g) [ (P2 – fp2x g) [ fpe , px , pd g, where px D p1x D p2x ; T D T1 [ T2 [ f", td g; F D F1 0 [F2 0 [ f(pe , "), (", p1e /, (", p2e /, (pd , td /, (td , p2e /g [f(p2e , t/jt 2 T1 g [ f(t, pd /jt 2 T1   p 1x g, where Fi 0 = Fi – f(t, pix /jt 2  p ix g [ f(t, px /jt 2  p ix g, i = 1, 2, Mc = M1c [ M2c , C D C1 [ C2 , W = W1 .F1 0 ) [W2 .F2 0 ) [ fw(pe , "), w(", p1e /, w(", p2e /, w(pd , td /, w(td , p2e /g [fw(p2e , t/jt 2 T1 g [ fw(t, pd /jt 2 T1   p 1x g, D D D1 [ D2 [fd ("), d (td /g. Disable B1 [iB2 models a potential interruption of process B1 by another process B2 during execution. According to LOTOS, there are three possibilities: (1) If B2 never begins, B1 will continue until termination. (2) If B2 begins while B1 is executing, B1 will be stopped and B2 continues its execution. (3) If B2 begins first, B1 will never begin and B2 will continue its execution until termination. One application of disable is to model task interruption in a JSSP. Although an operation

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will not be interrupted during execution, a process may be interrupted since it consists of several operations. If all initial transitions of B2 (i.e., outgoing transitions of p2e / are never fired, B1 will execute until completion. If any initial transition of B2 is fired when B1 is executing, B1 is stopped because any transition of B1 cannot be fired without a token in p2e and B2 will execute to completion. If B2 begins first, the token in p2e will immediately be removed. Then, it is obvious that B1 can never begin. Property 14 Suppose (B, Me / is induced from (B1 , M1e / and (B2 , M2e / by applying the operator disable. If M1e [1 > M1x in (B1 , M1e / and M2e [2 > M2x in (B2 , M2e /, then Me [" > Mx in (B, Me /, where  satisfies the following: (1)  D 1 , (2)  D 2 , and (3)  is a transition sequence consisting of the transitions in a firing sequence ˛ of B1 and 2 ; the appearance order is preserved and p2e  \ ˛ D . If 1 has the minimum makespan in (B1 , M1e / and 2 has the minimum makespan in (B2 , M2e /, then  has the minimum makespan in (B, Me / and #  equals to one of # 1 , # 2 , and # ˛+ # 2 . Definition 22 (Disable-Resume) For two processes Bi = (Ni , pi e , pix /, i = 1, 2, their composition by the operator disable-resume, in notation B1 [riB2 , is defined as the process B = (N , pe , px /, where P = (P1 [ P2 /[ fpe , pd g, where px = p1x , T D T1 [ T2 [ f"1 , "2 , td g; F = F1 [ F2 [ f(pe , "1 /, ("1 , p1e /, ("1 , p2e /, (p2x , "2 /, ("2 , p2e /, (pd , td /, (td , p2e /g [ f(p2e , t/jt 2 T1 g [ f(t, pd /jt 2 T1   p1x g, Mc D M1c [ M2c , C D C1 [ C2 , W D W1 [ W2 [ fw(pe , "1 /, w("1 , p1e /, w("1 , p2e /, w(p2x , "2 /, w("2 , p2e /, w(pd , td /, w(td , p2e /g [ fw(p2e , t/j t 2 T1 g [ fw(t, pd /jt 2 T1   p1x g, D D D1 [ D2 [fd("1 /, d ("2 /, d ( td /g. Disable-resume B1 [riB2 functions the same as the disable operator B1 [iB2 except that if B2 is ever executed and completed, then there will be three options: (1) The entire process terminates. (2) Execution on B1 resumes from wherever it is left. (3) B2 is repeated. Similar to the operator disable, there is also a non-pure version of disable-resume. Property 15 Suppose (B, Me / is induced from (B1 , M1e / and (B2 , M2e / by applying the operator disable-resume. If M1e [1 > M1x in (B1 , M1e / and M2e [2 > M2x in (B2 , M2e /, then Me ["1  > Mx in (B, Me /, where  satisfies the following: (1)  D 1 , (2) Suppose 1 D ˛ˇ, p2e  \ ˛ D  and p2e  \ ˇ ¤ . Then  D ˇ, where  is a transition sequence consisting of transitions in ˛ and 2 "2 such that the appearance order is preserved. If 1 has the minimum makespan in (B1 , M1e / and 2 has the minimum makespan in (B2 , M2e ), then  has the minimum makespan in (B, Me / and #  equals to one of # 1 , and # 1 C c # 2 , where c is a constant number. In JSSP, resource sharing is quite necessary; after the composition of the modules, sharing resource should be considered. Therefore, place-merging operator is applied for modeling and analyzing the resource-sharing problem.

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Qi

qi

B

B⬘

Fig. 12 B.Qi ! qi /: the operator place merging in a single process

Definition 23 (Place Merging (Fig. 12)) For a process B = (N , pe , px /, suppose N = (P0 [ Q1 [ : : : [ Qk , T , F; Mc , C, W, D) is an extended net satisfying the condition: 8i , j 2 f1, 2; . . . , kg and i ¤ j , P0 \ Qi D Ø and Qi \ Qj D Ø. Let B0 = (N0 , pe 0 , px 0 ), denoted as B.Qi ! qi /, where N 0 = (P0 [ Q0 , T 0 ,F 0 , Mc 0 , C0 , W0 , D0 ) and Q0 = fq1 , q2 , . . . , qk g, be obtained from (B, Me / by merging the places of each Qi into qi as follows: T 0 D T ; F 0 is obtained from F by replacing each set of arcs of the form f(t, p/jp 2 Qi g with (t, qi / (resp., f(p, t/jp 2 Qi g with (qi ; t// Mc0 .p/ D



Mc .p/ maxp2Qi fMc .p/g

p 2 P0 p D qi 2 Q0 ; i D 1; 2; : : :; k

C 0 = C, W 0 =W, D 0 = D. Property 16 Let (B, Me / and (B0 , Me 0 ) be defined in Definition 10. If Me [ > Mx in (B, Me /, then Me 0 [ > Mx 0 in (B0 , Me 0 ). Note that by the modeling technology, each transition t representing an operation forms a self-loop with the corresponding resource places. In such a case, Property 16 is true. Up to now, several integration operators, including the place-merging operators, are introduced. However, for the place-merging operators, the makespan of the resulted process cannot be obtained from the makespan of the original processes very easily. In order to simplify the computing process and get an exact makespan value, place-merging operator will be applied together with other integration operators if there exists resource sharing in the composition process. For example,

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when two processes are composed by enable operator, if some operations in these two processes share some common resources, place-merging operator will be applied together with enable operator. Notations M-enable, M-choice, M-interleave, etc. denote these mixed operators. Property 17 Suppose M1e [1 > M1x in (B1 , M1e / and M2e [2 > M2x in (B2 , M2e /. (B, Me / is resulted from B1 and B2 by applying mixed operators and Me [ > Mx in (B, Me /. (1) For operator M-enable, if 1 has the minimum makespan in (B1 , M1e /, and 2 has the minimum makespan in (B2 , M2e /, then  D 1 "2 has the minimum makespan in (B, Me / and # = # 1 + # 2 . (2) for M-choice, if 1 has the minimum makespan in (B1 , M1e / and 2 has the minimum makespan in (B2 , M2e /, then the firing sequence with the minimum makespan in (B, Me / is minf"1 1 , "2 2 g. (3) for M-interleave, denote 1 = ˛1 t1 ˛2 t2 : : : ˛k1 tk1 ˛k 0 0 0 0 and 2 D ˇ1 t1 ˇ2 t2 : : : ˇm1 tm1 ˇm , where t1 , t2 , . . . , tk1 and t10 , t20 ; : : : ; tm1 are the transitions sharing some resources in the operators, and ˛i and ˇj (i = 1, 2, . . . , k, j = 1, 2, . . . , m/ are partial firing sequences. Suppose # 1 # 2 . Then the upper k1 P #ti . bound for the makespan of (B, Me / is # 2 C i D1

The proofs for Property 17 (1) and (2) are trivial. For Property 17 (3), the worst case would be that when some resources are shared, it is always process B2 which waits. By Properties 12 and 16, for the mixed operator M-interleave, the firing sequence  of (B, Me / is a transition sequence consisting of all of the transitions in 1 and 2 such that the appearance order of the transitions conforms to the order in 1 and in 2 . Below, an algorithm is introduced for generating a firing sequence  of (B, Me / based on 1 and 2 with the minimum makespan. Makespan Generating Algorithm for M-Interleave Operator Input: Two firing sequences 1 D ˛1 t1 ˛2 t2 : : : ˛k1 tk1 ˛k and 2 D 0 0 0 ˇ1 t1 ˇ2 t2 : : : ˇm1 tm1 ˇm of (B1 , M1e / and (B2 , M2e /, respectively Output: A firing sequence  and the corresponding minimum makespan #  of (B, Me / 0 0 0 Step 1: Let 1 D ˛1 t1 ˛2 t2 : : : ˛k1 tk1 ˛k , 2 D ˇ1 t1 ˇ2 t2 : : : ˇm1 tm1 ˇm ,  D , and # = 0. Step 2: For i = 1, i < k; j = 1, j < m, if there exists a leftmost subsequence i D ˛i ti    ˛p tp of 1 and a leftmost 0 0 subsequence j D ˇj tj    ˇq tq of 2 satisfying # i  #tp  #j  #tq0  # 0 0 i , then  D  i .ˇj tj    ˇq /tq and #  = # + # i + #tq0 . If there exists a leftmost subsequence i D ˛i ti    ˛p tp of 1 and a leftmost 0 0 subsequence j D ˇj tj    ˇq tq of 2 satisfying # j - #tq0  # i - #tp  # j , then  D  j .˛i ti    ˛p /tp and #  = # + # j + #tp . Let 1 D 1  i , 2 = 2  j , i D p and j D q, i ++, j ++ Step 3. = maxf 1 ,  2 g and # = # + maxf# 1 , # 2 g.

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The above algorithm can also be applied to generate the firing sequence and compute the makespan for the mixed operators M-parallel, M-disable, and M-disable-resume. In the case of more than two composition processes, the algorithm can be improved easily such that the input is more than two firing sequences and the comparison is between more than two leftmost subsequences.

3.2

Modular-Based Design and Optimizatioan: A Case Study

This section shows how to apply the composition operators for designing JSSP and finding the optimal scheduling. In order to illustrate the technology more precisely, a real-time JSSP design will be explained as an example. Suppose there are three jobs to be processed in three workshops. Each job has two or three operations which will be processed in three workshops in a given sequence. In each workshop some machines can be used, and each operation only chooses one machine for processing. Four workers are responsible for these six machines in the three workshops. The details of the operation distribution, the processing time of each operation in different machines, and the worker distribution are shown in Table 3.

3.2.1 System Design For simplicity, in the figures of Petri net model, only those transitions representing the operations are labeled, and the resource places Rij represent machine Mi which are sponsored by worker Wj : By the requirement analysis, Job 1 has three operations. In operation 1, machine M3 is occupied which is sponsored by worker W2 . Hence, transition t1 , representing the operation 1, is associated with a resources place R32 in the Petri net model (Fig. 13). For operation 2, it can be processed either in machine M1 sponsored by worker W1 or in machine M2 sponsored by worker W1 or W2 . Choice operators are applied twice in this model, one is used for choosing machines (M1 and M2 /, and Table 3 Processing information of jobs Job

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(M3 , 12) (M1 , 9), (M2 ,10) (M4 ,5), (M5 ,6),(M6 ,8) (M3 ,6) (M4 ,6), (M5 ,9), (M6 ,8) (M1 , 4), (M2 ,5) (M1 , 5), (M2 ,4)

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M1 O M2 O O M3 O M4 O M5 O O M6 O Note: “O” means that the worker in the corresponding column is responsible for the machine in the corresponding row

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Fig. 13 Petri net model for Job 1

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another is used for choosing workers (W1 and W2 /. Similarly, for operation 3, three machines M4 , M5 , and M6 will be chosen, and machine M5 has a choice between the workers W3 and W4 . Since the three operations are in a sequence order, the three modules will be integrated by applying enable operator (Fig. 13). Similarly, the Petri net models for Jobs 2 and 3 can also be obtained (Figs. 14 and 15).

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Fig. 14 Petri net model for Job 2

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By Properties 8 and 9, the three jobs have the possible firing sequences listed in Fig. 16. In order for simplicity and clarity, only the transitions representing the operations are listed. The processing time and the resources for each operation are assigned to the corresponding transitions. By observing, there are 12 firing sequences for each Petri net model.

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Fig. 15 Petri net model for Job 3

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Since these three jobs are independent, they can be processed in an interleave relationship. In the three Petri net models, the resource places with the same labels represent the same resources. Hence, a mixed operator M-interleave is applied for composing these three modules, and the system model is obtained. Since the picture is too complex, it is omitted.

Property 18 The job shop scheduling system design is correct.

A system is correct if it is almost live, almost bounded, and almost reversible and terminates properly. Since each of the primary Petri net modules is correct, by the results in [3, 4], the system model resulted by combining the modules by applying the operators defined in this section is also correct. Hence, the design method can guarantee the correctness of the system.

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Fig. 16 The possible firing sequences for the three jobs

3.2.2 Scheduling Optimization By Property 12 and makespan generating algorithm for M-interleave operator, the minimum makespan and the corresponding firing sequence (i.e., schedule) can be obtained. Theoretically, the algorithm should run totally 123 times in order to search for the optimal solution for the example. In practice, quite a lot of firing sequences can be deleted from the list. In order to illustrate the method, two possible schedule examples are illustrated to run the algorithm. For example, let 1 = t1 t2 t5 , 2 D t9 t10 t14 , and 3 D t17 t20 . By makespan generating algorithm for M-interleave operator, the schedule is as shown in Fig. 17a, where machines 1, 3, and 4 are used and workers 1, 2, and 3 are needed to sponsor the machines. Rij means machine i sponsored by worker j . Jij means Job i ’s j th operation is being processed in the corresponding time slot. In this case, the number of machines is minimum and the makespan equals 29. In another case, let 1 D t1 t4 t6 , 2 = t9 t10 t14 , and 3 D t19 t21 . The schedule is shown in Fig. 17b. In this case, five machines are used and three workers are needed. The makespan is 28. There are some more schedules with makespan 28. For example, the following four firing sequences correspond to the optimal schedule: t1 t19 t20 t2 t9 t10 t7 t14 ; t18 t1 t20 t9 t3 t10 t6 t14 ; t17 t1 t23 t9 t4 t10 t6 t14 , t19 t1 t23 t9 t2 t10 t7 t14 . In order to reduce the time complexity, an upper bound for the makespan is given such that for those partial schedules with the possible makespan exceeding the upper bound, they can be deleted from the firing sequence list. For example, in the above example, the makespan of the system will not exceed 28. Hence, the firing sequences t1 t2 t8 , t1 t3 t8 , and t1 t4 t8 in the Job 1 (Fig. 13) will be deleted from the list since their makespan have exceeded 28. As for Job 2 (Fig. 14), before processing, it will wait for 12 time slots because of resource sharing; except the sequence t9 t10 t14 , the makespan for all of other partial schedule will not be less than 28. If an optimal solution is required, only the single firing sequence t9 t10 t14 needs to be considered. Hence, very possibly, the algorithm only needs to run no more than 9  12 times.

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a

b σ = t17 t1 t20 t9 t2 t10 t5 t14 0

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Fig. 17 Some possible schedules

Theoretically, JSSP design method can be applied to find an optimal scheduling and the idea can refer to [2, 4]. In practice, for large and complex systems, the main difficulty is to find the optimal scheduling since there are too many possible partial firing sequences. The makespan generating algorithm should run too many times in order to select the optimal one. In order to reduce the time complexity, one of the possible solutions is to combine the branch and bound method [45] with the algorithm of this chapter. The idea has been introduced in the above example. Another solution is to combine GA algorithm [38] with the algorithm of this chapter, and the idea is as follows: Let the partial firing sequences in each module be genes and the firing sequence of the system be used as the chromosomes. The makespan will be the fitness function and the operators will be newly defined according to the combination operators defined in this article. An efficient GA algorithm is introduced in the next section. The branch and bound method, GA method, and the methods of this chapter can be combined together to find the approximation optimal solution in a much more efficient way.

4

Genetic Algorithm for JSSP

In this section, genetic algorithm is applied to solve JSSP. Different from the existing genetic algorithms, the genetic algorithm applied in this chapter is based on the Petri net model. The chromosomes are composed by sequence of transitions, with crossover and mutation operations based on transition sequences. The experiment result shows that a definite solution to a specific JSSP can be found. Flexible job shop scheduling problem (FJSSP) is the extension of JSSP, and it is more complex than the classic JSSP. In FJSSP, each operation can be processed on different machines, desiring different period of time. FJSSP is more practical than classic job shop problem and hence attracts much more attentions. There are also many methods used to resolve single-objective flexible job shop problems, such as evolutionary algorithms [46], simulated annealing [47–49], and tabu search [50] which have been proved efficient for solving FJSSP. The combinations of local search and evolutionary algorithm have obtained good solutions for FJSSPs [51].

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For both JSSP and FJSSP in a practical application, there are many objectives, e.g., makespan, total tardiness, total earliness, workload of machines, cost, and so on. Sometimes there are conflicts among these objectives; it is nearly impossible to achieve optimization of more objectives simultaneously. Hence, searching for useful methods of finding a counterpoise of the objectives is a challenge and draws many researchers’ attention. For instance, Li-Ning Xing and Ying-Wu Chen presented a simulation model to solve the multi-objective FJSSP [52]. The simulation model is proposed for the FJSSP with multi-objectives of minimizing makespan, total workload of machines, and workload of the critical machine [53]. Kazi Shah Nawaz Ripon and Chi-Ho Tsang presented an evolutionary approach for solving multiobjective FJSSP using Jumping Gene Genetic Algorithm that heuristically searches for the near-optimal solutions and optimizes multiple criteria simultaneously [54]. They can achieve near-optimal or optimal solutions to some special problems in their research, but the methods are aimed to solve a given problem, not suitable for the general FJSSP. In this section, two objectives, that is, makespan and total tardiness, are considered. Firstly, the FJSSP is modeled with Petri nets, and then the problems are solved by a genetic algorithm. As a graphic and mathematic tool, Petri net has been used to model scheduling problems for a long time. Meanwhile, Petri net-based approach will face the state space explosion problem, which may cut down the efficiency of execution. In order to partially avoid the state explosion problem for Petri net-based approach, genetic algorithm has been used with Petri net to solve scheduling problem. Chung and Fu used timed place Petri net (TTPN) and genetic algorithm to model and schedule FMS [55]. A Petri net with controller is used to model discrete events in flexible job shop scheduling with results obtained based on genetic algorithm and simulated annealing algorithm [56]. Huang et al. applied QCPN (queuing colored Petri net) and GA to model and schedule wafer manufacture [57]. These methods all work effectively, but their encoding related with special case, which sacrificed their general use for good scheduling result. According to the topics talked above, Petri net model is proposed in this section that can visually describe FJSSP. The operations in genetic algorithm such as encode, crossover, and mutation are all operated on elements of Petri net. The method is useful for any FJSSP which can be modeled by Petri net. The results in this section is mainly derived from [58].

4.1

Petri Net-Based Modeling and Problem Formulation

In this section, how to model the scheduling problems with Petri nets is introduced. The FJSSP is described as follows: 1. J D fJ1 , J2 , J3 ; : : :; Jn g is a set of jobs to be scheduled. 2. Each job consists of a predefined sequence of operations; let Oij be operation j of jobi . 3. Each job has a deadline, D D fd1 ; d2 ; d3 ; : : : ; dn g is a set of deadline, and di is a deadline of jobi .

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4. M D fM1 , M2 , M3 , . . . , Mm g is a set of machines. Each operation is processed on some given machines for a fixed duration without interruption. 5. Each machine can process one operation at a time. Let C D fc1 , c2 , c3 , . . . , cn g be the set of completion time of jobs, ci is the completion time of job Ji . Two objectives are considered in this section: 1. Minimization of makespan, F1 =max(cP i ji =1,2,3, . . . , n/. 2. Minimization of total tardiness, F2 D i tardiness(i /, i =1,2,3,. . . , n  0; ci  di tardiness.i / D ci  di ; ci > di The task is to find a solution which is better than any others when both the objectives are considered. FJSSP, as one of the most difficult problems in combinational optimization, has been proved to be NP-hard problem [59]. Modeling the job shop problem with Petri nets can make the complex scheduling problem visualized. The scheduling problems can be modeled with Petri nets and the rules are as follows: 1. Each transition denotes an operation, and the input (resp., output) place of the transition can be interpreted as the beginning (resp., end) of the operation. 2. There are some initially marked places, called resource places, which denote machines. Each resource place is associated with some transitions when the operation is processed on this machine. 3. Each job can be modeled with a small Petri net, where machines are models as resource places in the model without sharing. Then all the small Petri nets are composed by merging their identical resource places if the corresponding machines are shared by different jobs. 4. For the Petri net model with m places and n transitions, an m-by-n matrix is generated as follows: Row i corresponds to place pi ; column j corresponds to transition tj ; if pi is the input place of tj , then the element in row i column j is 1; if pi is the output place of tj , then the element in row i column j is 1; otherwise, the element in row i column j is 0; if pi is a resource place connecting to tj , then the element in row i column j is the time pieces that the operation needs to be processed on the machine pi ; otherwise, the element in row i column j is 1,000. At the beginning, tokens only exist in the first place of each job and all the resource places. After firing a sequence of transitions in turn, the tokens are transferred to the last place of each job and resource places, which represents that the job has been finished. The following is a FJSSP example: three jobs (J1 , J2 , J3 / are to be processed on three machines (M1 ; M2 ; M3 ). Each job consists of three operations (O1 , O2 , O3 ), and each operation can be processed on some given machines. Meanwhile, there is a deadline of each job, and if a job is completed after the deadline, extra cost must be charged (Table 4). For example, the first operation (O1 / of J1 can be operated on machine M1 for 36-unit time or on machine M3 for 34-unit time. The second operation (O2 / of J1 can be operated on machine M3 for 34-unit time. The third operation (O3 / of J1 can be operated on machine M2 for 28-unit time or on machine M1 for 30-unit time. The deadlines of the three jobs are 98, 84, and 92, respectively.

Job Shop Scheduling with Petri Nets Table 4 The job shop scheduling problem

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O1 O2 O3

J1

J2

J3

M1 (36)/ M3 (34) M3 (34)

M2 (35)

M3 (33)

M1 (29)/ M3 (26) M3 (22)

M3 (34)

84

92

M2 (28)/ M1 (30) 98

Deadline

M1 (28)/M2 (30)

P11

36

34

t11

P12

m1

t12 34

30 P13

m3

28 m2

t13

Fig. 18 Petri net model for J1

P14

According to the rules of modeling, this section gets the Petri net model for each job and then merges the small Petri nets by sharing machines. Figures 18–20 are the Petri net model of J1 , J2 , and J3 , respectively. After merging the three small Petri nets by sharing machines, the whole Petri net (Fig. 21) is obtained. Take J1 as an example; t11 , t12 , and t13 are the three predefined operations of J1 . Add four places to connect the transitions such that except the first place, each place represents the completion state of the last operation and the beginning state of next operation. The first place of a job represents the beginning state of the job. M1 , M2 , and M3 are resource places. If an operation can be processed on a machine, there is an arc from the resource place to corresponding transition and also an arc from transition to the resource place. For instance, operation t11 can be operated on machine M1 for 36-unit time or on machine M3 for 34-unit time. So there is an arc from M1 to t11 and also an arc from t11 to M1 , both weights are 36, and there is an arc from M3 to t11 and also an arc from t11 to M3 , both weights are 34. If there are tokens in all the input places of an operation, the operation is ready to be processed. And if there are

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Fig. 19 Petri net model for J2

P21 35 t21 m2 m1

29

P22 26 t22

P23

m3 22

t23

P24

P31

t31

33 m2

P32 30

m1

28

t32

P33

t33

Fig. 20 Petri net model for J3

P34

34

m3

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P11

P21

P31

t11

t21

t31 m3

P12

m1

P22

P32

t12

t22

t32

P13

P23

P33

t13

t23

t33

P14

P24

P34

m2

Fig. 21 Petri net model for the FJSSP

tokens in output place of an operation, the operation has been operated. As shown in Fig. 18, operation t11 is ready to be operated. With the same method, J2 and J3 are modeled. For a Petri net model, a transition sequence is obtained by their firing time, which corresponds to a solution of the scheduling problem (e.g., t11 t21 t31 t32 t22 t33 t12 t13 t23 /.

4.2

Genetic Algorithm for FJSSP

Genetic algorithm (GA) has received much attention since it had been introduced, and it has been a powerful tool of solving some NP problems in many areas. This section uses Petri net to model the scheduling problem. Genetic algorithm is designed on the Petri net model, and a firing sequence as a chromosome, corresponding to a feasible solution.

Chromosome Encoding For a Petri net model, if there is a firing transition sequence  D t1 t2 t3    tn such that M0 [N , iMf (M0 is initial marking and Mf is final marking), then  is regarded as a chromosome.

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Fitness function The objectives are makespan and total tardiness, which are measured by time. For a chromosome , the makespan and the total tardiness of it are computed. Since the importance of the two objectives is equal, the weights are 0.5 and 0.5, respectively. F1 =max(c P i ji =1,2,3, . . . , n/ F2 D i tardiness(i /, i D 1; 2; 3; : : :; n  tardiness.i / D

0; ci  di ;

ci  di ci > di

The fitness function can be denoted as F D 0:5F1 C 0:5F2

Initial population Generally, the initial population is randomly produced. The initial population has great influence on finding final solution of JSSP using genetic algorithm. So the proper initial population should be selected in order to guarantee the quality of the final solution. Since there are orders among the operations of the same job, and unfeasible solutions may be generated when the initial population is generated, the illegal solutions can become feasible by changing some transitions’ positions. Selection operation Roulette Selection algorithm [60] is used to select parent chromosome to produce chromosome of the next generation by crossover and mutation operations. Crossover operation Crossover operation is vital for the genetic algorithm, and the possibility of it has a great influence on the final result. The process of crossover operation is as follows (Fig. 22): 1. Select two chromosomes 1 D t1 t2 t3 : : :ti ti C1 : : :tj 1 tj : : :tn , and 2 D t10 t20 t30 : : :ti0 ti0C1 : : :tj0 1 tj0 : : :tn0 ; and then fire the transitions one by one in both chromosomes. Suppose Mi ŒN; ti iMi C1 and Mi0 ŒN; ti0 iMi0C1 , i D 1; 2; : : :; n. Then, generate many corresponding temporary markings and get 1 D

t1 t2 t3 : : : ti ti C1 : : : tj 1 tj : : : tn M1 M2 M3 : : : Mi Mi C1 : : : Mj 1 Mj : : : Mn

and ¢2 D

t1 0 t2 0 t3 0 : : : ti 0 ti C1 0 : : : tj 1 0 tj 0 : : : tn 0 M1 0 M2 0 M3 0 : : : Mi 0 Mi C1 0 : : : Mj 1 0 Mj 0 : : : Mn 0

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τ1 = t1t2t3…titi+1…tj−1tj…tn τ2 = t1⬘t2⬘t3⬘…ti⬘ti+1⬘…tj−1⬘tj⬘…tn⬘ t2 M2

t3 . . . ti M3 . . . Mi

ti+1 . . . tj−1 Mi+1 . . . Mj−1

tj . . . tn Mj . . . Mn

σ1 =

t1 M1

σ2 =

t2⬘ t3⬘ . . . ti⬘ ti+1⬘ . . . tj−1⬘ tj⬘ . . . tn⬘ t1⬘ M1⬘ M2⬘ M3⬘ . . . Mi⬘ Mi+1⬘ . . . Mj−1⬘ Mj⬘ . . . Mn⬘ Mi=Mi⬘, Mj=Mj⬘

t1 σ1⬘ = M 1 σ2⬘ =

t1⬘ M1⬘

t2 M2 t2⬘ M2⬘

t3 . . . ti M3 . . . Mi

ti+1⬘ . . . tj−1⬘ Mi+1⬙ . . . Mj−1⬙

t3⬘ . . . ti⬘ M3⬘ . . . Mi⬘

tj . . . tn Mj . . . Mn

ti+1 . . . tj−1 Mi+1⵮ . . . Mj−1⵮

tj⬘ . . . tn⬘ Mj⬘ . . . Mn⬘

τ1⬘ = t1t2t3…titi+1⬘…tj−1⬘tj…tn τ2⬘ = t1⬘t2⬘t3⬘…ti⬘ti+1…tj−1tj⬘…tn⬘

Fig. 22 Process of crossover

τ = t1t2t3…ti…tj…tn

τ⬘ = t1t2t3…tj…ti…tn

Fig. 23 Process of mutation

2. Search for the identical markings in 1 and 2 such that Mi D Mi0 , randomly select two pairs such as Mi D Mi0 , Mj D Mj0 , then exchange the gene piece ti ti C1 : : :tj 1 tj of 1 and gene piece ti0 ti0C1 : : :tj0 1 tj0 of 2 , and get two child sequences ¢10 and 20 . At last two child chromosomes 10 and 20 are obtained.

Mutation operation Mutation operation is used to maintain the variety of populations, and it is operated on some genes of a chromosome. Mutation operation can renovate and make up genes that are lost in the selection and crossover operation. Crossing over mutation is applied to change two gene positions of a chromosome (e.g., exchange positions ti and tj in Fig. 23). For the chromosome , positions of ti and tj can be interchanged only when the two operations can be processed on the same machine, and they do not belong to the same job.

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Simulation Experiment

According to the topics talked above, the whole process of genetic algorithm is as follows: Step 1: Step 2: Step 3: Step 4:

Encode with the firing transition sequence. Initialize the populations. Calculate the fitness value of each chromosome of the population. Judge whether the population meets the given condition: YES, go to Step 5; NO, go to Step 7. Step 5: Run roulette selection algorithm to select parent chromosome. Take crossover operation with a possibility and obtain a new population. Step 6: Take mutation operation on the population achieved in Step 5, to optimize the population, and then go to step 3. Step 7: Output the best solution. For the example discussed in Sect. 4.1, firstly the Petri net model is transferred to a matrix F below (Fig. 24), and then all the genetic algorithm operations are all taken on the matrix. According to Table 4, there are three jobs and three machines and each job has three operations. So there are 9 transitions and 15 places (12 places and 3 resource places). A 15  9 matrix F is constructed. The elements of matrix F are defined as follows: F .p; t/ D 1, there is an arc from p to t; F .p; t/ D 1, there is an arc form t to p; F .p; t/ D 0, there is no arc between p and t; F .m, t/ D T; operation t can be processed on machine m for T -unit time; F .m, t/ D 1,000, operation t cannot be processed on machine m. Set the size of population 100 and the number of generation 50. Crossover possibility is 0.4 and mutation possibility is 0.2. The objective is to schedule

Fig. 24 Matrix of the 3  3 FJSSP

p11 p12 p13 p14 p21 p22 p23 p24 p31 p32 p33 p34 m1 m2 m3

t11 1 −1 0 0 0 0 0 0 0 0 0 0 36 1000 34

t12 0 1 −1 0 0 0 0 0 0 0 0 0 1000 1000 34

t13 0 0 1 −1 0 0 0 0 0 0 0 0 30 28 1000

t21 0 0 0 0 1 −1 0 0 0 0 0 0 1000 35 1000

t22 0 0 0 0 0 1 −1 0 0 0 0 0 29 26 26

t23 0 0 0 0 0 0 1 −1 0 0 0 0 1000 1000 22

t31 0 0 0 0 0 0 0 0 1 −1 0 0 1000 1000 33

t32 0 0 0 0 0 0 0 0 0 1 −1 0 1000 1000 34

t33 0 0 0 0 0 0 0 0 0 0 1 −1 28 30 1000

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Fig. 25 Solution of 3  3 FJSSP O32

O31

m3 m2

O21

m1

O11

O12

O23

O22

O13 O33

Table 5 5  4 FJSSP O1 O2 O3 Deadline

J1 M1 (4)/M2 (5)/ M3 (6) M3 (7)/M4 (6)

J2 M3 (8) M2 (3)/ M3 (4) M1 (8)/ M2 (9) 17

M2 (5)/M3 (4)/ M4 (7) 15

m4

O41

m3

O21

m2

O51

m1

J3 M2 (5)/ M3 (6) M1 (7)/ M4 (5) M1 (6)/ M3 (4)/M4 (7) 18

O52

O32

O12 O31

O11

O42 5

J5 M2 (7)/ M3 (9) M4 (5)

M3 (4)/M4 (5)

M1 (3)/ M2 (4) 15

14

O43 O33

O22 O53

10

J4 M2 (4)/M3 (4)/ M4 (5) M1 (6)/M4 (6)

15

O13 O23 20

25

Fig. 26 Solution of 5  4 FJSSP

the three jobs on the three machines to minimize makespan and total tardiness. Figure 25 is the solution for the 3-job-3-machine FJSSP. There is another case in Table 5: five jobs (J1 , J2 , J3 , J4 , J5 / (each has three operations (O1 , O2 , O3 // are scheduled on four machines (M1 , M2 , M3 , M4 /, each job has a deadline. The numbers in brackets are time needed on related machines. The objective is to complete the jobs before their deadline and minimize the makespan. The parameters stay the same as last example. Since genetic algorithm has randomicity, the experiment carried out 30 times, and there are 20 times such

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that the objective value is smaller than 70. The objective value of the optimal solution is 68. The final schedule is shown in Fig. 26.

5

Conclusion

Traditionally, Petri net-based methodologies for JSSP focused on two aspects: One is to model the system with Petri nets and then find the possible schedule by simulation. Another one is based on Petri net model, searching for algorithms for the optimization of the schedule. The common point of current methods is that the Petri net model is assumed to exist. For a real-time complex and large JSSP, generating the Petri net model is in fact a hard work. How to design a correct Petri net model such that it satisfies the requirement by itself is a research topic. Even when the model is given, the simulation and approximation algorithms are not applicable since the state space would be quite large. This chapter introduces a modular-based approach for JSSP design and optimization. It applies the operators in PPPA for specifying and verifying the JSSP such that a correct Petri net model is generated. Based on the approach, a deadlock-free JSSP is designed. Furthermore, the scheduling of a composite module can be induced from the partial scheduling of the constituent modules. Hence, the schedule of the whole system can be induced from the original modules step by step. Theoretically, the modular-based approach can be used to find the optimal solution for JSSP in a formal way. For real-time systems, due to its complexity, the algorithm may have a very large time complexity. In order to reduce the time complexity and to find an approximation optimal solution, the traditional optimization technologies such as GAs, branch and bound methods, hybrid methods, and rule-based methods can be applied together with the modular-based approach in this chapter. Based on the Petri net model, a genetic algorithm is introduced in this chapter to solve the problems. The firing sequence of transitions is regarded as a chromosome; genetic operations such as crossover and mutation are based on the elements of the Petri net model. In order to solve more general JSSP, more integration operators should be considered, and the conclusions about the scheduling-preservation and makespan calculation should also be considered in the future research. Acknowledgements This work was financially supported by the National Natural Science Foundation of China under grant Nos. 11071271 and 61100191 and the Fundamental Research Funds for the Central Universities under grant Nos. HIT.NSRIF.2009127 and HIT.NSFIR. 2011128.

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48. M. Mastrolilli, L.M. Gambardella, Effective neighborhood functions for the flexible job shop problem. J. Scheduling 3, 3–20 (2000) 49. P.J.M. Van Laarhoven, E.H.L. Aarts, K.J. Lenstra, Job shop scheduling by simulation annealing. Oper. Res. 40, 113–125 (1992) 50. M. Dell’Amico, M. Trubian, Applying tabu search to the job shop scheduling problem. Ann. Oper. Res. 40, 231–252 (1993) 51. N.B. Ho, J.C. Tay, Solving multiple-objective flexible job shop problems by evolution and local search. IEEE Trans. Syst. Man Cybern. C Appl. Rev. 38(5), 674–685 (2008) 52. L.N. Xing, Y.W. Chen, K.W. Yang, Multi-objective flexible job shop schedule: design and evaluation by simulation modeling. Appl. Soft Comput. 9, 362–376 (2009) 53. K.J. Shaw, P.L. Lee, H.P. Nott, Genetic algorithm for multi-objective scheduling of combined batch/continuous process plants, in Proceedings of the 2000 Congress on Evolutionary Computation, 2000, pp. 293–300, La Jolla, CA 54. K.S.N. Ripon, C.H. Tsang, S. Kwong, Multi-objective evolutionary job-shop scheduling using jumping genes genetic algorithm. International Joint Conference on Neural Networks, 2006, pp. 3100–3107, Vancouver, BC 55. Y.Y. Chung, L.C. Fu, M.W. Lin, Petri net based modeling and GA based scheduling for a flexible manufacturing system, in Proceedings of the 37th IEEE Conference on Decision and Control, vol. 4, 1998, pp. 4346–4347, Tampa, FL 56. Z. Tao, T.Y. Xiao, Petri net and GASA based approach for dynamic JSSP, in Proceedings of the IEEE International Conference on Mechatronics and Automation, 2007, pp 3888–3893, Harbin, China 57. T.C. Chiang, A.C. Huang, L.C. Fu. Modeling, scheduling, and performance evaluation for wafer fabrication: a queueing colored Petri-net and GA-based approach. IEEE. Trans. on Auto. Sci. and Eng. 3(3), 330–338 (2006) 58. H.J. Huang, T.P. Lu, Solving a multi-objective flexible job shop scheduling problem with timed Petri nets and genetic algorithm. Discr. Math. Algorithms Appl. 2(2), 221–237 (2010) 59. M. Garey, D. Johnson, R. Sethi, The complexity of flow shop and job shop scheduling. Math. Oper. Res. 24(1), 117–129 (1976) 60. J. Baker, Adaptive selection methods for genetic algorithms, in Proceedings of the 1st International Conference on Genetic Algorithms, 1985, pp. 100–111, Hillsdale, NJ, USA

Key Tree Optimization Minming Li

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Batch Update with Probability on Membership Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Uniform Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Loyal Users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Heterogeneous Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Batch Update with a Fixed Number of Membership Changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 1-Replacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 k-Replacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Other Variants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Minimize sequential update cost for a sequence of actions. . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Minimize Cost for the Next Update. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Optimization with Underlying Network Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In the applications where a group of people share some service such as teleconferencing and online TV, the most important security problem is to ensure that only the authorized users can enjoy the service. One popular way to enforce the security is to encrypt the service content using a group key and update it whenever users leave or join. The key tree model is proposed for the purpose of managing key updates. Sometimes, the system manager knows the user behavior related to the group and therefore hopes to reduce the number of encryptions needed to update keys. For example, in applications where the resource is limited and there are always users waiting to join the group, the system manager may

M. Li Department of Computer Science, City University of Hong Kong, Hong Kong, People’s Republic of China e-mail: [email protected] 1713 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 82, © Springer Science+Business Media New York 2013

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decide to update the keys whenever k users leave (and hence k new users join) the group. There are also scenarios where there are some users who never leave the group (loyal users), but others will leave the group with a certain probability before the system manager decides to rekey. In all these and other similar scenarios, finding a tree which requires minimum number of encryptions to update the keys will be of great interest to the system manager because encryptions usually take quite some time. In this chapter, the relevant results in this area will be surveyed.

1

Introduction

Secure group communication is one of the major problems in wireless security. How to distribute the keys over insecure wireless channels is quite a challenging problem. Thereareseveralfundamentally differentapproachesto dealing withkeymanagement. Key pre-distribution schemes refer to methods where a trusted authority distributes secret information to users so that only privileged users can get the group key. The concept of probabilistic key pre-distribution was proposed by Eschenauer and Gligor in their pioneering work [13]. Chan et al. [7] generalized this idea by considering q-compositeness, increasing the security and robustness against node loss. Liu and Ning [23], extending the ideas of [7, 13], later designed a polynomialbased framework by using random subsets and grids. Zhu et al. [46] designed LEAP protocol which uses different methods for different types of transmission. Contributory group key agreement schemes require that each member contribute an equal share of the group key. The Diffie-Hellman key exchange protocol [11] is the first important contributory key management protocol, and is also the pioneer of every other subsequent protocol of this kind. Later, the following generalizations are proposed. Ingemarsson et al. [19], Burmester and Desmedt [5], and Steiner et al. [34–36] arrange users in a logical ring or chain structure and accumulate the keying material while traversing group members one by one. Ateniese et al. [1] and Becker and Wille [3] also ask each node to contribute an input to establish a common secret through successive pairwise message exchanges among the nodes in a secure manner using two-party D-H exchange [11]. In [12, 39, 40], logical tree structures are introduced, and the number of rounds for establishing the group key is reduced to the logarithm of the group size. Group key management schemes use a trusted third party to perform all bookkeeping tasks, and only one group key is maintained and used by everybody. Regarding functionalities, they can be further divided into two types: broadcast encryption, in which join is not supported, and group key distribution, in which both join and leave are supported. Broadcast encryption schemes are created by Fiat and Naor [14] to handle dynamic membership changes in secure communication groups. Naor et al. [27] later designed an efficient stateless broadcast encryption scheme SD. SD users’ storage requirement O.log2 n/ was reduced by Halevy et al. [17], making the scheme more practical to large-scale networks. Aside from them, perfect hash families were also used to construct different broadcast encryption schemes, such as the work of Safavi-Naini et al. [30] and Fiat and Naor [14].

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Group key distribution protocols are mainly built on top of logical trees. We give more details below since it is the focus of the chapter. Other variants can be found in [2, 4, 6, 24–26, 29, 33, 37, 38]. In the group key distribution problem, there are n subscribers and a group controller (GC) that periodically broadcasts messages (e.g., a video clip) to all the subscribers over an insecure channel. To guarantee that only the authorized users can decode the contents of the messages, the GC will dynamically update the group key for the whole group. Whenever some user leaves or joins, the GC will generate a new group key and in some way notify the remaining users in the group. The update of keys basically provides two kinds of securities. One is Future Security which prevents a deleted user from accessing the future content; the other is Past Security which prevents a newly joined user from accessing the past content. Key tree model, which was proposed by Wong et al. [42] and Wallner et al. [41], is one of the most popular models used to achieve the goal above. In the key tree model, the group controller (GC) maintains a tree structure for the whole group. The root of the tree stores a Traffic Encryption Key (TEK) which is used to encrypt the content that should be broadcast to the authorized users. Every leaf node of the key tree represents a user and stores his individual key. Every internal node stores a Key Encryption Key (KEK) shared by all leaf descendants of the internal node. Every user possesses all the keys along the path from the leaf node (representing the user) to the root node. To prevent revoked users from knowing future message contents and also to prevent new users from knowing past message contents, the GC updates a set of keys, whenever a new user joins or a current user leaves, as follows. So long as there is a user change among the leaf descendants of an internal node v, the GC will (1) replace the old key stored at v with a new key and (2) broadcast (to all users) the new key encrypted with the key stored at each child node of v. Note that only users represented by leaf descendants of v can get useful information from the broadcast. Furthermore, this procedure must be done in a bottom-up fashion (i.e., starting with the lowest v whose key must be updated.) to guarantee that a revoked user will not know the new keys. The cost of the above procedure counts the number of encryptions used in step 2 (or equivalently, the number of broadcasts made by the GC). Because updating keys for each user change is too frequent in some applications, [21] proposed batch rekeying model where keys are only updated after a certain period of time. A good survey for key tree management can be found in [15]. In this chapter, only optimization problems in batch updating scenario will be considered. Other works can be found in [8, 18, 20, 31], etc.

2

Batch Update with Probability on Membership Change

When the service is not so frequently broadcasted but the user change is frequent, updating keys after every user change will be too costly and unnecessary. Thus, a batch rekeying strategy was proposed in [21] whereby rekeying is done only periodically instead of immediately after each member change. They designed a

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marking algorithm for processing the batch updates. It was shown by simulation that, using their algorithm, among the totally balanced key trees (where internal nodes of the tree have branching degree 2i ), degree 4 is the best when the number of requests (leave/join) within a batch is not large. For large number of requests, using a star (a tree of depth 1) to organize the users outperforms all the balanced key trees mentioned above in their simulation. According to how users behave, the following three scenarios are discussed here: uniform probability, loyal user, and heterogeneous probability.

2.1

Uniform Probability

Zhu et al. [45] proposed the uniform probability model where each of the n positions has the same probability p to independently experience subscriber change during a batch rekeying period. In this model, an internal node v with Nv leaf descendants will have probability 1  q Nv that its associated key kv requires updating, where q D 1  p. The updating incurs dv  .1  q Nv / expected broadcast messages by the procedure described above. P Thus, the expected updating cost C.T / of a key tree T is defined by C.T / D v dv  .1  q Nv / where the sum is taken over all the internal nodes v of T . C.T / can also be viewed as an edge weight summation: For each tree edge e D .u; v/ where u is a child of v, its edge weight is defined to be 1  q Nv . The optimization problem can be formulated as follows. Given two parameters 0  p  1 and n > 0, let q D 1  p. For a rooted tree T with n leaves and edge set E, define a weight function c.e/ on the tree edges as follows. For every edge e D .u; v/ where u is a child of v and P v has Nv leaf descendants, let c.e/ D 1  q Nv . Define the cost of T as C.T / D e2E c.e/. Find a T for which C.T / is minimized. We say that such a tree is .p; n/-optimal and denotes its cost by OP T .p; n/. Zhu et al. [45] characterized the optimal tree with the restriction that each internal node on level i has 2ai children and the total number of users is 2k . Graham et al. [16] extend the search of the optimal key tree for batch updates from restricted classes of trees to all trees. Below are some results and observations from [16]. The main idea of the algorithm to find the optimal tree is to prove some properties of the optimal tree by changing the structure of an arbitrary tree to one with better properties without increasing the total cost. This way, search space can be greatly reduced and make polynomial-time algorithm possible.

2.1.1 When the Star Is Optimal First, several definitions are given. A tree is of depth k if the longest leaf-root path consists of k edges. A tree of depth 2 is also referred to as a two-level tree. A tree of depth 1 is called a k-star if it has k leaves. A tree edge .u; v/ where u is a child of v is said to be at depth k if the path from u to the root consists of k edges. The branching degree of a node v is the number of children of v; the subtree size of v, denoted by Nv , refers to the number of leaf descendants of v.

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Lemma 1 If the n-star can achieve OP T .p; n/, then the (n–1)-star can also achieve OP T .p; n  1/. Proof The lemma can be proved by contradiction. Suppose the .n  1/-star cannot achieve OP T .p; n  1/. Let the degree of the root in a .p; n  1/-optimal tree be k where k < n  1. Write the optimal cost as OP T .p; n  1/ D k.1  q n1 / C C , where C represents the contribution to the cost by edges at depth  2. Thus, .n1/.1q n1 / > k.1q n1 /CC , which implies n.1q n / > .k C1/.1q n /CC . This means the cost of OP T .p; n/ can be reduced by adopting the same structure of the .p; n  1/-optimal tree but with root degree k C 1, a contradiction.  Lemma 2 When 0  q  31=3 , the n-star is strictly better than any two-level tree. Proof A two-level tree can be obtained from a star by successively grouping certain nodes together to form a subtree of the root. To prove the lemma, one only needs to show that the above operation always increases the cost of the tree, i.e., for any grouping size k where 1 < k < n, one needs to show that 1q n < k1 .1q n /C1q k . This is trivially true when k D 1 or q D 0, so one can assume that k  2 and q > 0. With fixed q > 0, define f .k/ for integer k by f .k/ D k logk .1=q/. Notice that for any fixed q where 0 < q < 1, the value of f .k/ is minimized when k D 3. Thus, when 0 < q  31=3 , one has k logk .1=q/  1 which implies kq k  1. Hence, for 0 < q  31=3 , the following deductions are correct:  1  qn 

1 .1  q n / C 1  q k k

 D

1 .kq k  1  .k  1/q n / k

1 .kq k  1/ k  0:
0. Proof The first part of the theorem follows from Lemmas 2 and 3. The cases of 2  n  4 can be verified easily. 

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2.1.2 Properties of an Optimal Tree By Theorem 1, the structure of an .p; n/-optimal tree is uniquely determined for 0  q  31=3 Ð 0:693 (or 1  p  0:307). In the following, some properties of the optimal trees will be derived which will be used for constructing an .p; n/optimal tree in the remaining range 1  q > 31=3 . Note that the following Lemmas 4 and 5 as well as Theorem 2 are true for all .p; n/-optimal trees where 0  p  1 and n > 0. For a tree T , associate a value tv D q Nv with every node v (thus, tv D q if v is a leaf). The subtree rooted at u is denoted by Tu . Tu is called a subtree of v if u is a child of v. Lemma 4 For a non-root internal node v with branching degree k in a .p; n/. optimal tree, every child u of v satisfies tu  k1 k , one can move u up to become a sibling of v, as shown in Fig. 1. Proof If tu < k1 k In this way, the total cost of the tree is increased by C D .1  q Nw / C .k  1/.1  q Nv Nu /  k.1  q Nv / < 1 C .k  1/.1  q Nv Nu /  k.1  q Nv Nu tu / D q Nv Nu .ktu  .k  1//

1=2. Thus, Nu < logq 12 . Since v has branching degree at most 4 by property 1 and also Nv  1, one has Nv  maxf4 logq 12 ; 1g. This completes the proof of the theorem. 

2.1.3 Algorithm for Constructing the Optimal Tree A .p; n/-optimal tree can be constructed by assembling a forest of suitable subtrees. Firstly, the cost function needs to be generalized from trees to forests as follows. Definition 1 For L  n, define a .p; n; L/-forest to be a forest of key trees with L leaves in total. The cost of the tree edges in the forest is defined as before, while the cost of the forest is the sum of individual tree costs plus k  .1  q n /, where k is the number of trees in the forest. The .p; n; L/-forest with minimum cost is referred to as the optimal .p; n; L/-forest, and the corresponding minimum cost is denoted by F .p; n; L/. Theorem 3 For any fixed p, Algorithm 1 computes the .p; n/-optimal tree cost in O.n/ time. Proof By Theorem 2, in a .p; n/-optimal tree, any subtree Tv where v is a child of the root satisfies (1) its size is at most maxf4 logq 12 ; 1g and (2) the branching degree of any internal node in Tv is at most 4. For fixed q, 4 logq 12 can be viewed as a constant K. For each i where 2  i  K, the minimum cost R.i / of any tree Tv with size i and subject to the degree restriction stated in (2) will be the focus of the calculations. The value of R.i / can be computed in constant time as follows. First, define .k1 ; k2 ; k3 ; k4 / to be an i -quadruple if k1 C k2 C k3 C k4 D i , 0  k1 ; k2 ; k3 ; k4  i , and k   2 where k  is the number of positive elements in .k1 ; k2 ; k3 ; k4 /. Then, R.i / is the minimum value of R.k1 /CR.k2 /CR.k3 /CR.k4 /C.1 q i / k  over all i -quadruples .k1 ; k2 ; k3 ; k4 /, and it can be computed in O.K 3 / time using dynamic programming. Now, one can obtain the true .p; n/-optimal tree also by dynamic programming, by computing optimal .p; n; L/-forests as subproblems of size L for 1  L  n as given in Algorithm 1 . Thus, the total running time of the algorithm is O.n  K C K 4 / which is O.n/ for fixed p. Algorithm 1 focuses on computing the optimal tree cost; the tree structure can be obtained by keeping track of the optimal branching at every dynamic programming iteration. 

2.1.4 Optimal Trees as p ! 0 Algorithm 1 has running time O.n  K C K 4 / where K is upper bounded by 4 logq 12 (and also by n). We can regard O.K 4 / as a constant term for fixed value of p, but its value gets large as p ! 0. Therefore, the structure of .p; n/-optimal trees as p ! 0 is worth exploring. The following theorem is proved in [16].

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Algorithm 1 Computing optimal key tree Input: n and p .0  p  1/ Output: Optimal tree cost OP T .n; p/ q D1p K D minf4 logq 21 ; ng if q  31=3 or 2  n  4 then OP T .p; n/ n  .1  q n / Return OP T .p; n/ end if R.1/ D 0 for i D 2 to 4 do R.i / D i  .1  q i / end for i D5 while i < K do Compute R.i /, cost of the restricted .p; i /-optimal tree. i Di C1 end while F .p; n; 0/ 0 for L D 1 to n do F .p; n; L/ min1j minfK;Lg .R.j / C 1  q n C F .p; n; L  j //. end for OP T .n; p/ F .p; n; n/ return OP T .n; p/

Theorem 4 As p ! 0, the .p; n/-optimal tree T  .n/ always has root degree 3 except for n of the form 4  3t , in which case T  .4  3t / has root degree 4, and for n D 2, when T  .2/ has root degree 2. When 2  3t  n  3t C1 , then T  .n/ is as close to a balanced ternary tree with n leaves as possible. Namely, all subtrees (as well as T  .n/ itself) have root degrees 3, except for the very bottom level, where subtrees of size 2 can occur. However, when 3t  n < 2  3t , T  .n/ can deviate substantially from a balanced ternary tree. For example, T  .39/ has three subtrees of sizes 9; 12; and 18 (see Fig. 4). It is observed by Graham et al. [16] to be difficult to predict the sizes of the subtrees in the optimal tree T  .n/ for certain values of n. For example, for n D 1;252, the sizes are 280; 486, and 486, while for n D 1;253, the sizes are 324; 443, and 486. In [22], the authors studied the approximate optimal tree. They design a simple O.n/ algorithm whose approximation ratio is nearly 2. They also design a refined algorithm and show that it achieves an approximation ratio of 1 C .

2.2

Loyal Users

The concept of “loyal users” is first proposed in [28]. They restructure the key tree scheme and design a modified protocol considering the membership duration. Two kinds of users, volatile and permanent members, are separately grouped into two

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Fig. 4 An optimal T  .39/ •••

LUN

•••

LUN

•••

normal leaves •••

Loyal leaves

•••

Loyal leaves

•••

normal leaves

Fig. 5 An example tree with LUN

key trees. The newly joined user is first regarded as a volatile member; once the duration of the user exceeds a threshold, it is moved to the permanent member group. Chan et al. [9] extends the uniform probability batch update key tree model by considering two types of users in the same key tree. Users belonging to the first type have probability p to issue a leave request (the vacant place will be taken by the newly joined user that belongs to this type), while users belonging to the second type are considered loyal (with probability zero to leave the group) to the group. The objective is to theoretically study the optimal key tree structure to minimize expected rekeying cost. Note that loyal users have zero probability to leave the group. To control the tree cost, the loyal users should be appropriately arranged. They define a special internal node, Loyal User Node (LUN), whose children all are loyal users. Figure 5 shows an example. For a key tree T with n leaves, Nv (and Lv ) is used to denote the number of “normal” (and “loyal”) leaf descendants of node v. The normal users have probability p to leave the group, while the loyal users have probability zero. Note that in this new model, the expected updating cost for every internal node v is dv  .1  .1  p/Nv /. Thus, one can apply the distribution trick again as follows. Let V be the node set of T , including all internal nodes, normal and loyal leaves. Given p and n and let q D 1p, the weight c 0 .u/ of node u over V can be defined as

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c 0 .u/ D

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0 if u is the root 1  q Nv if u is a child of internal node v

P The cost of T is defined as C 0 .T / D u2V c 0 .u/. Accordingly, T is a .p; n/0 optimal tree if C 0 .T / is minimum among all possible tree structures. The cost of the .p; n/0 -optimal tree is denoted as OP T 0 .p; n/. Our objective is to find the structure of .p; n/0 -optimal trees. They first showed the following property of loyal users. Lemma 6 For any tree with loyal users, removing loyal leaves strictly decreases the tree cost. Proof Consider the subtree Tv rooted at an internal node v which has lv (lv  1) children to be loyal leaves. Let Nv be the number of normal leaves on Tv . Removing all loyal children from v will decrease the tree cost by C D lv .1  q Nv /. It is obvious that C  0 as lv  1. The cost on the remaining nodes does not change. This finishes the proof.  Let r be the root of the tree T and Lr be the number of loyal leaves (Lr  1). One can obtain a tree with no larger cost by doing the following three steps: 1. Arbitrarily remove Lr  1 loyal leaves from T . 2. Replace the last loyal leaf u with a LUN v. 3. Add all loyal leaves removed in step 1 including u to be children of v. By Lemma 6, step 1 strictly decreases the tree cost. For steps 2 and 3, the new LUN v will not increase the tree cost because all nodes inserted to v are loyal leaves which do not issue any join/leave request. Thus, the transformation does not increase the tree cost. In fact, the newly joined loyal users will be placed under the LUN. This does not make any change to the tree structure excluding the LUN. In other words, the LUN can represent all loyal leaves. Thus, optimal trees can be found by focusing on trees with n normal leaves and one LUN. Similar properties can be proved about the optimal key tree based on the proof paradigm in [16]. The final dynamic programming algorithm to calculate the optimal key tree with loyal users is summarized below. Theorem 5 Algorithm 2 computes a .p; n/0 -optimal tree in O.n  K C K 4 / time where K D minf4.log q 1 /1 ; ng. Proof Define a .p; n; L/0 -f orest to be a forest of key trees with L normal leaves and one LUN. Similarly define .p; n; L/-f orest to be a forest of key trees with L normal leaves and no LUN. The cost of the forest of the trees is the cost of the individual trees plus k.1  q n / where k is the number of trees. Use F 0 .p; n; L/ to denote the minimum cost of a .p; n; L/0 -forest and F .p; n; L/ to denote the minimum cost of a .p; n; L/-forest. 

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Algorithm 2 Computing optimal key tree with a LUN Input: n: Number of normal users (n  1) p: Probability of normal users to update (0 < p < 1) Output: Cost of .p; n/0 -optimal tree OP T 0 .p; n/ q D 1  p;K D minf4.log q 1 /1 ; ng //Optimal .n/0 -star tree if q  0:57 then OP T 0 .p; n/ D .n C 1/.1  q n / return OP T 0 .p; n/ end if // Optimal tree for n  4.log q 1 /1 R.0/ D R.1/ D 0; R.2/ D 2.1  q 2 /; R0 .0/ D 0; R0 .1/ D 2.1  q/; R0 .2/ D 3.1  q 2 /; i D 3; while i  K do R.i / D minfR.k1 / C R.k2 / C R.k3 / C R.k4 / C .1  q i /k  g; R0 .i / D minfR0 .k1 / C R.k2 / C R.k3 / C R.k4 / C .1  q i /k  g; i D i C 1; end while if n  4.log q 1 /1 then return OP T 0 .p; n/ D R0 .n/. end if // Optimal tree when n > 4.log q 1 /1 F .p; n; L/ D 0;F 0 .p; n; L/ D 0; for L D 1 to n do F .p; n; L/ D min1j minfK;Lg fR.j / C 1  q n C F .p; n; L  j /g; F 0 .p; n; L/ D min1j minfK;Lg fR0 .j / C 1  q n C F .p; n; L  j /g; end for OP T 0 .p; n/ D F 0 .p; n; n/ return OP T 0 .p; n/

2.3

Heterogeneous Probability

When a different user has different probability of leaving the group, then the problem becomes much complicated. Lemmas 4 and 5 and Theorem 2 are still true. The size limit for each subtree connecting to the root is now only related to the smallest possible probability of leaving. However, with all these good features preserved, one still needs to guess the number of different probabilities in each branch, therefore making polynomial-time algorithm impossible if one still follows the dynamic programming framework.

3

Batch Update with a Fixed Number of Membership Changes

In this section, it is assumed that k users will leave the group during the batch period. A key tree which can minimize the worst case number of encryptions caused during the batch period is the target.

Key Tree Optimization

1725

a

b K10

K8

K9

K10

K6

K7

u6

u7

K8

K9

K11

K1

K2

K3

K4

K5

K1

K2

K3

K4

K5

K6

K7

u1

u2

u3

u4

u5

u1

u2

u3

u4

u5

u6

u7

Fig. 6 Two structures of a group with seven users

As shown in Fig. 6a, there are seven users in the group. We take the deletion of user u4 as an example. Since u4 knows k4 ; k9 , and k10 , the GC needs to update the keys k9 and k10 (the node that stores k4 disappears because u4 is already deleted from the group). GC will encrypt the new k9 with k5 and broadcast it to notify u5 . Note that only u5 can decrypt the message. Then, GC encrypts the new k10 with k6 ; k7 ; and k8 and the new k9 , respectively, and then broadcasts the encrypted messages to notify the users. Since all the users and only the users in the group can decrypt one of these messages, the GC can safely notify the users except user u4 about the new TEK. The deletion cost measured as the number of encryptions is 5 in this example. In popular servers, since there are always users on the waiting list, the batch rekeying model will update the TEK by replacing constant k revoked users with k waiting users. The newly joined k users will take the k positions which are vacant due to the leave of k users, thus keeping the structure of the tree unchanged for each update. This reduces the update frequency. The updating cost equals the summation of degrees of all ancestors of the k old users (leaves). Thus, the objective is to find an optimal tree where the worst case updating cost incurred by the kuser membership change is minimum. This problem is denoted as k-replacement problem. In fact, [32] showed that 2–3 tree is the optimal tree for 1-replacement problem. Wu et al. [43] further denote the problem to find the optimal tree when k users are deleted from the tree as k-deletion problem. As shown in Fig. 6b, the tree has a worst case cost 5 for 1-deletion problem and worst case cost 6 for 1-replacement problem. In the following, a node u is said to have degree/branching degree d if it has d children in the tree. Wu et al. [43] first define the k-replacement problem formally as follows. Definition 2 Given a tree T , the number of encryptions incurred by replacing P  ui1 ; : : : ; uik  with k new users is denoted as CT .ui1 ; : : : ; uik / D S dv where ANC.u/ is the set of u’s ancestor nodes and dv v2 ANC.u / 1j k

ij

is v’s degree. k-replacement cost refers to the maximum cost among all possible combinations and is written as Ck .T; n/ D maxi1 ;i2 ;:::;ik CT .ui1 ; ui2 ; : : : ; uik /.

1726

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An optimal tree Tn;k;opt (abbreviated as Tn;opt if the context is clear) for the k-replacement problem is a tree which has the minimum k-replacement cost over all trees with n leaves, i.e., Ck .Tn;opt ; n/ D min Ck .T; n/. This optimal cost is denoted T

as OP Tk .n/. The k-replacement problem is to find the OP Tk .n/ and Tn;k;opt . The k-deletion cost and the k-deletion problem are defined similarly by using the cost incurred by permanently deleting the leaves instead of the cost incurred by replacing the leaves. Note that some keys do not need to be updated if all its leaf descendants are deleted and the number of encryptions needed to update that key is also reduced if some branches of that node totally disappear due to deletion. Notice that the k-deletion problem and the k-replacement problem are not trivially equivalent. Wu et al. [43] showed that the k-deletion problem can be solved by handling the k-replacement problem. Below are major deductions from [43]. Definition 3 Define a node v to be a pseudo-leaf node if its children are all leaves. In the following two lemmas, t is used to denote the number of pseudo-leaf nodes in a tree T . Lemma 7 If t  k, then the k-deletion cost of T is at least n  k. Proof When t  k, in order to achieve k-deletion cost, at least one leaf needs to be deleted from each pseudo-leaf node. Suppose on the contrary there exists one pseudo-leaf node v where none of its children belong to the k leaves deleted. The discussion can be divided into two cases. First, if each of the k leaves is a child of the remaining t  1 pseudo-leaf nodes, then there exists one pseudo-leaf node u with at least two children deleted. In this case, a larger deletion cost can be achieved if one deletes one child of v while keeping one more child of u undeleted. Second, if some of the k leaves are not from the remaining t  1 pseudo-leaf nodes, then one can assume that u is one of them whose parent is not a pseudo-leaf. Then there exists one of u’s siblings w that has at least one pseudo-leaf w0 (w0 can be w itself) as its descendant. If no children of w0 belongs to the k leaves, then deleting a child of w0 while keeping u undeleted incurs larger deletion cost. If at least one child of w0 belongs to the k leaves, then deleting a child of v while keeping u undeleted incurs larger deletion cost. It can be seen that in the worse case deletion, each pseudo-leaf node has at least one child deleted, which implies that all the keys in the remaining nk leaves should be used once as the encryption key in the updating process. Hence the k-deletion cost of T is at least n  k.  Lemma 8 If t > k, then the k-deletion cost is Ck .T; n/  k where Ck .T; n/ is the k-replacement cost.

Key Tree Optimization

1727

Proof Using similar arguments as in the proof of Lemma 7, one can prove that when t > k, the k-deletion cost can only be achieved when the k deleted leaves are from k different pseudo-leaf nodes. Then it is easy to see that the k-deletion cost is Ck .T; n/  k where Ck .T; n/ is the k-replacement cost.  Although worst case costs between these two problems do not always differ by a constant k given some tree T , they show that the worst case costs between their optimal tree structures always differ by k. Theorem 6 When considering trees with n leaves, the optimal k-deletion cost is OP Tk .n/  k where OP Tk .n/ is the optimal k-replacement cost. Proof Note that in the tree where all n leaves have the same parent (denoted as one-level tree), the k-deletion cost is n  k. By Lemma 7, any tree with the number of pseudo-leaf nodes at most k has the k-deletion cost at least n  k. Hence, one only needs to search the optimal tree among the one-level tree and the trees with the number of pseudo-leaf nodes larger than k. Moreover, in the one-level tree T , the k-deletion cost is n  k D Ck .T; n/  k where Ck .T; n/ is the k-replacement cost. Further, by Lemma 8, all the trees in the scope for searching the optimal tree have k-deletion cost Ck .T; n/  k, which implies that the optimal k-deletion cost is OP Tk .n/  k where OP Tk .n/ is the optimal k-replacement cost.  The above analysis implies that the optimal tree for the k-deletion problem can be obtained by solving the k-replacement problem. Therefore, one only needs to consider the k-replacement problem in the following.

3.1

1-Replacement

In [32], the authors investigate the optimal tree structure with n leaves where the worst case single-deletion cost is minimum. Their result shows that the optimal tree is a special kind of 2–3 tree defined below. Definition 4 2–3 tree is a tree with n leaves constructed in the following way: 1. When n  5, the root degree is 3 and the number of leaves in three subtrees of the root differs by at most 1. When n D 4, the tree is a complete binary tree. When n D 2 and n D 3, the tree has root degree 2 and 3, respectively. When n D 1, the tree only consists of a root. 2. Each subtree of the root is a 2–3 tree defined recursively.

3.2

k-Replacement

The following discussion also comes from [43].

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M. Li

3.2.1 First Step: Loose Degree Bound for the k-Replacement Problem Given a constant k, k-replacement problem can be computed in polynomial time by deducing a loose degree bound. In the following proofs, the following procedure is often done: First, choose a template tree T and then construct a tree T 0 by removing from T some leaves together with the exclusive part of leaf-root paths of those leaves. Here, the exclusive part of a leaf-root path includes those edges that are not on the leaf-root path of any of the remaining leaves. In this case, T is called a template tree of T 0 . By the definition of the k-replacement cost, the following fact holds. Fact 1 If T is a template tree of T 0 , then the k-replacement cost of T 0 is no larger than that of T . Lemma 9 OP Tk .n/ is nondecreasing when n increases. Proof Suppose on the contrary OP Tk .n1 / > OP Tk .n2 / when n1  n2 , then there exist two trees T1 and T2 satisfying Ck .T1 ; n1 / D OP Tk .n1 / and Ck .T2 ; n2 / D OP Tk .n2 /. One can take T2 as a template tree and delete the leaves until the number of leaves decreases to n1 . The resulting tree T20 satisfies Ck .T20 ; n1 /  Ck .T2 ; n2 / < OP Tk .n1 / by Fact 1, which contradicts the definition of OP Tk .n1 /. The lemma is then proved.  Lemma 10 Tn;opt has root degree upper bounded by .k C 1/2  1. Proof The value of root degree d  .k C 1/2 can be divided into two sets, fd j.k C t/2  d < .k C t/.k C t C 1/; d; k; t 2 N; t  1g and fd j.k C t  1/.k C t/  d < .k C t/2 ; d; k; t 2 N; t  2g. Take k D 2, for instance, the first set is f9; 10; 11; 16; 17; 18; 19; 25; : : :g, while the other is f12; 13; 14; 15; 20; 21; 22; 23; : : :g. Case 1: .k C t/2  d < .k C t/.k C t C 1/ (t  1). Write d as .k C t/2 C r where 0  r < k C t. Given a tree T , one can transform it into a tree with root degree k C t as Fig. 7 shows. In the resulting tree T 0 , subtrees Tu1 ; : : : ; TukCt are k C t subtrees where the root u1 ; : : : ; ukCt are on level one. Among the k C t subtrees, there are r subtrees with root degree k C t C 1 and k C t  r subtrees with root degree k C t. Suppose that the k-replacement cost of T 0 is incurred by replacing k1 ; k2 ; : : : ; ks users from subtrees Ti1 ; Ti2 ; : : : ; Tis respectively where P k1 C k2 C : : : C ks D k and s  k. The corresponding cost is Ck .T 0 ; n/ D sj D1 Ckj .Tij ; nij / C D0 where nij is the number of leaves in Tij and D0 is the cost incurred in the first two levels. In the original Ps tree T , the corresponding Ps cost for replacing those 2leaves is Ck .T; n/ D C .T ; n / C d D k i i j j j j D1 j D1 Ckj .Tij ; nij / C .k C t/ C r. In the following, one can show Ck .T; n/  Ck .T 0 ; n/, i.e., D0  .k C t/2 C r when t  1.

Key Tree Optimization

1729 T⬘ d=k+t

T d = (k + t)2+r

u1

u2

uk+t−1

uk+t d=k+t

d=k+t+1 T1 T2 T3

Td−2 Td−1 Td T1

Tk+t+1

Td

Fig. 7 Transformation of the tree which has root degree d D .k C t /2 C r T⬘ d=k+t

T d = (k + t) − (k + t − r) 2

u1

u2

uk+t−1

d=k+t−1 T1 T2 T3

uk+t d=k+t

Td−2 Td−1 Td T1

Tk+t−1

Td

Fig. 8 Transformation of the tree which has root degree d D .k C t /2  .k C t  r/

Firstly, if r  k, the cost D0 is at most r.k C t C 1/ C .k  r/.k C t/ C k C t where there are r users coming from r subtrees with root degree k C t C 1 and k  r users coming from k  r subtrees with root degree k C t. Therefore, one has D0  .k C t C 1/r C .k C t/.k  r/ C k C t D .k C t/.k C 1/ C r  .k C t/2 C r: Secondly, if r > k, the cost D0 is at most .k C t/ C .k C t C 1/k where the k users are all from k subtrees which have root degree k C t C 1. Therefore, one has D0  .k C t/ C .k C t C 1/k  .k C t/.k C 1/ C k  .k C t/2 C r: Hence, in both situations, the condition t  1 ensures that the transformation from T to T 0 does not increase the k-replacement cost. Case 2: .k C t  1/.k C t/  d < .k C t/2 (t  2). In this case, d can be written as .k C t/2  .k C t  r/ where 0  r < k C t. One can transform T into a tree with root degree k C t where there are r subtrees with root degree k C t and k C t  r subtrees with root degree k C t  1, as Fig. 8 shows. Similarly for r  k, one has

1730

M. Li

D0  .kCt/rC.kCt1/.kr/CkCt D .kCt/.kC1/.kr/  .kCt/2 .kCtr/: The last inequality holds because t  2. While for r > k, one has D0  .k C t/k  .k C t/2  .k C t  r/: Thus, t  2 ensures that the transformation from T to T 0 does not increase the k-replacement cost. Therefore, for any root degree d  .k C 1/2 , through a series of transformations, the original tree can be transformed into a tree with root degree less than .k C 1/2 without increasing the k-replacement cost. For example, when d D 120 and k D 2, since 120 D .k C 9/2  1, firstly one can transform it into a tree with root degree k C 9 D 11. Since 11 D .k C 1/2 C 2 is still greater than .k C 1/2 , one can further transform the resulting tree into a tree with root degree k C 1 D 3. Because 3 < .2 C 1/2 , no further transformation will take place. The lemma is finally proved.  Lemma 10 suggests that one can find an optimal tree for the k-replacement problem among trees whose root degree is at most .k C t/2  1. Note that the degree bound in Lemma 10 is only for the root. One can also extend this property to all the internal nodes. This leads to a polynomial-time algorithm to compute the optimal k-replacement tree given a constant k. Theorem 7 Any internal node in Tn;opt has degree upper bounded by .k C 1/2  1. 2 Given a constant k, the k-replacement problem can be computed in O.n.kC1/ / time. Proof Suppose T is an optimal tree and has an internal node v which has degree dv  .k C 1/2 , as shown in Fig. 9. One can transform the subtree Tv rooted at v into Tv0 in the similar way as in Lemma 10 (The resulting tree is denoted by T 0 ). One can prove that the k-replacement cost in the resulting tree T 0 does not increase. Consider all the three possible relative positions of those k leaves whose membership change achieves k-replacement cost in T 0 . First, if none of the leaves are from Tv0 , then the replacement cost of the corresponding leaves in T will be the same as the replacement cost in T 0 . If there are m (m < k) leaves from Tv0 while the others are not, then replacing the corresponding leaves in T incurs no smaller replacement cost compared to that in T 0 , because replacing m leaves of Tv incurs no smaller cost than replacing the corresponding m leaves in Tv0 due to similar analysis shown in Lemma 10. If all the k leaves are from Tv0 , then replacing the corresponding leaves in T again incurs no smaller replacement cost than T 0 by Lemma 10. In all the above three cases, T 0 has replacement cost no greater than T , which enables us to transform the degree of every internal node to be equal to or less than .k C 1/2  1 without increasing the k-replacement cost. By enumerating all the possible degrees of the internal node, one can design a 2 O.n.kC1/ / time dynamic programming algorithm to iteratively compute the optimal

Key Tree Optimization

1731 T⬘ T T ⬘v d=k+t

Tv d = (k + t) + r 2

d=k+t+1 T1 T2 T3

d=k+t

Td−2 Td−1 Td T1

Tk+t+1

Td

Fig. 9 Transformation of tree T which has an internal node with degree d  .k C t /2

k-replacement cost OP Tk .i / (1  i  n). By keeping the branching information for each iteration, the optimal k-replacement tree can be constructed with the same complexity. 

3.2.2 Tight Degree Bound for 2-Replacement Problem From this subsection on, 2-replacement problem will be the major focus. Definition 5 Denote the maximum cost to delete a single leaf in a tree T as ST and the maximum cost to delete two leaves in T as DT , i.e., ST D C1 .T; n/ and DT D C2 .T; n/. According to Theorem 7, for the 2-replacement problem, Tn;opt has degree upper bounded by 8. Furthermore, root degree d D 1 is not possible in the optimal tree because merging the root and its single child into one node can strictly decrease the worst case updating cost. Fact 2 For the 2-replacement problem, suppose that a tree T has root degree d where d  2 and the d subtrees are T1 ; T2 ; : : : ; Td . One has DT D max fDTi C d; STi C STj C d g: 1i;j d

Proof It is easy to see that replacing any two leaves from a subtree Ti will incur a cost at most DTi C d , while replacing two leaves from two different subtrees Ti and Tj will incur a cost at most STi C STj C d . The 2-replacement cost comes from one of the above cases, and therefore the fact holds.  Wu et al. [43] further remove the possibility of degree 8 by the following lemma.

1732

M. Li T⬘ d=4

T d=8

T1

T2

T3

T7

d=2

T8 T1

T2 T3

T7

T8

Fig. 10 Transformation of tree T which has an internal node with degree 8

Lemma 11 For the 2-replacement problem, one can find an optimal tree among the trees with node degrees bounded between 2 and 7.

Proof By Theorem 7, one only needs to remove the possibility of degree 8 for any internal node. Firstly, any tree with root degree 8 can be transformed into a tree with a smaller root degree and no larger 2-replacement cost. Given a tree T with root degree 8 and subtrees T1 ; T2 ; : : : ; T8 , one can transform the structure at the root v into degree 4, and each child of v has degree 2, as shown in Fig. 10. In the original tree, 2-replacement cost is DT D maxfDTi C 8; STi C STj C 8g (Fact 2). In the resulting tree T 0 , the cost of replacing two leaves from different subtrees Ti and Tj is at most STi C STj C 8, and the cost of replacing two leaves from one subtree Ti is at most DTi C 6. Hence, 2-replacement cost in T 0 is at most maxfDTi C 6; STi C STj C 8g  DT . Note that the 1-replacement cost of T also does not increase in the transformation. One can then extend this transformation to any internal node as has been shown in the proof of Theorem 7. The lemma is then proved.  Based on this bound, the optimal tree can be computed in O.n8 / time by enumerating all possible degrees that are at most 7 using a dynamic programming algorithm. However, O.n8 / is still a bit unaffordable in real computations. Therefore, [43] reduced the complexity to O.n/ by accurately determining the structure of the optimal tree. They showed two important properties of the optimal tree (monotone property and 2–3 tree property) and then further remove the possibility of root degree 2 and 3 to reduce the scope of trees within which the optimal tree is searched for. Lemma 12 (monotone property) For the 2-replacement problem, suppose a tree T has root degree d where d  2 and d subtrees are T1 ; T2 ; : : : ; Td . Without loss of generality, assume that T has a nonincreasing leaf descendant vector .n1 ; n2 ; : : : ; nd /, where ni is the number of leaves in subtree Ti . Then, there exists an optimal tree where ST1  ST2  : : :  STd and Ti is a template of Ti C1 for 2  i  d  1.

Key Tree Optimization

1733

T

T1

T2

T3

T⬘

Td−1 Td

T1⬘ (T1)

T2⬘

T3⬘

Td−1⬘ Td⬘

Fig. 11 Transformation of a tree which has ST1 < ST2

T

T1

T2

T3

T⬘

Td−1 Td

T1⬘ (T1)

T2⬘ T3⬘ (T2)

Td−1⬘ Td⬘

Fig. 12 Transformation of a tree T which has STj < STj C1 where j  2

Proof If a given optimal tree contradicts the lemma, one can always change this tree into a tree satisfying the monotone property but with equal or less 2-replacement cost. Case 1: ST1 < ST2 . Let T10 be the same as T1 , and then choose T10 to be the template tree and construct a subtree T20 by removing n1  n2 leaves from T1 . Similarly, one can construct the subtrees Ti0 (3  i  d ) by removing ni 1  ni leaves from Ti01 , as shown in Fig.11. Since each subtree Ti0 is a template of Ti0C1 , the final set of new trees satisfies ST10  ST20  : : :  STd0 and DT10  DT20  : : :  DTd0 . Then one can prove that 2-replacement cost is not increased in the resulting tree T 0 . Note that ST10 D ST1 < ST2 and DT10 D DT1 . Since DT 0 D max1i c.u1 ; u2 / or c.u0 ; u2 / > c.u1 ; u2 /, which contradicts c.u1 ; u2 / D DT . Figure 13 shows the transformations for the following three cases: Case 1: u0 ; u1 ’s nearest common ancestor v0 2 ANC.v/. In this case, one has c.u0 ; u2 / D c.u0 / C c.u2 /  c.v0 /  dv0 , and therefore c.u0 ; u2 /  c.u1 ; u2 / D c.u0 /  c.u1 / C c.v/ C dv  c.v0 /  dv0 . Since c.u0 / > c.u1 / and c.v0 /Cdv0  c.v/ (v0 is v1 ’s ancestor), one has c.u0 ; u2 / > c.u1 ; u2 /; the lemma is proved in this case. Case 2: u0 ; u1 ’s nearest common ancestor v0 2 ANC.u1 / but v0 … ANC.v/. In this case, one has c.u0 ; u2 /  c.u1 ; u2 / D c.u0 /  c.u1 / > 0. Case 3: u0 ; u2 ’s nearest common ancestor v0 2 ANC.u2 / but v0 … ANC.v/. In this case, one has c.u0 ; u1 /  c.u1 ; u2 / D c.u0 /  c.u2 / > 0. In all the three cases, either c.u0 ; u1 / > c.u1 ; u2 / or c.u0 ; u2 / > c.u1 ; u2 /. Therefore, Item (1) of the fact is proved. Item (2) is correct because c.u2 /  ST .  Through a series of lemmas below, [43] proved the following theorem to further remove the possibility of root degrees 2 and 3 in the optimal tree. While removing large degree d  8 is easy, it is comparatively complicated to remove small degrees 2 or 3.

1736

M. Li T

Fig. 14 Transform a tree T where the two subtrees are 2–3 trees into a 2–3 tree T 0

T⬘

T1

T2 T1⬘

T2⬘

T3⬘

Theorem 8 For the 2-replacement problem, a tree T with root degree 2 or 3 can be transformed into a tree with root degree 4 without increasing the 2-replacement cost. Lemma 14 Given a tree T with root degree 2 where two subtrees T1 ; T2 are both 2–3 trees and jn1 n2 j  1, when T is transformed into 2–3 tree T 0 with root degree 3 as shown in Fig. 14, one has ST  ST 0 ; DT  DT 0 : Proof Without loss of generality, assume that n1  n2 . According to the definition and optimality of 2–3 tree, when the 1-replacement cost is r, the maximum possible number of leaves on the tree is f ./ where 8 < 3  3i 1 if f .r/ D 4  3i 1 if : 6  3i 1 if

r D 3i r D 3i C 1 r D 3i C 2

Notice that T1 and T2 are 2–3 trees which satisfy function f ./. When n1 2 .3  3i 1 ; 4  3i 1 , one has ST2  ST1 D 3i C 1 and ST D ST1 C 2 D 3i C 3. Since n1 C n2  8  3i 1 < 3  3i , one has ST 0  3i C 3 D ST . Similarly for n1 2 .4  3i 1 ; 6  3i 1 , one has ST D ST1 C 2 D 3i C 4 and ST 0  3i C 4. And for n1 2 .6  3i 1 ; 9  3i 1 , one has ST D ST1 C2 D 3i C5 and ST 0  3i C5. Therefore, the first inequality of the lemma holds. Suppose that the three subtrees of T 0 are T10 ; T20 ; T30 which are also 2–3 trees and satisfy ST10  ST20  ST30 . Since ST D ST1 C 2 and ST 0 D ST10 C 3, one has ST10  ST1  1 and DT 0  2ST10 C 3  2ST1 C 1. Note that one has ST2  ST1  1 because both subtrees are 2–3 trees with the number of leaves differed by at most 1. Hence, DT 0  ST1 C ST2 C 2  DT . The lemma is finally proved.  Lemma 15 For the 2-replacement problem, a tree T with root degree 2 can be transformed into a tree with root degree 3 or 4 without increasing the 2-replacement cost. Proof Lemma 13 implies that one can transform T into a candidate tree where T2 is a 2–3 tree. The discussion can be divided into three cases.

Key Tree Optimization

1737 T

Fig. 15 Transform a tree T from root degree 2 into degree 4

B1

T⬘

d=2 B1

T1 T2

T2⬘

T1⬘ (T1)

T' T T1

B1

B1

d=2

T2⬘ (T2)

T1⬘

T2 v

v

u2

u2 v0

v0 u1

v0

u1

u2

Fig. 16 Transform a tree T from root degree 2 to degree 3

Case 1: DT1 < ST1 C ST2 According to Fact 5, in subtree T1 , one has DT1 D ST1 C c.u2 /  c.v/  dv < ST1 C ST2 which implies c.u2 /  c.v/  dv  ST2  1 where v is the nearest common ancestor of u1 ; u2 . Two subcases are discussed below: c.u2 /  c.v/  dv  ST2  2 and c.u2 /  c.v/  dv D ST2  1. When c.u2 /  c.v/  dv  ST2  2, tree T can be transformed into a tree T 0 which is composed of the dominating branch B1 and T20 (T20 is the same as T2 ) as shown in Fig. 15. In T 0 , consider three subcases according to the positions of the two leaves whose replacement cost reaches DT 0 . For the subcase where two leaves are both from T20 , one has DT 0 D DT20 C 1 D DT2 C 1 < DT2 C 2  DT . For the subcase where one of the leaves is from T10 and the other is from T20 , one has DT 0 D ST10 C ST20 C 1 D ST1 C ST2 C 1 < ST1 C ST2 C 2  DT . For the subcase where both leaves u1 and u2 are from T10 , one has DT 0 D DT10 C 4 D DT1 C 4. Since c.u2 /  c.v/  dv  ST2  2, one has DT1  ST1 C ST2  2, which implies DT 0  ST1 C ST2 C 2  DT . When c.u2 /  c.v/  dv D ST2  1, consider another transformation for T . Notice that ST1 D c.u1 /  2  c.u2 /  c.v/ D ST2  1 C dv  ST2 C 1, which means ST2  ST1  1. Assume that Tv0 is the subtree of v which contains leaf u2 , then one can move Tv0 to the root to produce a tree T 0 , as Fig. 16 shows. One also needs to consider three subcases for T 0 according to the positions of the two leaves whose replacement cost reaches DT 0 . Firstly, if the two leaves are originally from T1 , then one has DT 0  DT1 C 3  ST1 C ST2 C 2 D DT . Secondly, if one of the two leaves is originally from T1 while the other is not, then one can

1738

M. Li T'

Fig. 17 u belongs to a leaf descendant of vO

B1

v0

T2 ⬘ (T2)

T1 ⬘ v

u2

v

v0

u

u1 T

T1 B11

Fig. 18 Transform a tree T from root degree 2 to degree 3

B1

T'

d=2 T2

B1d1

u2

B1d1

B1

T1 ⬘ B11

T2⬘ (T2) B1d1

show that replacing a leaf from T10 or Tv0 incurs cost at most ST1  1 which implies DT 0  .ST1  1/ C ST2 C 3 D DT . Suppose first the leaf u is from subtree Tv nTv0 which is rooted at v. Since dv decreases by 1 in the resulting tree, the cost to delete u in T10 is at most ST1  1. If the leaf u is from Tv0 , then the cost to delete u in Tv0 is at most ST1  dv . Otherwise, if the leaf is not from Tv , then suppose the leaf u is a leaf descendant of vO where vO is an ancestor of v, as shown in Fig. 17. Note that c.u1 / C c.u/  c.Ov/  dvO  c.u1 / C c.u2 /  c.v/  dv D c.u1 / C ST2  1; thus, one has c.u/  ST2  1 C c.Ov/ C dvO  ST2  1 C c.v/ D c.u2 /  dv  c.u1 /  dv D ST1 C 2  dv  ST1 , which implies c.u/  ST1 . Therefore, replacing u from T10 is at most ST1  2. Finally, if both leaves are originally from T2 , then one has DT 0 < 2ST2 C d C 1  ST1  1 C ST2 C d C 1 D ST1 C ST2 C d  DT . Hence, when DT1 < ST1 C ST2 , one can transform T into a better tree with root degree 3 or 4. Case 2: DT1 D ST1 C ST2 Suppose the root degree of T1 is d1 . One can move a branch B1d1 of T1 to the root (Fig. 18) to obtain a new tree T 0 . One can prove that 2-replacement cost of T 0 does not change compared to that of T . One also needs to consider three subcases for T 0 according to the positions of the two leaves whose replacement cost reaches DT 0 . If the two leaves are originally from T1 , one has DT 0  DT10 C d C 1 D DT1  1 C d C 1 D DT1 C d  DT . If one of the leaves is originally from T1 while the other is not, the cost DT 0 is ST10 C ST2 C d C 1 D ST1  1 C ST2 C d C 1 D ST1 C ST2 C d  DT or ST20 C ST1d1 C d C 1 D ST2 C ST1d1 C d C 1 < ST2 C ST1 C d  DT . If both leaves are originally from T2 , the cost DT 0 is no more than 2ST2 C d C 1. Furthermore, by Fact 5, DT1 D ST1 C c.u2 /  c.v/  dv  2ST1  1. Since DT1 D ST1 C ST2 , one

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1739 T

Fig. 19 Exchange two subtrees for the case that DT1 > ST1 C ST2

T1 v1

v

u1

B1

T'

d=2

B1 T1'

T2 v

v2 u2

T2⬘ (T2)

v2 u2

u1

has ST2  ST1  1. Therefore, DT 0  2ST2 C d C 1  ST1  1 C ST2 C d C 1 D ST1 C ST2 C d  DT . Hence, DT 0  DT . Case 3: DT1 > ST1 C ST2 . In this case, replacing two leaves from T1 incurs the cost DT . First, transform T into a tree satisfying case 1 or case 2. In subtree T1 , by Fact 5, if c.u1 ; u2 / D DT , one can assume c.u1 / D ST . Suppose v is the nearest common ancestor of u1 and u2 and v’s children v1 and v2 are ancestors of u1 and u2 , respectively (vi can be ui itself), then c.u1 /  c.v1 /  c.u2 /  c.v2 /. One can exchange subtree Tv2 which is rooted at v2 with subtree T2 , as shown in Fig. 19. Because ST1 C ST2 < DT1 D ST1 Cc.u2 /c.v2 / D ST1 CSTv2 , Tv2 has larger 1-replacement cost than T2 . After the exchange, one has DT 0  DT because replacing one leaf from Tv1 and one leaf from Tv2 incurs a cost at most ST10 C c.u2 /  c.v2 / C d D ST1 C c.u2 /  c.v2 / C d D DT ; replacing two leaves from T20 of T 0 incurs a cost at most 2ST2 C c.v/ C dv < ST2 C STv2 C c.v/ C dv  ST1 C ST2 C d < DT , and other combinations of two replaced leaves in T 0 incur at most the same cost as that of T correspondingly. Note that T 0 has degree 2 and belongs to cases 1 or 2. Hence, T 0 can be further transformed into a tree with degree 3 or 4 according to cases 1 and 2. Since in all cases, one can transform a tree with root degree 2 into a tree with root degree 3 or 4 without increasing 2-replacement cost; the lemma is finally proved.  Lemma 16 For 2-replacement problem, a tree T with root degree 3 can be transformed into a tree with root degree 4 without increasing the 2-replacement cost. Proof Case 1: DT1 < ST1 C ST2 . Lemma 13 implies that one can transform T into a candidate tree where T2 ; T3 are 2–3 trees without increasing 2-replacement cost. One can then reallocate the leaves in T2 ; T3 so that T2 is a template tree of T3 with 0  n2  n3  1 and both subtrees are still 2–3 trees as Fig. 20 shows. First, it is easy to see that this transformation will not increase 2-replacement cost. Then one can prove that such a tree can be further transformed into a tree with root degree 4 without increasing 2-replacement cost. Consider T ’s two branches B2 and B3 (Suppose these two

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Fig. 20 Transform a tree T from root degree 3 to root degree 4

d=3 T1

B1

T T2

T'

~ T

T3

B1

~ T'

T1⬘ (T1)

branches compose a new tree TQ ) in T ; according to the proof of Lemma 14, one can transform TQ into tree TQ 0 which is a 2–3 tree and denote the whole new tree as T 0 . The transformation ensures that STQ  STQ 0 ; DTQ  DTQ 0 . One can prove that the transformation does not increase the 2-replacement cost. One still needs to consider three subcases for T 0 according to the positions of the two leaves whose replacement cost reaches DT 0 . If two leaves are originally from T1 , one has DT 0 D DT10 C d C 1 D DT1 C d C 1  ST1 C ST2 C d D DT . If one of the leaves is originally from T1 while the other is from TQ , one has DT 0 D ST1 C STQ 0 C 1  ST1 C STQ C 1 D ST1 C ST2 C d D DT . If both leaves are from TQ , one has DT 0 D DTQ 0 C1  DTQ C1  DT . In all the subcases DT 0  DT . Case 2: DT1 D ST1 C ST2 The transformation and proof for this case is similar to the case 2 in the proof of Lemma 15. The only difference is that the original tree has root degree d D 3 instead of 2 in this case. Case 3: DT1 > ST1 C ST2 The transformation and proof for this case is similar to the case 3 in the proof of Lemma 15. The only difference is that the original tree has root degree d D 3 instead of 2 in this case. Since, in all cases, the tree with root degree 3 is transformed into a tree with root degree 4 without increasing 2-replacement cost, the lemma is finally proved. 

3.2.3

Linear Time Algorithm: Balanced Structure of 2-Replacement Problem Note that the optimal structure can be computed in O.n8 / time by enumerating all possible degrees that are at most 7 by the result of Sect. 3.2.2. Although the possibility of the root degrees 2 and 3 has been removed in Sect. 3.2.2 and the structure of the ordinary branches has been fixed, to exactly compute the optimal structure is still not efficient enough because one needs to enumerate all the possible structures of the dominating branch. In this subsection, [43] showed that among the candidate trees with n leaves, a balanced structure can achieve 2-replacement cost OP T2 .n/. This reduces the complexity of the algorithm substantially to O.n/. The basic idea is to first investigate the capacity g.R/ defined below for candidate trees with 2-replacement cost R (Theorem 9). Note that the optimal tree

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has the minimum 2-replacement cost with n leaves, which reversely implies that if one wants to find a tree with 2-replacement cost R and at the same time has the maximum possible number of leaves, then computing the optimal tree for increasing n until OP T2 .n/ > R will produce one such solution. Wu et al. [43] then analyze and calculate the exact value for the capacity (maximum number of leaves) given a fixed 2-replacement cost R (Theorem 10). Finally, they prove that certain balanced structure can always be the optimal structure that minimizes the 2-replacement cost (Theorem 11). Definition 6 Define capacity to be the maximum number of leaves that can be placed in a certain type of trees given a fixed replacement cost. According to [32], function f .r/ defined below is the capacity for 1-replacement cost r (among all the possible trees). Wu et al. [43] uses function g.R/ to denote the capacity for 2-replacement cost R (among all the possible trees). In other words, when g.R  1/ < n  g.R/, one has OP T2 .n/ D R: 8 < 3  3i 1 if f .r/ D 4  3i 1 if : 6  3i 1 if

r D 3i r D 3i C 1 r D 3i C 2

To facilitate the discussion, according to Fact 3, one can divide the candidate trees with 2-replacement cost R and root degree d into two categories as summarized in the following definition. Definition 7 Candidate trees of category 1: The two leaves whose replacement cost achieves 2-replacement cost are from different branches, i.e., DT D ST1 C ST2 C d , which implies ST1 C ST2  DT1 . Candidate trees of category 2: The two leaves whose replacement cost achieves 2-replacement cost are both from the dominating branch B1 , i.e., DT D DT1 C d , which implies ST1 C ST2 < DT1 . Correspondingly, denote the capacity of the candidate trees belonging to category 1 with 2-replacement cost R by g1 .R/, and denote the capacity of the candidate trees belonging to category 2 with 2-replacement cost R by g2 .R/. Note that one can find the optimal tree among the candidate trees according to Lemma 13, which implies that with the same 2-replacement cost R, the best candidate tree can always have equal or larger number of leaves than the general trees. That is, g.R/ D maxfg1 .R/; g2 .R/g. Thus, in the following discussions, only the candidate trees will be discussed. On the other hand, because trees with the maximum number of leaves are the targets, it is easy to see that one can assume the numbers of leaves in ordinary branches are all the same (Otherwise, the tree can be bigger without affecting the 2-replacement cost). In all candidate trees with 2-replacement cost R, by Fact 3, at most one of the two leaves whose replacement cost achieves 2-replacement cost is from an

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ordinary branch. Suppose each ordinary branch has 1-replacement cost r  , and correspondingly T1 has 1-replacement cost r C where r C  R  d  r  (otherwise, DT  r C C r  C d > R, a contradiction). For fixed cost R, Lemma 12 (monotonous property) implies that r C  r  . The following capacity bound can be proved. Theorem 9 gi .R/  .R  2r  /  f .r  /.i D 1; 2/. To prove this theorem, it is sufficient to prove g1 .R/  .R  2r  /  f .r  / and g2 .R/  g1 .R/. Wu et al. [43] used critical node defined below to denote a special kind of nodes in the following proofs. Definition 8 For a selected node u, define the path from u to the root to be a critical path. Correspondingly, u’s ancestor nodes on the path are named critical nodes. First, investigate the capacity of the candidate tree T in category 1 which has 2-replacement cost R. In Lemmas 17–20, assume that the number of leaves in T is the maximum possible and the goal is to fix the rmax ./ (Definition 9) of some critical nodes and their siblings to prove the capacity bound g1 .R/. Definition 9 For the subtree Tv rooted at node v, the maximum ancestor weight with respect to Tv is denoted as rmax .v/. Correspondingly, the maximum number of leaves Tv can hold given the 1-replacement cost rmax .v/ is denoted as cap.v/. Given a candidate tree in category 1 where DT D ST1 C ST2 C d , there exists at least one leaf u 2 B1 satisfying c.u/ D r C C d D R  r  . Choose such a leaf u to obtain a critical path; correspondingly u’s ancestors are critical nodes. Notice that critical node v has rmax .v/ D R  r   c.v/. Furthermore, without loss of generality, suppose that v1 is the nearest ancestor of u which satisfies rmax .v1 /  r  (which implies rmax .v0 / < r  where critical node v0 is v1 ’s child). The following discussions investigate the capacity for the critical node and their siblings. Lemma 17 If node v is a critical node, then rmax .v0 / D minfr  ; rmax .v/g for any sibling v0 of v. Proof By the definition of r C , one has rmax .v0 / C c.v0 /  d  r C D rmax .v/ C c.v/  d , which implies rmax .v0 /  rmax .v/. On the other hand, if one chooses to delete a leaf u in Tv and a leaf w in Tv0 , the replacement cost should be at most R, which means rmax .v/ C rmax .v0 / C c.v/  R. Note that rmax .v/ D R  r   c.v/. Therefore, rmax .v0 /  r  . The possibility rmax .v0 / < minf.r  ; rmax .v//g can be removed because otherwise the capacity of Tv0 is not maximum which contradicts the assumption that the number of leaves in T is the maximum possible. In other words, if rmax .v0 / < minfr  ; rmax .v/g, one can enlarge the tree by increasing rmax .v0 / to minfr  ; rmax .v/g while ensuring that the tree still belongs to candidate trees in category 1 and has 2-replacement cost R. This ends the proof. 

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Fig. 21 spi ne.v1 ; root / that belongs to candidate trees in category 1

r−

r−

r− v1 r− u1

Lemma 18 Given a critical node v which is on the path from v1 to the root, if v0 is a sibling of v, then rmax .v0 / D r  . Proof Firstly, rmax .v/ D R  r   c.v/ for the critical node v. Furthermore, since v is v1 ’s ancestor or v1 itself, one has rmax .v/  rmax .v1 /  r  . Hence, the sibling of v has rmax .v0 / D minfr  ; rmax .v/g D r  (Lemma 17). The lemma is proved.  Consider the structure composed by the critical nodes on the path from v1 to the root and the siblings of these critical nodes, as shown in Fig. 21. This structure is denoted as spi ne.v1 ; root/ according to the definition below. Suppose that w is a leaf in this structure. The value of rmax .w/ is r  by Lemma 18. Definition 10 Suppose that va and vb (va is a descendent of vb ) are critical nodes on the selected critical path. Denote the structure composed by the critical nodes from va to vb and their siblings as spi ne.va ; vb / (vb ’s siblings are not included). Consider two cases of spi ne.v1 ; root/ for candidate trees in category 1. One can prove g1 .R/  .R  2r  /  f .r  / in the following cases: Case 1: rmax .v1 / D r  . Case 2: rmax .v1 / > r  . Lemma 19 g1 .R/  .R  2r  /  f .r  / when rmax .v1 / D r  . Proof In this case, besides node v1 , any other leaf w of spi ne.v1 ; root/ also has rmax .w/ D r  according to Lemma 18. On the other hand, the total degrees from the parent of v1 to the root are c.v1 / D Rr  rmax .v1 / D R2r  , which equals the number of leaves on spi ne.v1 ; root/. Therefore, the capacity is bounded by .R  2r  /  f .r  /.  Lemma 20 g1 .R/  .R  2r  /  f .r  / when rmax .v1 / > r  .

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Proof Theorem 7 implies that dv1  7; thus, rmax .v1 / D rmax .v0 / C dv1  r  C 6 (because rmax .v0 / < r  ). Similar with the proof in the previous lemma, for any leaf v except v1 on spi ne.v1 ; root/, one has cap.v/  f .r  /, which results in at most .R  r   rmax .v1 /  1/  f .r  / leaves. To prove the capacity bound, one only needs to show that cap.v1 /  .rmax .v1 / C 1  r  /  f .r  /. If rmax .v1 / is written as r  C m where m  6, then it is equivalent to proving .m C 1/  f .r  /  f .r  C m/  0. When 1  m  5, one can verify that f .r  C m/ < .m C 1/f .r  /. When m D 6, the above relation does not hold because f .r  C 6/ D 9f .r  / > 7f .r  /. But when rmax .v1 / D r  C m D r  C 6, one has dv1 D 7. Lemma 17 implies that the sibling u of v0 has rmax .u/ D minfr  ; rmax .v0 /g D rmax .v0 / D r  C 6  7 D r   1. Hence, cap.v1 /  7f .r   1/  7f .r  /. This completes the proof.  By Lemmas 19 and 20, the capacity bound of candidate trees in category 1 is proved, i.e., g1 .R/  .R2r  /f .r  /. The following part is to show that candidate trees in category 2 have a capacity no greater than that of category 1. Note that the following fact holds for candidate trees T in category 2. Fact 6 Given a candidate tree T in category 2, suppose that u1 ; u2 are the two leaves (from subtree T1 ) whose replacement cost equals R, i.e., c.u1 / C c.u2 /  c.v/  dv D R where v is the nearest common ancestor of u1 ; u2 and c.u1 / D ST . In order to find an optimal tree, among candidate trees in category 2, one only needs to search among those trees satisfying c.u2 /  c.v/  dv < OP T1 .n1 / where n1 is the number of leaves in T1 . Proof If on the contrary in the optimal tree, one has c.u2 /c.v/dv  OP T1 .n1 /, then DT D c.u1 / C c.u2 /  c.v/  dv  2OP T1 .n1 / C d since c.u1 / D ST1 C d  OP T1 .n1 / C d . In this case, if T is transformed into T 0 by replacing subtree T1 with a 2–3 tree with the same number of leaves, one has DT 0  DT because ST10 D OP T1 .n1 /  ST1 and DT10 Cd < 2OP T1 .n1 /Cd  c.u1 /Cc.u2 /c.v/dv D R. Notice that T 0 is either a candidate tree in category 1 or a candidate tree in category 2 satisfying c.u2 /  c.v/  dv < OP T1 .n1 /. Thus, the fact follows.  Lemma 21 A candidate tree in category 2 has capacity g2 .R/  g1 .R/. Proof Given a candidate tree T in category 2, one has ST1 C ST2 C d < DT1 C d D R D DT . By the monotone property in Lemma 12, the d subtrees satisfy ST1  ST2  : : :  STd . If ST1 D ST2 , then DT1 C d  2ST1 C d D ST1 C ST2 C d , which leads to a contradiction. Thus, ST1  ST2 C1. Note that the ordinary branches are 2–3 trees; one can do a tree expansion by adding one leaf to each of the ordinary branches (where each resulting subtree is still a 2–3 tree). In this way, one can increase the number of leaves in the tree without affecting the 2-replacement cost. The resulting subtree’s 1-replacement cost will increase by at most 1. Suppose that two leaves in the resulting tree T 0 whose replacement cost achieves the 2-replacement cost are u1 and u2 . We discuss three cases. If u1 ; u2 are both from T1 ,

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then replacing u1 ; u2 incurs a cost DT1 C d D R. If one of the two leaves is from T1 while the other is from Ti , then replacing u1 ; u2 incurs a cost at most ST1 C ST2 C 1 C d  R. If u1 ; u2 are both from the ordinary branches, then replacing u1 ; u2 incurs a cost at most 2.ST2 C 1/ C d  ST2 C 1 C ST1 C d  R. The above tree expansion can be applied until ST1 C STi C d D R (The tree becomes a candidate tree of category 1), and the final tree is denoted as T 00 . One can prove that T 00 is a candidate tree of category 1. To show this, one only needs to prove that in T 00 subtree T200 has the number of leaves less than T1 . This follows because OP T1 .n002 / D ST200 D R  ST1  d D c.u2 /  c.v/  dv < OP T .n1 / implies n002 < n1 . Therefore, with 2-replacement cost R fixed, there is always a candidate tree of category 1 which has more leaves than any candidate tree of category 2. This finishes the proof of g2 .R/  g1 .R/.  Lemma 21 implies that the maximum capacity can always be achieved by the candidate trees in category 1. Thus, the capacity bound is .R  2r  /  f .r  / and Theorem 9 is then proved. In the following, among these candidate trees one can study the optimal structure which achieves the maximum capacity for different values of 2-replacement cost R. Only the proof for R D 6i C 6 will be elaborated due to the similarity of other proofs. The correctness of other proofs can be referred to in Table 1. The following fact is used in the proofs. Fact 7 8 ˆ ˆ
13 C 25 ) or two type-3 connections (with total weight > 23 ). The total IO-load for M must thus be at least 23 . Overall, the total IO-load that all crossbars in C3 carry must be at least 23 jC3 j. Finally, if no crossbar in C4 is

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available, then the total IO-load that crossbars in C4 are carrying must be at least 1 jC4 j. 2 Consequently, because the total IO-load overall is strictly smaller than 2n (at time t  1), the new request is not blocked if 4 2 1 jC2 j C jC3 j C jC4 j  2n: 5 3 2 The inequality would hold if 4 2 1 x2 C x3 C x4  2: 5 3 2 In a similar fashion, it is easy to argue that newly arriving type-3 requests can be routed if 2 3 x3 C x4  2 3 5 and a new type-4 request can be routed if 2 x4  2: 3 Overall, the routing algorithm works if the variables xi are feasible for the following linear program: min x1 Cx2 Cx3 Cx4 s.t. x1  2 4 2 1 x C x C x  2 2 3 4 5 3 2 2 3 x C x  2 3 3 5 4 2 3 x4  2 x1 ; x2 ; x3 ; x4  0: The solution x1 D 2; x2 D 3=8; x3 D 3=10; x4 D 3 is certainly feasible. The total number of middle crossbars used is 4 X

dxi ne  .x1 C x2 C x3 C x4 /n C 1:8 D 5:675n C 1:8:

i D1

To prove the better bound of 5:6355n, divide the rates into five types belonging to the intervals .1=2; 1, .2=5; 1=2, .1=3; 2=5, .11=43; 1=3, and .0; 11=43.  Remark 2 The above strategy can also give a better sufficient condition than the best known in [15] for the case when there is internal speedup in the Clos network.

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Strictly Nonblocking Unicast Multilog Networks for Photonic Switching Under the General Crosstalk Constraint

When the multilog architecture is employed to design a photonic switch, each 2  2 switching element (SE) needs to be replaced by a functionally equivalent optical component. For instance, when d D 2, the so-called directional couplers can be used as SEs [34, 48, 56]. However, directional couplers and many other optical switching elements suffer from optical crosstalk between interfering channels, which is one of the major obstacles in designing cost-effective switches [5, 11, 52]. To cope with crosstalk, the general crosstalk constraint was introduced by Vaez and Lea [53], which forces each new route to share at most c SEs with other active routes in the switch, where c is a parameter of the design. When c D 0, no two routes are allowed to share a common SE, and the switch becomes a crosstalkfree switch [5, 11, 25, 27, 33, 53, 54]. On the other hand, when c D n D logd N , a route can share any number of SEs with other routes (but still, no shared link) because each route in a logd .N; 0; m/ network contains exactly n SEs. In this case, only link blocking takes effect, and it is commonly referred to as the link-blocking case, which is relevant to electronic switches [4, 13, 14, 20, 21, 23, 26, 28, 43, 47]. A switching network which is strictly nonblocking (SNB) under this general crosstalk constraint shall be referred to as a c-SNB network. This section presents the linear programming analysis first derived in [40] for analyzing strictly nonblocking d -ary multilog networks under this general crosstalk constraint. Prior to [40], the only known results are sufficient conditions for the binary log2 .N; 0; m/ network to be c-SNB [53]. One important advantage of the LP-duality approach is the straightforwardness of verifying the sufficiency proofs. All one has to do is to check that a solution is indeed dual feasible, and the dual-objective value automatically gives us a sufficient condition. Earlier works on this problem relied on combinatorial arguments which are quite intricate and somewhat error-prone. Section 4.1 shows that the number of planes blocking a new request is bounded by the objective value of a general linear program which is dependent on the parameters of the problem (c; d; n) but independent from the request under consideration. By weak duality, in order to bound the number of planes blocking any new request, the objective value of any dual-feasible solution can be used. In Sects. 4.2 and 4.3, families of solutions which are dual feasible are constructed. Then, depending on the relationships between the problem’s parameters (c; d; n), the best dual-feasible solution from the constructed families is selected for bounding the number of blocking planes. Although verifying the LP-based proof is straightforward, constructing these families of dual-feasible solutions involves a good deal of experimentation, educated guesses, and even some Maple coding. This “cost” is hidden from the reader; thus, perhaps some insight is lost along the way as to why the bounds hold.

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The Linear Programming Duality Formulation

Let .a; b/ be an arbitrary request. For each i 2 f0; : : : ; n  1g, let Ai be the set of inputs x other than a, where x1::n1 shares a suffix of length exactly i with a1::n1 . Formally, define Ai WD fx 2 Znd  fag j SUF .x1::n1 ; a1::n1 / D i g : Similarly, for each j 2 f0; : : : ; n  1g, let Bj be the set of outputs y other than b which share a prefix of length exactly j with b, namely, Bj WD fy 2 Znd  fbg j PRE .y1::n1 ; b1::n1 / D j g : Note that jAi j D jBi j D d ni  d n1i for all 0  i  n  1. Suppose the network logd .N; 0; m/ already had some routes established. Consider a BY1 .n/plane which blocks the new request .a; b/ (i.e., .a; b/ cannot be routed through the plane). There can only be three cases for which this happens: Case 1: There is a request .x; y/ routed in the plane for which R.x; y/ and R.a; b/ share a link. By Proposition 2, it must be the case that .x; y/ 2 Ai  Bj for some i Cj  n. Such a request .x; y/ shall be called a link-blocking request with respect to .a; b/. The qualifier “with respect to” will be dropped when it is unambiguous from the current context which request is being blocked. Case 2: There is a request .x; y/ whose route R.x; y/ already intersects c other routes at c distinct SEs on the same plane, and adding .a; b/ would introduce an additional intersecting SE to the route R.x; y/. These c other requests shall be called secondary requests accompanying .x; y/, and the request .x; y/ shall be called a node-blocking request of type 1. In particular, by Proposition 1, it must be the case that .x; y/ 2 Ai  Bj for some i C j D n  1. Note that the common SE between R.a; b/ and R.x; y/ is at stage j C 1. The routes for secondary requests must thus intersect R.x; y/ at stages strictly less than j C 1 or strictly greater than j C 1. If the route R.u; v/ for the secondary request .u; v/ intersects R.x; y/ at stage s with 1  s < j C 1, then it follows that PRE.y; v/ D s  1 < j and SUF .x; u/ D n  1  .s  1/ D n  s > n  1  j D i: Hence, .u; v/ 2 Ai  Bs1 . Such a request .u; v/ shall be called a left secondary request. If the route R.u; v/ of secondary request .u; v/ intersects R.x; y/ at stage s where j C 1 < s  n, then it follows that PRE.y; v/ D s  1 > j and SUF .x; u/ D n  1  .s  1/ D n  s < n  1  j D i: Hence, .u; v/ 2 Ans  Bj . Such a request .u; v/ shall be called a right secondary request.

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Maximize

X

xij C

iCj n

Subject to X

xij C

j W iCj n

X

iW iCj n

X iCj Dn1

X

.yij C zij / C

j W iCj Dn1

xij C

X

yij C

1 cC1

X

zij

(4)

iCj Dn1

.lij C rij /  d ni  d n1i 8i

(5)

.lij C rij /  d nj  d n1j 8j

(6)

j W iCj .d  1/d 2t n1 . Thus, d t  k < d t  .d  1/d 2t n1  1 and c.k; pk ; qk /  C.t; ˘ f / follow straightforwardly. Case 4: t  n2 ; 2t  n  2 < r  n  t  1. Note that this case can only happen when t  2n=3. In particular, if t > ˘ and r  n  t  1, then case 5 applies.  2n=3 Set pk D n  t  r  1 and qk D nCx C 1. Proving c.k; pk ; qk /  C.t; f / is 2 almost identical to Case 3 where one considers different ranges of x D blogd kc.  ˘ Case 5: t  n2 ; r  min.2t  n  2; n  t  1/. Set pk D n  t  r  1 ˘  C 1. Showing c.k; pk ; qk /  C.t; f / is similar to Case 3. The and qk D nCx 2 only slight variation is, instead of bounding k  d xC1  1, the relation k  f is applied directly. The function h.k/ is then bounded by h.f /. Furthermore, it is not necessary to consider the cases when x  2t  n  1 because x  r  2t  n  2. t u

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H.Q. Ngo and T.-N. Nguyen

Some Quick Consequences of Theorem 9

By plugging in the parameters t and f and computing 1 C C.t; f /, the following results follow straightforwardly. Corollary 5 (Theorem 4 in [14]) Let r D blogd f c. The network logd .N; 0; m/ is f -cast strictly non-blocking if  nr  nr m  f d d 2 e1  1 C d nd 2 e : Proof This corresponds to the t D 0 case of the window algorithm, which becomes a strictly nonblocking condition as noted earlier:  nr  nr C.0; f / D f d d 2 e1  1 Cd nd 2 e 1:

t u

The following result took about 6 pages in [10] to be proved (in two theorems) with combinatorial reasoning. The result is on the general multicast case, without the fanout restriction f . In the current setting, one can simply set f D N D d n . In fact, even though the corollary states exactly the same results as in [10], the statement is simpler. Corollary 6 (Theorems 1 and 2 in [10]) The d -ary multilog network logd .N; 0; m/ is wide-sense nonblocking with respect to the window algorithm with window size d t if 8 n2t 1 ˆ C td nt 1 .d  1/ ˆ d n2 .d  1/, the network is f -cast strictly nonblocking if m  d n  d n2 .d  1/. Proof Routing using the window algorithm with window size, t D n is the same as routing arbitrarily in the network when the 1  m-SE stage cannot fanout. Thus, any sufficient condition for the window algorithm to work is a strictly nonblocking condition. Note that when t D n, the dual constraints (DC-1) do not exist. Consider a solution to the dual LP as follows: n2 n2 When Sf > d .d  1/, consider two cases. If k > d .d  1/, set ıv D 1 for all v 2 n1 B and all other variables to be 0. Then, the dual-objective value is j D0 j

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X v2

Sn1

j D0 Bj

ˇ ˇ ˇ n1 ˇ ˇ[ ˇ ıv D ˇˇ Bj ˇˇ D d n  k  d n  d n2 .d  1/  1: ˇj D0 ˇ

Thus, in this case d n  d n2 .d  1/, Banyan planes are sufficient. Next, suppose k  d n2 .d  1/, in which case kd < d n . Set u D 1 for all u with i.u/  1, ıv D 1 for all v 2 Bn1 , and all other variables to be 0. The solution is dual feasible with dual-objective value X

u C

X

ıv D

jAi j C jBn1 j

i D1

v2Bn1

uWi.u/1

n1 X



n1 X .d ni  d ni 1 / C minfd n  k; k.d  1/g i D1

D d n1  1 C k.d  1/  d n1 C d n2 .d  1/2  1 D d n  d n2 .d  1/  1 and thus again d n  d n2 .d  1/, Banyan planes are sufficient. Next, consider the case when f  d n2 .d  1/. In this case, r  n  2. Let  ˙ n1 ˙ nr1 . Then, 1  p  2 . Furthermore, pD 2 kd p  f d p < d rC1 d d

nr1 2

e D d d nCrC1 e  d n: 2

S Set u D 1 for all u with i.u/  p, ıv D 1 for all v 2 n1 j Dnp Bj , and all other variables to be 0. The solution is dual feasible because, for any pair .u; v/ for which i.u/ C j.v/  n  1, it must either S be the case that i.u/  p or that j.v/  n  p (which is the same as saying v 2 n1 j Dnp Bj ). Recalling Proposition 5 and the fact that kd p < d n shown above, the dual-objective value is X uWi.u/p

X

u C v2

Sn1

j Dnp

ˇ ˇ ˇ ˇ n1 ˇ ˇ [ ıv D jAi j C ˇˇ Bj ˇˇ ˇj Dnp ˇ i Dp n1 X

Bj



n1 X

.d ni  d ni 1 / C minfd n  k; k.d p  1/g

i Dp

D d np  1 C k.d p  1/  d np C f .d p  1/  1: Hence, in this case, d np C f .d p  1/ is a sufficient number of Banyan planes. 

Linear Programming Analysis of Switching Networks

7.2

1805

Specifying a Family of Dual-Feasible Solutions

The family of dual-feasible solution is specified with two integral parameters where 0  p  n  t  1 and n  t  q  n. The parameter p is used to set the variables u , ˛w , and ˇu;w , and the parameter q is used to set the variables u and ıv . As the variables are set, the feasibility of the constraints (DC-1) and (DC-2) is also verified, and the contributions of those variables to the final objective value are computed: • Specifying the u variables. The u are defined as follows: ( u D

1 i.u/  n  p

(46)

0 otherwise

The contribution of the u to the objective is X

f u D

n1 X i Dnp

u

f

X

1D

n1 X

f jAi j D f .d p  1/:

i Dnp

uWi.u/Di

• Specifying the ˛w and ˇu;w variables. Next, the ˛w and ˇu;w are defined. The constraints (DC-1) with i.u/  np are already satisfied by the u ; hence, the ˛w and ˇu;w only need to be set to satisfy (DC-1) when j.w/  p. (If j.w/  p  1, then for the constraint to exist, it must hold that i.u/  n  1  j.w/  n  p.) The variables ˛w and ˇu;w are set differently based on three cases as follows: ˙  Case 1. If t  n2 , then set ( ˇu;w D

1

p  j.w/  n  t  1; and n  1  j.w/  i.u/  n  p  1;

0

otherwise

(47) and set all the ˛w to 0. It can be verified straightforwardly that all constraints (DC-1) are satisfied. Thus, the contributions of the ˛w and ˇu;w to the dualobjective value are X

ˇu;w D

nt 1 X

jfw W j.w/ D j gj

j Dp

u;w pj.w/nt 1 n1j.w/i.u/np1

X

np1

D

jfu 2 A W i.u/ D i gj

i Dn1j

D

nt 1 X

np1

j Dp

i Dn1j

.d nj t  d nj t 1 /

X

jAi j

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D

nt 1 X

.d nj t  d nj t 1 /.d j C1  d p /

j Dp

D .n  t  p/.d nt C1  d nt /  d nt C d p : The second equality follows from the fact that the number of windows Ww for t which j.w/ D j is precisely˙d nj  d nj t 1 .  n Case 2. When p C 1  t  2  1, set ( ˇu;w D

1 p  j.w/  t  1 and n  1  j.w/  i < n  p 0 otherwise (

and ˛w D

1

t  j < nt

0

otherwise

All constraints (DC-1) are thus satisfied. The ˛w ’s and ˇu;w ’s contributions to the objective are X w t j.w/ n  t g.k; p; q/ D f .d p  1/ C .t  p/.d ntC1  d nt / C d nCp2t  d t C d q  d p C minfd t  k; k.d nq  1/g: For t  p and q D n  t , g.k; p; q/ D f .d p  1/ C d np  k: For t  p and q > n  t , g.k; p; q/ D f .d p  1/ C d np  d t C d q  d p C minfd t  k; k.d nq  1/g: Fig. 12 The dual-objective value of the family of dual-feasible solutions

for which i.u/ C j.v/  n  1 and i.u/ C j.w/  n  1 are being considered.) Consequently, for the network logd .N; 0; m/ to be crosstalk-free f -cast wide-sense nonblocking under the window algorithm with window size d t , it is sufficient that m1C

7.3

max

min g.k; p; q/:

1kmin.f;d t / p;q

(51)

Selecting the Best Dual-Feasible Solution

The proof of the following technical lemma is similar to that of Lemma 11, and thus the proof is omitted. Lemma 13 Let d; n; k be positive integers and x D blogd kc. Then, the following function   xCnC1 xCnC1 N h.k/ D d b 2 c C k d nb 2 c  1 (52) is nondecreasing in k.

Linear Programming Analysis of Switching Networks

1809

The upperbound G.t; f / To shorten the notations, let r D blogd f c. 8 ˆ d nt Œ.n  t /.d  1/  1 ˆ ˆ ˆ ˆ ˆ Cd t  d 2tnC1 .d  1/ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ f .d ntr  1/ C rd nt .d  1/  d nt ˆ ˆ ˆ ˆ ˆ b rCnC1 c C f .d nb rCnC1 c  1/ 2 2 ˆ ˆCd ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆd nt Œ.n  t /.d  1/  1 C d b rCnC1 c 2 ˆ ˆ ˆ ˆ rCnC1 ˆ n b c ˆCf .d 2  1/ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ntr ˆ  1/ C d nt Œr.d  1/  1 ˆ n=2; r  maxf2t  n  2; n  t C 1g t > n=2; r  minf2t  n  3; n  tg t > n=2; n  t C 1  r  2t n  3 t > n=2; 2t  n  2  r  nt t D n=2 t < n=2; r  n  2t and f  d n2t .d  1/ t < n=2; r  n  2t; f > d n2t .d  1/ t < n=2; n  2t C 1  r  nt t < n=2; n  t < r

Fig. 13 Theorem 11 shows that G.t; f /  maxk minp;q g.k; p; q/

Theorem 11 The logd .N; 0; m/ network is crosstalk-free nonblocking under the window algorithm with window size d t if m  1 C G.t; f / where G.t; f / is defined in Fig. 13. Proof Suppose t > n=2, i.e., 2t  n C 1. Four cases are considered as follows. In all cases, define an integral variable x D blogd kc: Case 1. r  maxf2t  n  2; n  t C 1g. If k  d 2t n2 .d  1/ C 1, then pick pk D 0 and qk D n  t. Then, g.k; pk ; qk / D .n  t/.d nt C1  d nt /  d nt C 1 C d t  k  .n  t/.d nt C1  d nt /  d nt C 1 C d t  d 2t nC1 .d  1/  1 D d nt Œ.n  t/.d  1/  1 C d t  d 2t nC1 .d  1/:

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˘  > On the other hand, if k  d 2t n2 .d  1/, then set pk D 0 and qk D xCnC1 2 nqk t N nt. Note that kd  d . Thus, recall from Lemma 13 that the function h.k/ defined in (52) is nondecreasing in k, it follows that xCnC1 g.k; pk ; qk / D .n  t/.d nt C1  d nt /  d nt C d b 2 c n  o xCnC1 C min d t  k; k d nb 2 c  1

D .n  t/.d nt C1  d nt /  d nt C d b

xCnC1 2

  c C k d nb xCnC1 c 2

N D .n  t/.d nt C1  d nt /  d nt C h.k/ N 2t n2 .d  1//  .n  t/.d nt C1  d nt /  d nt C h.d D d nt Œ.n  t/.d  1/  1 C d t  d 2t nC1 .d  1/: Case 2. r  minf2t  n  3; n  tg. ˘  > n  t. Similar to In this case, set pk D n  t  r  t  1 and qk D xCnC1 2 the above analysis, the following is obtained: g.k; pk ; qk / D f .d nt r  1/ C rd nt .d  1/  d nt C d b Ck.d nb

xCnC1 2

xCnC1 2

c

c  1/

N D f .d nt r  1/ C rd nt .d  1/  d nt C h.k/ N /  f .d nt r  1/ C rd nt .d  1/  d nt C h.f D f .d nt r  1/ C rd nt .d  1/  d nt C d b Cf .d nb

rCnC1 2

rCnC1 2

c

c  1/:

N For the inequality, note that the function h.k/ is nondecreasing in k and  k  ˘f . > Case 3. n  t C 1  r  2t  n  3. In this case, set pk D 0 and qk D xCnC1 2 n  t. Then, g.k; pk ; qk /  d nt Œ.n  t/.d  1/  1 C d b

rCnC1 2

c C f .d nb rCnC1 c  1/: 2

Case 4. 2t  n  2  r  n  t. In this case, set pk D n  t  r and consider two sub-cases as in case 1: k  d 2t n2 .d  1/ C 1 or k  d 2t n2 .d  1/. The eventual bound is g.k; pk ; qk /  f .d nt r  1/ C d nt Œr.d  1/  1 C d t  d 2t n2 .d  1/: When t D n=2, simply set qk D n  t and pk D 0. Then, g.k; pk ; qk / D d nt Œ.n  t/.d  1/  1 C d t :

Linear Programming Analysis of Switching Networks

1811

Finally, suppose t < n=2. The value pk D q  t will always be set in this situation. Also consider four cases:  ˙ Case 1. r  n  2t and f  d n2t .d  1/, then set pk D nr1  t. In this 2 case, g.k; pk ; qk / D f .d d

nr1 2

e  1/ C d nd nr1 e  1: 2

Case 2. r  n  2t and f > d n2t .d  1/, then set pk D t  1. In this case, g.k; pk ; qk / D f .d t 1  1/ C d nt 1 .d 2  d C 1/  1: Case 3. n  2t C 1  r  n  t, then set pk D n  t  r. In this case, g.k; pk ; qk / D f .d nt r  1/ C .2t  n  r/.d  1/d nt C d 2n3t r  1: Case 4. n  t < r, then set pk D 0. In this case, g.k; pk ; qk / D t.d  1/d nt C d n2t  1:

t u

Corollary 8 (Theorems 1 in [39]) The d -ary multilog network logd .N; 0; m/ is crosstalk-free wide-sense nonblocking with respect to the window algorithm with window size d t if 8 n2t ˆ C td nt .d  1/ ˆ n=2:

Conclusion

As illustrated with many examples in this chapter, the linear programming technique seems to be effective in deriving sufficient conditions for a switching network to be nonblocking. In many of the examples discussed above, the sufficient conditions turn out to be necessary also (Examples 1, 2, and many special cases of later theorems on the multilog networks). However, it is not clear how one can “quickly” prove corresponding necessary conditions. The main reason is that a dual-feasible solution may not naturally lead to an integral primal feasible solution, which is needed to combinatorially design a network configuration showing the necessary condition. Another line of open research questions is on multilog networks with extra stages [22]. For example, adding logd N extra stages to a Banyan plane, one obtains the

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Beneˇs network [1], a very important building block for many useful switching network architectures such as the Cantor network [2].

Cross-References Bin Packing Approximation Algorithms: Survey and Classification Combinatorial Optimization in Transportation and Logistics Networks Dual Integrality in Combinatorial Optimization Fractional Combinatorial Optimization Greedy Approximation Algorithms Network Optimization Optimization in Multi-Channel Wireless Networks

Recommended Reading 1. V.E. Beneˇs, Mathematical Theory of Connecting Networks and Telephone Traffic. Mathematics in Science and Engineering, vol. 17 (Academic, New York, 1965) 2. D.G. Cantor, On non-blocking switching networks. Networks 1, 367–377 (1971/1972) 3. G.J. Chang, F.K. Hwang, L.-D. Tong, Characterizing bit permutation networks [MR1630366 (99h:94088)]. Networks 33, 261–267 (1999). Selected papers from DIMACS Workshop on Switching Networks, Princeton, NJ, 1997 4. H.-B. Chen, F.K. Hwang, On multicast rearrangeable 3-stage clos networks without first-stage fan-out. SIAM J. Discret. Math. 20, 287–290 (2006) 5. V. Chinni, T. Huang, P.-K. Wai, C. Menyuk, G. Simonis, Crosstalk in a lossy directional coupler switch. J. Lightwave Technol. 13, 1530–1535 (1995) 6. S.-P. Chung, K.W. Ross, On nonblocking multirate interconnection networks. SIAM J. Comput. 20, 726–736 (1991) 7. C. Clos, Bell Syst. Tech. J. 32, 406–424 (1953) 8. P. Coppo, M. D’Ambrosio, R. Melen, Optimal cost/performance design of ATM switches. IEEE/ACM Trans. Netw. 1, 566–575 (1993) 9. J.G. Dai, B. Prabhakar, The throughput of data switches with and without speedup, in INFOCOM, Tel-Aviv, 2000, pp. 556–564 10. G. Danilewicz, Wide-sense nonblocking logd .n; 0; p/ multicast switching networks. IEEE Trans. Commun. 55, 2193–2200 (2007) 11. D. Li, Elimination of crosstalk in directional coupler switches. Opt. Quantum Electron. 25, 255–260 (1993) 12. J. Duato, S. Yalamanchili, L. Ni, Interconnection Networks: An Engineering Approach, 1st edn. (IEEE Computer Society, Los Alamitos, 1997) 13. F.K. Hwang, B.-C. Lin, Wide-sense nonblocking multicast log2 .n; m; p/ networks. IEEE Trans. Commun. 51, 1730–1735 (2003) 14. F.K. Hwang, Y. Wang, J. Tan, Strictly nonblocking f-cast logd .n; m; p/ networks. IEEE Trans. Commun. 55, 981–986 (2007) 15. B. Gao, F.K. Hwang, Wide-sense nonblocking for multirate 3-stage Clos networks. Theor. Comput. Sci. 182, 171–182 (1997) 16. R.R. Goke, G.J. Lipovski, Banyan networks for partitioning multiprocessor systems, in Proceedings of the 1st Annual Symposium on Computer Architecture (ISCA’73), Barcelona, Spain, 1973, pp. 21–28

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17. P.E. Haxell, A. Rasala, G.T. Wilfong, P. Winkler, Wide-sense nonblocking WDM crossconnects, in Algorithms—ESA’03, Rome, Italy. Lecture Notes in Computer Science, vol. 2461 (Springer, Berlin, 2002), pp. 538–549 18. H.S. Hinton, An Introduction to Photonic Switching Fabrics (Plenum, New York, 1993) 19. J. H. Hui, Switching and Traffic Theory for Integrated Broadband Networks (Kluwer Academic, Boston/Dordrecht/London, 1990) 20. F. Hwang, Choosing the best log2 .n; m; p/ strictly nonblocking networks. IEEE Trans. Commun. 46, 454–455 (1998) 21. F.K. Hwang, The Mathematical Theory of Nonblocking Switching Networks (World Scientific, River Edge, 1998) 22. F.K. Hwang, W.-D. Lin, Necessary and sufficient conditions for rearrangeable log/sub d/(n, m, p). IEEE Trans. Commun. 53, 2020–2023 (2005) 23. X. Jiang, A. Pattavina, S. Horiguchi, Rearrangeable f -cast multi-log2 n networks. IEEE Trans. Commun. 56, 1929–1938 (2008) 24. X. Jiang, A. Pattavina, S. Horiguchi, Strictly nonblocking f-cast photonic networks. IEEE/ACM Trans. Netw. 16, 732–745 (2008) 25. X. Jiang, H. Shen, M.M. ur Rashid Khandker, S. Horiguchi, Blocking behaviors of crosstalkfree optical banyan networks on vertical stacking. IEEE/ACM Trans. Netw. 11, 982–993 (2003) 26. W. Kabacinski, G. Danilewicz, Wide-sense and strict-sense nonblocking operation of multicast multi-log2 n switching networks. IEEE Trans. Commun. 50, 1025–1036 (2002) 27. C.-T. Lea, Muti-log2 n networks and their applications in high speed electronic and photonic switching systems. IEEE Trans. Commun. 38, 1740–1749 (1990) 28. C.-T. Lea, D.-J. Shyy, Tradeoff of horizontal decomposition versus vertical stacking in rearrangeable nonblocking networks. IEEE Trans. Commun. 39, 899–904 (1991) 29. S.C. Liew, M.-H. Ng, C.W. Chan, Blocking and nonblocking multirate clos switching networks. IEEE/ACM Trans. Netw. 6, 307–318 (1998) 30. Lucent Technologies Press Release, Lucent Technologies unveils ultra-high-capacity optical system; Time Warner Telecom first to announce it will deploy the system (2001), http://www. lucent.com/press/0101/010117.nsa.html 31. Lucent Technologies Press Release, Lucent Technologies engineer and scientists set new fiber optic transmission record (2002), http://www.lucent.com/press/0302/020322.bla.html 32. Lucent Technologies Website, What is dense wave division multiplexing (DWDM) (2002), http://www.bell-labs.com/technology/lightwave/dwdm.html 33. G. Maier, A. Pattavina, Design of photonic rearrangeable networks with zero first-order switching-element-crosstalk. IEEE Trans. Comm. 49, 1248–1279 (2001) 34. B. Mukherjee, Optical Communication Networks (McGraw-Hill, New York, 1997) 35. H.Q. Ngo, A new routing algorithm for multirate rearrangeable Clos networks. Theoret. Comput. Sci. 290, 2157–2167 (2003) 36. H.Q. Ngo, D.-Z. Du, Notes on the complexity of switching networks, in Advances in Switching Networks, ed. by D.-Z. Du, H.Q. Ngo. Network Theory and Applications, vol. 5 (Kluwer Academic, Boston, 2001) pp. 307–367 37. H.Q. Ngo, V.H. Vu, Multirate rearrangeable Clos networks and a generalized bipartite graph edge coloring problem. SIAM J. Comput. 32, (2003), pp. 1040–1049 38. H.Q. Ngo, D. Pan, C. Qiao, Constructions and analyses of nonblocking wdm switches based on arrayed waveguide grating and limited wavelength conversion. IEEE/ACM Trans. Netw. 14, 205–217 (2006) 39. H.Q. Ngo, T.-N. Nguyen, D.T. Ha, Crosstalk-free widesense nonblocking multicast photonic switching networks, in Proceedings of the 2008 IEEE Global Communications Conference (GLOBECOM), New Orleans, LA, USA (IEEE, Piscataway, 2008), pp. 2643–2647 40. H.Q. Ngo, Y. Wang, A. Le, A linear programming duality approach to analyzing strictly nonblocking d -ary multilog networks under general crosstalk constraints, in Proceedings of the 14th Annual International Computing and Combinatorics Conference (COCOON), Bejing, China, LNCS (Springer, 2008), pp. 509–519

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41. H.Q. Ngo, A. Rudra, A.N. Le, T.-N. Nguyen, Analyzing nonblocking switching networks using linear programming (duality), in INFOCOM, San Diego, 2010, pp. 2696–2704 42. T.-N. Nguyen, H.Q. Ngo, Y. Wang, Strictly nonblocking f -cast photonic switching networks under general crosstalk constraints, in Proceedings of the 2008 IEEE Global Communications Conference (GLOBECOM), New Orleans, LA, USA (IEEE, 2008) pp. 2807–2811 43. A. Pattavina, G. Tesei, Non-blocking conditions of multicast three-stage interconnection networks. IEEE Trans. Commun. 46, (2005), pp. 163–170 44. R. Ramaswami, K. Sivarajan, Optical Networks: A Practical Perspective, 2nd edn. (Morgan Kaufmann, San Francisco, 2001) 45. A. Rasala, G. Wilfong, Strictly non-blocking WDM cross-connects, in Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms SODA’2000, San Francisco, CA (ACM, New York, 2000), pp. 606–615 46. A. Rasala, G. Wilfong, Strictly non-blocking WDM cross-connects for heterogeneous networks, in Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing STOC’2000, Portland, OR (ACM, New York, 2000), pp. 513–524 47. D.-J. Shyy, C.-T. Lea, log2 .n; m; p/ strictly nonblocking networks. IEEE Trans. Commun. 39, 1502–1510 (1991) 48. T.E. Stern, K. Bala, Multiwavelength Optical Networks: A Layered Approach (Prentice Hall, Upper Saddle River, 1999) 49. Y. Tscha, K.-H. Lee, Yet another result on multi-log2 n networks. IEEE Trans. Commun. 47, 1425–1431 (1999) 50. J.S. Turner, R. Melen, Multirate Clos networks. IEEE Commun. Mag. 41, 38–44 (2003) 51. J.S. Turner, N. Yamanaka, Architectural choices in large scale ATM switches. IEICE Trans. Commun. E81-B, 120–137 (1998) 52. M.M. Vaez, C.-T. Lea, Wide-sense nonblocking Banyan-type switching systems based on directional couplers. IEEE J. Select. Areas Commun. 16, 1327–1332 (1998) 53. M.M. Vaez, C.-T. Lea, Strictly nonblocking directional-coupler-based switching networks under crosstalk constraint. IEEE Trans. Commun. 48, 316–323 (2000) 54. Y. Wang, H.Q. Ngo, X. Jiang, Strictly nonblocking f -cast d -ary multilog networks under fanout and crosstalk constraints, in Proceedings of the 2008 International Conference on Communications (ICC), Beijing, China (IEEE, 2008) 55. G. Wilfong, B. Mikkelsen, C. Doerr, M. Zirngibl, WDM cross-connect architectures with reduced complexity. J. Lightwave Technol. 17, 1732–1741 (1999) 56. J.-C. Wu, T.-L. Tsai, Low complexity design of a wavelength-selective switch using raman amplifiers and directional couplers, in GLOBECOM, San Francisco, California, 2006 57. Y. Yang, G.M. Masson, Nonblocking broadcast switching networks. IEEE Trans. Comput. 40, 1005–1015 (1981) 58. Y. Yang, J. Wang, Wide-sense nonblocking clos networks under packing strategy. IEEE Trans. Comput. 48, 265–284 (1999) 59. Y. Yang, J. Wang, Designing WDM optical interconnects with full connectivity by using limited wavelength conversion. IEEE Trans. Comput. 53, 1547–1556 (2004) 60. Y. Yang, J. Wang, C. Qiao, Nonblocking WDM multicast switching networks. IEEE Trans. Parallel Distrib. Syst. 11, 1274–1287 (2000)

Map of Geometric Minimal Cuts with Applications Evanthia Papadopoulou, Jinhui Xu and Lei Xu

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 On Geometric Minimal Cuts and Their Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Geometric Cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Identifying Geometric Minimal Cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Generating Map of Geometric Minimal Cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The MGMC Problem via Higher-Order Voronoi Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The MGMC Problem in a VLSI Layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Review of Concepts on L1 Voronoi Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Definitions and Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 A Higher-Order Voronoi Diagram Modeling Open Faults and the MGMC Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Computing the Opens Voronoi Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The L1 Hausdorff Voronoi Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definitions and Structural Complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Refined Plane Sweep Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 The work of the first author was supported in part by the Swiss National Science Foundation SNSF project 200021-127137. The work of the last two authors was supported in part by NSF through a CAREER Award CCF-0546509 and two grants IIS-0713489 and IIS-1115220.

E. Papadopoulou () Faculty of Informatics, Universit`a della Svizzera italiana, Lugano, Switzerland J. Xu • L. Xu Department of Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA 1815 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 27, © Springer Science+Business Media New York 2013

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Abstract

This chapter considers the following problem of computing a map of geometric minimal cuts (called the MGMC problem): Given a graph G D .V; E/ and a planar embedding of a subgraph H D .VH ; EH / of G, compute the map of geometric minimal cuts induced by axis-aligned rectangles in the embedding plane. The MGMC problem is motivated by the critical area extraction problem in VLSI designs and finds applications in several other fields. This chapter surveys two different approaches for the MGMC problem based on a mix of geometric and graph algorithm techniques that can be regarded complementary. It is first shown that unlike the classic min-cut problem on graphs, the number of all rectilinear geometric minimal cuts is bounded by a low polynomial, O.n3 /. Based on this observation, the first approach enumerates all rectilinear geometric minimal cuts and computes their L1 Hausdorff Voronoi diagram, which is equivalent to the L1 Hausdorff Voronoi diagram of axis-aligned rectangles. The second approach is based on higher-order Voronoi diagrams and identifies necessary geometric minimal cuts and their Hausdorff Voronoi diagram in an iterative manner. The embedding in the latter approach includes arbitrary polygons. This chapter also presents the structural properties of the L1 Hausdorff Voronoi diagram of rectangles that provides the map of the MGMC problem and plane sweep algorithms for its construction.

1

Introduction

This chapter considers the following problem called map of geometric minimal cuts (MGMC): Given a graph G D .V; E/ and a planar embedding of a subgraph H D .VH ; EH / of G, compute a map1 M of the embedding plane P of H so that for every point p 2 P , the cell in M containing p is associated with the “closest” geometric cut (in G) to p, where distance between a point p and a cut C is defined as the maximum distance between p and any individual element of C . A geometric cut C of G induced by a given geometric shape S is a set of edges and vertices in H that overlaps S in P and whose removal from G disconnects G. This chapter considers the case where geometric cuts are induced by axis-aligned rectangles and distances are measured in the L1 metric. The main objective of the MGMC problem is to compute the map M of all geometric minimal (or canonical) cuts (the exact definition of geometric minimal cuts will be given in the next section) of the planar embedding of H .

1

A map means a partition of the embedding plane (as in a trapezoidal map) into cells so that all points in the same cell share the same “closest” geometric minimal cut.

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The MGMC problem was formulated in [47]. It was introduced, as the geometric min-cut problem, in [34], in order to model the critical area computation problem for open faults in a VLSI design. Critical area is a measure reflecting the sensitivity of a VLSI design to random defects occurring during the manufacturing process. For a survey on VLSI yield analysis and optimization, see, for example, [16]. The critical area computation problem in a VLSI design for various types of fault mechanisms has been reduced to variants of generalized Voronoi diagrams; see, for example, [5, 32, 34–36, 46]. The Voronoi framework for critical area extraction asks for a subdivision of a layer A into regions that reveal, for every point p, the size of the smallest defect centered at p causing a circuit failure. For open faults, this results in the MGMC problem. In more detail, a VLSI net N can be modeled as a graph G D .V; E/ having a subgraph H D .VH ; EH / embedded on any conducting layer A; see, for example, [34]. The embedded subgraph H D .VH ; EH / is vulnerable to random manufacturing defects that are associated with the layer of the embedding and may create open faults. The MGMC problem computes, for every point p on layer A, the disk of minimum radius, which is centered at p and induces a cut on graph G resulting in an open fault. The subdivision of layer A that solves the MGMC problem corresponds to the Hausdorff Voronoi diagram of the geometric minimal cuts (see, e.g., [34, 47]). The MGMC problem finds applications in other networks, such as transportation networks, where critical area may need to be extracted for the purpose of flow control and disaster avoidance. This chapter surveys the literature on the MGMC problem and presents two complimentary approaches. These approaches are based on a mix of geometric and graph algorithm techniques, and they produce, by different means, the L1 Hausdorff Voronoi diagram of geometric minimal cuts on the embedding plane. In addition, this chapter presents structural properties and algorithms for the L1 Hausdorff Voronoi diagram, which provides a solution to the MGMC problem, and it is a geometric structure of independent interest. The first approach to the MGMC problem presented in this chapter is based on the results of [47] for a rectilinear embedding of the embedded subgraph H . This approach first classifies geometric cuts into two classes – 1-D cuts and 2-D cuts – and shows that the number of all possible geometric 1-D and 2-D minimal cuts, given a rectilinear embedding of H , is O.n2 / and O.n3 /, respectively, where n is the size of H . The O.n3 / bound on the number of geometric minimal cuts makes the enumeration of geometric cuts feasible. In contrast, the corresponding bound of the classic min-cut problem in graphs is exponential. Based on interesting observations and dynamic connectivity data structures, the approach in [47] directly computes all geometric minimal cuts in O.n3 log n.log log n/3 / time. A special case in which the inducing rectangle of a cut has a constantly bounded edge length is also considered, in which case the time complexity of the algorithm can be significantly improved to O.n log n.log log n/3 /. Once all geometric minimal cuts are identified, the solution to the MGMC problem is reduced to computing their Hausdorff Voronoi diagram. The method in [47] revisits the plane sweep construction of the L1 Hausdorff Voronoi diagram of rectangles and presents the first output-sensitive algorithm for

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its construction, which runs in O..N CK/ log2 N log log N / time and O.N log2 N / space, where N is the number of rectangles and K is the complexity (or size) of the Hausdorff Voronoi diagram. The algorithm is based on a plane sweep construction in 3-D and uses several interesting data structures that can achieve an output-sensitive result. The second approach is based on results in [34], and it is based on higherorder Voronoi diagrams. The embedded subgraph needs not be rectilinear, it can be arbitrary; moreover, it corresponds to polygonal shapes. Given a geometric graph with a subgraph embedded in the plane, the method in [34] iteratively identifies geometric minimal cuts and their generators,2 based on an iterative construction of order-k Voronoi diagrams in increasing order of k. In the worst case, the method requires time O.n3 log n/ to compute higher-order Voronoi diagrams, plus time O.n3 log n.log log n/3 / to answer connectivity queries on the graph G, assuming that the dynamic connectivity data structures of [20, 45] are used for this purpose. The resulting map provides a solution to the MGMC problem and reveals the Hausdorff Voronoi diagram of all geometric minimal cuts. In practice, typically, it is not necessary to guarantee the identification of all possible geometric cuts, in which case the iterative construction of [34] can be considerably sped up. Once a sufficient set of cut generators are identified, their weighted Voronoi diagram can be computed in a simple manner, providing a map that reliably approximates the solution to the MGMC problem. In the case of rectilinear embeddings, generators are simple axisparallel line segments of constant weights. For arbitrary embeddings, generators correspond to portions of farthest Voronoi diagrams of cuts in the final map, and their weights correspond to farthest distance functions. Assuming that the number of iterations is kept to a small constant, as experimental results in [34] suggest, this results in an O.n log n.log log n/3 / algorithm for an approximate map solving the MGMC problem. The Hausdorff Voronoi diagram of a set S of point clusters in the plane is a subdivision of the plane into regions, such that the Hausdorff Voronoi region of a cluster P is the locus of points closer to P than to any other cluster in S, where distance between a point t and a cluster P is measured as the farthest distance between t and any point in P . The Hausdorff Voronoi diagram first appeared in [11] as the cluster Voronoi diagram, where several combinatorial bounds were derived by means of a powerful geometric transformation in three dimensions and an O.n2 /-time algorithm for its construction. A tight combinatorial bound on its structural complexity in the Euclidean plane, as well as a plane sweep construction were given in [33]. The L1 version of the problem was introduced in [32] for the VLSI critical area extraction problem for a defect mechanism called a via-block. In [32], the problem was reduced to an L1 Voronoi diagram of additively weighted line segments, and it was computed by a plane sweep procedure. The plane sweep construction was revisited in [47] and [39]. Additional work on Hausdorff Voronoi

2

The generator of a cut is a portion of the farthest Voronoi diagram of the elements constituting the cut.

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diagrams is included in [1,9,38]. In this chapter, we first present an output-sensitive version of the plane sweep in three dimensions for the L1 Hausdorff Voronoi diagram of rectangles as given in [47]. We also present the structural properties of this diagram and a simple two-dimensional plane sweep as given in [32, 39], which is not output sensitive; nevertheless, it achieves optimal O.n log n/-time and O.n/-space in the case of non-crossing rectangles. In case of a large number of crossings, the size of the Hausdorff Voronoi diagram may be .n2 / [11], and the O.n2 / algorithm of [11] may result in a better approach. This chapter is organized as follows. Section 2 refers to [47], Section 3 refers to [34], and Section 4 to [39]. The sections are presented independently and they can be read in any order. In more detail, Sect. 2.1 introduces the concepts of 1-D and 2-D geometric minimal cuts; Sect. 2.2 presents the algorithms of [47] for computing the geometric minimal cuts; and Sect. 2.3 presents the output-sensitive plane sweep algorithm in 3-dimensions of [47]. Section 3 presents an iterative approach to the MGMC problem via higher-order Voronoi diagrams based on [34]. Section 4 analyzes the structural complexity of the L1 Hausdorff Voronoi diagram and summarizes the simple two-dimensional plane sweep algorithm for its construction presented in [32, 39]. In Section 5 we provide concluding remarks.

2

On Geometric Minimal Cuts and Their Map

2.1

Geometric Cuts

Let G D .V; E/ be the undirected graph in an MGMC problem and H D .VH ; EH / be its planar subgraph embedded in the plane P with jV j D N , jEj D M , jVH j D n, and jEH j D m. Due to the planarity of H , m D O.n/. In this chapter, unless explicitly mentioned otherwise, all edges in H are either horizontal or vertical straight-line segments. A pair of vertices u and v in a graph are connected if there is a path in this graph from u to v. Otherwise, they are disconnected. A graph is connected if every pair of its distinct vertices is connected. Without loss of generality, G is assumed to be connected in this chapter. A cut C of G is a subset of edges in G whose removal disconnects G. A cut C is minimal if removing any edge from C no longer forms a cut. Definition 1 Let R be a connected region in P , and C D R \ H be the set of edges in H intersected by R. The cut C is called a geometric cut induced by R if the removal of C from G disconnects G. When there is no ambiguity of the region R, the cut induced by R is called a “geometric cut” for simplicity. For a given cut C , its minimum inducing region R.C / is the minimum axis-aligned rectangle which intersects every edge of C (i.e., if we shrink the width or length of R.C / by any arbitrarily small value , some edge in C will no longer be intersected by R.C /). For some geometric cut C , its minimum inducing region R.C / could be degenerated into a horizontal or vertical

1820 Fig. 1 (a) 1-D cuts C1 ; C2 ; and C3 with C3 being the minimal cut. (b) A 2-D cut bounded by 4 edges

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a

b

Cleft C1

Cright C2 C3

line segment, or even a single point. It is easy to see that if R.C / is not degenerate, it is unique. Definition 2 A geometric cut C is called a 1-D geometric cut (or a 1-D cut) if R.C / is a line segment. If R.C / is an axis-aligned rectangle, then C is called a 2-D geometric cut (or a 2-D cut). Definition 3 A geometric cut C is a geometric minimal cut if the set of edges intersected by any region obtained by shrinking R.C / no longer forms a geometric cut. The following lemma easily follows from the above definitions and the problem setting. Lemma 1 Let C be any geometric minimal cut. If C is a 1-D cut, then each endpoint u of R.C / is incident to either an endpoint of an edge e in H with the same orientation as R.C / and e \ R.C / D u or an edge in H with different orientation as R.C / (see Fig. 1a). If C is a 2-D cut, each bounding edge s of R.C / is incident to either an endpoint v of an edge e in H of different orientation with s and R.C / \ e D v, or an edge in H with the same orientation as s (see Fig. 1b; note that s could be incident to multiple edges). From the above lemma, it is clear that each 1-D geometric minimal cut is bounded by up to two edges in H and each 2-D geometric minimal cut is bounded by up to 4 edges in H . For a cut C , let B.C / denote the set of edges bounding C . Due to the minimality nature of C , removing any edge in B.C / will lead to a noncut. This means that any edge in B.C / is necessary for forming the cut. However, this is not necessarily true for edges in C nB.C /. Thus, a geometric minimal cut may not be a minimal cut. Note that for a given graph, the number of minimal cuts could be exponential. However, as shown later, the number of geometric minimal cuts is much less (i.e., bounded by a low degree polynomial). For a 1-D geometric minimal cut C , it is possible that all edges in C have different orientation with R.C /. In this case, there may exist an infinite number of 1-D geometric minimal cuts, all cutting the same set of edges C (in other words, the minimum inducing region for such a 1-D geometric C is not unique). For example, sliding R.C / along B.C / will obtain an infinite number of 1-D minimal cuts.

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Among these cuts, there are two extreme cuts, Clef t and Cright (or Ct op and Cbot t om if R.C / is horizontal), incident with the left and right (or up and bottom) endpoints of some edges in C , respectively. Since all these geometric cuts cut the same set of edges, they are counted as one cut.

2.2

Identifying Geometric Minimal Cuts

To solve the MGMC problem, the main idea in [47] is to first identify all possible geometric minimal cuts and then construct a Hausdorff Voronoi diagram of these cuts as the map M. Thus, three major problems need to be solved: 1. Finding all 1-D minimal cuts; 2. Finding all 2-D minimal cuts; and 3. Efficiently constructing the Hausdorff Voronoi diagram for the geometric cuts. This section discusses the main ideas for problems 1 and 2. The approach for problem 3 will be discussed in the next section.

2.2.1 Computing 1-D Geometric Minimal Cuts As discussed in the previous section, a 1-D geometric minimal cut can be induced by either horizontal or vertical segments. The following lemma bounds from above the total number of 1-D geometric minimal cuts. Lemma 2 There are O.n2 / 1-D geometric minimal cuts in H . Proof Only the 1-D cuts induced by vertical segments are considered. Cuts induced by horizontal segments can be proved similarly. If we place a vertical line through each vertex (or endpoint), then the plane P is partitioned into O.n/ vertical slabs, with each slab containing no endpoint in its interior. For a particular slab S containing k edges, say e1 ; e2 ;    ; ek , it could be shown that there are at most O.k/ 1-D geometric minimal cuts. To see this, all 1-D minimal cuts formed by the k edges are considered. A cut C is owned by an edge ei if ei 2 B.C /. Clearly, each 1-D minimal cut has at least one owner. Now consider each edge ei , it can only be the owner of up to two 1-D minimal cuts, one bounded by ei from the bottom and one bounded by ei from the top. Note that due to the fact that the cuts are all minimal, it is impossible to have two cuts bounded by ei from the top (or bottom) simultaneously. Thus, each edge in S owns at most two 1-D geometric minimal cuts. Hence, the total number of 1-D geometric minimal cuts in S is O.k/. Summing over all slabs, the total number of 1-D geometric minimal cuts is O.n2 /. Thus, the lemma follows.  To compute the O.n2 / 1-D geometric minimal cuts, since each cut is a set of edges C 2 H which appear consecutively in some slab and whose removal disconnects G, it is necessary to first (a) Identify all possible cuts in H , and then (b) For each possible cut, determine whether it is indeed a cut (such a test is called a cut query). To overcome the two difficulties, a straightforward way is to enumerate

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all possible cuts, and for each possible cut C , use graph search algorithms (such as depth first search (DFS) or breadth first search (BFS)) to check the connectivity of GnC . As shown in the proof of Lemma 2, the embedding plane P can be partitioned into O.n/ slabs, and for each slab with k edges, there are O.k 2 / subsets of consecutive edges. Thus, a total of O.n3 / possible subsets need to be checked, and the connectivity checking for each subset takes O.N C M / time. Therefore, this will lead to a total of O.n3 .N C M // time. To speed up the computation, the idea is to first simplify the graph G. Since the cuts involve only the edges in H , the connectivity in GnEH will not be affected no matter which subset of edges in H is removed. To make use of this invariant, the connected components of GnEH are firstly computed. For each connected component C C , it is contracted into a supernode vC C . For each vertex u 2 H \C C , an edge .u; vC C / is added. Let the resulting graph be G 0 D .V 0 ; E 0 / (called contracted graph). The following lemma gives some properties of this graph. Lemma 3 The number of vertices in G 0 is jV 0 j D O.n/ and the number of edges in G 0 is jE 0 j D O.n/. Furthermore, a subset of edges in H is a cut of G if and only if it is a cut of G 0 . Proof Follows from the construction of G 0 .



The above lemma shows that the size of G 0 could be much smaller than G. Thus, the time needed for answering a cut query is significantly reduced from O.N C M / to O.n/. Converting Cut Queries to Connectivity Queries As discussed previously, to compute all 1-D geometric minimal cuts, it is necessary to check O.n3 / possible subsets of edges in H . Many of them are quite similar (i.e., differ only by one or a small number of edges). Thus, it would be highly inefficient to handle them independently and answer each cut query by searching the graph G 0 . To yield a better solution, it is better to consider all these cut queries simultaneously and share the connectivity information among similar cut queries. To share information among slightly different cut queries, one possible way is to use persistent data structure [10]. However, due to the fact that inserting (or removing) an edge could cause a linear, instead of a constant, number of 1-D geometric minimal cuts to be destroyed or generated, persistent data structures require substantial amount of updating after each insertion or deletion, thus they do not lead to a highly efficient solution. To overcome this difficulty, the main idea is to further decompose each cut query into a set of connectivity queries with each connectivity query involving only one edge. The reduced granularity in query enables us to better share information among related cut queries. Connectivity Query Before continuing the discussion on connectivity queries, some existing algorithms and data structures for the dynamic connectivity problem

Map of Geometric Minimal Cuts with Applications Fig. 2 An example for 1DSlab, where the dotted lines are the edges in G 0 nH . f2; 3g and f5g are the 1-D geometric minimal cuts

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are briefly reviewed. In the fully dynamic connectivity problem, the input is a graph G whose vertex set is fixed and whose edges can be inserted and deleted. The objective is to construct a data structure for G so that the connectivity of any pair of vertices can be efficiently determined. Most fully dynamic connectivity data structures support the following three operations: (1) Insert.e/, (2) Delete.e/, and (3) Connectivity.u; v/, where the Insert.e/ operation inserts edge e into G, the Delete.e/ operation removes edge e from G, and the Connectivity.u; v/ operation determines whether u and v are connected in the current graph G. Extensive research has been done on this problem, and a number of results were obtained [12, 13, 17, 18, 20, 45]. In [20], Thorup et al. gave a simple and interesting solution for this problem which answers each connectivity query in O.log n/ time and takes O.log2 n/ time for each update. Later Thorup gave a near-optimal solution for this problem [45] which answers each connectivity query in O.log n= log log log n/ time and completes each insertion or deletion operation in O.log n.log log n/3 / expected amortized time. The approach described in this chapter uses the data structure in [45] for the MGMC problem. In practice (e.g., critical area computation), the simpler algorithm in [20] may be more practical. When the choice of the connectivity data structure is unclear, M axQU is used to represent the maximum of the connectivity query time and the update operation time. Enumerating 1-D Geometric Minimal Cuts in a Slab The problem of identifying 1-D geometric minimal cuts in a vertical slab S with k edges (see Fig. 2) is to make use of the connectivity data structure. Clearly, there are O.k 2 / possible 1-D geometric cuts and O.k/ 1-D geometric minimal cuts. To find out the O.k/ minimal cuts from O.k 2 / possible locations, edges are sorted based on the y coordinates, and let e1 D .u1 ; v1 /; e2 D .u2 ; v2 /;    ; ek D .uk ; vk / be the k edges. A fully dynamic connectivity data structure FDC.G 0 / for the contracted graph G 0 is built, and then the algorithm for a slab (called 1DSlab) is run on it. The main steps of 1DSlab are the follows.

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1. Initialize a variable r D 1 to represent the index of the next to-be-deleted edge. 2. Starting from er , repeatedly delete edges of S from G 0 according to their sorted order and store them in a queue Q. For each deleted edge ei , query FDC.G 0 / about the connectivity of ui and vi . Stop the deletion until encountering the first edge ej whose two endpoints uj and vj are disconnected or the last edge. In the latter case, insert all deleted edges back and stop. 3. Insert the deleted edges in Q back in the same order as they are deleted and updating FDC.G 0 /. After inserting each edge ei , query the connectivity of the two endpoints of ej , uj , and vj . 4. If uj and vj are disconnected, add a forward pointer from ei to ej and insert edges in Q back to G 0 . 5. If uj and vj are connected, add a forward pointer from ei to ej , set r D j C 1, and repeat Steps 2–5 until encountering the last edge ek . In this case, insert all remaining edges in Q back to G 0 and FDC.G 0 /. 6. Reverse the order of the k edges and repeat the above procedure. In this step the added pointers are backward pointers. 7. For each edge ej , find the nearest edge ei which has a forward pointer to ej and the nearest edge ei0 with a backward pointer to ej . Output the set of edges between ei and ej (including ei and ej ) as a 1-D geometric minimal cut and edges between ej and ei0 (including ej and ei0 ) as another 1-D geometric minimal cut. The correctness of the above algorithm is shown below. Lemma 4 Assume G 0 is originally a connected graph. In Step 2 of the 1DSlab algorithm, if ui and vi (the endpoints of the last deleted edge ei ) are connected, the set of deleted edges (i.e., er to ei ) does not form a cut in G 0 . On the other hand, if ui and vi are disconnected, then the set of deleted edges does form a cut in G. Proof Firstly, if ui and vi are disconnected, obviously the set of deleted edges forms a cut. Thus, only the case that ui and vi are connected is considered. In this case, the set of deleted edges does not form a cut. This could be proved by induction on the number i of deleted edges. The base case is i D 1. In this case, since G 0 is originally a connected graph and ui and vi are still connected in G 0 nfei g, obviously there exists no cut. Assume that after deleting i  1 edges, the set of edges Si 1 D fe1 ; e2 ;    ; ei 1 g does not form a cut. Then since G 0 nSi 1 is a connected graph, after deleting edge ei , the case becomes the same as the base case. Thus, the lemma follows.  Lemma 5 In the 1DSlab algorithm, the disconnectivity of uj and vj in Step 4 or the connectivity of uj and vj in Step 5 implies that the set Si;j D fei ; ei C1 ;    ; ej g forms a cut in G 0 . Proof The first case (i.e., uj and vj are disconnected in Step 4) is obvious. For the second case (i.e., uj and vj are connected in Step 5), since uj and vj are disconnected before the insertion of ei , it follows that Si;j is a cut. 

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The above lemmas indicate that the cut query can be answered by a sequence of connectivity queries.

Theorem 1 The 1DSlab algorithm generates all 1-D geometric minimal cuts in the slab S in O.kM axQU / time, where M axQU is the maximum of the query time and the updating time of the FDC.G 0 / data structure. Proof The reason that the 1DSlab algorithm generates all 1-D geometric minimal cuts is shown below. By Lemmas 4 and 5, it is known that the forward pointers to ej indicate all cuts whose edge set appears consecutively and ends at edge ej . Similarly, the backward pointers to ej indicate all cuts whose edge set appears consecutively and starts from ej . By finding the nearest edges with forward and/or backward pointers to ej , the algorithm identifies the (up to) two 1-D geometric minimal cuts starting from and ending at ej . Since the computation is for every edge in S , it generates all 1-D geometric minimal cuts. For the running time, it is clear that the sorting takes O.k log k/ time. After sorting, in each of the forward and backward computations, every edge in S is deleted and inserted once. As for the connectivity queries, the endpoints of each edge can be queried multiple times (one in Step 2 and others in Step 3). To bound the total query time, the query on uj and vj in Step 3 is charged to edge ei . Since ei is inserted only once, it will be charged only once. Thus, each edge in S will be responsible for at most two queries. Therefore, the total number of connectivity queries is O.k/. Since the insertion, deletion, and query operations take no more than O.M axQU / time, the theorem follows.  From the above theorem, it is known that even though there are O.k 2 / consecutively appeared subsets of edges that could be 1-D geometric minimal cuts, the connectivity data structure can reduce the time to near linear. Generating 1-D Geometric Minimal Cuts The algorithm 1DSlab is combined with a plane sweep algorithm to generate all possible 1-D geometric minimal cuts in H . A straightforward way of using 1DSlab is to partition the plane P into O.n/ slabs and apply the 1DSlab algorithm in each slab. This will lead to a O.n2 M axQU /time solution. A more output-sensitive solution is to use the following plane sweep algorithm. In the plane sweep algorithm, a vertical sweeping line L is used to sweep through all edges in H to generate those 1-D geometric minimal cuts induced by vertical segments. Similarly, those 1-D geometric minimal cuts induced by horizontal segments can be generated by sweeping a horizontal line. For the vertical sweeping line algorithm, the event points are the set VH of endpoints in H . At each event point, the set of edges intersecting the sweeping line L forms a slab. However, instead of applying the 1DSlab algorithm to the whole set of intersecting edges, only a subset of edges are considered.

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Let u be the event point. If u is the left endpoint of an edge e, then e is the new edge just encountered by L. Thus, it is only needed to identify all 1-D geometric minimal cuts which contain e. This means that the 1DSlab algorithm needs only to consider the minimal cut C1 bounded by e from the bottom, the minimal cut C2 bounded by e from the top, and all minimal cuts between the top edge et of C1 and the bottom edge eb of C2 . To deal with this case, the 1DSlab algorithm can be easily modified. Particularly, the modified algorithm can start from e, have a backward computation, and stop at the first edge et , whose removal disconnects its two endpoints, to generate C1 . Similarly, the algorithm can start from e and have a forward computation to generate C2 . In this way, dealing with possibly many edges beyond et and eb is avoided. If u is the right endpoint of e, it is necessary to check those cuts containing e to see whether they are still geometric minimal cuts. It is also needed to check whether new cuts can be generated. Clearly, it is only needed to consider the edges between et0 and eb0 , where et0 is the top edge of the 1-D geometric minimal cut bounded by e from the bottom and eb0 is the bottom edge of the 1-D geometric minimal cut bounded by e from the top. Thus, such information from the previous step (i.e., the step before encountering u) are first obtained and then the 1DSlab algorithm on the subset of edges between et0 and eb0 is applied directly. Theorem 2 All 1-D geometric minimal cuts of H can be found in O.n  M axC  M axQU / time, where M axC is the maximum size of a 1-D geometric minimal cut. Proof Follows from the above discussion.



2.2.2 Computing 2-D Geometric Minimal Cuts This section extends the algorithm in Sect. 2.2.1 to compute 2-D geometric minimal cuts. To compute 2-D geometric minimal cuts, it is easy to see that a 1-D geometric minimal cut is a degenerate version of a 2-D geometric minimal cut. The only difference is that the minimum inducing region of a 1-D cut has two opposite sides degenerated to points. Thus, if two opposite sides of the minimum inducing region R.C / of a 2-D cut C are conceptually “contracted” into “points,” the plane sweep algorithm for 1-D cuts can be applied to generate 2-D cuts. To implement this idea, two parallel sweeping lines L1 and L2 (called primary and secondary sweeping lines) are used to bound the “contracted” sides of R.C /. By Lemma 1, it is known that each 2-D geometric minimal cut is bounded by the endpoints (or edges) of up to four edges. This suggests that the possible locations of L1 and L2 are the endpoints of the input edges. Similarly to the plane sweep algorithm for 1-D cuts, the edges in H are swept twice, one time vertically and the other time horizontally. The discussion below is focused on horizontal sweeping (i.e., using vertical sweeping lines). The double plane sweep algorithm first sorts all edges in H based on the x coordinates of their left endpoints (for vertical segments, their upper endpoints are viewed as the left endpoints and their lower endpoints as the right endpoints).

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Fig. 3 An example for illustrating the double plane sweep algorithm for 2-D cuts

L1

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Let S1 D fe1 ; e2 ;    ; en g be the sorted order. The primary sweeping line L1 stops at the left endpoint of each edge e in the order of S1 . If the right endpoint of e is to the left of the left endpoint of the next edge e 0 2 S1 , the sweep line L1 is moved to the right endpoint of e (see Fig. 3). If e is a cut by itself, L1 is moved to the next edge in S1 . If e is not a cut, L1 is fixed and the secondary sweeping line L2 is swept from the left endpoint of the next edge e 0 in S1 to the last edge in S1 . When L2 stops, the plane sweep algorithm for 1-D cuts (in Sect. 2.2) is used to compute all 2-D geometric minimal cuts formed by the set S of edges intersecting the vertical region VR bounded by L1 and L2 . For edges in S , there are two sorted orders SL and SU . SL is sorted based on the y coordinates of the lower endpoints, and SU is based on the y coordinates of the upper endpoints. (For horizontal edges, their left endpoints are viewed as upper endpoints and right endpoints as the lower endpoints. It is not difficult to see that the two sorted lists can be dynamically maintained in the double plane sweep algorithm.) SL is used to generate cuts for edges above e and SU is used to generate cuts for edges below e. Each 2-D geometric minimal cut C in the vertical region VR is bounded by L1 from left, L2 from right, an edge eu from above, and an edge ed from below. Since all edges in S are in sorted order and R.C / intersects edges in the same order in either SL or SU (depending on their relative locations to e), the 2-D geometric minimal cuts in VR can be viewed as 1-D geometric minimal cuts induced by vertical “segments” with a segment width equal to the horizontal distance of L1 and L2 . Thus, they can be generated by using the plane sweep algorithm for 1-D cuts. Furthermore, similar to the plane sweep algorithm for 1-D cuts, in VR it is only needed to consider those 2-D geometric minimal cuts containing edge e (see Fig. 4). Once all the cuts in VR are identified, L2 is moved to the next edge in S1 . After sweeping L2 , L1 is moved to the next edge in S1 and the above procedure is repeated until L1 sweeps every edge. The following lemma is used to analyze the double plane sweep algorithm. Lemma 6 There are at most O.n3 / 2-D geometric minimal cuts in H .

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Fig. 4 Example of the double plane sweep algorithm for 2-D cuts. The region bounded by the dotted lines is the actual region where the algorithm searches for all 2-D geometric minimal cuts bounded by the leftmost segment from the left.

Proof By Lemma 1, it is known that the inducing region of each 2-D geometric minimal cut is bounded by an edge on each side. There are O.n2 / pairs of edges to bound the left and right sides of the inducing region. For each pair, the vertical region is similar to a slab (according to the above discussion). By a similar argument in the proof of Lemma 2, it is easy to see that in each vertical region, there at most O.n/ 2-D geometric minimal cuts. Thus, the lemma follows.  For the running time of the double plane sweep algorithm, the primary sweeping line L1 stops at O.n/ location. For each fixed location of L1 , L2 sweeps all edges not yet encountered by L1 , whose number can be O.n/. For each vertical region bounded by L1 and L2 , it takes O.M axC  M axQU / time (by Theorem 2). Thus, the total time is O.n2  M axC  M axQU /. Theorem 3 All 2-D geometric minimal cuts in H can be found in O.n2  M axC  M axQU / time. Proof Follows from the above discussion.



Corollary 1 All geometric minimal cuts in the MGMC problem can be found in O.n3 log n.log log n/3 / time in the worst case. Proof Follows from Theorems 2 and 3, and article [45].



Corollary 2 If the maximum size of a cut is bounded by a constant, then all geometric minimal cuts in H can be found in O.n log n.log log n/3 /-time.

Proof If the maximum size of a defect is bounded by some constant, the size of an inducing rectangle is bounded by a constant. Furthermore, since edges in VLSI design are separated by some constant distance, thus the total number of edges in an inducing rectangle is also a constant. In this case, the secondary sweeping line L2 needs only to sweep a constant number of edges. Thus, the running time of both plane sweep algorithms is O.n  M axQU /. Also from Theorems 2 and 3, article [45], the lemma follows. 

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This section shows that Problem 3 can be solved by using the Hausdorff Voronoi diagram.

2.3.1 From Geometric Minimal Cuts to Hausdorff Voronoi Diagram Given two sets A and B, the directed Hausdorff distance from A to B is h.A; B/ D fmaxa2A minb2B d.a; b/g; and the undirected Hausdorff distance between A and B is dh .A; B/ D maxfh.A; B/; h.B; A/g; where d.a; b/ is the distance between the points a and b. In this chapter, the L1 metric is used to measure the distance. Given a set C of geometric minimal cuts of H , the Hausdorff Voronoi diagram of C is a partition of the embedding plane P of H into regions (or cells) so that the Hausdorff Voronoi cell of a cut C 2 C is the union of all points whose Hausdorff distance to C is closer than to any other cut in C. This means that for any point p 2 P , if an L1 ball is grown from p (i.e., an axis-aligned square centered at p), the ball entirely contains R.C.p// earlier than the minimum inducing region of any other cut, where C.p/ is the cut owning the Hausdorff Voronoi cell containing p. Thus, if such a map M exists for the critical area extraction problem, then for each point p, one can easily determine the minimum size of a defect centered at p and disconnecting the circuit. In the MGMC problem, two types of objects exist, the minimum inducing regions of 1-D geometric minimal cuts and the minimum inducing regions of 2-D geometric minimal cuts. For every 2-D cut C , the rectangle R.C / is fixed. However, for a 1-D cut C , the location of R.C / is not fixed, since there may be an infinite number of 1-D cuts cutting the same set of edges. In this case, the union of the infinite number of inducing segments R.C / is used to represent the cut, which is a rectangle (denoted by UR.C /) bounded by the two extreme 1-D cuts Clef t and Cright (or Ct op and Cbot t om ) and the two bounding edges B.C / of C . Thus, from thereafter, the objects of the Hausdorff Voronoi diagram are a set of axis-aligned rectangles. As it is well known, the Hausdorff Voronoi diagram construction can be viewed as propagating a wave from each rectangle with unit speed. The Hausdorff distance from an arbitrary point p to an axis-aligned rectangle R.C /, corresponding to the minimum inducing region of a 2-D cut C , is considered to better illustrate the wave propagation. The Hausdorff distance of dh .p; R.C // is achieved at one of the four corner points, v1 .C /; v2 .C /; v3 .C /; and v4 .C /, of R.C /. Thus, dh .p; R.C // D maxfd1 .p; v1 .C //; d1 .p; v2 .C //; d1 .p; v3 .C //, d1 .p; v4 .C ///g. To propagate a wave W .C / from R.C /, the initial wavefront @W .C / is the set of points whose Hausdorff distance to R.C / is the minimum. Notice that when R.C / has positive size (i.e., R.C / is not a point), the minimum Hausdorff distance is positive (i.e., maxfa; bg=2, where a; b are the length and width of R.C / respectively) and it is achieved when d1 .p; v1 .C //; d1 .p; v2 .C //; d1 .p; v3 .C // and d1 .p; v4 .C /// are equal. The wavefront then expands to points having larger Hausdorff distance to R.C /. Since the Hausdorff distance to R.C / is determined by the four corner points, an equivalent view is to propagate four distinct waves from the four corner points of R.C / with each being an L1 ball. Let B1 .C /; B2 .C /; B3 .C /, and B4 .C / be the

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z y

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four L1 balls. The common intersections of the four balls constitute the wave of R.C / (i.e., W .C / D \4iD1 Bi .C /). Initially, @W .C / is empty. Once the size of the four balls Bi .C / reaches the minimum Hausdorff distance to R.C /, their common intersection forms a segment sC located at the center of R.C / and parallel to the shorter side of R.C /. As Bi .C / grows, @W .C / becomes a rectangle. To visualize the whole growing process, the waves can be lifted to 3-D with time being the third dimension. Thus, the wavefront of R.C / becomes a 4-sided facet cone in the 3-D space and apexed at sC (i.e., the apex is not in the xy plane due to its positive minimum Hausdorff distance; see Fig. 5). Each facet of @W .C / forms a 45ı angle with the xy plane. For a 1-D cut C , its wavefront is slightly different. Let UR.C / be the rectangle of C . Since UR.C / is the union of an infinite number of inducing segments R.C /, the Hausdorff distance to an arbitrary point p is calculated differently. For a fixed inducing segment R.C / 2 UR.C /, let u1 .R.C // and u2 .R.C // be its two endpoints. The Hausdorff distance from R.C / to p is achieved at one of the two endpoints (i.e., dh .p; R.C // D maxfd.p; u1.R.C ///; d.p; u2.R.C ///g). Thus, the wavefront of R.C / is the common intersection of two L1 balls centered at the two endpoints, respectively, which is also a facet cone in 3-D space. The Hausdorff distance from p to UR.C / is the minimum distance from p to one of the inducing segments in UR.C /, that is, dh .p; UR.C // D

min

R.C /2UR.C /

dh .p; R.C //:

Thus, the wavefront of UR.C / is the union of an infinite number of wavefronts, which is still a facet cone with similar shape to the wavefront of a 2-D cut. The difference between the wavefront of UR.C / and that of a 2-D cut with exactly same-shaped R.C / is that their respective sC may orient differently and locate at different heights. Lemma 7 Let C be a 1-D or 2-D geometric minimal cut. At any moment, the wavefront of C is either empty or an axis-aligned rectangle. Furthermore, the

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wavefront in 3-D is a facet cone apexed at a segment and with each facet forming a 45ı angle with the xy plane. Proof Follows from the above discussion.



With the 3-D wavefronts of all cuts, the Hausdorff Voronoi diagram can be constructed by computing the vertical projection of the lower envelope of the set of 3-D wavefronts. Lemma 8 The Hausdorff Voronoi diagram can be obtained by projecting the lower envelope of the 3-D facet cones to the xy plane. Proof Follows from the definitions of the 3-D wavefronts and the Hausdorff Voronoi diagram. 

2.3.2

Plane Sweep Approach and Properties of 3-D Cones and Hausdorff Voronoi Diagram To efficiently construct the Hausdorff Voronoi diagram H VD.C/, the approach follows the spirit of Fortune’s plane sweep algorithm for points [15] and sweeps along the x axis direction a tilted plane Q in 3-D which is parallel to the y axis and forms a 45ı with the xy plane (see Fig. 6a). Q intersects the xy plane at a sweep line L parallel to the y axis. Since every facet of a 3-D facet cone forms a 45ı angle with the xy plane and apexed at either a horizontal or vertical segment, the intersection of Q and a cone @W .C / is either a V -shape curve (i.e., consisting of a 45ı ray and a 135ı ray on Q) or a U -shape curve (i.e., consisting of a 45ı ray, a segment parallel to L, and a 135ı ray; see Fig. 6b). When the cone is first encountered, it introduces either a V -shape curve or a U -shape curve to Q. When L (or Q) moves, the curve grows and its shape may change from a V -shape to a U -shape. Accordingly, the lower envelope (or beach line) of all those V - or U -shape curves forms a monotone polygonal curve, instead of parabolic arcs as in Fortune’s algorithm. With the beach line, one might think of directly applying Fortune’s algorithm to sweep the objects. However, due to the special properties of this problem, quite a few significant differences exist between the MGMC problem and the ordinary Voronoi diagram problem, which fail the Fortune’s algorithm. Below is a list of major differences. 1. When a cone is first encountered by Q, its corresponding initial V - or U -shape curve may not necessarily be part of the beach line. 2. The initial V - or U -shape curve may affect a number of curves in the beach line. 3. Once a V shape moves away from the beach line, it may re-appear in the beach line in a later time. The differences indicate that it is not sufficient to maintain only the beach line in order to extract all possible event points and produce the Voronoi diagram. More information of the arrangement of the V - and U -shape curves is needed. This seemingly suggests that an algorithm with running time of the order of the

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Fig. 6 (a) The intersections of 3-D plane Q and cones. (b) The Voronoi diagram with sweep line and beach line

complexity of the arrangement might be unavoidable. With several interesting observations and ideas, an output-sensitive plane sweep algorithm could be achieved. First, some basic properties of the Hausdorff Voronoi diagram of C are presented below. Definition 4 A 3-D facet cone @W .C / is a U cone (or V cone) if its apex segment sC is parallel to the y (or x) axis. For U cones, let @W .C / be any U cone with apex segment sC , and v1 and v2 be the two endpoints of sC . When the sweep plane Q first encounters @W .C /, it introduces a U -shape curve Cu to Q. Let rl , rr , and sm be the left and right rays and the middle segment of Cu , respectively. Initially, sm is the apex segment sC , and rl and rr are the two edges of facet cone. When Q (or L) moves, Cu grows and always maintains its U -shape. Lemma 9 Let @W .C /, Cu , rl , rr , and sm be defined as above. When Q moves in the direction of the x axis, Cu is always a U -shape curve. The two supporting lines of rl and rr remain the same on Q, and the two endpoints of sm (also the fixed points of rl and rr ) move upwards in unit speed along the two supporting lines. Proof Follows from the shape and orientation of a U cone.



For an arbitrary V cone @W .C 0 /, let sC 0 be its apex segment and v01 and v02 be its two endpoints (or left and right endpoints). When Q first touches @W .C 0 / at v01 ,

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Fig. 7 (a) The V -shape curve changes to U -shape curve. (b) The V cone is hidden by another V cone

it generates a V -shape curve Cv0 . Cv0 remains a V -shape curve before encountering v02 . After that, Cv0 becomes a U -shape curve (see Fig. 7a). Lemma 10 Let rl and rr be the two rays of Cv0 , and sm be the middle segment of the U -shape curve Cv0 after Q visits v02 . During the whole sweeping process, the supporting lines of rl and rr are fixed lines on Q. Cv0 remains the same V-shape curve on Q before encountering v02 . sm moves upwards in unit speed along the supporting lines of rl and rr after Q encounters v02 . Proof Follows from the shape and orientation of a V cone.



As mentioned earlier, the apex segment of each 3-D cone is located at different height (i.e., its minimum Hausdorff distance). The heights and shapes of the 3-D cones are the main reasons which cause the three differences in the MGMC problem. For example, due to the existence of height in a 3-D cone, the initial curve created by a newly encountered cone may be above the beach line (i.e., Difference (1)). In addition due to the different size of the initial curve (unlike the ordinary Voronoi diagram in which the initial curve is a vertical ray), it may intersect a number of segments or rays of the beach line (i.e., Difference (2)). More importantly, due to the existence of U - and V -shape curves, a V -shape curve which is not part of the beach line could become part of the beach line in a later time (i.e., Difference (3)). It is shown in the following lemma. Lemma 11 Let @W .C1 / be either a U or V cone and @W .C2 / be a V cone with its left endpoint v1 of sC2 being inside of @W .C1 / and its right endpoint v2 being outside of @W .C1 /. If @W .C2 / is not entirely contained by the union [Ci 2CICi ¤C2 @W .Ci /, the V -shape curve C2 introduced by @W .C2 / will be hidden by the beach line at the

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beginning and will become part of the beach line later. This is the only case in which a hidden U - or V -shape curve can appear in the beach line. Proof To simplify the proof, @W .C1 / and @W .C2 / are assumed to be the only two cones in C (see Fig. 7b). The multiple cones case can be proved similarly by induction. By the definition of the two 3-D cones, it is known that Q first encounters @W .C1 / and generates a curve C1 on Q. C1 is the only curve in the beach line. When v1 of @W .C2 / is first encountered by Q, it introduces a V -shape curve C2 on Q. Since v1 is inside @W .C1 / and every facet of the two cones forms a 45ı angle with the xy plane, the two rays of C2 will be inside the region defined by the two rays of C1 . This means C2 will not be part of the beach line (or lower envelope). Since @W .C2 / is not entirely inside @W .C1 /, v2 must be outside of @W .C1 /. Let vi be the intersection point of sC2 and the wall of @W .C1 /. When Q sweeps through vi , the apex point of the V -shape curve C2 will intersect the U -shape curve C1 (note that at this moment C1 can only be a U -shape curve even if @W .C1 / is a V cone), since C1 is the intersection of Q and @W .C1 / and vi is the intersection of @W .C1 / and sC2 . Thus, C2 becomes part of the beach line. To show that this is the only case where a hidden curve could appear in the beach line, first it is noted that the apex segment sC2 of @W .C2 / cannot be completely outside of @W .C1 /, otherwise the initial curve of C2 will be part of the beach line. Second, @W .C1 / cannot be a U cone. If this is the case, then sC2 is either partially or entirely inside @W .C1 /. For the first case, the middle segment sm of the initial U shape curve C2 will intersect one of the two rays of C1 , which makes C2 be part of the beach line. For the second case, the initial U -shape curve C2 will be hidden by C1 . When Q moves, only the middle segment sm of C2 moves upwards in unit speed along the two rays of C2 (by Lemma 9). If @W .C1 / is a U cone, then by Lemma 9, the middle segment of C1 will also move upwards in unit speed. Thus, it will never catch up sm . Therefore C2 will never be part of the beach line. Similarly the same for the case @W .C1 / is a V cone can be proved. Thus, @W .C2 / has to be a V cone. In this case, if sC2 is completely inside of @W .C1 /, then by the same argument, it can be shown that C2 will never be part of the beach line. Thus, the lemma follows.  In the above lemma, the point vi indicates that when Q sweeps it, the beach line is having a topological structure change. Thus, vi needs to be an event point for the plane sweep algorithm. However, since vi is the intersection point of an apex segment and a 3-D cone, it has to be computed on the fly. This indicates that in the MGMC problem there is a new type of event points. The ideas for constructing the Hausdorff Voronoi diagram H VD.C/ are discussed below. First, the bisector of two rectangles (or cuts) is considered. Let C1 and C2 be two axis-aligned rectangles in C. The bisector of C1 and C2 is a segment with two rays as shown in Fig. 8. Each bisector contributes two vertices to the Hausdorff Voronoi diagram. Hence, the Hausdorff Voronoi diagram consists of two types of vertices: (a) The intersection points of the bisectors; and (b) The vertices of the bisectors.

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Fig. 8 Bisector is composed by a segment with two half-lines

Lemma 12 Let C be a set of N rectangles. The edges of H VD.C/ are either segments or rays, and the vertices of the H VD.C/ are either the vertices of bisectors or the intersections of bisectors. Proof Follows from the above discussion.



To obtain a plane sweep algorithm, it is needed to design data structures to maintain the beach line and the event points. In the MGMC problem, the beach line is the lower envelope of the set of V - and U -shape curves and is a y-monotone polygonal curve. For non-disjoint 3-D cones, the complexity of the beach line may not be linear in the number of the rectangles in C. Figure 9b shows a newly generated U -shape curve intersecting the beach line a number of times and contributing multiple edges to it. Consequently, the complexity of H VD.C/ is not linear. The following lemma is a straightforward adaptation of Theorem 1 in [33] for the L1 metric. Lemma 13 The size of the L1 Hausdorff Voronoi diagram of N rectangles is O.N C M 0 /, where M 0 is the number of intersecting pairs of rectangles. In the worst case, the bound is tight.

2.3.3 Events For event points, it is needed to detect all events that cause the beach line to have topological structure changes. More specifically, it is necessary to identify all the moments when a U - or V -shape curve is inserted or deleted from the beach line. There are two ways by which a curve could appear in the beach line: (A) A newly generated U - or V -shape curve becomes part of the beach line. (B) A hidden V -shape curve appears in the beach line. There are also two ways for a curve or a portion of a curve to disappear from the beach line: (C) A curve (or part of the curve) becomes hidden by a newly generated curve. (D) A U -shape curve (or part of the U -shape curve) moves out of the beach line.

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Fig. 9 The beach line L is intersected by a new U cone (dashed line) or V cone (dotted line)

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Fig. 10 (a) The bottom segment of U -shape curve C1 reached Vi . (b) The hidden V -shape curve C2 appeared on the beach line L

For (A), it is known that this is caused by the sweep plane Q encountering a new 3-D cone @W .C /. Such events can be detected in advance and are called site events. A site event, however, does not necessarily lead to a topological structure change to the beach line, since the new curve C can be hidden by the beach line. If @W .C / is a U cone, the two endpoints of sC are encountered by Q at the same time and either of them can be treated as a site event. If the corresponding U -shape curve C is not hidden, it may intersect the beach line multiple times as shown in Fig. 9. In this case, it is necessary to update the beach line for each breakpoint introduced by C . Consequently, the U -shape curve C will be partitioned into multiple pieces. Each piece is either a part of the beach line (called unhidden portion of C ) or hidden by other U - or V -shape curves in the beach line (called a hidden portion of C ). If @W .C / is a V cone, then the left endpoint v1 of sC is encountered first and can be viewed as a site event. When Q sweeps the right endpoint v2 of sC , the C changes from a V -shape curve to a U -shape curve. To distinguish from the site event, v2 is called as a U event. For unhidden V -shape curve C , it intersects the beach line at most twice (see Fig. 9). For (B), it occurs when an unhidden portion of the bottom segment sm of a U -shape curve C1 moves upwards and encounters the apex point of a hidden V shape curve C2 . This kind of events is called V events (see Figs. 10 and 7b). The V events are not known in advance and need to be computed by using the saved information in the data structures. Note that when sm moves upwards, its hidden portions may also encounter the apex point of some hidden V -shape curve. In this case, it is not viewed as an event since the beach line has no topological structure

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Fig. 11 Dominating regions of unhidden portions of U -shape curves

change, thus generating no new Voronoi vertex or edge. To distinguish the two cases, each unhidden portion of a U -shape curve is associated with a region, called dominating region (see Fig. 11), which is the region swept by the unhidden portion when it moves upwards. The dominating regions could have a few different shapes (see the shaded regions in Fig. 11), with each of them bounded by zero, one, or two 45ı rays, zero, one, or two 135ı rays, and an unhidden portion of the bottom segment sm of some U -shape curve. Clearly, for a hidden V -shape curve to cause a V event, its apex point has to fall in a dominating region of an unhidden portion of a U -shape curve. Thus, to capture all V events, it is only needed to focus on those hidden V -shape curves whose apex points fall in the dominating regions. For (C), it is obviously caused by a site event and thus can be detected in advance. For (D), the disappearing U -shape curve C (or its unhidden portion) is caused by the upward movement of the bottom segment of C . It involves three curves, C and its immediate left and right neighbors C 0 and C 00 in the beach line. Since this event is similar to the circle event in the computation of the standard Voronoi diagram, it is also called here a circle event. The circle events cannot be detected in advance and have to be computed on the fly. Thus, there are in total four types of events, site events, circle events, U events, and V events. The ideas on how to handle these events are discussed in the next section.

2.3.4 Data Structures and Events Handling To construct H VD.C/, doubly connected edge lists are used to store H VD.C/. Two data structures for the plane sweep algorithm are also needed: an event queue and a sweep plane status structure representing the beach line. The status structure for the beach line consists of three balanced binary search trees T , T=4 , and T3=4 . T stores the y-monotone polygonal curve of the beach line. Each part of the curve corresponds to a V -shape curve or an unhidden portion of a U -shape curve. The leaves of T correspond to the V -shape curves and the unhidden portions of the U -shape curves on the beach line sorted by their y coordinates. Each leaf also stores location information of the corresponding 3-D cone. The internal nodes of T adjacent to the leaves represent the breakpoints (i.e., the intersection of a pair of U or V curves) on the beach line. A breakpoint is stored at an internal node by an ordered tuple of curves < Ci ; Cj >, where Ci is the left curve of the breakpoint and Cj is the right curve of the breakpoint. T=4 is used to maintain the

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orders (along the norm direction) of the 45ı rays of all U - or V -shape curves which appear in the beach line. Similarly, T3=4 maintains the orders of the 135ı rays of U - or V -shape curves which appear in the beach line. Each leaf node in T=4 or T3=4 represents a 45ı or 135ı ray, and each ray corresponds to a U - or V -shape curve (represented by a leaf node in T ) in the beach line. For each leaf node of T=4 and T3=4 , a pointer pointing to the corresponding leaf node in T is maintained. In this case, by doing binary search on T=4 and T3=4 and the pointers between the trees, the positions in the beach line for the apex points of each newly encountered cone at a site event could be located in O.log N / time, and the beach line is updated in O.k log N / time, where k is the number of breakpoints destroyed and created after inserting the newly encountered U - or V -shape curve into the beach line. The event queue Q is implemented by a priority queue, where the priority of an event is the x coordinate of the corresponding event point. If two points have the same x-coordinate, the one with larger y coordinate has the priority. All the site events and U -events are known in advance and are stored in Q. The main challenge is to detect the V events. For a V event, it is known that it occurs when the apex point of a hidden V-shape curve C2 appears in the beach line. To detect such events, it is necessary to store in the data structure the information of all hidden V -shape curves. This could potentially require us to maintain the whole arrangement of all curves on Q and therefore results in unnecessarily high computational cost. To efficiently detect all possible V events, the main idea is not to maintain the arrangement, but rather to use the properties of V events to convert the problem into a query problem in 3-D. To achieve this, first it is easy to see that a V event is always caused by the upward movement of an unhidden portion of the bottom segment sm of some U -shape curve C1 and occurs when sm coincides with the apex point v of a hidden V -shape curve C2 (by Lemma 11). Thus, in order to detect all possible V events, it is needed to solve the following two difficulties: (1) Identify the next V event associated with each unhidden portion of a U -shape curve, and (2) for each newly encountered hidden V -shape curve (in a site event), find the unhidden portion of a U -shape curve which will later cause a V event involving this V -shape curve. There is another one (Difficulty (3)) related to the two difficulties: How to find the exact boundary of the dominating region for a given unhidden portion c of a U -shape curve. First Difficulty (1) is considered. For this difficulty, it is known that the next V event associated with an unhidden portion c of a U -shape curve C1 is the hidden V-shape curve C2 whose apex point v lies inside the dominating region of c and is the closest to the bottom segment sm of C1 . However, as it is noticed in the last section, the dominating region could have various shapes which seemingly suggest that it is costly to find the next V event even if the dominating region is known. To overcome this difficulty, first it is observed that the dominating region is bounded by 45ı and 135ı rays and the bottom segment. From Lemmas 9 and 10, it is known that the rays of any U or V curve have fixed directions (e.g., forming 45ı and 135ı angles with the y axis) and their supporting lines remain the same during the whole sweeping process. This suggests that the dominating regions can be orthogonalized in 3-D space. The three dimensions of the new

Map of Geometric Minimal Cuts with Applications Fig. 12 A dominating region is converted to a 3-D box in the orthogonalized 3-D space for MD

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space are the orthogonal directions of the 45ı and 135ı lines in the sweep plane Q and the orthogonal direction of the bottom segment (or the y axis). In this way, each dominating region is converted into a (possibly unbounded) box in 3-D (see Fig. 12). To efficiently find the next V event in the orthogonalized dominating region, the apex points of all V -shape curves are processed into a 3-D dynamic range search tree data structure MD [29, 31]. In MD, the apex point of each hidden V -shape curve is mapped into a 3-D point. Thus, for a particularly dominating region R, its orthogonalized box can be used as the query range to find the closest apex point (among all hidden V -shape curves whose apex points fall in R) to the unhidden portion of the bottom segment of a U -shape curve in O.log2 N log log N / time [29]. To efficiently maintain the MD, it can be built in advance for all possible V -shape curves. In each node p of the MD tree, a mark is stored to indicate whether there is any active V -shape curve in the subtree rooted at p. A V -shape curve C is active if its corresponding 3-D cone has already been encountered by the sweep plane Q, and C has not yet changed to a U -shape curve due to a U event. In this way, it is only needed to change the marks when the status of a V -shape curve changes and therefore avoid complicate updating (such as tree rotation) to the MD. To make use of MD, it is necessary to identify all scenarios in which it is needed to either update MD or query MD for detecting potential V events. First, it is noticed that a site event or a U event could introduce (a) a new U -shape curve C to the beach line and generate a set of unhidden portions of C and (b) a hidden V -shape curve C 0 . Thus, for (a), in each such event, a 3-D range query in MD is performed for each unhidden portion ci to find the closest hidden V -shape curve to ci in its dominating region and insert a V event into the event queue Q. For (b), it is necessary to find out the exact dominating region R which contains the apex point of the newly encountered hidden V -shape curve C 0 (i.e., Difficulty (2)) and then insert the hidden V event into MD, since the V event of C 0 might be the new next V event of the unhidden portion c of R. (The idea for this challenging problem will be discussed later.) Second, after a V event, it is also necessary to perform a 3-D range query in MD to find the closest hidden V -shape curve to the new U shape curve converted from the V -shape curve of the V event. Third, if an unhidden portion c of a U -shape curve C disappears from the beach line (e.g., a circle event), its associated V event is deleted from Q, since c will never appear in the beach line again by Lemma 11.

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p

Fig. 13 Finding the dominating region of p

L

L  q L

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The ideas for Difficulty (2) (i.e., finding the unhidden portion of the dominating region containing the apex point of a hidden V -shape curve in a site event) are discussed below. Let p be the apex point of the hidden V -shape curve. As shown in Fig. 13, finding the four rays (two 45ı rays and two 135ı rays) bounding p does not necessarily give us the correct dominating region. This is because the rays bounding p could be stopped by the rays bounding the actual dominating region. For example, L is stopped by L00 at point q. Thus, a ray inside a dominating region may not be a ray bounding that dominating region. Definition 5 A ray (or a portion of a ray) is active if it bounds some dominating region and inactive otherwise. Let l be any ray in the beach line. If starting from the bottom (i.e., the endpoint) of l and walking along it, l is active until it is stopped by some other ray and thus becomes inactive. It is easy to see that once a ray becomes inactive, it will never be active again. In Fig. 13, L is active until stopped by L00 and will no longer be active. A ray has two sides and it may not be active on both sides simultaneously. Also one side of a ray may bound more than one dominating region (L00 bounds the dominating regions generated by U -shape curves with bottom segments a and c). It is easy to see that for any ray, there is always one side bounding at most one dominating region. Otherwise, there will be an intersection between two dominating regions, thus contradicting the definition of dominating regions. With these observations, to find out the actual dominating region of p, first the four rays bounding it are found by searching the T=4 and T3=4 trees. Further, it is needed to find out whether or not these rays are active on the sides containing p. To determine this, it is needed to know whether each of the four rays has been stopped by other rays. If the ray L which stops these rays can be found out, then it means that p belongs to the dominating region bounded by L. This means that for a given ray, it is needed to have a way to efficiently determine which ray stops it. To achieve this, the T=4 and T3=4 trees are augmented. For each node of the trees, the z-coordinate of the lowest bottom (segment) of all rays in the subtree rooted at that node is stored. The z-coordinate of the bottom segment of a V -shape curve is the z-coordinate of its apex point minus the length of bottom segment of the

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3-D V cone. With the augmented information, the ray L00 which stops a ray L can be searched in the two trees in O.log N / time. To better understand the idea, consider the example in Fig. 13. Let L be a 45ı ray with its endpoint v. A ray L00 which stops L is a 135ı ray if it exists. To find L00 , another 135ı ray L0 with endpoint v is created. First the position of L0 is searched in T3=4 . Then it is only needed to find the ray between L0 and p (in the direction of L) with (1) lower z-coordinate than v and (2) the closet z-coordinate. This can be done by following the searching path of L0 in T3=4 upwards until finding a node satisfying the two conditions and then moving downwards to locate the ray L00 . In Fig. 13, the dominating region with bottom segment b dominates p. Lemma 14 It takes O.log N / time to find out the exact dominating region of an unhidden portion c for the apex point p of the hidden V -shape curve C 0 generated by the site event. Proof Follows from the above discussion.



It is not difficult to see that the augmented information can be maintained during the whole sweep process within the same time bound. With the augmented information, the exact boundary of a dominating region R for an unhidden portion c of a U -shape curve (i.e., Difficulty (3)) can also be found in O.log N / time following the same idea (i.e., finding the rays stopping the bounding rays of R). Circle events can be handled in a way similar to the standard Voronoi diagram. More specifically, every new triple of consecutive U - or V -shape curves that appear in the beach line is checked. If such a new triple has converging breakpoints, the event is inserted into the event queue Q. Furthermore, for all disappearing triples, the corresponding event is de-queued from Q if it has been inserted.

2.3.5 Algorithm and Analysis The entire plane sweep algorithm is described below. The main steps of the algorithm are as follows. Algorithm HAUSDORFF-VORONOI-DIAGRAM.C/ Input. A set C of axis-aligned rectangles (or geometric minimal cuts) in the plane. Output. The Hausdorff Voronoi diagram in a doubly connected edge list D. 1. Initialize the event queue Q with all site events and U events; initialize empty sweep plane status structures T , T=4 , and T3=4 , and an empty doubly-connected edge list D; initialize a 3-D range search tree MD for all possible V -shape curves with all nodes marked as inactive. 2. while Q is not empty 3. do Remove an event with the smallest x-coordinate from Q. 4. if the event is a site event, then HANDLE-SITE-EVENT. 5. if the event is a circle event, then HANDLE-CIRCLE-EVENT. 6. if the event is a U -event, then HANDLE-U-EVENT. 7. else the event is a V event; HANDLE-V-EVENT.

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HANDLE-SITE-EVENT 1. Let C be the new curve. If the status structure is empty, insert C into it and mark the apex point in MD as active if it is a V cone. Otherwise, continue with steps 2–8. 2. If C is a V -shape curve, mark its apex point in MD as active and continue with steps 3–5. 3. Locate the position of the apex point of C in the beach line by searching T ,T=4 , and T3=4 . 4. If C is not hidden, insert C to the beach line by updating T ,T=4 , and T3=4 . This includes inserting C into the beach line, computing new breakpoints, inserting possible circle events into Q, and removing all curves hidden by C from the beach line. If an unhidden portion of a U -shape curve is removed, delete its corresponding V event from Q. If an unhidden portion of a U -shape curve is partially blocked by C , search for its closest hidden V -shape curve in the reduced dominating region if necessary. 5. Else if (the apex of) C is inside the dominating region an unhidden portion c of a U -shape curve C 0 , update the associated V event of c if needed. 6. Else if C is a U -shape curve, continue with steps 7–8. 7. Locate the position of C in the beach line by searching T , T=4 , and T3=4 . 8. If C is not fully hidden, insert C into the beach line by updating T , T=4 , and T3=4 . This includes computing possibly multiple breakpoints and the corresponding circle events for all its unhidden portions, removing blocked curves (similarly to Step 4), and finding the possible V event for each unhidden portion of C and partially blocked unhidden portion. HANDLE-CIRCLE-EVENT 1. Update the beach line by updating T , T=4 , and T3=4 . Delete unnecessary circle events from Q involving the part disappearing from the beach line. If an unhidden portion of a U -shape curve moves out of the beach line, delete its associated V event. 2. Add the vertex to D. Two new breakpoints of the beach line will be traced out. 3. Check the new triples involving the left or right neighbor of the disappearing part and insert the corresponding circle event into Q, if necessary. HANDLE-U-EVENT 1. If the V -shape curve C introduced by the V cone appeared on the beach line, the corresponding part of the beach line is changed from a V -shape curve to a U -shape curve. Update T , T=4 , and T3=4 , and add vertex to D. Also find the possible V event for C by querying MD, and insert it into Q. 2. Mark the node in MD containing C as inactive, and update its ancestors if needed. 3. Detect the circle events and insert them into Q if necessary. HANDLE-V-EVENT 1. Update T , T=4 , and T3=4 by inserting the V -shape curve C into the beach line. Create new breakpoints, detect the possible circle event, and insert them into Q. 2. For the corresponding unhidden portion c of a U -shape curve C 0 , break it into two unhidden portions, c 0 and c 00 , and find possible new V event for c 0 and c 00 , respectively by querying MD.

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Theorem 4 The L1 Hausdorff Voronoi diagram H VD.C/ of a set C of geometric minimal cuts (or axis-aligned rectangles) can be constructed by a plane sweep algorithm in O..N C K/ log2 N log log N / time, where N D jCj and K is the complexity of the Hausdorff Voronoi diagram.

Proof The discussion of the algorithm shows that H VD.C / can be constructed by a plane sweep algorithm. Thus, the proof focuses on the time complexity. First, site events are considered. Let C be a newly encountered U - or V -shape curve. If C is a V -shape curve, it is either part of the wavefront or a hidden V -shape curve. For the former case, the computation cost includes inserting C into the beach line and updating MD and D. The cost for these is O.log2 N log log N C k log N /, where k is the total number of breakpoints hidden by C . Since each breakpoint corresponds to a Voronoi vertex in H VD.C /, the cost of O.k log N / can be charged to the breakpoints (and their corresponding Voronoi edges). Thus, each will be charged a cost of O.log N /. The cost of .log2 N log log N / can be charged to each V -shape curve. For the latter case, if C is not in the dominating region of any unhidden portion of a U -shape curve, the only cost is O.log2 N log log N /, which can be charged to C . If C is inside the dominating region of some unhidden portion c of a U -shape curve which can be checked in O.log N / time, it is also needed to check whether C is the closest V -shape curve to c, and this can be done in O.1/ time. Thus, in the latter case, C is charged a cost of O.log2 N log log N /. If C is a U -shape curve, it could be fully hidden by the beach line, and thus, will never appear in the beach line in a later time. In this case, the total cost is O.log N /, which can be charged to C . If C appears in the beach line and contributes some unhidden portions, then it is needed to update the sweep plane status structure, which takes O.k log N / time, and find the closest V -shape curve for each unhidden portion c. The cost of O.k log N / can be evenly charged to all hidden and newly created breakpoints. Thus, each of them will be charged a cost of O.log N /. The closest V -shape curve to each unhidden portion c can be found by a query to MD, which takes O.log2 N log log N / time and can be charged to the breakpoint bounding c. In a site event, up to two unhidden portions can be partially hidden by the newly encountered U - or V -shape curve C . In this case, each of them may need to find a new closest hidden V -shape curve in its reduced dominating region. For this, the cost of O.log2 N log log N / is charged to the new breakpoint created by the two rays of C and the two unhidden portions. Thus, after processing all site events, each V -shape curve will be charged a cost of .log2 N log log N /, and each U -shape curve will be charged a cost of O.log N /. Some breakpoints will be charged a cost of O.log2 N log log N /. Clearly, each circle event can be handled in O.log N /-time, and the total number of such events is bounded by O.K/. Thus, the total cost for all circle events is O.K log N /. For U events, there are only O.N / such events. Each event takes O.log2 N log log N / time to update MD, and find the closest hidden V -shape curve to the new U -shape curve. Again, all the cost is charged to the V -shape curve.

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For V events, it is clear from the algorithm, that each such event takes O.log2 N log log N / time. To bound the total cost of all V events, it is needed to bound their total number. For this, it is noticed that at any moment, (1) Each unhidden portion of a U -shape curve is associated with only one hidden V -shape curve; and (2) The association of a hidden V -shape curve C and an unhidden portion c of a U -shape curve can change for only two reasons: (a) The dominating region of c changes (in a site event), and (b) a new hidden V -shape curve C 0 is activated and C 0 is closer to c than C . For (a), it means that part of c is hidden by other U - or V -shape curve C 00 and the cost of de-association can be charged to the edges or vertices introduced by C 00 . For (b), the cost of de-association can be charged to C 0 . Each V -shape will be charged no more than once. This means that computation cost of each V event can be charged to the Voronoi edge or vertex corresponding to the two breakpoints bounding the unhidden portion. Each Voronoi vertex and edge is charged a constant times with a total cost of O.log2 N log log N /. Thus, the total cost for all V events is O.K log2 N log log N /. Putting all together, the total cost is O..N C K/ log2 N log log N /. Thus, the theorem follows. 

3

The MGMC Problem via Higher-Order Voronoi Diagrams

In this section, we present the iterative approach of [34] to the MGMC problem via higher-order Voronoi diagrams. A more general version of the MGMC problem is addressed in this section, where the embedded subgraph H corresponds to arbitrary polygonal shapes. The polygonal shapes are not assumed rectilinear, but they may consist of edges in arbitrary orientation. A polygonal shape is reduced to a collection of (additively) weighted line segments by means of its L1 medial axis. The approach is complementary to the one presented in Sect. 2 and it does not precompute any minimal cuts. On the contrary, it discovers those geometric minimal cuts that are guaranteed to participate in the final map, on the fly, by iteratively constructing variants of higher-order Voronoi diagrams. At the end of the iteration, the derived subdivision is the Hausdorff Voronoi diagram of all geometric cuts, that is, the solution to the MGMC problem. Definitions, results, and most figures in this section are reproduced from [34].

3.1

The MGMC Problem in a VLSI Layout

In a VLSI setting, the geometric min-cut problem is as follows [34]: We are given a collection of geometric graphs that have portions embedded on a plane (a VLSI layer) A. The embedded portions on plane A are vulnerable to random defects that may form cuts on the given graphs. A defect of size r is a disk of radius r. The size of a geometric cut C at a given point t is the size of the smallest defect centered at t that overlaps all elements in C , as opposed to the number of edges in C in the classic min-cut problem. Compute, for every point t on the vulnerable plane A, the

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Fig. 14 (a) A net N spanning over two layers. (b) Dark defects create opens while transparent defects cause no faults

size of the minimum defect causing a geometric cut at t. The size of the minimum geometric cut at a point t is the critical radius for open faults at t. The resulting subdivision is referred to as the opens Voronoi diagram, and it corresponds to the map of the MGMC problem. Figure 14a, reproduced from [34], illustrates an example of a simple VLSI net N spanning over two metal layers, say M1 and M2, where M2 is illustrated shaded. The two contacts illustrated as black squares are designated as terminal shapes. A VLSI net remains functional as long as terminal shapes remain interconnected. In Fig.14b, defects that create open faults are illustrated as dark squares, and defects that cause no fault are illustrated hollow in dashed lines. Hollow defects do break wires of layer M1; however, they do not create opens as no terminals get disconnected. A compact graph representation for a VLSI net N , denoted G.N /, can be obtained as follows: Each graph node of G.N / represents a connected component of a net N on a conducting layer, and two nodes are connected by an edge if there exists at least one contact or via connecting the respective components of N . Some of the shapes constituting net N are designated as terminal representing the entities that the net must interconnect. A node containing terminal shapes is designated as a terminal node. Given a layer A of interest, the extended graph of N on A, G.N; A/, is derived from G.N / by expanding the components of N on layer A by their medial axis. Contact points between neighboring layers are approximated as points on the medial axis of the corresponding shape, called via points. Any via point or any portion of the medial axis corresponding to terminal shapes is identified as terminal. Figure 15a illustrates G.N; A/, where A D M1, for the net of Fig. 14; terminal points are indicated by hollow circles; dashed lines represent the original M1 polygon and they are not part of G.N; A/. G.N; A/ can be cleaned up from any trivial parts [34] by computing biconnected components, bridges, and articulation points. In the following, we assume that G.N; A/ is free from any trivial parts, as shown in Fig. 15b. The collection of G.N; A/ for all nets N involved on a layer A is the graph

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Fig. 15 The net graph of Fig. 14 before (a) and after (b) clean-up of trivial parts

of the MGMC problem, and its portion on layer A is the planar embedded subgraph H . For more details on the derivation of G.N; A/, see [34].

3.2

Review of Concepts on L1 Voronoi Diagrams

The Voronoi diagram of a set of polygonal sites in the plane is a partitioning of the plane into regions, one for each site, called Voronoi regions, such that the Voronoi region of a site s is the locus of points closer to s than to any other site. The Voronoi region of s is denoted as reg.s/ and s is called the owner of reg.s/. In the interior of a simple polygon, the Voronoi diagram is known as the medial axis of the polygon (a minor difference in the definition is ignored (see [26])). The boundary between two Voronoi regions is called a Voronoi edge and it consists of portions of bisectors between the owners of the neighboring regions. The bisector of two polygonal objects (such as points, segments, polygons) is the locus of points equidistant from the two objects. A point where three or more Voronoi edges meet is called a Voronoi vertex. The L1 distance between two points p D .xp ; yp / and q D .xq ; yq / is d.p; q/ D max fjxp  xq j; jyp  yq jg. In the presence of additive weights, the (weighted) distance between p and q is dw .p; q/ D d.p; q/ C w.p/ C w.q/, where w.p/ and w.q/ denote the weights of points p and q, respectively. In case of a weighted line l, the (weighted) distance between a point t and l is dw .t; l/ D minfd.t; q/ C w.q/; 8q 2 lg. The (weighted) bisector between two polygonal elements si and sj is b.si ; sj / D fy j dw .si ; y/ D dw .sj ; y/g. In L1 , any Voronoi edge or vertex can be treated as an additively weighted line segment. For brevity and in order to differentiate from ordinary line segments, the term core segment, more generally core element, is used to denote any portion of interest along an L1 Voronoi edge or vertex. The term standard-45ıs is used

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Fig. 16 The regions of influence of the core elements of a core segment: two endpoints and an open line segment. (a) an axis-parallel core segment, (b) a non-axis-parallel core segment

to refer to Voronoi edges of slope ˙1 that correspond to bisectors of axis-parallel lines. Figure 16 illustrates examples of core segments. Let s be a core segment induced by the polygonal elements el ; er . Every point p along a core segment s is weighted with w.p/ D d.p; el / D d.p; er /, where el ; er are the polygonal elements inducing s. The 45ı rays3 emanating from the endpoints of s partition the plane into the regions of influence of either the open core segment portion or the core endpoints. In Fig.16, the L1 distance is indicated by straight-line arrows emanating from various points t. In the north and south (resp. east and west) regions, the L1 distance simplifies to a vertical (resp. horizontal) distance. In the region of influence of a non-axis-parallel core segment, it is measured by the side of a square touching ei as shown in Fig. 16b. In the region of influence of a core point p, distance is measured in the ordinary weighted sense, that is, for any point t, dw .t; p/ D d.t; p/Cw.p/. In the region of influence of an open core segment s, distance, in essence, is measured according to the farthest polygonal element defining s, that is, dw .t; s/ D d.t; ei /, where ei is the polygonal element at the opposite side of s than t; see, for example, the small arrows in Fig. 16. In L1 , this is equivalent to the ordinary weighted distance between t and s. For more details see [34]. The (weighted) bisector between two core elements is defined in the ordinary way, always taking the weights of the core elements into consideration. Similarly, the (weighted) Voronoi diagram of a set of core elements is defined as above,

3

A 45ı ray is a ray of slope ˙1.

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Fig. 17 The L1 farthest Voronoi diagram of axis parallel segments t

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with the difference that distance between a point t and a core element s is always measured in an additive weighted sense, dw .t; s/. The (weighted) Voronoi diagram of core medial axis segments was first introduced in [32] as a solution to the critical area computation problem for a simpler notion of an open, called break, that was based solely on geometric information. For Manhattan geometries, core segments are simple (additively weighted) axis-parallel line segments and points. The farthest Voronoi diagram of a set of polygonal sites is a partitioning of the plane into regions, such that the farthest Voronoi region of a site s is the locus of points farther away from s than from any other site. For typical cases (e.g., points, line segments), it is a tree-like structure consisting only of unbounded regions (see, e.g., [4, 6]). In the L1 metric, when sites are points or axis-parallel segments, its structure is particularly simple, consisting of exactly four regions as shown in Fig. 17. In each region, the L1 distance to the farthest element is measured as the vertical or horizontal distance to an axis-parallel line marked by t; b; l; r, where t (resp. b) is the horizontal line through the topmost (resp. bottommost) lower (resp. upper) endpoint of all core segments and l (resp. r) is the vertical line through the leftmost (resp. rightmost) right (resp. left) endpoint of all core segments. In Fig. 17, the thin arrows indicate the farthest L1 distance of selected points.

3.3

Definitions and Problem Formulation

Let core.N; A/ denote the collection of all core elements of a net N on a layer A, that is, the collection of all medial axis vertices and edges of G.N; A/ on A that are of interest. For simplicity, all standard 45ı edges are excluded from core.N; A/. The union of core.N; A/ for all nets N on layer A is denoted as core.A/. Core segments and points in core.A/ represent all wire segments vulnerable to defects on layer A and correspond to weighted line segments. Core segments are assumed to consist of three distinct core elements: two endpoints and an open line segment. Open faults are determined based on the connectivity information of G.N; A/ and the geometry information of core.N; A/.

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Fig. 18 Generators for strictly minimal opens

Definition 6 A minimal open is a defect D of minimal size that breaks a net N , that is, if D is shrunk by  > 0, then D no longer breaks N . An open is any defect that entirely overlaps a minimal open. A minimal open is called strictly minimal if it contains no other open in its interior. Definition 7 The center point of an open D, is called a generator point for D and it is weighted with the radius of D. The generator of a strictly minimal open is called critical. A segment formed as the union of generator points is called a generator segment or simply a generator. In Fig. 14, strictly minimal opens are illustrated by dark shaded disks, other than the original via and contact shapes. Figure 18 illustrates the generators for strictly minimal opens for the net of Fig. 14, thickened; the shaded squares indicate strictly minimal opens. For brevity, we say that a defect D overlaps a core element c 2 core.A/, but we mean that D overlaps the entire width of the wire segment induced by c. Definition 8 A cut for a net N is a collection C of core elements, C  core.N; A/, such that G.N; A/nC is disconnected leaving nontrivial articulation or terminal points in at least two different sides. A cut C is called minimal if C nfcg is not a cut for any element c 2 C . A defect of minimal size that overlaps all elements of cut C is called a cut-inducing defect. The centerpoint p of a cut-inducing defect that encloses no other defect in its interior is called a generator point for cut C . If, in addition, the cut-inducing defect is a strictly minimal open, then p is called critical. The collection of all generator points of a cut C is referred to as the generator(s) of C .

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The generator of a cut C can consist of critical and noncritical portions. Critical portions correspond to generators of strictly minimal opens. Noncritical portions correspond to centers of cut-inducing disks that, in addition to overlapping C , may also overlap some additional cut on a layer A, and thus, although they break C , they do not correspond to strictly minimal opens. Definition 9 Generators of minimal cuts that consist of a single core element are called first-order generators. Generators of minimal cuts that consist of more than one core element are called higher-order generators. The set of all critical generators on layer A is denoted as G.A/. In Fig. 18 first-order generators are illustrated as the thickened core segments in the interior of polygons; the vertical thick segment in the exterior of polygons is a higher-order generator that involves a pair of core elements. Given G.N; A/, we can detect biconnected components,4 bridges, and articulation points5 using depthfirst search (DFS) as described in [21, 42]. By definition, we have the following property. Lemma 15 The set of first-order generators on a layer A, denoted as G1 .A/, consists of all the bridges, terminal edges, articulation points, and terminal points of G.N; A/ \ core.N; A/, for all nets N . All first-order generators are critical. The generator of a minimal cut C that consists of more than one core element must be a subset of the L1 farthest Voronoi diagram of C , derived by ignoring the standard-45ı edges of the diagram. For Manhattan geometries, the generator of any cut is always a single axis-parallel segment (that can degenerate to a point); see, for example, Fig. 17. Any generator point p of a cut C is weighted with w.p/ D maxfdw .p; c/; 8c 2 C g. The disk D centered at p of radius w.p/ is clearly an open. If in addition D is strictly minimal, then p is a critical generator. Definition 10 The Voronoi diagram for opens on a layer A is a subdivision of A into regions, such that the critical radius of any point t in a Voronoi region is determined by the owner of the region. The critical radius of a point t, rc .t/, is the size (radius) of the smallest defect centered at t and causing an open. The following theorem is easy to see by properties of Voronoi diagrams. For a proof, see [34].

4

A biconnected component of a graph G is a maximal set of edges, such that any two edges in the set lie on a common simple cycle. 5 An articulation point (resp. bridge) of a graph G is a vertex (resp. edge) whose removal disconnects G.

Map of Geometric Minimal Cuts with Applications Fig. 19 The Voronoi diagram for open faults on layer A. The shaded region illustrates the Voronoi region of the higher-order generator g

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Theorem 5 The Voronoi diagram for open faults on a layer A corresponds to the (weighted) Voronoi diagram of the set G.A/ of all critical generators for strictly minimal opens on A, denoted as V.G.A//. Figure 19 illustrates the opens Voronoi diagram for the net of Fig. 14. The shaded region illustrates the Voronoi region of a higher-order generator g, reg.g/. Generator g is the critical generator of a cut consisting of two core segments as indicated by two small arrows. The critical radius of a sample point t in reg.g/ is indicated by the arrow emanating from t. Corollary 3 Given the opens Voronoi diagram, the critical radius of any point t in the region of a generator g is rc .t/ D dw .t; g/. If g is a higher-order generator of cut C , then rc .t/ D dw .t; g/ D maxfdw .t; c/; 8c 2 C g. In the following section, the Voronoi diagram for opens is formulated as a special higher-order Voronoi diagram of elements in core.A/.

3.4

A Higher-Order Voronoi Diagram Modeling Open Faults and the MGMC Problem

Let V.A/ denote the (weighted) Voronoi diagram of the set core.A/ of all core elements on a plane (layer) A. If there were no loops associated with A, then V.A/ would provide the opens Voronoi diagram on A, and core.A/ would be the set of all critical generators. V.A/ for Manhattan layouts has been defined in detail in [32]. Figure 20 illustrates V.A/ for the net graph of Fig.14. The arrows in Fig.20 illustrate several minimal radii of defects that break a wire segment. Given a point t in the region of generator s, dw .t; s/ gives the radius of the smallest defect centered at t

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Fig. 20 The L1 Voronoi diagram of core.A/ on layer A, V .A/

that overlaps the wire segment induced by s. Assuming no loops, dw .t; s/ would be the critical radius of t. Once loops are taken into consideration, only bridges, articulation, and terminal points, among the elements of core.A/, correspond to critical generators. Let us augment V.A/ with information reflecting critical generators. In particular, the regions of first-order generators get colored red reflecting the regions of critical generators. The critical radius of point t in a red region of owner s is rc .t/ D dw .t; s/. In Fig. 21, red regions are shown shaded and critical generators are shown thickened. Let us now define the order-k Voronoi diagram on the plane A, denoted as V k .A/. For k D 1, V k .A/ D V.A/. Following the standard definition of higherorder Voronoi diagrams, a region of V k .A/ corresponds to a maximal locus of points with the same k nearest neighbors among the core elements in core.A/. The open portion of a core segment and its two endpoints count as different entities. A kth-order Voronoi region, k > 1, belongs to a k-tuple C representing the k nearest neighbors of any point in the region of C . The region of C is denoted reg.C /, and it is further subdivided into finer subregions by the farthest Voronoi diagram of C . For any point t 2 reg.C /, d.t; C / D maxfd.t; c/; 8c 2 C g. If C constitutes a cut of a net N , then the region of C is colored red. In order to appropriately model open faults, the above standard definition is slightly modified, and in certain cases fewer than k elements are allowed to own a Voronoi region of order k. In the following, the term k-th order Voronoi diagram implies the modified version of the diagram as follows: – A red region corresponds to a maximal locus of points with the same r, 1  r  k, nearest neighbors, C , among the core elements in core.A/, such that C constitutes a minimal cut for some net N . – Any time a core segment s and one of its endpoints p participate in the same set C of nearest neighbors, s is discarded from C ; this is because d.t; p/  d.t; s/

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Fig. 21 The first-order opens Voronoi diagram on layer A, V 1 .A/. Shaded regions belong to first-order generators and the critical radius of any point within is determined by the region owner

8t 2 reg.C /. Intuitively, a defect that destroys a core endpoint automatically also destroys all incident core segments, but not vice versa. Figures 22 and 23 illustrate V 2 .A/ and V 3 .A/, respectively, for the net of our example. kth-order Voronoi regions are illustrated in solid lines; red regions are illustrated shaded. The darker shaded region in V 2 .A/ shows the 2nd-order red region of a pair of core segments that constitute a cut. The thick dashed lines indicate the farthest Voronoi diagram subdividing a kth-order region. In a red region, the thick dashed lines (excluding standard 45ı s) correspond to critical generators. All critical generators are indicated thickened; solid ones are first-order generators and dashed ones in red regions are higher-order generators. All thin dashed lines in Figs. 21–23 can be ignored. Due to our conventions, the Voronoi region of any core endpoint p in V 1 .A/ remains present in V 2 .A/ and expands into the regions of the core segments incident to p. Theorem 6 The Voronoi diagram for opens on a layer A is the minimum-order m Voronoi diagram of core.A/, V m .A/; m  1, such that all regions of V m .A/ are colored red. Any region reg.H /, where jH j > 1, is subdivided into finer regions by the farthest Voronoi diagram of H . The critical radius for any point t in reg.H / is rc .t/ D dw .t; H / D maxfdw .t; h/; h 2 H g. For a proof of Theorem 6, see [34]. Figure 24 illustrates the opens Voronoi diagram, for our example; arrows indicate the critical radius of several points; all critical generators are indicated in thick solid lines. Note that in L1 , the Voronoi subdivision is not unique but depends on the conventions used on how to distribute regions that are equidistant from collinear elements on axis-parallel lines. Critical area calculations are immune to such differences, however, they may have an effect on the number of iterations needed to compute the opens Voronoi diagram. The adapted convention is that critical generators have priority over non-critical ones and regions equidistant from a critical and a noncritical generator are assigned to the critical one and get colored red.

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Fig. 22 The second-order opens Voronoi diagram, V 2 .A/. The darker shaded region belongs to a pair of core segments forming a cut

Fig. 23 The third-order opens Voronoi diagram, V 3 .A/

Corollary 4 The higher-order critical generators on a layer A are exactly the farthest Voronoi edges and vertices, excluding the standard-45ı Voronoi edges, constituting the farthest Voronoi subdivisions in the interior of each region in V m .A/. All higher-order critical generators are encoded in the graph structure of V k .A/, for some k, 1  k < m. Let G.A/ denote the set of all critical generators on layer A, including first-order and higher-order generators. Let us classify higher-order critical generators according to the minimum-order-k Voronoi diagram they first appear in. In particular,

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Fig. 24 The Voronoi diagram for open faults on a layer A. Arrows illustrate the critical radius of several points

higher-order generators encoded in V k .A/ are classified as .k C1/-order generators and they are denoted as GkC1 .A/; 1  k < m. Let G.A/ D [1i m Gi .A/. By Theorems 5 and 6, V.G.A// and V m .A/ are identical.

3.4.1 An Approximate Opens Voronoi Diagram Given any subset G 0 .A/ of the set of critical generators G.A/, the (weighted) Voronoi diagram of G 0 .A/ can be used as an approximation to V.G.A//. Clearly, the more critical generators are included in G 0 .A/, the more accurate the result is. In practice, we can derive G 0 .A/ as [1i k Gi .A/, including all i th-order generators up to a small constant k. Since the significance of critical generators reduces drastically with the increase in their order, V.G 0 .A// should be sufficient for critical area computation for all practical purposes. Corollary 5 Let G 0 .A/ D [1i k Gi .A/ be a subset of critical generators including all generators up to order k for a given constant k. The (weighted) Voronoi diagram of G 0 .A/, V.G 0 .A//, can serve as an approximation to the opens Voronoi diagram. If G 0 .A/ D G.A/, then the two diagrams are equivalent. Figure 25 illustrates the (weighted) Voronoi diagram of G1 .A/. V.G1 .A// reveals critical radii for opens under the (false) assumption that all loops are immune to open faults. In Fig. 25, solid arrows indicate selected critical radii as derived by V.G1 .A//, while dashed arrows indicate true critical radii. Several critical radii can be overestimated in V.G1 .A//, resulting in underestimating the total critical area for open faults. As k increases, however, V.[1i k Gi .A// converges fast to V.G.A// (see, e.g., Sect. 8 of [34] for experimental results).

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Fig. 25 V .G1 .A// as an approximate opens Voronoi diagram

3.5

Computing the Opens Voronoi Diagram

In this section, we give algorithmic details on how to compute GkC1 .A/ and V kC1 .A/, given V k .A/, for 1  k < m. We also discuss how to compute V.[1i m Gi .A// and V.G.A//. The iterative process to compute higher-order generators and higher-order opens Voronoi diagrams. Let’s first discuss how to identify the set GkC1 .A/ of .k C 1/-order generators, given V k .A/, for k  1. The following property is shown in [34]. Lemma 16 A Voronoi edge g that bounds two non-red Voronoi regions reg.H / and reg.J / in V k .A/ corresponds to a critical generator if and only if both the core elements h 2 H and j 2 J that induce g (g 2 b.h; j /) are part of the same biconnected component B, and in addition, H [ J corresponds to a cut of B, that is, removing H [ J from B disconnects B, leaving articulation points in at least two sides. No Voronoi edge bounding a red region can be a critical generator. To determine if Voronoi edge g is a critical generator, we need to pose a connectivity query to biconnected component B after removing H [ J . To perform connectivity queries efficiently, we can use the fully dynamic connectivity data structures of [20], which support edge insertion and deletions in O.log2 n/-time, while they can answer connectivity queries fast. For simplicity in the implementation, instead of employing dynamic connectivity data structures, reference [34] gives a simple (almost brute force) algorithm as follows: Remove the elements of H from B and determine new nontrivial bridges, articulation points, and biconnected components of BnH . For any Voronoi edge g bounding reg.H /, where g is portion of b.h; j /; h 2 H; j 2 J , g is a critical generator if and only if j is a new non-trivial

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bridge or articulation point of BnH . Generator g gets associated with the tuple of core elements H [ J , simplified, in case j or h are core endpoints, by removing any core segment incident to j; h. The above process can be considerably simplified in the special case where the biconnected component B is a simple cycle. In this case, a simple coloring scheme in the DFS tree of B can efficiently identify all cuts of B that may be associated with a second-order generator. The time complexity of determining GkC1 .A/, given V k .A/, is summarized in the following lemma. Note that the size of V k .A/ is O.k.n  k// (see [26]). Lemma 17 The .k C 1/-order generators can be determined from V k .A/ in time O.k n log2 n/ using the dynamic connectivity data structures of [20] or in time O.k n2 / using the simple algorithm presented above. In case of biconnected components forming simple cycles, second-order generators can be determined from V.A/ in O.n/ time. Let us now discuss how to obtain V kC1 .A/ from V k .A/, k  1. The following is an adaptation of the iterative process to compute higher-order Voronoi diagrams of points [26], to the case of (weighted) segments. Let reg.H / be a non-red region of V k .A/. Let N.H / denote the set of all core elements that induce a Voronoi edge bounding reg.H / in V k .A/. 1. Compute the (weighted) L1 Voronoi diagram of N.H / and truncate it within the interior of reg.H /; this gives the .k C 1/-order subdivision within reg.H /. Each .k C 1/-order subregion of reg.H / is attributed to a tuple J D H [ fcg, c 2 N.H /. In case c is a core endpoint incident to a core segment s in H , J simplifies to J D H nfsg [fcg. In case c is part of a cut C owning a neighboring red region of V k .A/, the subregion of J gets colored red and gets as owner the cut C . 2. Once the .k C 1/-order subdivision within all non-red regions neighboring reg.H / has been performed, merge any incident .k C 1/-order subregions that belong to the same tuple of owners J into a maximal .k C 1/-order region, reg.J /. The edges of V k .A/ included within reg.J / constitute the finer subdivision of reg.J / by its farthest Voronoi diagram. All .k C 1/-order red subregions are merged into the neighboring red regions of V k .A/ forming the maximal red regions of V kC1 .A/. Using established bounds for higher-order Voronoi diagrams of points (see, e.g., [26]) we conclude the following. Lemma 18 V kC1 .A/ can be computed from V k .A/ in time O.k.n  k/ log n/, plus the time T .k; n/ to determine the .k C 1/-order generators, where T .k; n/ is as given in Lemma 17.

3.5.1 Computing the Opens Voronoi Diagram from Critical Generators The iterative process of Sect. 3.5 can continue until all regions are colored red and the complete opens Voronoi diagram (i.e., the map of the MGMC problem) is guaranteed to be available. By Lemmas 17 and 18, this results in

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an O.n3 log n.log log n/3 /-time algorithm. In practice, however, this would be unnecessarily inefficient. Note that the iterative process may continue for several rounds without any new critical generators being identified, only the regions of existing critical generators keep enlarging into neighboring non-red regions. Note also that as the number of iterations k increases, the weight of order-k critical generators (if any) increases as well, and their contribution to the total critical area drastically reduces. In practice, we can restrict the number of iterations to a small predetermined constant k, or to a small number determined adaptively, and compute only a sufficient set of critical generators G 0 .A/ D [1i k Gi .A/. We can then use Theorem 5 to report V.G 0 .A// as an approximate opens Voronoi diagram. The overall algorithm can be broken into two independent parts: – Part I: Compute the set of critical generators G 0 .A/ D [1i k Gi .A/, up to a given (or adaptively determined) order k. – Part II: Compute the (weighted) Voronoi diagram of G 0 .A/, V.G 0 .A//, as the opens Voronoi diagram. Part I can be performed using the iterative process of Sect. 3.5. Experimental results in [34] suggest that k D 2 is often adequate and no k > 4 is ever needed. Alternatively, k can be determined adaptively, for example, to the first round such that no new critical generators are determined. Part II can be performed using a plane sweep algorithm for computing V.A/. Critical generators have similar properties to the core elements of core.A/, and the same plane sweep algorithm can be used to compute either (see [32, 36]). The computations of Parts I and II can be synchronized: once a generator is discovered in Part I, it can be immediately scheduled for processing in Part II. For more details on the plane sweep construction and the synchronization of Parts I and II that is important in maintaining the locality of the computation see [34]. We thus conclude: Theorem 7 Assuming that G.N / is available for all nets under consideration and given a small constant k, the approximate opens Voronoi diagram V.G 0 .A//, G 0 .A/ D [1i k Gi .A/ can be computed in time O.n log n/ plus the time needed to answer connectivity queries. The latter can be done in time O.n log2 n/. The original implementation of this method, whose experimental results are reported in [34], used a slightly different approach in order to guarantee accuracy while the locality property was preserved. Namely, the iterative process of Sect. 3.5 was applied to each biconnected component independently. The advantages of considering each biconnected component independently were locality as well as the ability to run the process on each individual component to completion and guarantee the accuracy. The disadvantage was that generators produced in this manner, G 00 .A/, did not need all be critical. Including noncritical generators in the set G 00 .A/ complicates the algorithm of Part II because it amounts to computing the Hausdorff Voronoi diagram of corresponding geometric minimal cuts on layer A. Note that critical generators can be treated as simple additively weighted segments, having special weights, such that Voronoi regions remain connected and the entire generator is always enclosed in its Voronoi cell. This property considerably simplifies the construction of their (weighted) Voronoi diagram, which is, the Hausdorff Voronoi

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diagram, which remains similar to the construction of ordinary Voronoi diagrams of line segments. Once non-critical generators are present, however, this property no longer holds and the algorithms is equivalent to constructing the Hausdorff Voronoi diagram of the corresponding cuts as presented in Sects. 2.3 and 4. For information on critical area computation once the solution to the MGMC problem is available, see, for example, [32, 34, 36].

4

The L1 Hausdorff Voronoi Diagram

In this section, we review structural properties of the L1 Hausdorff Voronoi diagram of clusters of points, or equivalently, the L1 Hausdorff Voronoi diagram of rectangles, presented in [39]. The L1 Hausdorff Voronoi diagram of rectangles provides a solution to the MGMC problem after geometric minimal cuts have been identified. Results in this section are reproduced from [39]. Given a set S of point clusters in the plane, the Hausdorff Voronoi diagram of S , denoted HVD(S), is a subdivision of the plane into regions, such that the Hausdorff Voronoi region of a cluster P , denoted hreg(P), is the locus of points closer to P than to any other cluster in S , where distance between a point t and a cluster P is measured as the farthest distance between t and any point in P , df .t; P / D maxfd.t; p/; p 2 P g, hreg.P / D fx j df .x; P / < df .x; Q/; 8Q 2 S g. It is subdivided into finer regions by the farthest Voronoi diagram of P , FVD(P). The farthest distance df .t; P / is equivalent to the Hausdorff distance6 dh .t; P / between t and P . In the L1 metric, df .t; P / (equiv. dh .t; P /) is equivalent to df .t; P 0 / (equiv. dh .t; P 0 /), where P 0 is the minimum enclosing axis-aligned rectangle of P . Thus, the L1 Hausdorff Voronoi diagram of S is equivalent to the L1 Hausdorff Voronoi diagram of the set S 0 of the minimum enclosing rectangles of all clusters in S . In the following, the terms cluster and rectangle are used interchangeably. In this section, we review the tight bound on the structural complexity of the L1 Hausdorff Voronoi diagram given in [39]. It is shown that the structural complexity of the L1 Hausdorff Voronoi diagram is ‚.n C m/, where n is the number of input clusters (equiv. rectangles) and m is the number of essential pairs of crossing clusters (see Definition 11). We also review a simple plane sweep construction in two dimensions given initially in [32] and improved in [39]. The improved algorithm consists of an O..nCM / log n/-time preprocessing step, based on point dominance in R3 , followed by the main plain sweep algorithm that runs in O..n C M / log n/time and O.n C M /-space, where M reflects special crossings that are potentially essential (see Definition 12); m; M are O.n2 /, m  M , but m D M , in the worst case. In practice, typically, m; M P s . If, in addition, there is a minimum enclosing square of Q that is crossing P , P is said to strongly dominate Q. To eliminate source-2 of invalid V-vertex events, a preprocessing step can be added to the algorithm that precomputes dominating sequences of vertical

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Fig. 30 The dominating sequence of rectangle P consisting of rectangles R1 ; R2 ; R3

R2

P

P

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rectangles, which is described in [39]. Given the preprocessing information, the number of all V events generated is bounded by O.n C M / as shown in [39]. Definition 13 The dominating sequence of a vertical rectangle P , denoted ds(P), is a maximal collection of vertical rectangles Ri ; i D 1; : : : ; k, such that every Ri is entirely enclosed in a minimum enclosing square of P , R1 is the rectangle with the rightmost west edge among all rectangles dominated by P , and if Ri is crossing P , i  1, then Ri C1 is the vertical rectangle with the rightmost west edge dominated by square.P; Rie1 /. Every Ri in the dominating sequence of P , except possibly the last rectangle Rk , is crossing P . Rectangle R1 in the dominating sequence of P is referred to as the rightmost rectangle dominated by P . In the case of non-crossing rectangles, the dominating sequence of P is R1 (if not empty). Figure 30 shows the dominating sequence of a vertical rectangle P consisting of three rectangles R1 ; R2 ; R3 . Considering only those four rectangles, the region of P is alternating with regions of Ri on the core segment of P , as shown in Fig. 30. Lemma 3 in [39] shows that the dominating sequence of P (except the last rectangle Rk , if Rk is non-crossing with P ) forms a maximal staircase of vertical rectangles crossing P that is potentially essential. No vertical rectangle, other than the dominating sequence of P , may induce a right Voronoi V-vertex on the core segment of P , even when only ds.P / is considered. Given the dominating sequences of vertical rectangles, every time a V event is generated, source-2 of invalid V events can be completely eliminated. The exact algorithmic details are given in [39]. Using the information of dominating sequences of vertical rectangles, the total number of V events that may be produced throughout the algorithm is shown in [39] to be O.n C M /. The problem of computing the dominating sequence of every vertical rectangle can be transformed into a point dominance problem in R3 which can be solved by plane sweep in O..n C M / log n/-time, as described in Sect. 4 of [39]. Combining with Lemma 3 of [39] we conclude.

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Theorem 9 HVD(S) can be computed by plane sweep in O..n C M / log n/-time and O.n C M /-space. In the case of non-crossing rectangles, this results in optimal O.n log n/-time and O.n/-space. For the plane-sweep algorithm to compute dominating sequences of vertical rectangles, see [39] (Sect. 4). It is based on a simple transformation to a variant of the standard point dominance problem in R3 (see, e.g., [40]) and the use of an auxiliary standard priority search tree [28].

5

Conclusion

Given a graph G D .V; E/ with a subgraph H embedded in the plane, this chapter presented the problem of computing the map of geometric minimal cuts (MGMC) of H , as induced by axis-aligned rectangles in the embedding plane. It surveyed two different approaches for the solution of the MGMC problem [34, 47], that were based on a mix of geometric and graph-algorithm techniques. The chapter also surveyed results on the related Hausdorff Voronoi diagram, which provides a solution to the MGMC problem.

Recommended Reading 1. M. Abellanas, G. Hernandez, R. Klein, V. Neumann-Lara, J. Urrutia, A combinatorial property of convex sets. Discret. Comput. Geom. 17, 307–318 (1997) 2. E. Aurenhammer, Voronoi diagrams – a survey of a fundamental data structure. ACM Comput. Surv. 23(3), 345–405 (1991) 3. F. Aurenhammer, R. Klein, Voronoi diagrams, in Handbook of Computational Geometry, ed. by J. Sack, G. Urrutia (Elsevier, Amsterdam, 2000), pp. 201–290 4. F. Aurenhammer, R. Drysdale, H. Krasser, Farthest line segment Voronoi diagrams. Inf. Process. Lett. 100, 220–225 (2006) 5. S.C. Braasch, J. Hibbeler, D. Maynard, M. Koshy, R. Ruehl, D. White, Model-based verification and analysis for 65/45nm physical design, CDNLive! September 2007 6. M. de Berg, O. Schwarzkopf, M. van Kreveld, M. Overmars, Computational Geometry: Algorithms and Applications, 2nd edn. (Springer, Berlin/New York, 2000) 7. J.P. de Gyvez, C. Di, IC defect sensitivity for footprint-type spot defects. IEEE Trans. Comput. Aided Des. 11, 638–658 (1992) 8. F. Dehne, R. Klein, “The big sweep”: on the power of the wavefront approach to Voronoi diagrams. Algorithmica 17, 19–32 (1997) 9. F. Dehne, A. Maheshwari, R. Taylor, A coarse grained parallel algorithm for Hausdorff Voronoi diagrams, in Proceedings of the 2006 International Conference on Parallel Processing, Columbus (2006), pp. 497–504 10. J.R. Driscoll, N. Sarnak, D. Sleator, R. Tarjan, Making data structures persistent, in STOC, Berkeley, CA (1986), pp. 109–121 11. H. Edelsbrunner, L.J. Guibas, M. Sharir, The upper envelope of piecewise linear functions: algorithms and applications. Discret. Comput. Geom. 4, 311–336 (1989) 12. D. Eppstein, Z. Galil, G.F. Italiano, A. Nissenzweig, Sparsification – a technique for speeding up dynamic graph algorithms. J. ACM 44(5), 669–696 (1997) 13. G.N. Frederickson, Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput. 14(4), 781–798 (1985)

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39. E. Papadopoulou, J. Xu, The L1 Hausdorff Voronoi diagram revisited, in IEEE-CS Proceedings, ISVD 2011, International Symposium on Voronoi Diagrams in Science and Engineering, Qingdao (2011) 40. F.P. Preparata, M.I. Shamos, Computational Geometry: An Introduction (Springer, New York, 1985) 41. D. Sleator, R. Tarjan, A data structure for dynamic trees. J. Comput. Syst. Sce. 26(3), 362–391 (1983) 42. R. Tarjan, Depth-first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972) 43. R.E. Tarjan, Efficiency of a good but not linear set union algorithms. J. ACM 22, 215–225 (1975) 44. M. Thorup, Decremental dynamic connectivity, in Proceedings of the 8th SODA, New Orleans (1997), pp. 305–313 45. M. Thorup, Near-optimal fully-dynamic graph connectivity, in STOC, Portland (2000), pp. 343–350 46. “Voronoi CAA: Voronoi Critical Area Analysis”, IBM CAD Tool, Department of Electronic Design Automation, IBM Microelectronics, Burlington, VT. Initial patents: US6178539, US6317859. Distributed by Cadence 47. J. Xu, L. Xu, E. Papadopoulou, Computing the map of geometric minimal cuts, in Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009), Hawaii, USA. Lecture Notes in Computer Science, vol. 5878, 16–18 Dec 2009, pp. 244–254

Max-Coloring Juli´an Mestre and Rajiv Raman

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notations and Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Summary of Known Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Max-Coloring Paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Max-Coloring Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Max-Coloring Bipartite and k-Colorable Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Planar Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Restricted Bipartite Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Split Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Max-Coloring Interval Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Max-Coloring on Hereditary Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Online Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Max-Edge-Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Bounded Max-Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1872 1872 1874 1875 1881 1886 1888 1894 1897 1897 1898 1898 1900 1902 1903 1906 1909 1909 1910

J. Mestre () School of Information Technologies, The University of Sydney, Sydney, NSW, Australia e-mail: [email protected] R. Raman Indraprasta Institute of Information Technology, Delhi, India e-mail: [email protected] 1871 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 32, © Springer Science+Business Media New York 2013

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Abstract

The max-coloring problem is a natural generalization of the classical vertex coloring problem. The input is a vertex-weighted graph. The objective is to produce a proper coloring such that the overall weight of color classes is minimized, where the weight of each class is defined to be the maximum weight of vertices in that class. Max-coloring has received significant attention over the last decade. Approximation algorithms and hardness results are now known for a number of graph classes in both the off-line and online setting. The objective of this chapter is to survey the algorithmic state of the art for the max-coloring problem.

1

Introduction

The max-coloring problem is a natural generalization of the classical vertex coloring problem. The input is a vertex-weighted graph. The objective is to produce a proper coloring such that the overall weight of color classes is minimized, where the weight of each class is defined to be the maximum weight of vertices in that class. More formally, let G D .V; E/ be an undirected graph and w W V ! N a weight function. The problem is to find a proper vertex coloring C1 ; C2 ; : : : ; Ck of G P that minimizes kiD1 maxv2Ci fw.v/g. Note that the special case of this problem in which w.v/ D 1 for all v 2 V is simply the classical minimum coloring problem. Max-coloring has received significant attention over the last decade. Approximation algorithms and hardness results are now known for a number of graph classes in both the off-line and online setting. The objective of this chapter is to survey the algorithmic state of the art for the max-coloring problem. The rest of this section reviews some of the applications that motivated the study of max-coloring, introduces some notation, and describes briefly known results. Later sections discuss in more detail some key results.

1.1

Applications

The max-coloring problem arises naturally in the context of buffer management in operating systems and batch scheduling in certain manufacturing settings.

1.1.1 Buffer Minimization Programs that run with stringent memory or timing constraints use a dedicated memory manager that provides better performance than the general purpose memory management of the operating system. An example of such programs is protocol stacks like GPRS and 3G that run on mobile devices. These protocol stacks have stringent memory requirements as well as soft real-time constraints, so a dedicated memory manager is a natural design choice. A dedicated memory manager for these

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stacks must have deterministic response times and use as little memory as possible. The most commonly used memory manager design for this purpose is the segregated buffer pool. This consists of a fixed set of buffers of various sizes with buffers of the same size linked together in a linked list. As each memory request arrives, it is satisfied by a buffer whose size is large enough. The assignment of buffers to memory requests can be viewed as an assignment of colors to the requests – all requests that are assigned a buffer are colored identically. Thus, the problem of determining whether a given segregated buffer pool suffices for a particular sequence of allocation requests can be formalized as follows: Let G D .V; E/ be a graph whose vertices are objects that need memory and whose edges connect pairs of objects that are alive at the same time. Requests which are live at the same time have to be satisfied by different buffers. Let w W V ! N be a weight function that assigns a natural number weight to each vertex in V . For any object v, w.v/ denotes the size of memory it needs. Suppose the segregated buffer pool contains k buffers with weights w1 ; w2 ; : : : ; wk . The problem is to determine if there is a k-coloring of G into color classes C1 ; C2 ; : : : ; Ck such that for each i , 1  i  k, maxv2Ci w.v/  wi . The optimization version of this problem is the max-coloring problem: Solving the max-coloring problem and selecting buffers according to the optimal solution lead to a segregated buffer pool that uses minimum amount of total memory. Note that this is an off-line problem. In the online version, buffers have to be allocated to requests as they arrive and without knowledge of future requests. The off-line version is also useful because designers of protocol stacks would like to estimate the size of the total memory block needed by extracting large traces of memory requests and running off-line algorithms on these traces. When memory allocation requests are generated by straight-line program, (i.e., programs without loops or branching statements), each memory allocation request is made for a specific duration of time. Thus, these requests can be viewed as intervals on the real line. In this restriction to straight-line programs, the underlying graph is an interval graph. See Sect. 6 for a description.

1.1.2 Batch Scheduling In several scenarios involving the scheduling of a set of jobs on a set of machines, the jobs are partitioned into batches. Jobs in a batch are processed simultaneously, but batches are processed sequentially: Processing of the next batch of jobs starts only after all the jobs in the current batch are complete. Batched scheduling is common in manufacturing processes where a machine is capable of handling several types of jobs but requires a different setup for each job. In this case, all jobs of the same type can be scheduled on the machine, before changing the setup for a different type of job. For example, materials in a factory may have to be processed in a kiln, where all materials that need to be processed at the same temperature can be processed simultaneously, before starting a next batch with a different temperature. Batching thus leads to a higher utilization of the machine and hence more efficient schedules. Another application where batch scheduling is used is in scheduling jobs

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on a cluster machine, where jobs that need to communicate to accomplish a task are scheduled together. A paper by Potts and Kovalyov [33] surveys recent results in batched scheduling. Batch scheduling problems with conflicting jobs can be modeled as a maxcoloring problem as follows: The input to a batch scheduling problem is a set of jobs J D fJ1 ;    ; Jn g with processing times w W J ! N, and a graph G, whose vertex set is the set of jobs and edge between a pair of jobs, implies the two jobs are not compatible and hence cannot be scheduled simultaneously. A proper vertex coloring of this graph corresponds to a schedule where each color class constitutes a batch. The processing time for a batch is the maximum processing time of any job in this batch, and the processing time for the entire set of jobs is the sum of processing times of the batches. Thus, if S1 ;    ; Sk are the independent sets in a coloring, the weight of an independent set w.Si / D maxv2Si w.v/, and the cost of this coloring is P the sum of the weights of the independent sets; That is, kiD1 w.Si /. The problem of minimizing the makespan of this batched schedule is precisely the max-coloring problem. In some settings, in addition to the time it takes to process the batches themselves, there is a setup time or delay d incurred by each individual batch. This is easily modeled by increasing the weight of every vertex by d .

1.2

Notations and Definitions

In this chapter, all graphs are assumed to be undirected and connected, and in all but for edge-coloring, the graphs are assumed to be simple. A graph G D .V; E/ is made up of a vertex set V .G/ and an edge set E.G/. The notation n.G/ and m.G/ is used to denote the cardinality of V .G/ and E.G/, respectively. The maximum degree of G will be denoted by .G/. (For the sake of succinctness, .G/ is omitted from these quantities whenever the graph G is clear from the context.) A vertex weight function w W V ! N is simply an assignment of positive integer values to the vertices of the graph; similarly, an edge weight function assigns a positive integer value to every edge in the graph. Unless otherwise stated, the weight functions will be of the vertex type. The notion of an induced subgraph is used in several places throughout the survey. For S  V , GŒS  is the vertex-induced subgraph or simply induced subgraph, of G that has S for vertex set and f.u; v/ 2 E W u; v 2 S g for edge set. Similarly, one can define an edge-induced subgraph, or simply subgraph as follows: Let F  E, then GŒF  is the edge-induced subgraph of G that has F for edge set and endpoints of edges in F for vertex set. A proper k-coloring of the graph G D .V; E/ is a function  W V ! f1; : : : ; kg such that for every edge .u; v/ 2 E, it is the case that .u/ ¤ .v/. The color classes C1 ; : : : ; Ck of  are defined as Ci D fv 2 V W .v/ D i g for each i 2 f1; : : : ; kg. Given a weight function w, the weight of color class Ci is defined to be

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w.Ci / D max w.v/ v2Ci

and the weight of coloring  to be w./ D

k X

w.Ci /:

i D1

The chromatic number of a graph G is the smallest k such that G admits a proper k-coloring; this quantity is denoted by .G/. The max-chromatic number, denoted mc .G/, denotes the smallest number k such that there is an optimal max-coloring using k colors, and OPTmc .G/ denotes the weight of an optimal max-coloring of G. Finally, !.G/ denotes the size of the largest clique in the graph; notice that !.G/  mc .G/. Let A be an algorithm for a given minimization problem, and let   1. The algorithm A is said to be a -approximation or that A attains an approximation ratio of  if for all instances of the optimization problem, the solution returned by A has cost at most  times the optimal solution. In the online context, where the instance is not revealed all at once but rather progressively, over time, here, algorithm A is said to be -competitive if it returns a solution whose cost is no more than  times the optimal solution that can be constructed after the whole instance has been revealed. A polynomial time approximation scheme (PTAS) is an algorithm such that for all  > 0, the algorithm returns a 1 C  approximate solution in O.nf ./ /, where f is an arbitrary function. Let G be a graph, and let C1 ; : : : ; Ck be some proper coloring of G. The chapter always follows the convention of listing color classes in non-increasing order of their weights; that is, w.C1 /  w.C2 /  : : :  w.Ck /. Further, it is assumed without loss of generality that Ci is a maximal independent set in the graph induced by [kj Di Ci ; indeed, otherwise, vertices can be migrated to lower indexed color classes without increasing the weight of the coloring.

1.3

Summary of Known Results

The majority of papers on max-coloring focus on solving the problem on different graph classes. These results are summarized in Table 1 for the off-line setting and in Table 2 for the online setting, where vertices are revealed one by one; in which case, a vertex must be assigned a color as soon as it arrives. Another line of work has studied a variant of max-coloring where the objective is to color the edges of the graph; these results are summarized in Table 3. Yet another line of work has studied a variant of all these problems where there is an additional constraint on the size of the color classes used; these results are summarized in Table 4. The rest of this section briefly describes the results in Tables 1–4. Later sections highlight and elaborate in more detail a selected subset of these results.

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Table 1 Table for off-line max-coloring vertices of graphs. A lowerbound of c > 1 means that there is no .c  /-approximation unless P D NP . All algorithms run in polynomial time unless stated otherwise in the notes column Graph class

Lowerbounds

Upperbounds

Path Tree

– –

Low treewidth Bipartite

– 8=7 [30]

k-Colorable

–

Split Interval Perfect

NP-hard [8] NP-hard [31] 8=7 [30]

1 [14, 18, 19, 22] 1 [30] .1 C / [18, 30] .1 C / [14] 8=7 [14, 30] k3 [14] 3k 2  3k C 1 .1 C / [7] .1 C / [28] e [34]

Hereditary

–

e  c [12, 34]

Notes

Section

nO.log n/ time nO.1=/ time nO.1=Ctw/ time

 Sect. 3  Sect. 4  Sect. 4 –  Sect. 5

Assumes k-coloring is computable

 Sect. 5

n

O.1=/

time

–  Sect. 6 –

Assumes c-approx. for regular coloring

 Sect. 7

Table 2 Table for online max-coloring vertices of graphs. All results appear in the paper of Epstein and Levin [12]. Each row is subdivided into results for deterministic and randomized algorithms. A lowerbound of c > 1 means that there is no .c  /-competitive algorithm. For interval graphs, two models are considered: In the list model, intervals arrive at in arbitrary order; in the time model, intervals arrive sorted by their left endpoint. These results are covered in Sect. 7.1 Graph class Hereditary Interval (list model) Unit interval (list model) Interval (time model)

Lowerbounds – – 4 4 2 11=6   1:618 4=3

Table 3 Table for max-edge-coloring graphs. A lowerbound of c > 1 means that there is no .c  /-approximation unless P D NP

Upperbounds 4c ec 12 3e 8 2e 4 e

Graph class Paths Trees Bipartite General simple graphs General multigraphs

Notes Assumes c-comp. for regular online coloring Intervals arrive in arbitrary order Unit intervals arrive in arbitrary order Intervals arrive sorted by left endpoint

Lowerbounds – – 7/6 [7] 4=3 [20] 4=3 [20]

Upperbounds 1 [14, 18, 19, 22] PTAS [26] 1.74 [26] 2 2  C2 [4] 2 2  1:5C1 [4]

1.3.1 Paths The simplest graph class imaginable is that of simple paths. It should not be surprising then to learn that max-coloring paths can be solved optimally in polynomial time. Indeed, it is not hard to see: First, note that an optimal solution

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Table 4 Table for bounded max-coloring and bounded max-edge-coloring. A lowerbound of c > 1 means that there is no .c  /-approximation unless P D NP Coloring type

Graph class

Lowerbounds

Upperbounds

Notes

Vertex

Trees Bipartite Hereditary

– 4=3 [3] –

PTAS [2] 17=11 [2] c C 1 [6]

O.n2 / time

Trees Bipartite General

NP-hard [2] 7=8 [7] 4=3 [20]

2 [2] e [2] 3 [2, 6]

Edge

1=

Assumes c-approx. for max-coloring

uses at most three colors;otherwise, one can recolor the vertices in the lightest color class and decrease the cost of the solution. The algorithm guesses the weights of these three color classes (there are O.n3 / possibilities to consider) and for each guess uses dynamic programming to check in linear time whether a 3-coloring with these weights exists. This is the basis of the O.n4 / time algorithm of Guan et al. [18]. The running time was subsequently improved to O.n2 / by Escoffier et al. [14], then to O.n log n/ by Halldorsson and Schachnai [19], and finally to O.n C S.n// by Kavitha and Mestre [22], where S.n/ is the time it takes to sort the weights. The last paper also provided a matching lowerbound of .n log n/ in the algebraic computation tree model on the running time required for max-coloring a path. The algorithm of Kavitha and Mestre is covered in Sect. 3.

1.3.2 Trees While the classic graph coloring problem is trivial on trees (any tree can be colored with at most two colors), the max-coloring problem on trees has turned out to be a challenging problem. It can be shown that on trees, an optimal max-coloring may use as many as dlog2 ne colors, and the best algorithm to compute an optimal max-coloring runs in quasi-polynomial time, that is, in time 2poly log n . This quasipolynomial time algorithm can also be used as a basis for obtaining a PTAS for the problem, and these results extend directly to graphs of bounded treewidth, provided a tree decomposition is given. These results are presented in Sect. 4. No hardness result is known for max-coloring trees. Indeed, settling its computational complexity remains an intriguing open problem. 1.3.3 Bipartite Graphs While the max-coloring problem on trees is yet to be fully resolved, the situation is different for the class of bipartite graphs that includes trees. On bipartite graphs, it can be shown that an optimal max-coloring may use as many as O.n/ colors on a graph with n vertices. Pemmaraju and Raman [30] and independently Escoffier et al. [14] gave tight results for the max-coloring problem on bipartite graphs. The authors show that the problem is hard to approximate beyond 87   for any  > 0

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unless P D NP, and they give a combinatorial algorithm that achieves a ratio of 87 . The surprising aspect of this algorithm is that it uses at most four colors, while an optimal max-coloring may use as many as O.n/ colors. Several authors have further tried to delineate the boundary between polynomial time solvable cases and the NP-hard case for bipartite graphs. DeWerra et al. [7] prove that on P 5-free bipartite graphs (i.e., bipartite graphs that do not have a path on five vertices as an induced subgraph), the max-coloring problem admits a polynomial time solution, while on P 8-free bipartite graphs, the authors show that the problem becomes NP-hard.

1.3.4 Split Graphs A Split graph is a graph G D .K [ S; E/, where the vertex set V can be partitioned into a complete graph K and an independent set S . Between K and S is an arbitrary bipartite graph. Split graphs are a subclass of perfect graphs, and hence, the classic coloring problem admits a polynomial time algorithm. DeWerra et al. [7] study the max-coloring problem on split graphs. They prove the problem to be NP-complete and give a PTAS. 1.3.5 k-Colorable Graphs Even though coloring general graphs is a very difficult problem, certain graph classes can be optimally colored or good approximations exist. It makes sense then to study how well max-coloring can be approximated in these special classes. Max-coloring is APX-hard even for bipartite graph, where an optimal coloring can be found in linear time, so one should not expect an approximation preserving reduction between the two problems. Nevertheless, being able to compute a k-coloring of the input graph is useful for approximating max-coloring. Indeed, any proper k-coloring is a k-approximation for max-coloring. Can this trivial observation be improved? Bourgeois et al. [4] considered this question k3 and showed how to get a 3k 2 3kC1 approximation for max-coloring provided a k-coloring for the input graph can be computed. This result is described in Sect. 5. Here, two applications to planar graphs and triangle-free planar graphs are discussed. The celebrated four-color theorem states that every planar graph can be vertexcolored using four colors. Robertson et al. [35] provide an O.n2 / time algorithm for 4-coloring any planar graph. Using the above result, one can get a 64 37 < 1:73 approximation for max-coloring planar graphs. On the negative side, the hardness of graph coloring planar graphs [16] means it is NP-hard to approximate max-coloring better than 4=3 > 1:33. Gr¨otzsch theorem states that every triangle-free planar graph can be 3-colored. Dvorak et al. [10] give a linear time algorithm to compute such a coloring. Using the above result, one can get a 27 19 < 1:43 approximation for max-coloring triangle-tree planar graphs. On the negative side, De Werra et al. [7] showed that in this class of graphs, max-coloring cannot be approximated better than 76 > 1:16.

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1.3.6 Interval Graphs Interval graphs are intersection graphs of intervals on the real line. In other words, a graph is an interval graph if and only if each vertex can be associated with an interval on the real line (closed or open) such that two intervals intersect if and only if their corresponding vertices are adjacent. Interval graphs arise in several applications from scheduling to computational biology. They are a sub-class of perfect graphs, and hence, .H / D !.H / holds for any interval graph G and induced subgraph H  G. Another feature that will be useful is that given a graph, one can decide in polynomial time whether it is an interval graph and, if so, construct an interval representation [5]. Hence, for these algorithms, one can assume that the interval representation is given. Interval graphs being perfect, minimum coloring can be solved in polynomial time. However, one can compute a coloring much easier than for perfect graphs by sorting the intervals in order of their left end-points and coloring them in a greedy manner with the smallest available color at each step. (See the book by Golumbic [17] for further details.) However, the max-coloring problem is NP-hard on interval graphs. A 2-approximation was presented by Pemmaraju et al. [31]; this result appears in Sect. 6. Very recently, a PTAS has been proposed by Nonner [28], which settles the computational complexity of max-coloring in interval graphs. 1.3.7 Hereditary Graph Classes A class of graphs G is called hereditary if it is closed under taking vertex-induced. Hence, for all graphs G 2 G, if H  G is an induced subgraph of G, this implies H 2 G as well. Most of the graph classes considered in this survey are hereditary graph classes. Clearly, the max-coloring problem is NP-hard on hereditary graphs. However, one can show that max-coloring can be approximated as well (up to constant factors) as regular coloring. More precisely, a c-approximation algorithm for classic coloring can be used to produce an e  c-approximation algorithm for max-coloring on this class of graphs; here, e is the base of the natural logarithm. This result is presented in Sect. 7. 1.3.8 Online In some application domains, the instance is not given to us all at once,but rather, vertices are revealed one by one. For instance, this is the case in the buffer minimization application described in Sect. 1.1, where memory requests depend on the actual execution path of the program that is running. Epstein and Levin [12] studied max-coloring in the online setting. Vertices arrive one by one. When a vertex u arrives, its neighborhood (restricted to the vertices that have already arrived) is revealed. Upon u’s arrival, the algorithm must assign a valid color to u that cannot be changed later on. They showed a reduction from online max-coloring to online coloring in hereditary graph classes. Given a c-competitive algorithm for online coloring, they design a deterministic 4c-competitive algorithm for max-coloring and a randomized e c-competitive algorithm. This result and some applications thereof are described in Sect. 7.1.

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Epstein and Levin also considered the interesting class of interval graphs. Here, they assumed that the interval representation, rather than the graph itself, is revealed. They considered two different interval arrival models: In the list model, intervals arrive in some arbitrary order, while in the time model, intervals arrive sorted by their left endpoint. In the time model, a simple greedy algorithm solves the online coloring problem optimally. Thus, the reduction for general hereditary graph classes yields 4-competitive deterministic and e-competitive randomized algorithms for max-coloring interval graphs in the time model. Similar results are obtained by using the 3-competitive algorithm of Kierstead and Trotter [24] for coloring interval graphs in the list model and the 2-competitive algorithm of Epstein and Levy [13] for coloring unit interval graphs in the list model. On the negative side, the authors show a number of lower bounds for online max-coloring interval graphs under these two models. The results are summarized in Table 2.

1.3.9 Max-Edge-Coloring An important application of max-coloring arises in the context of scheduling transmissions in optical switches. Optical switches can be modeled as a complete bipartite graph connecting n input ports and n output ports. In each time step, the switch transmits packets along a perfect matching between input and output ports. A scheduler procedure is responsible for deciding which matching to use at each time step in order to route a given traffic demand pattern. In the language of graphs, the problem is defined by an edge-weighted bipartite graph G D .A; B; E/, where vertices in A and B represent input and output ports, respectively; edges represent data transfers that must be carried out; and weights represent the time it takes to carry out the transfer. Transmissions occur in rounds or batches. The set of edges associated with the transmissions carried out during a single round must form a matching. The objective is to minimize the makespan of the schedule. In a sense, this is just another example of batch scheduling. However, the difference here is that colors are assigned to edges of the input graph instead of the vertices. The problem was first studied by Kesselman and Kogan [23], who showed that a simple greedy algorithm is a 2-approximation for the problem. This was later slightly improved by Bourgeois et al. [4], who showed that taking the best between 2 greedy and an arbitrary k-coloring is a 2  kC1 approximation. These algorithms are described in Sect. 8. It is worth noting that for general graphs, max-edge-coloring inherits the . 43  /-hardness of regular edge-coloring [20]. Max-edge-coloring remains NP-hard in planar bipartite of degree 3 with edges weights in f1; 2; 3g in fact, this case is hard to approximate better than 76 [7]; this is in stark contrast with regular edge-coloring, which is polynomially solvable for general bipartite graphs. Bourgeois et al. [4] give two approximation algorithms for bipartite 2 graphs: The first one has an approximation ratio of 2 C1 , while the second one has 3

2.C1/ attained an approximation ratio of 3 C52 C5C3.1=/  . Very recently, Lucarelli and Milis [26] gave a 1.74-approximation algorithm, the current best factor for the

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problem. It is worth mentioning that this last algorithm can be adapted for general graph, however, yielding an asymptotic approximation factor of 1.74. The complexity of max-edge-coloring trees is open, as in the case of max(vertex)-coloring trees. As in this other case, there is PTAS for max-edge-coloring trees [26]. Max-edge-coloring can be reduced to max-(vertex)-coloring by creating a new graph, where every new vertex corresponds to an old edge and two new vertices are connected if the corresponding old edges share an endpoint; the new graph is called the line graph of the old graph. Since the line graph of a path is again a path, max-edge-coloring can be solved optimally in paths.

1.3.10 Bounded Max-Coloring Consider the batch scheduling application, where each batch consists of jobs that are non-conflicting. In the language of max-coloring, each color class consists of a batch of jobs that can run simultaneously. In many applications, however, there is a fixed number of machines, m, and each job in a batch runs on a separate machine. In such applications, the number of machines is typically fixed, so the size of the batches should not exceed m; in other words, each color class must have at most m vertices. Coloring unweighted graphs with a bounded size on each color class is well studied, and there are results on several restricted graph classes. The problem was introduced by Baker and Coffmann [1] under the title “Mutual Exclusion Scheduling.” For permutation graphs that are a subclass of comparability graphs, Jansen [21] proved that the problem is NP-hard for m  6. For interval graphs, Bodlaender and Jansen [3] have shown that the problem on interval graphs is NP-hard for m  4. For m D 2, the problem can be solved easily in polynomial time via matching techniques, while Gardi [15] gave linear time algorithms for interval and circular-arc graphs for m D 2. Clearly all the hardness results for the max-coloring problem carry over to the bounded case. The bounded case, however, seems to be harder: For some graph classes for which the max-coloring problem can be solved in polynomial time, the bounded max-coloring problem is NP-hard. On the positive side, there is an algorithm for hereditary graph classes such that given an ˛-approximation algorithm for the max-coloring problem, it is possible to compute an ˛ C 1-approximation for the bounded max-coloring problem. Bampis et al. [2] obtained a number of results for different graphs classes, which are summarized in Table 4. These results are presented in Sect. 9.

2

Tools

This section develops some tools and techniques that have proved useful for attacking the max-coloring problem on various graph classes. First, consider the simple bound on mc .G/ for any graph G. Assume C1 ; C2 ; : : : ; Ck is an optimal max-coloring of a given graph G.

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Lemma 1 ([18]) For any graph G and any weight function w W V ! N, mc .G/  C1 Proof Let k D mc .G/. If k >  C 1, then for each vertex v 2 Ck , there is a color class Ci , i < k, such that v has no neighbors in Ci . Note that since w.Ci /  w.Ck /, when v is moved into Ci , the weight of Ci does not increase. Furthermore, when Ck becomes empty, the weight of the coloring decreases, contradicting the optimality of C1 ; C2 ; : : : ; Ck .  Consider an optimal max-coloring of a graph G such that .G/ D l. The reason max-coloring uses more colors than l to color G is that coloring heavy weight vertices with the same color offsets the increase in weight due to the use of many more color classes. This relation is captured in the following lemma, which is a key property of any optimal max-coloring. Lemma 2 Let G be an l-colorable graph, and let C1 ; : : : ; Ck be an optimal maxcoloring with k > l colors. Then, for each i D 1; : : : ; k, .l  1/w.Ci / > Pk w.C j /. j Di C1 Proof The vertices in [kj Di Cj can be l-colored for a cost of at most l  w.Ci /. P Since k colors are used, it must be that kj Di w.Ci / < l  w.Ci /, and the result follows.  In the max-coloring problem, it makes sense to color the large weight vertices with the same color to avoid paying for them several times in each color class. A natural approach is to partition the vertices into groups such that within each group either the number of distinct weights is small or the variance between the largest and smallest weights is small (or both). Then one can develop algorithms that work well in this restricted setting, apply the algorithm separately to each set of the partition, and obtain a max-coloring for the entire graph. The next lemma shows that such a partition can be always found at the expense of only a small increase in the approximation factor. Lemma 3 ([28]) Let G D .V; E/ be a vertex-weighted graph with weight function w W V ! N, and let  > 0 be a constant. Then, G can be partitioned into induced subgraphs G0 ; G1 ; : : : such that within each induced subgraph, the ratio wmax =wmi n is O. 2 /, the number of distinct weights is 2O.1=/ , and optimally max-coloring each subgraph independently leads to a .1 C / approximation for G. Proof Let ˛ D ˛./ > 1 be a constant that depends on . Define a new weight function as follows: Let d.v/ D dlog˛ w.v/e, and let w0 .v/ D ˛ d.v/ for each v 2 V . Hence, each vertex weight is rounded up to the nearest power of ˛. The graph with the new weights is called the rounded instance. Let OPT˛mc .G/ denote the weight of an optimal coloring with the new weights w0 . Then, the following is true:

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OPTmc .G/  OPT˛mc .G/  ˛  OPTmc .G/

(1)

The first inequality clearly holds since the weight of each vertex increases. The second inequality holds since each vertex weight increases by at most a factor of ˛, so using the coloring with weight OPTmc .G/ with the new weights has cost at most ˛  OPTmc .G/. Now, let ı D ı./ > 0 be a constant that depends on . Let m D maxv2V d.v/. Hence, m is the number of distinct weights in the rounded instance. Partition the set of vertices into l D ı  m groups, each group consisting of 1ı contiguous weights. Assume that 1ı 2 N and that ı  m  1; otherwise, m is already constant. The partitioning will be randomized. To this end, pick a value z uniformly at random from the range f1; 2; : : : ; 1ı g. Using this value z define the partition of the vertex set as follows: o n i i 1 C z < d.v/  C z ; i D 0; 1; : : : ; l  1: Vi D v 2 V W ı ı The graph Gi is the graph induced by the vertices in Vi for i D 0; 1; : : : ; l  1. The idea is to compute an optimal max-coloring for each group separately, using a fresh palette for each Gi , and return the union as a max-coloring of G. It can be shown that this solution has weight that is not much more than OPTmc .G/. Note that the colorings produced by partitioning and solving the max-coloring problem on each group separately has the feature that each color class consists only of vertices from the same set Vi . In order to show that not much is lost by partitioning, consider an optimal max-coloring of the rounded instance of weight OPT˛mc .G/ with color classes C1 ; : : : ; Ck . Let d.Ci / D dlog˛ w.Ci /e. Define Dij D Ci \ Vj , for i D 1; : : : ; k and j D 0; : : : ; l  1. Then, the color classes Dij consist only of vertices from the set Vj . Let S ˛ .G/ denote the weight of the new coloring. Then, we have l1 k X hX i EŒS ˛ .G/ D E w.Dij / ; i D1 j D0

D

k h X E ˛ d.Ci / C i D1



k X

D

i D1

i w.Dij / ;

j

j W ı Cz ˇ  w2 . Let z be the vertex transferred by the algorithm from L to H in going from Œw1 ; w2 ; w3  to Œw01 ; w02 ; w03 , and let p be the subpath in H that z belonged to after the transfer. Notice that every vertex in p has weight w.z/ or more, and w3 D w.z/. If < w.z/, then all vertices in p such be colored using f1; 2g in the coloring Œ˛; ˇ; , but then, this implies that ˇ  w02 , since optimally, 2-coloring p alone requires this. This contradicts the assumption; thus,

 w.z/ D w3 .  To finish the proof, let Œ˛; ˇ;  be an optimal max-coloring of the input path. By Lemma 2, the algorithm must have recorded a signature Œw01 ; w02 ; w03  such that w01 C w02 C w03  ˛ C ˇ C . The algorithm return the coloring fA1 ; A2 ; A3 g associated with the best signature Œw1 ; w2 ; w3 ; that is, w1 Cw2 Cw3  w01 Cw02 Cw03 . Furthermore, by Lemma 1, w.A1 / C w.A2 / C w.A3 /  w1 C w2 C w3 . Therefore, the coloring returned by the algorithm is optimal.  It is natural to wonder whether the sorting time S.n/ is really necessary in the running time of MAX-COLORING-PATHS or whether perhaps there is a linear time algorithm. Kavitha and Mestre [22] showed a lowerbound of .n log n/ time in the algebraic computation tree model; this is a model where the operations fC; ; ; ; p; D 0;  0g on reals can be performed at unit cost. Theorem 2 ([22]) The complexity of max-coloring a path with real vertex weights is ‚.n log n/ in the algebraic computation tree model.

4

Max-Coloring Trees

The max-coloring problem on trees has turned out to be surprisingly difficult. This section starts with some simple upper bounds on mc .T / for any tree T and then gives a quasi-polynomial time algorithm for max-coloring trees exactly. Finally, it

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is shown how this algorithm can become the basis for obtaining a PTAS for the problem. From Lemma 2 on the property of optimal max-colorings of k-colorable graphs, one can obtain the following corollary for 2-colorable graphs and hence trees. Corollary 1 Let G be a 2-colorable graph with fC1 ; C2 ; : : : ; Ck g an optimal maxcoloring of G. Then, w.Ci /  2w.Ci C2 /, for i D 1;    ; k  2, and hence, w.C1 /  i 1 2b 2 c  w.Ci /. Corollary 1 states that the weights of the color classes decrease geometrically. Hence, one can expect that mc .T / may not be too large for any tree T , and this is captured by the following lemma. Lemma 6 Let T be a vertex-weighted tree on n vertices. Then, mc .T /  blog2 nc C 1. Proof Let fC1 ; : : : ; Ck g be an optimal max-coloring of T with mc .T / colors. For each vertex v 2 C1 , let T .v/ denote the rooted tree with one vertex, namely, v. For each v 2 Ci , i > 1, define T .v/ as the tree rooted at v, such that: 1. The children of v in T .v/ are exactly the neighbors of v in T belonging to color classes C1 ; C2 ; : : : ; Ci 1 . 2. For each child u of v, the subtree of T .v/ rooted at u is simply T .u/. For each i , 1  i  k, let Si D minfjT .v/j j v 2 Ci g. In other words, Si is the size of a smallest tree T .v/ rooted at a vertex v in Ci . Recall that each Ci is a maximal independent set in the graph induced by [kj Di Cj . This means that every vertex u 2 Ci has at least one neighbor in Cj for all j < i . Therefore, S1 D 1; and Si 

i 1 X

Sj C 1; for each i > 1:

j D1

This implies that Si  2i 1 for 1  i  k. Using the fact that Sk  n, one gets mc .T /  blog2 nc C 1.  The number of colors required by an optimal max-coloring also depends on the number of distinct weights. Indeed, if all vertex weights are the same, then the optimal max-coloring uses at most two colors. Hence, the optimal max-coloring can be bounded in terms of the number of distinct weights. This relation is stated precisely below. Lemma 7 Let T be a vertex-weighted tree on n vertices, and let W be the ratio of the weight of heaviest vertex to the weight of the least heavy vertex. Then, mc .T /  dlog2 W e C 1.

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Proof Let k D mc .T /, and let C1 ; C2 ; : : : ; Ck be an optimal max-coloring of T ; as usual, assume w.C1 /  w.C2 /      w.Ck /. Notice that w.C1 / equals the weight of the heaviest vertex in the tree. Let ` be the smallest integer such that w.v/  w1 =2` for all v 2 V ; in other words, ` D dlog2 W e. Consider the following collection of disjoint intervals I D fI0 ; I1 ; : : : ; I`1 g: 

w.C1 / w.C1 / Ii D ; 2i C1 2i





 w.C1 / ; w.C1 / : for i D 1; : : : ; `  1; and I0 D 2

Because of the choice of `, for each vertex v 2 V .T /, its weight w.v/ belongs to exactly one interval Ij . Let Vj D fv 2 V .T / j w.v/ 2 Ij g, j D 0; 1; : : : ; `  1. Vertex v is said to contribute to a color class Ci if v 2 Ci , and w.v/ D w.Ci /. The contribution of an interval Ij is the maximum number of vertices in Vj that contribute to distinct color classes. Corollary 1 says that w.Ci /  2  w.Ci C2 / for i D 1;    k  2. Therefore, no interval Ij , j D 1; 2; : : : ; `  1 has a contribution of more than two. If the contribution of I0 is three or more, then it must be the case that one can construct a 2-coloring with the same or smaller weight, compared to fC1 ; C2 ; : : : ; Ck g. This contradicts the fact that k D mc .T /, and C1 ; C2 ; : : : ; Ck is an optimal max-coloring of T . Now suppose that intervals Ii1 ; Ii2 ; : : : ; Iit , 0  i1 < i2 <    < it  `  1 are the sequence of all intervals in I, each of whose contribution is two. The following claim can be shown. Claim 3 For any pair of consecutive intervals Ip , p D ij and Iq , q D ij C1, where j < t; it is the case that there is an interval in fIpC1 ; IpC2 ; : : : ; Iq1 g with contribution zero. Given this claim, one can charge the “extra” contribution of each Iij to an interval between Iij and Iij C1 , whose contribution is zero. This implies that the contributing vertices in all intervals except It can be reassigned to a distinct interval. Since there are ` intervals and since the contribution of It is at most two, there is a total contribution of at most ` C 1, implying that there are at most ` C 1 color classes. The claim is proved by contradiction, assuming that the contribution of every interval in fIpC1 ; IpC2 ; : : : ; Iq1 g is one. Let fxp ; xpC1 ; : : : ; xq g [ fyp ; yq g be vertices such that: (1) For each j D p; p C 1; : : : ; q, xj 2 Vj and xj contribute to some color class. (2) For each j 2 fp; qg, yj 2 Vj and xj and yj contribute to distinct color classes. 1/ Since xj 2 Vj , w.xj /  w.C , j D p; p C 1; : : : ; q. Also, since yq 2 Vq , 2j C1 w.C1 / w.yq /  2qC1 . Therefore,

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q X w.C1 / w.C1 / w.xj / C w.yq /  C qC1 j C1 2 2 j Dp j Dp

D w.C1 / D

2qpC1  1 w.C1 / C qC1 qC1 2 2

w.C1 / > w.yp / 2p 

This contradicts Lemma 2 and proves the claim.

The upperbounds in the preceding lemmas are all tight, as the following example shows. Let T0 ; T1 ; : : : be a sequence of trees, where T0 is a single vertex, with weight 1, and Ti , i > 0 is constructed from Ti 1 as follows: Let V .Ti 1 / D fu1 ; u2 ; : : : ; uk g. To construct Ti , start with Ti 1 and add a set of new vertices fv1 ; v2 ; : : : ; vk g, each with weight 2i , and edges fui ; vi g for all i D 1; 2; : : : ; k. Thus, the leaves of Ti are fv1 ; v2 ; : : : ; vk g, and every other vertex in Ti has a neighbor vj for some j . Now consider a tree Tn in this sequence. Clearly, jV .Tn /j D 2n and the maximum vertex degree of Tn , .Tn / D n1. See Fig. 1 for T0 ; T1 ; T2 ; and T3 . Now consider the coloring of Tn defined by Ci D fleaves of TnC1i g, 1  i  n C 1. Note that w.Ci / D 2nC1i and therefore the weight of the coloring is 2nC1  1. It is not hard to see that any coloring using fewer than n C 1 colors has weight at least 2nC1 . Therefore,  mc .T / D n C 1 D log2 jV .Tn /j C 1 D  C 1 D log2

2n 1

 C 1:

Everything is in place to describe the algorithm to find an optimal max-coloring. From Lemma 6, it follows that mc .T /  log2 n C 1. Hence, there are at most nO.log n/ candidate weight sequences corresponding to feasible max-colorings. A candidate weight sequence consists of a sequence .W1 ; : : : ; Wk / of weights, with k  dlog2 ne, W1  W2  : : :  Wk , and each Wi is the weight of some vertex in T . For a given candidate weight sequence .W1 ; : : : ; Wk /, the algorithm needs to decide if there is a coloring of T that respects this weight sequence. A coloring of T respects the given weight sequence if the coloring is proper and uses at most k colors C1 ; : : : ; Ck , and maxv2Ci w.v/  Wi for each i D 1; : : : ; k. The feasible-k-coloring problem asks is to find, given a weighted tree T and a sequence .W1 ; W2 ; : : : ; Wk /, a coloring satisfying the sequence .W1 ; : : : ; Wk /, or if such a coloring does not exist, to report so. A simple dynamic programming algorithm solves feasible-k-coloring on trees in O.nk/ time. Let T be rooted at an arbitrary vertex r. Let children.v/ denote the set of children of v, and let parent.v/ denote the parent of v. For a vertex v, let T .v/ denote the subtree of T rooted at v. Let Œk denote the set of colors f1;    ; kg. For a vertex v, let F .v/ D fj j w.v/ > Wj g be the set of forbidden colors, and let S.v/ D fi W there exists a feasible coloring of T .v/ with .v/ D i g :

(2)

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T1

T2

T3

Fig. 1 Sequence of trees which show that the upperbounds of Lemmas 1, 6 and 7 are all tight. Ti is obtained by adding the light-colored vertices to Ti1 . The figure above shows the trees T0 ;    ; T3

Algorithm 3 computes a feasible coloring, if one exists; we call this algorithm FKC. The correctness of the algorithm and the running time are easily established and are described in the lemma below. Lemma 8 Algorithm FKC solves the feasible-k-coloring problem in O.nk/ time.

Proof Assume that the sets S.v/ computed by the algorithm FKC follow Eq. (2). Given this, it is trivial to see that the algorithm indeed solves the feasible-k-coloring problem. If there exists some vertex v such that S.v/ D ;, then S.r/ D ;, and clearly, there is no feasible coloring. Otherwise, for each v ¤ r, one must argue that Line 14 can assign a valid color to v. Indeed, if jS.v/j > 1, this can always be done, and if jS.v/j D 1, then the single color present in S.v/ does not belong to S.parent.v//. Thus, Line 14 can always be carried out.

Algorithm 3 FKC.T; w; W / 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

 ˚ Let F .v/ D j W w.v/ > Wj , 8 v 2 T for each vertex v in a post-order traversal of T do Let S.v/ D Œk  F .v/ for each u 2 children.v/ do if S.u/ D ; then Set S.v/ D ; if jS.u/j D 1 then Set S.v/ D S.v/  S.u/ if S.r/ D ; then return no feasible color exists else Pick an arbitrary i 2 S.r/ and set .r/ D i for each v ¤ r in a pre-order traversal of T do Pick an arbitrary j 2 S.v/  .parent.v// Set .v/ D j return 

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Consider the following inductive argument on the height of T .v/ to show that the sets S.v/ obey (2): If the height of T .v/ is 0, then v is a leaf, and so, S.v/ D Œk  F .v/, which is precisely what the algorithm returns. Assume the claim is true for all subtrees of height < m. Let v be a vertex such that T .v/’s height is m. Suppose that v has a child u such that S.u/ D ;. By inductive hypothesis, T .u/ does not admit a feasible coloring and so neither does T .v/. Hence, the assignment S.v/ D ; in Line 6 is correct. Now suppose that v has a child u such that S.u/ D fcg. By inductive hypothesis, S.u/ obeys (2), and so c is the only color that can be assigned to u in any feasible coloring of T .u/. Hence, the c cannot be given to v, namely, c … S.v/, and so Line 8 correctly removes c from S.v/. For each color that remain in S.v/, v can be assign color and then extend the coloring to the remaining vertices in T .v/ by following for loop of Line 13. This finishes the proof that the S.v/ sets obey Eq. (2). 

Theorem 3 There is quasi-polynomial time algorithm for max-coloring trees that runs in nO.log n/ time. Proof Since mc .T / is at most log2 n C 1 for any tree T , the algorithm only has to consider weight sequences .W1 ; W2 ; : : : ; Wk / with k  log n C 1. There are nlog nC2 possibilities. Each sequence can be checked for feasibility with FKC in O.n log n/ time. An optimal max-coloring is simply a feasible sequence minimizing W1 CW2 C    C Wk .  The main idea underlying the PTAS is the reduction of the number of distinct weights of the vertices down to a constant. Candidates for the weights of the color classes are chosen; then for each such choice, using the algorithm for feasible-kcoloring, it can be tested if there is a legal coloring of the tree with the chosen weights for the color classes. Let T be the input graph. Using Lemma 3, one can, with a loss of 1 C  in the objective, reduce the general problem to a number of smaller instances T0 ; T1 ; : : : such that the number of distinct to the special problem where the number of weights used in each Ti is at most O.1= 2 /. Together with the FKC algorithm, this means 2 that a .1 C / approximate solution can be found in O.nO. / / time. Below is an alternative proof that improves the dependency on 1= in the exponent. Theorem 4 There is a PTAS for max-coloring trees. Proof Let T be a tree, with weight function w W V ! N and an  > 0. Let c > 0 be an integer such that .2 log c C 3/=c  , and let ˛ D .w1  1/=c. Let I1 ; I2 ;    ; Ic be a partition of the range Œ1; w1 /, where Ii D Œ1 C .i  1/˛; 1 C i  ˛/, 1  i  c. Let T 0 be a tree that is identical to T , except in its vertex weights. The tree T 0 has vertex weights w0 W V ! N defined by the rule for any v 2 V , if w.v/ 2 Ij , then w0 .v/ D 1 C .j  1/  ˛, and if w.v/ D w1 , then w0 .v/ D w1 . In other words, except for vertices with maximum weight w1 , all other vertices have their weights “rounded” down. As a result, T 0 has c C 1 distinct vertex weights. Now let OPT0mc

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denote the weight of an optimal max-coloring of T 0 and let C 0 D C10 ; C20 ; : : : ; Ck0 be the color classes corresponding to OPT0mc . Since the weights of vertices have fallen in going from T to T 0 , clearly, OPT0mc  OPTmc . The coloring C 0 for T has weight, is at most OPT0mc C k˛. Substituting .w1  1/=c for ˛ and noting that w1  OPT0mc , one can see that weight of C 0 used as a coloring for T 0 is at most .1 C kc /OPT0mc . It was shown that given the distribution of vertex weights of T 0 , k D O.log c/. To 0 0 see this, first observe that the weights of last three color classes Ck0 , Ck1 , and Ck2 cannot all be identical, by Lemma 2. Also, observe that the possible vertex weights 0 of T 0 are 1; 1 C ˛; 1 C 2˛; : : :. Therefore, w.Ck2 /  1 C ˛. From Corollary 1, 1 C ˛  w.Ck2 / 

w1 : 2b.k3/=2c

Solving this for k yields k  2 log2 .c/ C 3. Therefore, by the choice of c, 2 log2 .c/ C 3 k   : c c Thus, .1 C /OPT0mc is an upperbound on the weight of C 0 used as a coloring for T . Since OPT0mc  OPTmc , it follows that the weight of OPT0mc used as a coloring for T is at most .1 C /OPTmc . To construct OPT0mc in polynomial time, for each k D 1; : : : ; 2 log c C 3, generate all O.c k / possible sequences of weights and call algorithm FKC for each subsequence and pick the coloring with the minimum weight. This gives OPT0mc . Each call to FKC takes O.nk/ time, and we have O.c k / sequences, for k D 1; : : : ; 2 log c C 3. Using the fact that .2 log2 c C 3/=c  , a little bit of algebra yields a running time that is linear in n and exponential in 1=.  The same algorithm can be extended to coloring graphs of bounded treewidth. Theorem 5 ([14]) There is a PTAS for max-coloring graphs of bounded treewidth, provided a tree-decomposition is given.

5

Max-Coloring Bipartite and k-Colorable Graphs

The max-coloring problem on bipartite graphs, unlike the case with trees, is fully resolved. There is a 87 -approximation algorithm and is inapproximable beyond 87   for any  > 0 unless P D NP. Further, the approach used to max-color bipartite graphs can be extended to an approximation algorithm for coloring k-colorable graphs for which a k-coloring can be obtained in polynomial time. Figure 2 shows that an optimal max-coloring may use as many as O.n/ colors. The example is a graph G D .U [ V; E/, where jU j D jV j D n. The vertices in U

Max-Coloring Fig. 2 An instance of max-coloring of a bipartite graph on 2n vertices that requires n colors in an optimal max-coloring. The weights of the vertices are powers of 2 and are shown next to the vertices. A k-coloring for any k < n has weight at least 2n , while a coloring with n colors has weight 2n  1

1895

1

1

2

2

... 2n−1

... 2n−1

are labeled 1; : : : ; n. and the vertices in V are labeled 1; : : : ; n. Vertex i has weight 2i 1 . Each vertex i 2 U is adjacent to all vertices in V except the vertex labeled i . A 2-coloring of G with all vertices in U colored 1 and all vertices in V colored 2 has a weight of 2n1 C 2n1 D 2n , while an optimal coloring consists of assigning color i to the vertices labeled i in both U and V , which has a weight of 2n  1. The PTAS for the max-coloring problem on trees relied on the fact that the feasible-k-coloring problem on trees can be solved in polynomial time for any k. However, feasible-k-coloring is NP-complete for bipartite graphs for k  3 as shown by Kratochvil et al. [25]. Hence, a different approach for k-colorable graphs is needed. Consider the following algorithm that uses at most k C 2 colors for maxcoloring a k-colorable graph. Assume that a polynomial time algorithm is given to compute an arbitrary proper k-coloring of the input graph. The algorithm exploits the fact that an optimal max-2coloring of a bipartite graph can be found efficiently. The basic idea is to optimally 2-color the i heaviest vertices in the graph (if possible) and then use the k-coloring algorithm to color the rest. We repeat this for each i D 0; : : : ; n, and return the best coloring found. Call this algorithm MAX-COLORING-K-COLORABLE. Its pseudocode is given in Algorithm 4 . Theorem 6 Provided with a procedure for finding a proper k-coloring of the input k3 graph, algorithm MAX-COLORING-K-COLORABLE is a 3k 2 3kC1 -approximation for max-coloring. Proof First consider the issue of finding an optimal 2-coloring of GŒVi . Notice that the graph to be colored is a collection of connected components. Each component is connected bipartite graph, which admits only two proper 2-colorings. Color each component so that the heaviest color class is 1; the resulting coloring of GŒVi  is an optimal max-2-coloring. Thus, provided a k-coloring of the input graph can be computed, the algorithm MAX-COLORING-K-COLORABLE runs in polynomial time.

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Algorithm 4 MAX-COLORING-K-COLORABLE.G; w/ 1. Sort the vertices in non-increasing weight order v1 ; v2 ; : : : ; vn 2. for i 2 f0; ng do fv1 ; : : : ; vi g 3. Vi 4. if GŒVi  is bipartite then 5. optimally 2-color GŒVi  6. arbitrarily k-color GŒV n Vi  using k new colors 7. let i be the resulting coloring 8. return best coloring among the i for i D 1; : : : ; n

Second, consider the quality of the solution returned. Three different upper bounds on the value of the solution returned by the algorithm are derived. Later, it will be argued that the best of these bounds attains the desired approximation k3 ratio of 3k 2 3kC1 . The value of the solution found is denoted by MCKC. Assume the vertices are sorted in non-increasing weight order v1 ; v2 ; : : : ; vn . Let fC1 ; : : : ; Ck  g be an optimal max-coloring for G. Let ji be the smallest index such vji 2 Ci for each i D 1; : : : ; n. Notice that when i D 0, the graph GŒVi  is empty, so there is no cost incurred by the 2-coloring step. Hence, MCKC  kw.C1 /: (3) Let i be the largest index such that w.vi / > w.vj2 /. Notice that the set Vi is independent in G. Therefore, the graph GŒVi  can be colored with a single color class at the cost of w.C1 /. The rest of the graph GŒV n Vi  can be colored with additional k colors at a cost of at most kw.C2 /. Hence, MCKC

 w.C1 / C kw.C2 /:

(4)

Finally, let i be the largest index such that w.vi / > w.vj3 /. In this case, the graph GŒVi  is bipartite, and the coloring fC1 \ Vi ; C2 \ Vi g has a cost of at most w.C1 / C w.C2 /. The rest of the graph GŒV n Vi  can be colored with additional k colors at a cost of at most kw.C3 /. Hence, MCKC

 w.C1 / C w.C2 / C kw.C3 /:

(5)

Now combine these three bounds using the right multipliers: Take .k1/ k3 1 times (3), plus k.k1/ times (4), plus times (5) to obtain the following equation: 3 k k MCKC



2

3k 2  3k C 1  w.C1 / C w.C2 / C w.C3 /: k3

Recall that fC1 ; : : : ; Ck  g was an optimal max-coloring. Thus, it was shown that k3 is a 3k 2 3kC1 approximation. 

MAX - COLORING - K - COLORABLE

Max-Coloring

1897

For bipartite graphs, the algorithm yields an 87 -approximation that uses at most four colors. One might be tempted now to continue this and try to come up with an algorithm that uses five or more colors and obtains a better approximation ratio for bipartite graphs. Surprisingly, no such improvement is possible in polynomial time as there is a gap introducing reduction from the problem of pre-coloring extension on graphs which was shown to be NP-complete by [25]. Theorem 7 ([9], [30]) For any  > 0, there is no .8=7  /-approximation algorithm for max-coloring on bipartite graphs, unless P D NP.

5.1

Planar Graphs

The four-color theorem states that every planar graph can be colored at most four colors. Robertson et al. [35] provide an O.n2 / time algorithm for 4-coloring any planar graph. Using Theorem 6, one gets a 64 < 1:73 approximation for max37 coloring planar graphs. On the negative side, the hardness of graph coloring planar graphs [16] means it is NP-hard to approximate max-coloring better than 4=3 > 1:33. Theorem 8 Max-coloring in planar graphs can be approximated within within 43   for any  > 0 unless P D NP .

64 37

but not

Gr¨otzsch theorem states that every triangle-free planar graph can be 3-colored. Dvorak et al. [10] give a linear time algorithm to compute such a coloring. Using Theorem 6, one gets a 27 19 < 1:43 approximation for max-coloring triangle-free planar graphs. On the negative side, De Werra et al. [7] showed that in this class of graphs, max-coloring cannot be approximated better than 76 > 1:16. Theorem 9 Max-coloring in triangle-free planar graphs can be approximated 7 within 22 19 but not within 6   for any  > 0 unless P D NP .

5.2

Restricted Bipartite Graphs

DeWerra et al. [7] have attempted to further delineate the boundary between polynomial time solvable and NP-hard cases of the max-coloring problem. The authors consider bipartite graphs that do not contain an induced path on l vertices. Theorem 10 ([7]) Max-coloring on P8 -free bipartite graphs is NP-complete. Further, they show that for P5 -free bipartite graphs, the problem can be solved in polynomial time. There are several characterizations of P5 -free bipartite graphs (See [7] and references therein). In particular, a bipartite graph G is P5 -free if and

1898

J. Mestre and R. Raman

only if each connected component of G is 2K2 -free, that is, it does not contain edges whose vertices are nonadjacent. The authors then use this characterization to prove that any optimal max-coloring of a P5 -free bipartite graph uses at most three colors. Theorem 11 ([7]) Max-coloring can be solved in polynomial time on P5 -free bipartite graphs. Further, the authors observe that P5 -free bipartite graphs have clique-width at most 5 and leave as an open question the problem of max-coloring on graphs of bounded clique-width.

5.3

Split Graphs

A graph class similar to bipartite graphs is that of split graphs. A graph G D .K [ S; E/ is a split graph if its vertex set can be partitioned into two sets K and S such that K is a complete graph on n1 vertices, and S is an independent set on n2 vertices. The edge set of G includes an arbitrary bipartite graph between the vertices of K and S . Split graphs are a subclass of perfect graphs, and hence, classic coloring is easy. It is also easy to see that n1  .G/  n1 C 1 for any split graph. The maxcoloring problem on split graphs, however, is NP-hard even with just two distinct weights as shown by Demange et al. [8]. However, a PTAS can easily be obtained as follows: Consider an optimal maxcoloring of a split graph G, C1 ; : : : ; Ck where k 2 fn1 ; n1 C 1g. There is some i0  k C 1 such that all color classes with index i < i0 have at least one vertex from K, and the remaining color class i0 has all the remaining vertices of S . For a fixed  > 0, trying all bijections between K and the first d 1 e color classes, one obtains a solution which is at least as good as the first d 1 e color classes. Theorem 12 ([7]) Max-Coloring on Split Graphs admits a PTAS.

6

Max-Coloring Interval Graphs

An interval graph is an intersection graph of intervals on the real line. Recall that given an interval graph, an interval representation of it can be produced in polynomial time. Hence, assume that the graph is given as a set of intervals on the x-axis. Each interval has a weight w. If two intervals overlap, their corresponding vertices are adjacent. Interval graphs arise in several applications including resource allocation and scheduling and hence have been the subject of intensive study over several years. In particular, the online coloring problem on interval graphs is well studied, and it is known that the first-fit coloring algorithm is at most 8-competitive ([27,31], G. Brightwell, H.A. Kierstead and W.T. Trotter, 2003, Firstfit coloring of interval graphs, Personal communication). Applying Lemma 4 form

Max-Coloring

1899

Algorithm 5 BETTER-MCA.G; w/ 1. Sort the vertices of G in non-increasing order of weights // Let .v1 ;    ; vn / be this ordering of the vertices of G ; 2. Let k 1; Sk 3. for j 1 to n do 4. Insert vj into set Si 5. where i  k is the smallest value such that 6. S1 [ S2 [ : : : [ .Si [ fug/ does not contain an .i C 1/-clique 7. If no such i exists, 8. set k k C 1 and insert vj into Sk 9. Use color 1 to color vertices in S1 10. for i 2 to k do 11. Use colors 2i  2 and 2i  1 to 2-color the vertices in Si 12. return coloring

Sect. 2 immediately gives an approximation algorithm for max-coloring with an approximation factor of 8, after noting that interval graphs form a hereditary graph class. The best-known algorithm for online coloring interval graphs is a slight modification of the first-fit algorithm. This algorithm by Kierstead and Trotter [24] has a competitive ratio of 3, and now with Lemma 4, one immediately gets a 3-approximation algorithm for max-coloring. What follows is a description of the Kierstead-Trotter algorithm, which is the basis for the 2-approximation algorithm. The Kierstead-Trotter algorithm maintains sets S1 ; S2 ; : : : such that when a vertex u is presented, it finds the smallest i such that S1 [ S2 [ : : : [ .Si [ fug/ does not contain an .i C 1/-clique. (Recall that a maximum clique can be found in polynomial time on interval graphs.) The vertex u is then inserted into Si . The fact that the Kierstead-Trotter algorithm is 3-competitive is a consequence of the following observations: 1. Let k be the largest index of a nonempty set Si . Then, an interval I was placed in Sk since it induced a k-clique with intervals presented before. Hence, k  !.G/. 2. Let I1 ; I2 ; I3 be a triple of intervals in Si that pairwise intersect. Without loss of generality, assume that I3  I1 [ I2 . Then, it follows that I3 induces an i C 1clique with intervals presented before it and hence cannot have been placed in Si . Hence, each Si is a disjoint union of paths. Let k be the largest index of a nonempty set Si . Then, k  !.G/ D .G/. Since each Si is a disjoint union of paths, each Si can be online 3-colored. Hence, the total number of colors used is at most 3  k  3  .G/. However, since the max-coloring problem is an off-line problem, color each set Si off-line using at most two colors. The algorithm is described in detail in Algorithm 5 . Theorem 13 BETTER-MCA is a 2-approximation algorithm for the max-coloring problem on interval graphs.

1900

J. Mestre and R. Raman

Proof Let C1 ; C2 ; : : : ; Ck be a coloring of G that is optimal for the max-coloring problem. Let wi D maxv2Ci w.v/, and without loss of generality, assume that w1  P w2      wk . Now note that k  .G/ and OPTmc .G/ D kiD1 wi . Suppose that at the end of Step (6) in BETTER-MCA, we have sets S1 ; S2 ; : : : ; St . An element is inserted into St only because S1 [ S2 [ : : : [ .St 1 [ fug/ has a t-clique. Therefore, .G/  t, and it follows that t  k. Let si D maxv2Si w.v/. The claim is that for each i , 1  i  t, si  wi . It is clear that s1 D w1 . To obtain a contradiction, suppose that for some i , si > wi , and let i be the smallest such value. This implies any vertex x with weight si or larger is in an earlier color class, Cj , for some j < i in the optimal coloring. Therefore, vertices with weight si or larger are .i  1/-colorable, so they induce a clique of size at most i  1, and therefore, in Step (6), no vertex with weight si will get inserted into Si – a contradiction. In Steps (9) and (10), the vertex partition S1 ; S2 ; : : : ; St is converted into a P coloring whose weight is at most s1 C 2  ti D2 si . The following inequalities s1 C 2 

t X i D2

si  w1 C 2 

t X

wi  2  OPTmc .G/

i D2

give the result sought.



Pemmaraju et al. [31] showed via a reduction from E4-set splitting that maxcoloring on interval graphs is NP-hard. A simpler proof of NP-hardness was discovered by Escoffier et al. [14] and independently by Pemmaraju et al. [32] via a reduction from coloring circular-arc graphs. Theorem 14 ([14, 32]) Max-Coloring interval graphs is NP-hard. The results on NP-hardness of max-coloring interval graphs left open the question of whether there exists a PTAS for the problem. This has been answered in the affirmative recently by Nonner [28].

7

Max-Coloring on Hereditary Graphs

Recall that a class of graphs closed under taking vertex-induced subgraphs is called hereditary. Let G be a hereditary class of graphs for which the classic vertex coloring admits a c-approximation algorithm. In this section, we describe approximation algorithms for the max-coloring problem on this class of graphs. The algorithm is quite natural and works by partitioning the vertices of the input graph G into weight classes, where each weight class consists of vertices of nearly equal weight, and solving the classic coloring problem in the subgraph induced by each weight class separately using a fresh palette for each. The section first describes how this approach leads to a 4-approximation algorithm for the max-coloring problem and

Max-Coloring

1901

later shows how the approximation factor can be improved to e, the base of the natural logarithm. Let G D .V; E/ be the input graph with weight function w. Let wmin and wmax denote the minimum and maximum weight vertices, respectively, and assume wmin D 1 without loss of generality. The algorithm partitions the vertices of G into weight classes, R0 ; R1 ; : : : ; Rdlog2 wmax e , where Ri D fv W w.v/ 2 Œ2i ; 2i C1 /g, and applies the c-approximate coloring algorithm for each weight class ignoring the weights. This algorithm is called WP, for weighted partition. Theorem 15 Let G be a hereditary class of graphs which admit a c-approximation algorithm for coloring. Then WP is a deterministic 4c-approximation algorithm for max-coloring for the class G of graphs. Proof Let C1 ; : : : ; Ck be the color classes of OPTmc . Consider a fixed color class Ci , and let jmax D maxfj W Rj \ C ¤ ;g. Partition C into at most jmax C 1 classes Dji D C \ Rj , j D 0; : : : ; jmax . Then, since this partition colors vertices in each Ri with the same color,

WP



jmax k X X

c  w.Dji /  c 

i D1 j D0

c

k X

 4c 

k X

2j C1

i D1 j D0

2jmax C2  2c 

i D1

jmax k X X

k X

2jmax C1

i D1

w.Ci / D 4c  OPTmc



i D1

To improve the approximation factor, the algorithm has to randomize the partitioning of the vertices into weight classes. Let q be a constant to be chosen later, and let ˛ be chosen uniformly at random from Œ0; 1. We now redefine the weight classes as Ri D fv W w.v/ 2 Œq i C˛ ; q i C1C˛ /g Using this new partitioning and applying the c-approximation algorithm within each weight class, one obtains an ec-approximation algorithm. The algorithm, called RWP, is described in Algorithm 6 . Theorem 16 Let G be a hereditary class of graphs which admit a c-approximation algorithm for coloring. Then RWP is an ec-approximation algorithm for maxcoloring for the class G of graphs. Proof Let C1 ; : : : ; Ck be the color classes of OPTmc . For color class Ci , let w.Ci / D q xi . Ri D fi W w.v/ 2 Œq i C˛ ; q i C1C˛ /g, for i D 0; 1; : : : ; dlogq wmax e. Let jmax D maxfj W Rj \ Ci ¤ ;g. Then, partitioning Ci into at most jmax C 1

1902

J. Mestre and R. Raman

Algorithm 6 RWP.G; w/ 1. Choose ˛ uniformly at random from Œ0; 1/ 2. for i D 0; 1;    do 3. Let Ri D fvj q iC˛ < w.v/  q iC1C˛ g 4. Color each Ri using the c-approximation algorithm 5. return Union of the individual colorings

color classes, Dj , where w.v/ 2 Rj , for all v 2 Dj , there is a coloring such that each color class contains vertices from only one set Ri . However, since RWP colors each set Ri using a c-approximation algorithm, jmax jmax k X k h X i hX i X EŒRWP   E c  w.Dji /  c E q j C1C˛ ; i D1 j D0

i D1

j D0

c

Z k k h q jmax C1C˛C1 i X X q xC1 1 1C˛x Dc E q d˛; q1 q1 0 i D1 i D1

Dc

k k X qc X q xC1 q  1 qc D  OPTmc : w.Ci /  q  1 ln q ln q i D1 ln q i D1

Since lnqq is minimized at q D e, the base of the natural logarithm, the algorithm is an ec-approximation for max-coloring.  It is worth noting that RWP can be de-randomized by running the algorithm on some critical set of ˛ values and then keeping the best solution produced. For each vertex v, there is a unique value ˛v at which the vertex changes from Ri to Ri C1 for some i . For values of ˛ in between two consecutive critical values, the algorithm behaves the same way, so the minimum cost solution produced at these critical value is no more than the average case.

7.1

The Online Case

Even though the proof of Theorem 15 provided above focused on the off-line setting, the same algorithms and the same arguments can be used to prove the following theorem about the online setting. The only difference is that one cannot derandomize the second algorithm, so in this case, there a gap between deterministic and randomized performance Theorem 17 Let G be a hereditary class of graphs which admits a c-competitive algorithm for online coloring. Then WP is a deterministic 4c-competitive algorithm

Max-Coloring

1903

and RWP is a randomized ec-competitive algorithm for online max-coloring graphs in the class G. Two applications to max-coloring interval graphs are discussed to exemplify the usefulness of the above theorem. Assume that the interval representation, rather than the graph itself, is revealed. Two interval arrival models are considered: In the list model, intervals arrive in some arbitrary, while in the time model, intervals arrive sorted by their left endpoint. First fit is a simple heuristic that assign, the lowest color available to an incoming interval. It is well known that first fit is optimal in the time model, namely, first fit is a deterministic 1-competitive algorithm in the time model. Plugging this result into Theorem 17, one gets the following: Corollary 2 There is a 4-competitive deterministic algorithm and a e-competitive randomized algorithm for max-coloring interval graphs in the time model. In the list model, a different algorithm by Kierstead and Trotter [24] is 3-competitive for online coloring interval graphs. Plugging this result into Theorem 17, one gets the following: Corollary 3 There is a 12-competitive deterministic algorithm and a 3ecompetitive randomized algorithm for max-coloring interval graphs in the list model. Finally, it should be noted that better algorithms are known for restricted classes of interval graphs. Unit interval graphs are a special class of interval graph where every interval has unit length. Epstein and Levy [13] designed a 2-competitive algorithm for this special class. Thus, one gets the following result: Corollary 4 There is an 8-competitive deterministic algorithm and a 2ecompetitive randomized algorithm for max-coloring unit interval graphs in the list model.

8

Max-Edge-Coloring

This section first describes a simple approximation algorithm for max-edge-coloring a graph G. Later, it is shown how the approximation ratio can be improved slightly. The algorithm processes the edges in non-increasing weight order. When an edge is considered, it assigns the lowest color possible so as not to create a conflict with adjacent previously colored edges. This algorithm is called GREEDY-MAX-EDGECOLORING . Its pseudo-code is given in Algorithm 7 . Every edge-coloring  is just a collection of matchings fM1 ; M2 ; : : : ; Mk g. As usual, these matchings are listed in non-increasing weight order; that is,

1904

J. Mestre and R. Raman

Algorithm 7 GREEDY-MAX-EDGE-COLORING.G; w/ 1. 2. 3. 4. 5. 6.

A.u/ f1; 2; : : : ; 2  1g, for all u 2 V sort edge in non-increasing weight for .u; v/ 2 E in sorted order do .u; v/ D min A.u/ \ A.v/ remove .u; v/ from A.u/ and from A.v/ return 

w.M1 /  w.M2 /      w.Mk /. In the proofs, we will say that a coloring fM1 ; : : : ; Mk g is nice if for all .u; v/ 2 Mi there is no j < i such that both u and v are unmatched in Mj . For a given input instance .G; w/, denote by fC1 ; : : : ; Ck  g an optimal maxcoloring of G; assume w.C1 /      w.Ck  /. In addition, OPTmc is used to denote P  the cost of this optimal solution, that is, OPTmc D ki D1 w.Ci /. ˚ Theorem o18 The algorithm GREEDY-MAX-EDGE-COLORING is a min 2  2

wmax OPTmc

1 ;

approximation for general multigraphs, where wmax D maxe w.e/.

Proof Start by showing that the algorithm indeed computes a coloring. The only subtlety that needs to be justified is that Line 4 indeed assigns a color to .u; v/. At the beginning of the algorithm, A.u/ \ A.v/ D f1; : : : ; 2  1g, so initially, this is trivially feasible. By the time .u; v/ is considered in the for loop, at most 1 edges incident on u and another 1 edges incident on v could have been colored already. Each edge can potentially disallow a distinct color; however, even after removing these 2.  1/ colors from f1; : : : ; 2  1g, there must remain at least one color that can be used for .u; v/. Let fM1 ; : : : ; Mk g be the coloring computed by the algorithm. For each i D 1; : : : ; k, let ei be the edge in Mi attaining the maximum weight in the matching (i.e., w.ei / D w.Mi /) and let Ei be the edges that were colored before ei and ei itself. Since the coloring computed by the schedule is nice, it follows that the ˙ for each i D 1; : : : ; k. All edges in Ei subgraph Ei has degree at least i C1 2 ˙ matching in the have weight w.ei / or larger. Hence, there must be at least i C1 2 optimal solution whose weight is at least w.ei /. In other words, w.Cd i C1 e /  w.ei /. 2 Therefore, k X i D1

w.Mi / D

k X

w.ei / 

i D1

 w.C1 / C 2

21 X i D1

 X i D2

w.Cd i C1 e /

w.Ci /

2

Max-Coloring

1905 

D 2  OPTmc  w.C1 /  2

k X

w.Ci /

i DC1

D OPTmc

 OPTmc

! P  w.C1 / C 2 ki DC1 w.Ci / 2 Pk  i D1 w.Ci / ! w.C1 / 2  P i D1 w.Ci /

The proof is completed by noting that wmax .

P

i D1

w.Ci /  w.C1 / and w.C1 / D 

Recall that when edge weights are uniform, max-edge-coloring reduces to regular edge-coloring. In the uniform case, GREEDY-MAX-EDGE-COLORING returns an arbitrary nice edge-coloring. Simple examples exist showing that such a coloring can be 2  1 away from the optimal max-coloring. However, for instances where the weights are uniform, or close to uniform, one can simply ignore the weights and approximate the underlying edge-coloring instance. The next lemma states that taking the best between these two strategies leads to an improved approximation algorithm. Lemma 9 Let  be the edge-coloring returned by GREEDY-MAX-EDGE-COLORING and b  be an arbitrary edge-coloring that uses b k colors. Then the best solution between these two is a 2  2 approximation for max-edge-coloring. b kC1 Proof Recall that by Theorem 18   wmax w./  OPTmc 2  : OPTmc On the other hand,

w.b /  b k  wmax :

Consider taking a convex combination of these two inequalities with coefficients b k and 1 , respectively. The result is b b kC1 kC1   b b b k  w./ w.b 2 / k  wmax b k  wmax k C  2OPTmc   C D OPTmc 2  b b b b b kC1 b kC1 kC1 kC1 kC1 kC1 This proof is finished by noting that any convex combination of w./ and w.b / can only be larger than min fw./; w.b /g. 

1906

J. Mestre and R. Raman

Thus, if one can efficiently edge-color the input graph with few colors, then we can obtain a slightly better approximation than that offered by Theorem 18. For bipartite multigraphs a -edge-coloring can be computed in polynomial time [36, Ch. 20]. For general simple graphs, Vizing’s algorithm finds an . C 1/-edgecoloring in polynomial time [36, Ch. 28]. For general multigraphs, Shannon’s algorithm finds a 32 -edge-coloring in polynomial time [36, Ch. 28]. From these facts, one obtains the following theorem. 2 Corollary 5 There is a 2  C1 approximation for bipartite multigraphs, a 2 2 approximation 2  C2 approximation for general simple graphs, and a 2  1:5C1 for general multigraphs.

On the negative side, max-edge-coloring inherits the hardness of regular edgecoloring in general graphs. Unlike, regular edge-coloring, it remains hard even for bipartite graphs. Theorem 19 ([7, 20]) For any  > 0, max-edge-coloring cannot be approximated within 43   in general graphs or within 76   in bipartite graphs.

9

Bounded Max-Coloring

This section studies the bounded max-coloring problem that is defined as follows Given a graph G D .V; E/ with weight function w W V ! N on the vertices and a number m 2 N, the bounded max-coloring problem is to produce a proper vertex coloring of P G such that if C1 ; : : : ; Ck are the color classes, then jCi j  m for i D 1; : : : ; k, and kiD1 maxv2Ci w.v/ is minimized. An algorithm for hereditary graph classes is presented that transforms an ˛-approximation algorithm for the max-coloring problem into an ˛ C 1approximation for the bounded max-coloring problem. The section first describes this generic reduction and then proceeds to discuss special graph classes. Theorem 20 ([6]) For a class of graphs G if there is an ˛-approximation algorithm for the max-coloring problem, then for any m 2 N, there is an ˛ C 1-approximation algorithm for the bounded max-coloring problem with a bound of m on the size of any color class. Proof The proof is algorithmic. Given an ˛-approximate solution to the maxcoloring problem, an algorithm converts this solution into a feasible solution to the bounded max-coloring problem. The algorithm simply sorts the vertices in each color class in non-increasing order of weights and partitions each color class into sets of size at most m in this order. The algorithm is described in Algorithm 8 . The goal is to bound the performance of Algorithm B-MAX-COLOR which we do as follows. Notice that

Max-Coloring

1907

Algorithm 8 B-MAX-COLOR.G; w; m/ 1. 2. 3. 4. 5. 6. 7.

Let C1 ; : : : ; Ck be a max-coloring of G Let ci D jCi j for i D 1; 2; : : : ; k do Sort Ci in non-increasing weight order // Let v1 ; : : : ; vci be this order for j D 1; 2; : : : ; dci =me do Let Cij D fvk W bk=mc D j g return Cij for all i; j as the coloring

OPTbmc .G/ D

X

w.Cij /

i;j

D

k X i D1

ci

w.Ci1 / C

k bX mc X

w.Cij /

i D1 j D2 ci

 ˛  OPTmc .G/ C

k bX mc X

w.Cij /

(6)

i D1 j D2

Let V 0 D V n [kiD1 Ci1 . The goal now is to bound the second term in the righthand side of (6). Let H D .V; ;/ be the weighted graph with no edges, where the vertices have the same weights as in the original graph G. An optimal bounded max-coloring of H follows by sorting the vertices in non-increasing weight order and greedily partitioning this order into sets of size m. The color classes of an optimal bounded max-coloring of H induce a permutation   of V , defined as follows: The color classes of H are ordered in non-increasing weight order, and within each color class, we orders the vertices again in nonincreasing weight order. This is equivalent to ordering all the vertices in nonincreasing weight order by the discussion in the previous paragraph. Similarly, define a permutation  on the vertices in V 0 as follows. The color classes Cij , i D 1; : : : ; k, j D 2; : : : ; bci c are ordered in non-increasing weight order, where the weight of Cij D maxv2Cij w.v/, and within each color class, order the vertices in non-increasing weight order. Since a greedy partitioning is used in algorithm B- MAX - COLOR, all but at most one color class Cij has size < m for each i . Assume that all color classes have size m by adding dummy elements equal to the smallest element in the color class. Further, assume that ties are broken identically in   and . Consider the position .u/ of a vertex u. Next, it is shown that .u/    .u/ for all vertices u 2 V 0 . Suppose vertex u 2 V 0 is in color class Ci in the original coloring. Let t.u/ denote the number of color classes Cs , s ¤ i such that there is a vertex v 2 Cs that preceds u in the permutation , that is, .v/ .u/. Let S< .u/ and S .u/ denote the sets of vertices that have weight at most u and greater than u, respectively, in positions 1; 2; : : : ; .u/. Then,

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.u/  jS< .u/j C jS .u/j

(7)

Consider the number of elements in S< .u/. Clearly, each color class Cs , s ¤ i contributes at most m  1 elements to S< .u/, and hence, jS< .u/j  t.u/  .m  1/

(8)

To count jS .u/j, all the vertices in   that appear before u are in S .u/. However, at least t  m vertices have been eliminated that appear in color classes Ci1 for all i . Hence, jS .u/j    .u/  t  m

(9)

Now, substituting the inequalities (8) and (9) in the inequality (7), one gets

.u/  t.u/  .m  1/ C   .u/  t.u/  m D   .u/  t.u/    .u/ Since .u/    .u/ for each element u 2 V 0 , it follows that ci

k dX me X

w.Cij /  OPTbmc .H /

i D1 j D2

 OPTbmc .G/

(10)

Hence, plugging this into inequality (6), one gets OPTbmc .G/  .˛ C 1/OPTbmc .G/



Theorem 20 gives a constant factor approximation algorithm for bounded maxcoloring whenever max-coloring has a constant factor approximation. In particular, there is a constant approximations for max-coloring bipartite and interval graphs and for max-edge-coloring general graphs. Bampis et al. [2] considered the bounded max-coloring and bounded max-edgecoloring problem in bipartite graphs and trees. In doing so, they were able to obtain better bounds than those provided by Theorem 20. In particular, for bipartite graphs, they get the following results. Theorem 21 ([2]) For bipartite graph, bounded max-coloring can be approximated within 17 , and bounded max-edge-coloring can be approximated within e. 11

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On the negative side, Gardi [15] showed that bounded coloring in bipartite graphs (even when all weights are uniform) cannot be approximated within 43   for any  > 0 unless P D NP . For bounded max-edge-coloring, the best hardness is 76   for any  > 0, and it is inherited from regular max-edge-coloring [7]. In the case of trees, they gave a PTAS for bounded max-coloring and a 2-approximation for bounded max-edge-coloring. Like the regular max-coloring, it is unknown whether bounded max-coloring trees is hard or not. Interestingly, bounded max-edge-coloring trees is hard. Theorem 22 ([2]) For trees, bounded max-coloring admits a PTAS, and bounded max-edge-coloring can be approximated within 2. The latter is NP-hard.

10

Conclusion

This chapter has described some results on the max-coloring problem, a natural generalization of classic graph coloring. There have been several results on this problem for various graph classes, in particular, for perfect graphs and subclasses thereof. While this survey is not meant to be exhaustive, the presentation here was restricted to some representative graphs and simple algorithms. Besides the natural question of closing the gap between the known lower and upper bounds, there is one intriguing open question that deserves mentioning. The max-coloring problem on trees admits an exact algorithm running in 2poly log n time, and hence, there is no hope of showing NP-hardness. There are a few natural problems known that have such a property, for example, computing a minimum dominating set in a tournament or satisfiability with n clauses and log2 n variables. For such problems, Papadimitriou and Yannakakis define the complexity classes log NP and log SNP [29]. It would be interesting to put the max-coloring problem on trees into one of these classes. The max-coloring problem aims to minimize the makespan of a batched schedule of jobs. Another natural objective is to minimize the average completion time of the jobs. This objective has been considered by Epstein et al. [11], who obtain approximation algorithms for various graph classes, most notably, the class of perfect graphs and subclasses thereof. There is a third objective that is more natural and that is of the flow-time of the jobs. Each job comes with a release date and cannot be processed before this start date. The cost of a job is the difference between its completion time and release time, that is, the amount of time the job stays in the system. Almost nothing is known about this objective when there is an underlying conflict graph on the jobs.

Cross-References Equitable Coloring of Graphs Greedy Approximation Algorithms On Coloring Problems

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Recommended Reading 1. B.S. Baker, E.G. Coffmann Jr., Mutual exclusion scheduling. Theor. Comput. Sci. 162, 225–243 (1996) 2. E. Bampis, A. Kononov, G. Lucarelli, I. Milis, Bounded max-colorings of graphs, in Proceedings of the 21st International Symposium on Algorithms and Computation (Springer, 2010), pp. 353–365 3. H.L. Bodlaender, K. Jansen, Restrictions of graph partition problems. Part I. Theor. Comput. Sci. 148(1), 93–109 (1995) 4. N. Bourgeois, G. Lucarelli, I. Milis, V.Th. Paschos, Approximating the max-edge-coloring problem. Theor. Comput. Sci. 411(34–36), 3055–3067 (2010) 5. D.G. Corneil, S. Olariu, L. Stewart, The ultimate interval graph recognition algorithm? (extended abstract), in Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms (SIAM, 1998), pp. 175–180 6. J. Correa, N. Megow, R. Raman, K. Suchan, Cardinality constrained graph partitioning into cliques with submodular costs, in Proceedings of the 8th Cologne Twente Workshop on Graphs and Combinatorial Optimization (2009), pp. 347–350 7. D. de Werra, M. Demange, B. Escoffier, J. Monnot, V.Th. Paschos, Weighted coloring on planar, bipartite and split graphs: Complexity and approximation. Discret. Appl. Math. 157(4), 819–832 (2009) 8. M. Demange, D. de Werra, J. Monnot, V.Th. Paschos, Weighted node coloring: When stable sets are expensive, in Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science (Springer, 2002), pp. 114–125 9. M. Demange, D. de Werra, J. Monnot, V.Th. Paschos, Time slot scheduling of compatible jobs. J. Sched. 10(2), 111–127 (2007) 10. Z. Dvorak, K.-I. Kawarabayashi, R. Thomas, Three-coloring triangle-free planar graphs in linear time. ACM Trans. Algorithms 7(4), 41 (2011) 11. L. Epstein, M. M. Halldorsson, A. Levin, H. Shachnai, Weighted sum coloring in batch scheduling of conflicting jobs. Algorithmica 55(4), 643–665 (2009) 12. L. Epstein, A. Levin, On the max coloring problem, in Proceedings of the 2th Workshop on Approximation and Online Algorithms (Springer, 2007), pp. 142–155 13. L. Epstein, M. Levy, Online interval coloring and variants, in Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (Springer, 2005), pp. 602–613 14. B. Escoffier, J. Monnot, V.Th. Paschos, Weighted coloring: further complexity and approximability results. Inf. Process. Lett. 97(3), 98–103 (2006) 15. F. Gardi, Mutual exclusion scheduling with interval graphs or related classes. Part I. Discret. Appl. Math. 157(1), 19–35 (2009) 16. M.R. Garey, D.S. Johnson, L.J. Stockmeyer, Some simplified np-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976) 17. M.C. Golumbic, Algorithmic graph theory and perfect graphs (Academic, New York, 1980) 18. D.J. Guan, X. Zhu, A coloring problem for weighted graphs. Inf. Process. Lett. 61(2), 77–81 (1997) 19. M.M. Halld´orsson, H. Shachnai, Batch coloring flat graphs and thin, in Proceedings of the 11th Scandinavian Workshop on Algorithm Theory (Springer, 2008), pp. 198–209 20. I. Holyer, The np-completeness of edge-coloring. SIAM J. Comput. 10(4), 718–720 (1981) 21. K. Jansen, The mutual exclusion scheduling problem for permutation and comparability graphs. Inf. Comput. 180(2), 71–81 (2003) 22. T. Kavitha, J. Mestre, Max-coloring paths: Tight bounds and extensions, in Proceedings of the 20th International Symposium Algorithms and Computation (Springer, 2009), pp. 87–96 23. A. Kesselman, K. Kogan, Nonpreemptive scheduling of optical switches. IEEE Trans. Commun. 55(6), 1212–1219 (2007) 24. A. Kierstead, W.T. Trotter, An extremal problem in recursive combinatorics. Congr. Numerantium 33, 143–153 (1981)

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25. J. Kratochvil, Precoloring extensions with a fixed color bound. Acta Math. Univ. Comen. 62, 139–153 (1993) 26. G. Lucarelli, I. Milis, Improved approximation algorithms for the max edge-coloring problem. Inf. Process. Lett. 111(16), 819–823 (2011) 27. N.S. Narayanswamy, R. Subhash Babu, A note on first-fit coloring of interval graphs. Order 25, 49–53 (2008) 28. T. Nonner, Clique-clustering yields PTAS for max-coloring interval graphs, in Proceedings of the 38th International Colloquium on Automata, Languages and Programming (Springer, 2011), pp. 183–194 29. C.H. Papadimitriou, M. Yannakakis, On limited nondeterminism and the complexity of the v-c dimension. J. Comput. Syst. Sci. 53(2), 161–170 (1996) 30. S.V. Pemmaraju, R. Raman, Approximation algorithms for the max-coloring problem, in Proceedings of the 32th International Colloquium on Automata, Languages and Programming (Springer, 2005), pp. 1064–1075 31. S.V. Pemmaraju, R. Raman, K. Varadarajan, Buffer minimization using max-coloring, in Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SIAM, 2004), pp. 562–571 32. S.V. Pemmaraju, R. Raman, K. Varadarajan, Max-coloring and online coloring with bandwidths on interval graphs. ACM Trans. Algorithms 7(3) (2011) 33. C.N. Potts, M.Y. Kovalyov, Scheduling with batching: A review. Eur. J. Oper. Res. 120, 228–249 (2000) 34. R. Raman, Chromatic Scheduling. PhD thesis, University of Iowa, 2007 35. N. Robertson, D.P. Sanders, P.D. Seymour, R. Thomas, The four-colour theorem. J. Comb. Theory B 70(1), 2–44 (1997) 36. A. Schrijver, Combinatorial Optimization (Springer, Berlin/Heidelberg, 2003) 37. S. Vishwanathan, Randomized online graph coloring. J. Algorithms 13(4), 657–669 (1992)

Maximum Flow Problems and an NP-Complete Variant on Edge-Labeled Graphs Donatella Granata, Raffaele Cerulli, Maria Grazia Scutell`a and Andrea Raiconi

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Maximum Flow Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Augmenting Path-Based Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preflow-Push Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Edge-Labeled Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Maximum Flow with the Minimum Number of Labels (MF-ML). . . . . . . . . . . . . . . . . . . . . . . . 4.1 Basic Notations and Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Time Complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A Mathematical Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Skewed Variable Neighborhood Search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Computational Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1914 1915 1918 1922 1927 1933 1933 1935 1936 1938 1939 1945 1945 1945

D. Granata () Department of Statistics, Probability and Applied Statistics, University of Rome “La Sapienza”, Rome, Italy e-mail: [email protected] R. Cerulli • A. Raiconi Department of Mathematics, University of Salerno, Fisciano (SA), Italy e-mail: [email protected]; [email protected] M.G. Scutell`a Department of Computer Science, University of Pisa, Pisa, Italy e-mail: [email protected] 1913 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 64, © Springer Science+Business Media New York 2013

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Abstract

The aim of this chapter is to present an overview of the main results for a well-known optimization problem and an emerging optimization area, as well as introducing a new problem which is related to both of them. The first part of the chapter presents an overview of the main existing results for the classical maximum flow problem. The maximum flow problem is one of the most studied optimization problems in the last decades. Besides its many practical applications, it also arises as a subproblem of several other complex problems (e.g., min cost flow, matching, covering on bipartite graphs). Subsequently, the chapter introduces some problems defined on edge-labeled graphs by reviewing the most relevant results in this field. Edge-labeled graphs are used to model situations where it is crucial to represent qualitative differences (instead of quantitative ones) among different regions of the graph itself. Finally, the maximum flow problem with the minimum number of labels (MF-ML) problem is presented and discussed. The aim is to maximize the network flow as well as the homogeneity of the solution on a capacitated network with logic attributes. This problem finds a practical application, for example, in the process of water purification and distribution.

1

Introduction

Network flow problems arise in several fields of inquiry, including applied mathematics, engineering, management, operations research, and computer science. One of the most relevant network flow problems is the maximum flow problem (MFP) that can be stated as follows: given an arc-capacitated graph and two special nodes, namely, a source s and a sink t, the problem is to send as much flow as possible from s to t, by respecting the arc capacities. In the standard maximum flow problem formulation, capacities are real values, while integral and unit capacity cases reflect special scenarios. There is a wide presence of flow networks in the real world. For instance, think about a network of pipes used to transport fluids like gas, water, or oil, in which the arcs represent the pipes and the nodes are their junctions. Other examples can be found in transportation networks, where arcs and nodes represent roads and crosses, or in communication networks, where the arcs represent cables with different bandwidths connecting a set of terminals. Whatever is the kind of the considered network, the maximum flow on them represents the maximum rate at which it is possible to ship commodities from the source to the sink. Besides its practical applications, the MFP arises as a subproblem of more complex optimization problems, such as minimum cost flow, matching, and covering on bipartite graphs as well as NP-hard network design and routing problems. This justifies the continuous efforts carried out throughout the years by researchers to improve the efficiency of the existing algorithms and to design new ones.

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Extensive discussion on the maximum flow problem and its applications can be found in the books of Even [29], Ford and Fulkerson [33], Lawler [43], Papadimitriou and Steiglitz [52], Ahuja et al. [7], Tarjan [59], and in Goldberg et al., survey [38]. Recently, an NP-hard variant of the classical MFP has been introduced, namely, the maximum flow with minimum number of labels (MF-ML). MF-ML belongs to the class of problems defined on arc-labeled (or colored) graphs, which has met a growing interest in the last decade. Labeled graphs are used to model situations where it is necessary to represent qualitative differences among different regions of the graph itself. Typically, each label represents a different property (or a set of properties) to model. The MF-ML problem consists in finding the maximum flow solution that minimizes the number of different labels on a graph where a label and a capacity are assigned to each arc. As for the original version of the problem, this variant finds real-world applications in several types of networks, although this chapter just shows an application to water purification systems. The work is organized as follows: In Sect. 2, the maximum flow problem is formally defined, and the most relevant solution approaches for the problem are introduced. Section 3 presents an overview of the literature and the most important contributions in the emerging field of the edge-labeled graph problems. Finally, Sect. 4 addresses the maximum flow with the minimum number of labels (MFML), by presenting both mathematical formulations and metaheuristic algorithms to solve it.

2

The Maximum Flow Problem

The maximum flow problem can be defined both on directed and undirected graphs, but it is necessary to distinguish between them, although the directed and undirected variants are closely related. Ford and Fulkerson [33] provided a reduction from undirected graphs to directed graphs with comparable size and capacities. Picard and Ratli [53] gave the reduction from directed to undirected graphs, allowing to find the maximum flow value in a directed graph by solving a minimum cut problem on an undirected graph. However, a significant drawback occurs when this reduction is applied to a unit capacity instance, since it produces an instance with capacities that are proportional to the number of nodes. No efficient algorithms are available, so far, for the reduction from the unit capacity undirected problem to the unit capacity directed problem. For the sake of simplicity, from now on, only the maximum flow problem defined on directed graphs is considered. Let G D .V; A/ be a directed graph where V is the set of nodes and A is the set of arcs, with n D jV j and m D jAj. Let uij  0 be a capacity assigned to each arc .i; j / 2 A, and let U D maxfuij W uij < 1 8.i; j / 2 Ag. It is now made the assumption that whenever an arc .i; j / belongs to A, arc .j; i / also belongs to it; this assumption is not a limiting one, since arcs with zero capacity are allowed. Let ıGC .i / D f.i; j / W .i; j / 2 Ag and ıG .i / D f.k; i / W .k; i / 2 Ag be the forward

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star and the backward star of node i 2 V , respectively. A graph G 0 D .V 0 ; A0 / is a subgraph of G if V 0  V and A0  A. A path P D .v1 ; : : : ; vk / is a sequence of distinct nodes belonging to V such that, for 1  i  k  1, .vi ; vi C1 / belongs to A. Finally, two special nodes s and t are distinguished (called source and sink, respectively). Every other node is called intermediate. A function x W A ! R is a feasible flow of G with value fO  0 if the following constraints are satisfied: 8 ˆ < fO i D s X X xij  xki D (1) 0 8i 2 V n fs; tg ˆ : fO  C .k;i /2ıG .i / i Dt .i;j /2ıG .i / 0  xij  uij

8.i; j / 2 A

(2)

The variable xij represents the amount of flow sent on arc .i; j /. Flow conservation constraints (1) make sure that for each intermediate node the total ingoing flow is equal to the total outgoing flow, while the net outgoing flow of the source is equal to the net ingoing flow of the sink. Capacity constraints (2) impose that the flow sent on each arc is nonnegative and does not exceed its capacity. The maximum flow problem consists in finding a feasible flow with the maximum value f  , and can therefore be formulated as max

fO

(3)

s:t: (1) and (2) Note that the feasibility region of this LP problem is nonempty since xij D 0 8.i; j / 2 A is a feasible solution and is bounded by constraints (2). From now on, the expressions feasible flow and flow will be indistinguishably used since only feasibility will be considered unless differently specified. Given a flow x of G, let the residual network of G induced by x, denoted as G x D .V; Ax /, be a graph composed of the same set of vertices of G and the set of arcs Ax such that: • For each .i; j / 2 A such that xij D 0, .i; j / 2 Ax and has capacity rij D uij . • For each .i; j / 2 A such that 0 < xij < uij , .i; j / 2 Ax and has capacity rij D uij  xij ; moreover .j; i / 2 Ax with capacity rj i D xij . • For each .i; j / 2 A such that xij D uij , .j; i / 2 Ax and has capacity rj i D uij . Each arc .i; j / 2 Ax has residual capacity rij > 0; this means that it is possible to send (or subtract) a positive amount of flow on (i; j ). An example of residual network is shown in Fig. 1. An s-t cut of G, denoted by ŒVs ; Vt , is a partition of its set of vertices V in two subsets Vs and Vt such that s 2 Vs and t 2 Vt . Moreover, let .Vs ; Vt / denote the set of forward arcs going from Vs to Vt , and let .Vt ; Vs / be the set of backward arcs going from Vt to Vs . The capacity of an s–t cut .Vs ; Vt /, denoted by uŒVs ;X Vt , is given by the sum of the capacities of its forward arcs, that is, uŒVs ; Vt  D uij . .i;j /2.Vs ;Vt /

Maximum Flow Problems and an NP-Complete Variant on Edge-Labeled Graphs

a 3/3

1 4

s 4/4

2

2 3/5 3

4/5

b

3

3/5

3

t

3

4

4

s 4

4/6

5

1

2

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2 3 1 4 1

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3 2 4

t

2

Fig. 1 (a) An example of a flow network with a maximum flow value f  D 7. The source is s, and the sink t. The numbers denote flow and capacity, respectively. (b) Corresponding residual network

Now, consider the linear dual of the maximum flow formulation (10)–(12): min

X

uij hij

(4)

.i;j /2A

s:t: wt  ws D 1

(5)

wi  wj C hij  0

8.i; j / 2 A

(6)

hij  0

8.i; j / 2 A;

(7)

Where each variable wi corresponds to the flow conservation constraint for node i belonging to (1), and each variable hij corresponds to constraint xij < uij . The dual constraint (5) is associated with the primal flow variable fO, while constraints (6) are related to the variables xij . It can be noted that this dual formulation models the problem of finding the s–t cut with minimum capacity. Indeed, given a feasible solution ŒVs ; Vt , let  wi D  hij D

1 8i 2 Vt 0 8i 2 Vs

1 8.i; j / 2 .Vs ; Vt / 0 8.i; j / 2 A n .Vs ; Vt /

This assignment of values provides a feasible solution for the dual formulation, and the objective function value (4) is equal to the capacity value uŒVs ; Vt . Hence, from the weak duality theorem, it can be derived that the capacity of any cut of G is an upper bound to the maximum flow that can be sent from s to t, while as a consequence of the strong duality theorem, the following result can be stated: Theorem 1 (Maximum Flow – Minimum Cut Theorem) The maximum flow value in G is equal to the minimum capacity of the s–t cuts of G.

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In the following sections, two general classes of algorithms to solve the maximum flow problem are presented, namely: • Augmenting path-based algorithms: During their execution, these algorithms maintain the flow conservation constraints for each node of the network, and iteratively increase the value along augmenting paths from s to t. • Preflow-push algorithms: Algorithms belonging to this class do not guarantee that flow conservation constraints are respected during their execution. Therefore, there could be flow excesses in some intermediate nodes. Preflow-push algorithms send these flow excesses toward nodes close to the sink, or close to the source when the former is not possible anymore.

2.1

Augmenting Path-Based Algorithms

The first known augmenting path procedure was introduced by Ford and Fulkerson in [31]. This algorithm is based on the concepts of residual graphs, cuts (as defined above) and augmenting paths. Given a graph G and a flow x, an augmenting path is a path .v1 ; v2 ; : : : ; vk / from the source s to the sink t in the residual network G x , that is, v1 D s, vk D t. The residual capacity r of an augmenting path P is defined as the minimum among all capacities of the arcs in P . From the definition of residual network, r is always a positive value. Therefore, if there is an augmenting path in G x , then it is possible to send additional flow from s to t. Indeed, it is possible to prove that: Theorem 2 (Ford-Fulkerson) A flow x is maximum if and only if there is no augmenting path; that is, t is not reachable from the source s in the residual graph G x . From Theorem 2, the basic scheme for augmenting path algorithms is derived; that is, find an augmenting path P into the residual graph and send r flow units along P , were r is its capacity. This operation is repeated until there are no augmenting paths available in the residual graph. The steps of this generic augmenting path algorithm are subsumed in the Algorithm 1 . Algorithm 1 does not specify how to compute the augmenting path at each iteration. This is a crucial aspect on which the time complexity of the algorithm depends. One of the first algorithms developed to compute the augmenting path is the labeling algorithm, proposed by Ford and Fulkerson in [32]. This algorithm explores the residual network starting from the source s in order to find all the reachable vertices. During the exploration, vertices are divided into labeled (i.e., reachable from s) and non-labeled. The algorithm iteratively selects one of the labeled vertices, checks its forward star in the residual network, and labels all its neighbors. When the sink t is labeled, an augmenting path P is found and a flow equal to the residual capacity of P is sent along it; then, all the labels are deleted for the next iteration. If after checking the forward stars of all labeled vertices the sink

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Algorithm 1 Generic augmenting path algorithm Procedure Augmenting Path for each .u; v/ 2 A do x.u; v/ 0 end for while there is a path P from s to t in G x do r 1 for each .u; v/ in the path P do r mi nfr, ruv g end for for each .u; v/ in the path P do x.u; v/ x.u; v/ C r x.v; u/ x.v; u/  r end for end while return x

is not labeled, the algorithm stops because no augmenting paths are available in the residual network. In Fig. 2 an execution of the algorithm is shown. A worst-case time complexity analysis of the labeling algorithm is now provided. Note that each iteration except the last one increases the flow along an augmenting path, and requires O.m/ time. In fact, since each node is visited only if it is not labeled and is labeled after that, each forward star (and therefore each arc) is explored at most once. Sending the flow, as well as deleting all the labels, requires O.n/ time. Therefore, the complexity of the algorithm depends on the number of augmenting paths found. If all capacities are integer values and limited by an upper bound U , the capacity of an s–t cut can be at most .n  1/U . The augmenting path algorithm will send at least one flow unit at each iteration; therefore, the overall time complexity of the algorithm is O.nmU /. Unfortunately, there are some cases in which the Ford-Fulkerson algorithm may loop. In particular, if all the arc capacities are rational, then the Ford-Fulkerson algorithm eventually halts, but if they are irrational, the algorithm may loop forever, since it could perform an infinite sequence of small flow augmentations, and in the worst case it may not even converge to the maximum flow. Zwick [68] showed three small acyclic sample networks on which the Ford-Fulkerson procedure may fail to terminate. Edmonds and Karp in [28] suggested two specializations of the Ford-Fulkerson algorithm. The first one consists in computing, at each iteration, the augmenting path of minimum length, which can be found by means of a breadth-first search (BFS) visit of the graph starting from s. It was proved that using shortest augmenting paths, the algorithm performs O.nm/ augmentations and, therefore, the whole procedure runs in O.nm2 / time. The second one is based on the idea of scaling that is based on computing augmenting paths with a residual capacity which is at least equal to a parameter . In this way, it is guaranteed that at each iteration the flow value is increased of at

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Fig. 2 Execution of the Ford-Fulkerson algorithm. (a)–(d) The left side figures show the augmenting path P on the residual network G x , represented by bold arrows. The notation f low=capaci ty is used for each arc. The figures on the right side show the obtained residual graph from adding f to the current flow. The case (d) shows the residual network when there does not exist any augmenting path, and the flow x shown in (d) is therefore a maximum flow

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least  units. To this end, the algorithm starts with  D 2blog U c and builds the residual network G x; containing only arcs with a residual capacity greater than or equal to . Then, it computes an augmenting path in G x; . If this path exists, then a quantity of flow greater than or equal to  is sent along this path, and the residual graph is updated. Otherwise, the algorithm sets  D =2 and rebuilds G x; . The algorithm stops when on the residual network G x; , with  D 1, no augmenting paths are found. Its time complexity is equal to O.m2 log U /. A refinement of the shortest augmenting path algorithm was proposed by Ahuja and Orlin in [5], and it relies on the concept of distance labels. Given a flow x, a distance function d assigning a nonnegative integer to each node is valid if it satisfies the following conditions: d.t/ D 0 d.v/  d.w/ C 1 8.v; w/ 2 Ax

(8) (9)

As a consequence of the distance function definition, the following results can be stated: Corollary 1 If the distance labels are valid, the distance label d.u/ of a node u is a lower bound on the length of the shortest directed path from u to the sink t in the residual graph. Corollary 2 If the distance labels are valid and d.s/  n, there are no paths from s to t in the residual graph. Corollary 1 can be easily proved by considering a path to t of length k in G x , .u1 ; u2 ; : : : ; uk1 ; uk ; t/. By conditions (8) and (9), it follows that d.uk /  d.t/ C 1 D 1 d.uk1 /  d.uk / C 1  2 ::: d.u1 /  d.u2 / C 1  k Corollary 2 is an immediate consequence of Corollary 1. Distance labels are defined exact if, for each node u, d.u/ is exactly the length of the shortest directed path from u to t in the residual graph. Exact distance labels can be determined in O.m/ time by means of a reverse breadth first search starting from t. Moreover, an arc .u; v/ 2 Ax is defined admissible if d.u/ D d.v/ C 1, and a path is admissible if it is only composed of feasible arcs. It is straightforward to observe that an admissible s–t path is an augmenting path with minimum length. The improved shortest augmenting path algorithm starts by computing the exact distance labels for the nodes of G. At each iteration, it tries to build at each iteration

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an admissible s–t path, gradually extending an admissible path from s to some intermediate node u. If an admissible arc .u; v/ exists, the path is extended (advance step); otherwise, the algorithm backtracks on the decision of including u in the path, and its distance label d.u/ is updated to 1 C minfd.v/ W .u; v/Ax g (retreat step). If the admissible path reaches t, an augmentation is performed using the path. The algorithm stops when d.s/  n, by returning a maximum flow in O.n2 m/. In 1970, Dinitz [27] introduced his algorithm for the MFP, which became known as the Dinic’s algorithm. Dinic’s algorithm is based on the observation that many consecutive steps in the shortest path version of the augmenting path algorithm involve augmenting paths of the same length. Therefore, its aim is to push flow along all the shortest augmenting paths simultaneously. In fact, Dinic’s algorithm considers the graph as a layered network, in which a given layer k contains all the vertices whose distance from the source s is k; for example, layer 0 contains just s. The algorithm works in phases and uses the classical breadthfirst search (BF S ). At the beginning of each phase, the algorithm constructs the residual network G x induced by the current flow x and runs the BF S on it, starting from s. If the sink t is not reached, then x is maximal, and the algorithm stops. Otherwise, the BF S constructs a layered network Lx . When the algorithm reaches t, at distance k from s, it puts t in layer k and discards all remaining vertices. The procedure keeps in Lx all those arcs that go from one layer to the next one (discarding arcs within the same layer and backward arcs). The 2 algorithm p runs in O.n m/ time and in networks with unit capacities it terminates in O.m n/ time. In conclusion, it is important to emphasize that the search for more efficient maximum flow algorithms led to the development of sophisticated data structures for implementing Dinic’s algorithm. Some data structures were suggested by Shiloach [54] and Galil and Namaad [36]; the resulting implementations run in O.nm log2 n/ time, but the most efficient one is the dynamic tree data structure introduced by Sleator and Tarjan [55, 56], which yields an O.nm log n/ time maximum flow algorithm.

2.2

Preflow-Push Algorithms

Given a graph G, a preflow x is a flow such that constraints (2) arePsatisfied whereas the net flow at node v 2 N n fs; tg can be positive, that is e.v/ D .v;w/2ıC .v/ xvw  G P  .k;v/2ıG .v/ xkv  0, where e.v/ is called the excess flow of node v. That is, in this class of algorithms the capacity constraints are satisfied, but the total amount of incoming flow for an intermediate vertex can exceed the total amount outgoing from it. A node with a strictly positive flow excess is defined active. The first algorithm in this class is due to Karzanov [41]. The algorithm proposed by Karzanov considers layered networks, in the sense already introduced by describing the Dinic’s algorithm. It starts by sending as much preflow as possible from s to its adjacent nodes (i.e., nodes in layer 1). The preflow is then propagated

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Algorithm 2 Generic preflow-push algorithm Procedure Preflow-Push Preprocess while there is at least an active node do Select active node u Push/Relabel(u) end while

to t: each intermediate node considers its arcs to the next layer according to a predefined order and saturates them one at a time (with the possible exception of the last considered arc), until its excess becomes zero. After this phase, defined advance, the algorithm considers the highest layer containing active nodes. For each of these nodes, the incoming flow is reduced until the excess is equal to 0, and all its incoming arcs are marked as closed; this phase is defined balance. Subsequently, a new advance phase is executed, sending as much preflow as possible on arcs that are not closed. The algorithm iterates advance and balance phases until there are no intermediate nodes with positive excess. Karzanov’s algorithm runs in O.n3 / time. Cherkassky [21] and Galil [35] presented further improvements of Karzanov’s algorithm, which allow one to improve the time complexity to O.n2 m1=2 / and O.n5=3 m2=3 /, respectively. Goldberg and Tarjan [37] proposed a class of preflow-push algorithms using the concept of distance labels. These algorithms work by examining the active intermediate nodes and pushing excess from them to nodes that are estimated to be closer to the sink t, by sending flow along admissible arcs in the residual network (push operation). If the currently considered active node does not have any admissible arc, then its distance label is increased (relabel operation), in order to refine the distance approximation and to generate, as a consequence, new admissible arcs. The algorithms terminate when there are no active nodes. The steps of the generic preflow-push algorithm by Goldberg and Tarjan are presented in Algorithm 2 , while Algorithm 3 shows the preprocess and the push/relabel subroutines. Implementing the procedure requires to specify in which order the active vertices will be visited. The preprocess phase performs some initialization operations. More in detail, it sets the initial flow to 0 for each arc of the network, computes the exact distance labels, and sends flow from the source to its neighbors. Since all the arcs outgoing from the source are saturated, the procedure also updates the value of d.s/ to n, which by Corollary 2 implies that the residual graph does not contain any directed path from s to t. The push-relabel operation on a generic active node u checks whether there is an admissible arc in the residual graph on which a push operation can be performed. A push along .u; v/ decreases both ruv and e.u/ and increases both e.v/ and rvu . It is straightforward to observe that if the push operation does not saturate the selected arc, then the excess of u is reduced to 0. The relabel operation aims at creating at least one admissible arc on which the algorithm will be able to

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Algorithm 3 Preprocess and push/relabel operations Procedure Preprocess for each .u; v/ 2 A do x.u; v/ 0 end for for each u 2 V do compute the exact distance label d.u/ end for for each .s; j / 2 ıGC .s/ do x.s; j / usj end for d.s/ n Procedure Push/Relabel(u) if there is an admissible .u; v/ 2 ıGCx .u/ then push: ı minfe.u/; ruv g x.u; v/ x.u; v/ C ı e.u/ e.u/  ı e.v/ e.v/ C ı ruv  ı ruv rvu C ı rvu else relabel: d.u/ minfd.v/ C 1 W .u; v/ 2 ıGCx .u/g end if

perform new pushes. Eventually, no more flow can be sent to the sink, and relabeling will send flow back to the source. When the algorithm terminates, there is no excess in the intermediate nodes, and therefore the preflow is a feasible flow. Since the algorithm maintains valid distance labels and d.s/ D n throughout the algorithm execution, then there are no augmenting paths from s to t in the residual graph. Therefore, the termination criterion of the augmenting path algorithm is met, and the flow sent to t is a maximum flow. Goldberg and Tarjan in [37] proved that the generic version of the preflow-push algorithm terminates in O.n2 m/ time. The computational time is dominated by nonsaturating pushes. In [38], the authors have proposed an efficient implementation where each vertex v maintains a list of incident arcs, in a fixed but arbitrary order; the vertex also keeps track of its current arc, which will be the next one used for pushing operations originating from v. The implementation iteratively performs the discharge operation, shown in Algorithm 4 . The operation attempts iteratively to push excess of flow from an active node along its current arc. If the arc is saturated, the current arc is replaced with the next one in the list; if the last arc of the list has

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Algorithm 4 Discharge operator procedure Discharge(u) let .u; v/ be the current edge for u ti me to relabel f alse while e.v/ > 0 or ti me to relabel DD f alse do if .u; v/ is admissible then push on .u; v/ else if .u; v/ is not the last edge of the list of u then update the current edge with the next one else update the current edge with the first one t rue ti me to relabel end if end if end while if ti me to relabel DD t rue then relabel u end if

been saturated, the current arc is updated with the first one of the list and the node is relabeled. The operation terminates when either the node is no more active or it has been relabeled. A crucial factor for the performance of a push-relabel algorithm is the order in which active nodes are examined. Two possible orders are suggested and analyzed in [37] and [44]. In the First In, First Out selection strategy (FIFO), the data structure implemented for the set of active vertices is a queue Q; therefore, the front vertex of Q is chosen for discharge operations, and all the new active vertices are pushed to the back of it. The other selection strategy is called largest label. According to this strategy, the vertex with the bigger distance label is selected for the discharge operation. The FIFO algorithm runs in O.n3 / time, as shown by Golderg and Tarjan in [37], while the largest label algorithm runs in O.n2 m1=2 / time as proved by Cheriyan and Maheshwari [44]. The implementation of the largest label algorithm uses an array of sets Li , 0  i  2n  1 and an index l into the array. The set Li is composed of all the active vertices labeled with i , which are represented as a doubly linked list. Therefore, the insertion and deletion operations take O.1/ time. The index l is the largest label of an active vertex. After the initialization phase, all arcs outgoing from the source are saturated, and all the resulting active vertices (i.e., the vertices adjacent to s) are inserted into L0 , and l is set to 0. At each iteration, the algorithm removes a vertex from Ll , processes it using the discharge operation, and updates l. The algorithm loops until l becomes negative, that is, when all the vertices are not active. The steps of the process-vertex procedure by Goldberg et al. [38] are shown in Algorithm 5.

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Algorithm 5 Process-vertex procedure Procedure Process-vertex remove vertex u from the current Ll last label d.u/ Di scharge.u/ add to Ld.w/ each vertex w that became active after the discharge operation if d.u/ ¤ last label then l d.u/; add u to Ll ; else if Ll D ; then l l  1; end if end if

This algorithm correctly maintains l, since for each node v d i scharge.v/ either relabels v or reduces to zero its excess, but doesn’t do both at the same time. In the first case, v is largest distance label active vertex, and then l must be increased to d.l/. In the other case, if the excess at v is zero, then the excess quantity has been moved to vertices with distance labels equal to l  1, so if Ll is empty, l must be decreased by one. Cheriyan and Mehlhorn [20] presented an alternative proof that the complexity of the preflow-push method by Goldberg and Tarjan using the largest label policy is O.n2 m1=2 /. The proof is based on a potential function argument, and unlike the one in [44] does not depend on the implementation of the algorithm using current edges. Ahuja and Orlin [4] proposed a variant of the algorithm that incorporates the concept of excess scaling in order to limit the number of nonsaturating pushes. This is done by pushing flow from nodes with a sufficiently large excess to nodes with a sufficiently small one, and never letting excesses to become too large. The algorithm makes use a parameter  called excess dominator, which is a power of 2 larger than any excess in the residual graph, and guarantees that each nonsaturating push sends at least =2 units of flow. The preflow-push algorithm with excess scaling computes a maximum flow in O.nm C n2 log U / time, where U is a bound on arc capacities. The authors also presented a parallel implementation using the PRAM model and requiring p D dn=me processors that runs in O.n2 log U log p/ time. Subsequently, Ahuja et al. [6] developed two improved versions of the excess-scaling algorithms, namely, the stack-scaling algorithm that runs in O.nm C n2 log U= log log U / and the wave-scaling algorithm that runs in O.nm C n2 .log U /1=2 /. They also demonstrated that the time complexity of the latter algorithm can be lowered to O.nm log..n=m/.log U /1=2 C 2// by using dynamic trees in the implementation. Mazzoni et al. [45] proposed an algorithm that first finds a maximum preflow (the so-called first phase), and then converts the preflow in a maximum flow (the second phase). In the first phase, the algorithm selects an active node v, detects

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an augmenting path from v to t, and moves some amount of flow along such a path. Then, it repeats this process until no augmenting path exists from any active node to the sink. Once a maximum preflow is found, in the second phase, the generic algorithm sends to the source the excess flow of each active node along augmenting paths from active nodes to s. Cerulli et al. [18] analyzed a potential drawback of the preflow-push algorithm, defined ping pong effect. This issue, deriving from the local nature of the choices made by the procedure, occurs when flow units are iteratively exchanged between a node u and a node v along a sequence of push-relabel operations. Interrupting this sequence might require the distance label of one of the two nodes to become greater than the one of the source, which can correspond to .n/ flow exchanges between the nodes and .n2 / overall push operations. To overcome the ping pong effect, the authors proposed a procedure called budget algorithm. The algorithm combines the preflow-push and the augmenting path approaches in order to make sure that, starting from a node u, instead of sending flow to node v with d.u/ D d.v/ C 1, it is sent to a node v0 such that d.u/  d.v0 /  k for a given constant k.

3

Edge-Labeled Graphs

In the graph theory literature, a considerable amount of work has been spent to face problems related to colored (labeled) graphs. Labeled graphs, as opposed to weighted graphs, are used to model situations where it is crucial to represent qualitative differences (instead of quantitative ones) among different regions of the graph itself. Each label represent a different property (or set of properties) to model. More in detail, considering colors or labels in association to the edges of a graph, it is possible to divide the literature contributions in two main classes: The works related to the assignment of such colors (edge coloring problems), and the ones whose aim is to find homogeneous or heterogeneous substructures in the case in which such an assignment is already given (problems on edge-labeled graphs). The maximum flow problem variant presented in Sect. 4 belongs to the second class. Some of the main existing results related to this class are summarized in this section. Problems defined on edge-labeled graphs usually are stated as an input graph G D .V; E; L/ such that V is the set of the vertices, E is the set of the edges, and L is the set of labels in the graph. Each edge e 2 E has an associated label le 2 L. Moreover, let n D jV j and m D jEj. Problems belonging to this field of research have application in many fields, like communication networks, multimodal transportation networks, and molecular biology, among others. One of the most widely studied problem in this area is the minimum labeling spanning tree problem (MLST). Given an undirected edge-labeled graph G D .V; E; L/, the problem consists in finding the spanning tree with the minimum number of colors. The problem was initially addressed by Broersma and Li [9]. They proved, on the one hand, that the MLST is NP-hard by reduction from the minimum dominating set problem, and, on the other hand, that the “opposite” problem of looking for a spanning tree with the maximum number of colors is polynomially solvable. Independently, Chang and Leu [19] provided a different NP-hardness proof

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of the problem by reduction from the Set Covering problem. They also developed two heuristics to find feasible solutions of the problem and tested the performance of these heuristics by comparison with the results of an exponential exact approach based on an A* algorithm, reporting very good performances for one of the heuristics (the maximum vertex covering algorithm, or MVCA). Krumke and Wirth [42] presented a modification of MVCA and demonstrated that the algorithm approximates the optimal solution with a worst case ratio of at most 2 ln n C 1. They also showed that the other heuristic in [19] can perform arbitrarily bad and that the problem cannot be approximated within a constant factor unless P = NP. Wan et al. [60] provided a better analysis of the MVCA variant given in [42] by showing that its worst case performance ratio is at most ln.n  1/ C 1. More recently, Xiong et al. [62] obtained the better bound of 1 C ln.b/, where each color appears at most b times. The pseudocode for the procedure is given in Algorithm 6 . It starts from a subgraph H composed by all the nodes of G and an empty set of edges. Iteratively, it adds to the subgraph all the edges with a given label, until H is connected. At each iteration, the label whose insertion would produce the smallest number of connected components is selected. There also exist some works in literature presenting metaheuristic comparisons for the resolution of the MLST problem, including genetic, pilot method, simulated annealing, VNS, tabu search, and GRASP [15, 23, 61, 63]. Captivo et al. [11] presented a mixed integer linear formulation for MLST based on single-commodity flow. Moreover, the authors demonstrated that by relaxing the binary variables related to the selection of the edges, an optimal solution for the original problem is always found, while the computational time required to solve the formulation by means of a solver is greatly reduced. A variant of the problem has been studied by Br¨uggemann et al. [10], where the MLST with bounded color classes has been addressed, in which each color appears a maximum of r times in the graph. This special case of the MLST is polynomially solvable for r D 2, and NP-hard and APX-complete for r greater than or equal to 3; moreover, they showed that the problem can be approximated with a factor equal to r=2 through local search. A different variant, where the edges are both labeled and weighted, was proposed by Xiong et al. [65]. The authors investigated the label-constrained minimum spanning tree problem (LCMST), where given a positive integer k, the aim to find a minimum weight spanning tree that uses at most k distinct labels. The problem was demonstrated to be NP-complete. Moreover, the authors proposed two different local search schemes, a genetic algorithm and two ILP formulations based on singlecommodity and multi-commodity flow. The authors also introduced the related problem consisting in finding the spanning tree with the minimum number of labels which respects a bound on the total weight, defined cost-constrained minimum label spanning tree (CCMLST). They presented an algorithmic framework based on bisection search that, given a generic algorithm for LCSMST, can be used to find heuristic solutions for CCMLST. The minimum labeling Steiner problem is a variant of the classical Steiner problem and is an extension of the MLST that has also been object of study; in

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Algorithm 6 Modified maximum vertex covering algorithm Procedure MVCA Let L0 ¿ be a set of labels Let E 0 ¿ be a set of edges Let H .V; E 0 / be an undirected edge-labeled graph while H is not connected do mi ncomp number of connected components in H for all i 2 L n L0 do Hi .V; E 0 [ fedges labeled with i g/ number of connected components in Hi compi if compi < mi ncomp then mi ncomp compi mi nlabel i end if end for H .V; E 0 [ fedges labeled with mi nlabelg/ 0 L L0 [ fmi nlabelg end while return L0

this problem, a subset of basic nodes is considered as in the case of the Steiner tree problem, and the problem consists in finding a subgraph covering all the basic nodes using the minimum number of labels. The problem was originally proposed by Cerulli et al. [17], where the authors presented a heuristic algorithm based on MVCA and several metaheuristics, namely, VNS, reactive tabu search, simulated annealing, and pilot method. Consoli et al. [22, 24, 25] proposed GRASP, discrete particle swarm, and VNS metaheuristics, as well as a hybrid local search combining VNS and simulated annealing. Carr et al. [12] analyzed a possible generalization of MLST, which is the problem of finding a subgraph with the minimum number of nodes that satisfies certain connectivity constraints. They called this problem the min-color generalized forest and showed that even the simplest case, in which given a couple of nodes fs; tg the aim is to find the s–t path with fewer labels (MLP), is at least as hard to approximate as the red-blue set cover problem. In the red-blue set cover problem, given two set of elements R and B and a family S  2R[B , the aim is to find a subfamily C  S that covers all elements in B and the minimum number of elements in R. Unless P = NP, 1ı the red-blue set cover problem cannot be approximated within O.2log jS j /, where ı D 1= log logc jS j for any c < 1=2. In Yuan et al. [66] the authors took into account reliability issues related to mesh networks in scenarios when several connections can fail due to the same event, by associating colors to failures. Therefore (assuming that each failure event could occur with the same probability), they also proposed the problem of finding the s–t path with the minimum number of labels, that they called minimum-color

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single-path (MCSiP). Furthermore, they introduced two problems in which the aim is to find two link-disjoint s–t paths that minimize either the total number of colors (minimum total color disjoint-path or MTCDiP) or the number of overlapping colors (minimum overlapping color disjoint-path or MOCDiP). MTCDiP aims at reducing the overall possibility of a failure, while MOCDiP minimizes the possibility that both paths could fail due to a single failure. The authors proved that the problems are NP-complete and proposed integer linear formulations as well as heuristic algorithms for each of the them. Other results regarding MLST and MLP can be found in Hassin et al. [39]. In this work, the authors proposed a variant of MLST where a weight is associated to each label and showed an extension of the modified MVCA that yields an approximation guarantee equal to Hn1 , that is, the n  1th harmonic number. Moreover, by terminating prematurely MVCA and switching to the exact algorithm proposed in [10] when each label does not appear more than twice, they obtained an algorithm with H.r/  1=6 approximation for the MLST p when each label appears at most r times in the graph. They also proposed an O. m/ approximation algorithm for MLP and proved that the algorithm cannot be approximated within a polylogarithmic factor unless P = NP. In Zhang et al. [67], the authors investigated the minimum label s–t cut problem (MLC). Given a couple of nodes fs; tg, the objective of the problem is to select a subset of labels with minimum cardinality such that the removal of pthe corresponding edges would disconnect s from t. The authors presented an O. m/ approximation algorithm and showed that the problem cannot be approximated 11= log logc n n within 2log . They also proved that the same approximation hardness holds for MLP, improving previous approximation results. A variant of the classical Hamiltonian cycle problem, in which the aim is find a Hamiltonian cycle with the minimum number of colors, has also been object of studies. The problem can be considered, for instance, a variant of the traveling salesman problem in cases in which connections belong to different transportation providers adopting fixed-rate charging. The problem was introduced by Cerulli et al. [16] with the name minimum labelling Hamiltonian cycle problem (MLHC). They proved the problem to be NP-complete both on general and complete edge-colored graphs and proposed a heuristic algorithm as well as a Tabu Search metaheuristic. Independently from their work, Xiong et al. [64] introduced the same problem limiting their scope to complete graphs, with the name colorful traveling salesman problem (CTSP). This work also includes a constructive heuristic and a metaheuristic, namely, a genetic algorithm. Cou¨etoux et al. [26] presented approximation algorithms and hardness approximation results for the TSP with both the maximum and the minimum number of colors on complete edge-colored graphs. Regarding the maximization version, they introduced an algorithm with 12 -approximation based on local improvements and showed the problem to be APX-hard. The minimization version was proved to be not approximable within n1 for any fixed  > 0, where n is the number of nodes in the graph. Moreover, the authors demonstrated that if each color does not appear more than r times, where r is an increasing function of n, the problem cannot be approximated within a factor that is less than O.r 1 / for any  > 0. Finally,

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for the special case where color frequency is a constant, a greedy algorithm with r approximation factor equal to rCH was shown. For each algorithm, the authors 2 demonstrated the tightness of their analysis by means of worst-case test instances. Similarly to what was done in [65] for the MLST problem, in [40] the authors introduced two variants of MLHC defined on graphs that have both costs and labels associated to the edges. In the first variant, a maximum cost for the tour is imposed as constraint, and the aim is to minimize the number of labels. In the second variant, the budget constraint is on the maximum number of labels that can be used, and the Hamiltonian cycle with minimum cost has to be found. The authors presented branch-and-cut optimal algorithms for MLHC and both the variants. Monnot [50] proposed the labeled perfect matching problem in bipartite graphs. Given an input graph G D .V; E; L/, with jV j D 2n and such that E contains a perfect matching of size n, the problem consists in finding the perfect matching with either the minimum (labeled MinPM) or the maximum (labeled MaxPM) number of colors. The author proved that on two-regular bipartite graph (i.e., consisting in a collection of pairwise disjoint cycles with even length), if each label appears at most r times, both the problems are APX-complete for each r  2. Using reductions from MinSAT and MaxSAT, he also proved that the minimization variant is 2-approximable and the maximization one is 0.7846approximable. On complete bipartite graphs, it was proven that the labeled MinPM is APX-complete for r  6 and can’t be approximated within a factor of O.log n/. The author also proposed a greedy algorithm for labeled MinPM which selects in each iteration a monochromatic matching with maximum size and showed that it r provides a rCH -approximation. Subsequently, in Monnot [51] the author showed 2 that labeled MinPM is not in APX whenever it is defined on a bipartite graph with maximum degree equal to 3 and that unless P = NP the problem is not in polyLog-APX. Carrabs et al. [13] presented the labeled maximum matching (LMM) problem, in which the aim is to determine the maximum matching with the minimum number of different colors. They proved that the problem is NP-complete and presented four mathematical formulations, as well as a branch-and-bound approach. In [30] several of the above mentioned problems were analyzed in terms of parameterized complexity. More in detail, the authors considered MLST, MLHC, MLP, MLC, labeled MinPM, LMM, and the minimum label edge dominating set (MLEDS), in which the objective is the edge-dominating set that uses the minimum number of labels. When parameterized by the number of used labels, all the problems are W[2]-hard. On the other hand, when parameterized by the solution size, MLP and MLC are W[1]-hard, while all the other problems are fixed-parameter tractable. Several other works in this area are focused on the determination of properly edge-colored subgraphs respecting a specific pattern. A subgraph G 0 D .V 0 ; E 0 ; L0 / of G D .V; E; L/ is properly edge-colored if and only if jE 0 j  2 and any two successive edges differ in color. A properly edge-colored path does not allow that two adjacent edges have the same color and vertex repetitions. A properly edgecolored trail does not allow edge repetitions and every successive couple of edges must differ in color.

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The determination of properly edge-colored s–t paths is known to be solvable in polynomial time for general graphs with two colors as showed by Bang-Jensen and Gutin [8]. Their results were extended by Szeider [58], who proved that the problem of finding a properly colored path in a graph with any number of colors can be solved in linear time. Abouelaoualim et al. proved in [1] the equivalence between properly edge-colored path and properly edge-colored trails. They also stated that the existence of properly edge-colored s–t trails can be checked in polynomial time (as a straightforward consequence of the results in [58]), presenting a polynomial procedure for both of them. Finally, they proved that is NP-complete to check the existence of k pairwise vertex/edge disjoint properly edge-colored s–t paths or trails in and edge-colored graph G D .V; E; L/ even for k D 2 and jLj D O.jV j2 /, and that these problems are still NP-complete on graphs with jLj D O.jV j/ that do not contain properly edge-colored cycles. Abouelaoualim et al. [2] gave sufficient degree conditions for the existence of properly edge-colored cycles and paths in edge-colored graphs, multigraphs and random graphs. In particular, they proved that an edge-colored multigraph of order n with at least three colors and with minimum color degree greater or equal to d.n C 1/=2e has properly edge-colored cycles of all possible lengths, including Hamiltonian cycles. Agueda et al. [3], introduced the properly edge-colored spanning tree problem. There exist two types of properly edge-colored tree, called weak (WST) and strong (SST), respectively; the main difference among them is that in a weak tree adjacent edges may have the same color. More formally, a tree T in an edge-colored graph is a SST if every two adjacent edges differ in color, while given a root node r it is a WST if any path in T from r to a leaf is a properly edge-colored one. The authors presented sufficient conditions for the existence of spanning trees with both patterns. Finally, it’s worth reporting the main results related to problems involving edgecolored graphs and connection (or reload) costs applied to each vertex, which depend on the colors of its adjacent edges. The reload cost model was introduced by Wirth and Steffan [57]. In a given undirected edge-colored graph G, with edges of the same color constituting a subgraph, let a reload cost matrix R represent the costs of moving from one subgraph to another, that is, the cost of switching between the corresponding colors. The reload cost of a path in G is the sum of the reload costs that arise at its internal nodes from passing from the color of an incident edge to the next one. Wirth and Steffan [57] proved that the M i n-Di am problem (i.e., the problem of looking for a spanning tree of G having a minimum diameter and taking into account the reload cost) is not approximable at all, even when the maximum degree of the graph is five. If, instead, the reload costs satisfy the triangle inequality, the problem is named -M i n-Di am. They proved that this problem is not approximable within a logarithmic bound. Galbiati [34] showed that M i nDi am on graphs having maximum degree 4 cannot be approximated within any constant  < 2, and -M i n-Di am cannot be approximated within any constant  < 5=3.

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Maximum Flow with the Minimum Number of Labels (MF-ML)

In this section, a variant of the classical maximum flow problem on edge-labeled graphs, which has been introduced by Cerulli and Granata [14], will be presented, namely, the maximum flow with the minimum number of labels (MF-ML). Given a directed capacitated and labeled graph G D .N; A; L/, where a capacity and a label belonging to the set L are associated with each arc, the aim is to look for a maximum flow on G, from a source s to a sink t, using the minimum number of different labels. A practical application of MF-ML concerns the purification of water during the distribution process. Contamination in distribution systems may result from many factors including cross connections to sewers, contamination at service reservoirs and poor hygiene practice during repairs. All the water distribution systems can be contaminated by ingress of pollutants through leaks present in the pipes, pumps, valves, or storage tanks with consequences for the public health. Therefore, it is crucial to maintain water quality and reduce the possibility to deliver water with pollutants to households. To this end, the water is collected in cisterns, before the distribution to customers, where it is subjected to a decontamination process. Now, suppose to identify each pollutant with a different color and to associate a color with the pipe of the distribution network according to the pollutant, it could be subjected to (in case of more pollution contaminations associated with a single pipe, it is possible to model the pipe with a chain of arcs labeled with different pollution colors). Minimizing the number of colors used to bring the maximum flow from sources to destinations (cistern) means to minimize the pollutions eventually involved in a possible contamination and then minimizing the number of treatments to use during the decontamination process. In the sequel of the section, a more formal definition of the problem and an analysis of its time complexity are initially presented. Then, a mathematical model and some possible enhancements are proposed. Finally, since the problem is NPhard, a metaheuristic procedure based on the skewed V NS scheme is described, and related preliminary computational results are reported.

4.1

Basic Notations and Definitions

Let G D .N; A; L/ be a directed graph where a label lij 2 L and a capacity uij are assigned to each arc .i; j / 2 A. Two special nodes s and t in G are given. Given a feasible flow x D fxij g on G, denote with G x the subgraph of G containing only edges .i; j / with xij > 0, that is G x D .N; Ax ; Lx / with Ax D f.i; j / 2 A W xij > 0g and Lx D flij W .i; j / 2 Ax g. Moreover, given a label l 2 L, let Al D f.i; j / 2 A W lij D lg. The couple .x; fO/ is used to denote a feasible flow x with value fO. From now on, denote by f  the value of the maximum flow on G from s to t. The MF-ML problem consists in looking for, among all the admissible

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flows with value f  , a flow x such that jLx j is minimized. In the following, two useful definitions are introduced. Definition 1 (Necessary label) A label k 2 L is a necessary label if and only if there is no feasible solution .x; f  / in the subgraph Gk D fN; Ak ; L n fkgg with Ak D f.i; j / 2 A W lij 2 L n fkgg. Definition 2 (Necessary node) A node i 2 N n fs; tg is a necessary node if and only if the value of the maximum flow fi on the subgraph of G induced by the nodes in N n fi g is strictly lower than f  . Denote by CL  L the set of the necessary labels and by CN  N the set of the necessary nodes. Moreover, let fi D f   fi 8i 2 CN ; it is straightforward to understand that for any max flow solution at least fi  units of flow have to pass through any necessary node i . In the example in Fig. 3, labels f3; 4g as well as nodes f1; 2g are necessary. Moreover, f1 D 1 and f2 D 4.

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Time Complexity

In this section, a proof that the decisional version of problem MF-ML is NP-complete is given. Consider the decisional version of the problem, namely, the bounded labeled maximum flow problem (BLMFP): Bounded Labeled Maximum Flow Problem: Given a directed, capacitated, and edge-labeled graph G D .N; A; L/, with a source node s 2 N , a sink node t 2 N , and a positive integer k, is there a maximum flow x from s to t in G such that the total number of different labels corresponding to arcs .i; j / with xij > 0 is less than or equal to k? Theorem 3 The bounded labeled maximum flow problem is NP-complete. Proof It is easy to see that BLMFP is NP since it can be checked in polynomial time whether a given flow is feasible and maximum and whether the number of different labels of the arcs with a positive flow is less than or equal to k. The theorem is proved by reduction from the minimum labeled spanning tree problem (MLST [9]). Let G D .N; A; L/ be an undirected and edge-labeled graph. Given an instance of MLST, let us generate a directed and arc-labeled graph G 0 D .N 0 ; A0 ; L0 / with a capacity associated with each arc and show that there exists a spanning tree of G with at most k labels if and only if there exists a maximum flow x from s to t in G 0 such that the total number of different labels of the arcs with a positive flow is at most k 0 D k C 1 (Fig. 4). Let G 0 be the directed graph obtained from G in the following way: • Replace any edge .i; j / 2 A with two directed arcs .i; j / and .j; i / having the same label of the original arc and associate a capacity n  1, where n D jN j, with both of them. • Add two new nodes in G 0 , respectively, the source node s and the sink node t (N 0 D N [ fs; tg). • Connect the source s to a generic node, say node w, by adding the arc .s; w/ with capacity n  1 and a new label z. • Connect any node, but node w, to the sink node t, with z-labeled arcs, .j; t/ whose capacity is equal to 1. Therefore, G 0 contains jN 0 j D n C 2 vertices, jA0 j D 2jAj C n arcs, and jL0 j D jLj C 1 labels. This construction can be accomplished in polynomial time. Note that the maximum flow value in G 0 is equal to n  1, and all the z-labeled arcs must have a positive flow in each maximum flow solution. Consider now a spanning tree T of G with at most k labels. A maximum flow in G 0 from s to t can be defined by sending a positive flow on all the z-labeled arcs and along the arcs of G 0 generated starting from the edges of T (see Fig. 4). Note that such a flow has at most k 0 D k C 1 labels. Vice versa, consider a maximum flow from s to t in G 0 with at most k 0 labels. This flow is such that all the z-labeled arcs have a positive flow and all the arcs with a positive flow whose label other than z form a spanning tree of G with at most k D k 0  1 labels. 

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4.3

A Mathematical Formulation

Given a capacitated and arc-labeled directed graph G D .N; A; L/ and a feasible flow .x; f  /, for each label k 2 L, introduce a binary variable yk whose value is equal to 1 if k 2 Lx and 0 otherwise. Let B be an input matrix where the entry bij k (.i; j / 2 A and k 2 L) is equal to 1 if l.i; j / D k and 0 otherwise. An integer linear programming formulation (ILP1) for MF-ML is the following: .ILP1/

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(10)

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Constraints (11) force the variable yk to be equal to 1 if there exists at least one arc with positive flow whose associated label is k. Otherwise the variable yk is not constrained and will be set to 0 to minimize the objective function. Constraints (12) and (13) are the classical flow conservation and capacity constraints once the flow value is fixed to f  . The (ILP1) formulation can be strengthened by incorporating information about the necessary nodes and labels, obtaining (ILP2): (ILP2)

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By introducing necessary labels, it is possible to reduce the cardinality of the set the binary variables, since variables corresponding to such labels would be always set to 1. Moreover, as already discussed, positive flow will pass through each necessary node, which brings to constraints (17) and (18). A further improvement could be obtained by considering an additional class of valid inequalities defined as follows. A label cutting set is a set of labels CS such that at least one of them is in every optimal solution. A cutting set CS is minimal if there does not exist another cutting set CS0 such that CS0  CS. For example, in Fig. 3 the label set f1,2g is a minimal cutting set. Note that, by definition, a single necessary label is a minimal cutting set. Moreover, it can be also observed that for each necessary node, the set of labels belonging to its ingoing arcs as well as the ones of the outgoing arcs define two cutting sets. P For each cutting set CS, the valid inequality k2CS yk  1 can be derived, which can be further strengthened by considering minimal cutting sets. In order to obtain good lower bounds using the above presented formulation, one possibility is to relax the binary variables y. However, it is experimentally verified that this continuous relaxation brings to trivial lower bounds. Better lower bounds can be obtained by keeping integrality for variables yk and by relaxing the flow conservation constraints for intermediate nodes, replacing them with

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4.4

Skewed Variable Neighborhood Search

Skewed variable neighborhood search (SVNS) [49] is a metaheuristic, introduced as an improvement of the simple variable neighborhood search (VNS). The idea was driven by the necessity to explore more fully valleys which are far away from the incumbent solution. Recall that the VNS is based on dynamically changing neighborhood structures during a local search process, without following a trajectory, but searching new solutions in increasingly distant neighborhoods from the current solution, jumping to the next local minimum only if it is a better solution than the incumbent one [46–48]. On the contrary, the SVNS accepts a solution close to the best one known but not necessarily better. In particular, let z be the objective function of a minimization problem (i.e., z.x/ D jLx j in the MF-ML case), x be the current solution, x  be the incumbent solution, and x 00 be the local optimum of a given iteration. Then, SVNS accepts x 00 if it improves the incumbent solution (i.e., z.x 00 / < z.x  /) or if z.x 00 /  ˛.x; x 00 / < z.x/, where .x; x 00 / denotes a distance between the solution x and the new one x 00 and ˛ is a given parameter. Whenever the solutions are described by boolean vectors (as in the MF-ML case, by considering the boolean variables yk ), the Hamming distance can be used as a metric for evaluating the distance between the solutions. As for the value of the ˛ parameter, after a tuning phase, in the SVNS algorithm the parameter ˛ is set as follows: ˛ D 0:5 if d  0:5 and ˛ D 1:5 otherwise, where d is a measure of the density of G (the computation of this measure will be discussed in Sect. 4.5). A starting feasible solution is determined by solving the following problem, which minimizes the sum of the flows along the arcs of the network, by then recovering the corresponding number of used labels zUB : .UB/

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s:t: (12) and (13) The underlying idea of this upper bound method is to find a maximum flow of G with a low number of used arcs since, intuitively, the number of used colors tends to decrease if the number of used arcs decreases as well.

Maximum Flow Problems and an NP-Complete Variant on Edge-Labeled Graphs Fig. 5 In this graph example, the maximum flow f can be sent using: (i) a path of two arcs (.s; 1; t / with zUB D 2); (ii) a path of four arcs (.s; 2; 3; 1; t / with zUB D 4); (iii) using both the paths (i) and (ii) (with (zUB D 5))

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P Figure 5 shows that minimizing .i;j /2A xij is likely to reduce the number of used arcs. Moreover, in order to implement a SVNS procedure for the problem, a set of parametric neighborhoods have been defined. These neighborhoods are based on two integer parameters k and ˇ: 0 • Nk;ˇ .x/: a solution x 0 belongs to the neighborhood if the label set Lx can be x obtained by removing up to k labels in L n CL and adding up to k new labels, and moreover, the new labels associated with arcs which are incident to at least ˇ nodes belonging to CN . Let lmax 2 L be a label such that Almax has the maximum cardinality. The SVNS algorithm considers a neighborhood for each value of k in f1; : : : ; jL n CLjg and ˇ D 0; moreover, it considers the neighborhood obtained by the previous definition by choosing k D bjAlmax j=2c and ˇ D 3. That is, the SVNS algorithm takes into account a set of neighborhoods Nh , h D 1; : : : ; hmax such that N1  N1;0 , : : :, NjLnCLj  NjLnCLj;0 , NjLnCLjC1  NbjAlmax j=2c;3 . In Algorithm 7 , the pseudocode of the SVNS procedure is given. In particular, in (a) a random solution is chosen from the current neighborhood; in (b) the algorithm tries to improve this solution by using a simple local search as described in the next subsection; in (c) is checked if the local optimum found is an improvement of the current incumbent, while in (d) it is checked if the found solution respects the SVNS criterion of acceptance.

4.4.1 Local Search Phase Given a solution x 0 , at most 200 neighbors belonging to the current neighborhood Nk .x 0 / are examined. The local search phase tries to find a better solution by removing k labels and adding up to k  1 new ones one at time until a new feasible solution is obtained and the ˇ parameter is satisfied. The algorithm stops as soon as a neighbor x 00 such that z.x 00 / < z.x 0 / is found; if no such solution is found after the considered iterations limit, the solution x 00 D x 0 is returned.

4.5

Computational Results

The instances used in the tests have been randomly generated by using three parameters: the number of nodes n, the density of the graph d , and the number of

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Algorithm 7 Skewed variable neighborhood search for MF-ML Procedure Initialize 1. Select the neighborhood structures Nk , for k D 1; : : : ; kmax , where kmax D jL n CLj C 1 that will be used in the search; 2. find an initial solution z and its value z.x/; x 3. x  4. z z.x/ Procedure SVNS 1. k 1 2. Repeat the following steps until k D kmax Random solution 2 Nk .x/ (a) x 0 (b) x 00 LocalSearch.x 0 / (c) if z.x 00 / < z then x x 00 z z.x 00 / end if (d) if z.x 00 /  ˛.x; x 00 / < z.x/ x x 00 k 1 else k kC1 end if

labels jLj. Two collections of scenarios have been generated by considering two different ranges for the value of n. In the first one (small instances), the value of n ranges from 20 to 80 with a step equal to 20, while in the second one (big instances), the value of n ranges from 100 to 200 using the same step. Parameter d varies in the set 0:2; 0:5; 0:7; the total number of arcs in a given graph is m D d n.n  1/. As for parameter jLj, it is set to 20, 30, and 40 %m. For each scenario, three instances have been generated, for a total of 90 scenarios (36 small and 54 big scenarios, respectively) and 270 instances (108 small and 162 big instances, respectively). Tables 1 and 2 report average results for each scenario. In the tables, the first three columns report the scenario characteristics: number of nodes, number of arcs, and number of labels. Columns 4 and 5 report average values for two instance characteristics, that is, maximum flow value f  and number of necessary labels jCLj. In Table 1, columns with headers RF, ILP1, and ILP2 represent the average optimum values of the relaxed model and the of integer programming formulations, respectively, computed using the CPLEX solver. The average computational time for ILP1 and ILP2 are also reported. For each run, the time limit has been set to 3,600 s. Since some of the small instances are not solved by the solver in the considered time limit, in those cases the best upper bounds found by the solver are used to compute the averages. In particular, it is used the notation sol0 , sol1 or sol2 in

n 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 60 60

76 76 76 190 190 190 265 265 265 312 312 312 780 780 780 1,091 1,091 1,091 708 708

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f 42 26 8 91 106 99 162 167 172 198 78 98 374 498 283 481 440 591 308 329

jCLj 5:00 4:67 1:00 8:33 9:67 18:33 21:33 13:33 29:33 12:00 4:00 6:00 20:33 38:00 13:67 28:67 25:67 27:00 10:67 20:67

Table 1 Computational result on small instances RF Sol 6:33 6:33 2:00 13:33 16:00 20:33 24:00 24:00 29:67 19:67 9:67 15:67 32:00 42:67 27:67 44:00 41:67 51:00 23:33 28:33

ILP1 Sol 7:33 6:67 2:00 13:67 18:00 22:33 24:00 25:00 30:67 21:33 10:67 17:67 32:00 44:00 29:33 44:33 42:33 52:33 24:00 30:00 Time 0.00 0.00 0.00 0.01 0.21 0.39 0.01 0.01 0.01 0.53 1.81 1.21 0.07 0.48 5.34 0.06 0.36 7.51 10.17 69.03

ILP2 Sol 7:33 6:67 2:00 13:67 18:00 22:33 24:00 25:00 30:67 21:33 10:67 17:67 32:00 44:00 29:33 44:33 42:33 52:33 24:00 30:00 Time 0.00 0.00 0.00 0.01 0.02 0.26 0.01 0.01 0.01 0.11 1.00 4.51 0.06 0.2 3.39 0.06 0.2 2.78 6.48 32.7

SVNS Sol 7:33 6:67 2:00 13:67 18:00 22:33 24:00 25:00 30:67 21:33 10:67 17:67 32:00 44:00 30:00 44:33 42:33 53:33 24:33 30:67 Time 0.00 0.00 0.00 1.28 0.49 0.84 2.16 0.37 0.59 1.12 5.20 3.76 3.42 2.98 3.23 3.15 3.40 4.63 2.50 3.22

Gap % 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.26 0.00 0.00 1.89 1.36 2.22 (continued)

Maximum Flow Problems and an NP-Complete Variant on Edge-Labeled Graphs 1941

m

708 1,770 1,770 1,770 2,477 2,477 2,477 1,264 1,264 1,264 3,160 3,160 3,160 4,423 4,423 4,423

n

60 60 60 60 60 60 60 80 80 80 80 80 80 80 80 80

283 354 531 708 495 743 991 252 379 505 632 948 1,264 884 1,327 1,769

jLj

Table 1 (Continued)

335 778 820 826 1,198 1,177 1,278 492 448 645 1,525 1,656 1,169 2,024 2,231 2,014

f

RF Sol 25.33 43.67 50.00 56.00 62.67 77.00 85.67 25.00 27.00 37.67 67.67 71.67 62:671 94.00 95.00 92.67

jCLj

10:33 27:00 27:67 30:67 33:67 59:67 73:33 13:67 11:00 15:00 62:67 37:00 34:00 52:67 53:00 51:67 27.00 44.00 50.67 57.67 62.67 78.00 87.00 26.67 28.33 40:001 67.67 73:672 65:670 94.00 95.67 94:672

ILP1 Sol 19:03 0:35 46:62 145:38 0:36 6:61 122:98 446:95 1; 781:07 1; 240:09 0:83 2; 155:64 3; 600 0:72 37:56 1; 980:49

Time 27.00 44.00 50.67 57.67 62.67 78.00 87.00 26.67 28.33 39.67 67.67 73.00 64:672 94.00 95.67 94:672

ILP2 Sol 12:42 0:23 55:43 51:41 0:36 1:35 19:7 129:67 793:42 1; 536:47 0:38 558:06 3; 205:36 0:87 52:16 2; 278:04

Time 27.33 44.67 51.00 60.00 62.67 78.33 87.67 27.00 29.00 41.67 67.67 75.00 67.00 94.00 96.67 97.67

SVNS Sol

3:43 4:96 8:79 9:44 10:60 17:09 21:18 5:03 8:05 16:12 20:94 32:34 27:16 44:60 41:52 50:65

Time

1.26 1.52 0.63 3.95 0.00 0.42 0.75 1.26 2.30 4.92 0.00 2.70 3.56 0.00 1.04 3.09

Gap %

1942 D. Granata et al.

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Table 2 Computational result on big instances n 100 100 100 100 100 100 100 100 100 120 120 120 120 120 120 120 120 120 140 140 140 140 140 140 140 140 140 160 160 160 160 160 160 160 160 160 180 180 180 180 180 180

m 1,980 1,980 1,980 4,950 4,950 4,950 6,930 6,930 6,930 2,856 2,856 2,856 7,140 7,140 7,140 9,995 9,995 9,995 3,892 3,892 3,892 9,730 9,730 9,730 13,621 13,621 13,621 5,088 5,088 5,088 12,720 12,720 12,720 17,807 17,807 17,807 6,444 6,444 6,444 16,110 16,110 16,110

jLj 396 594 792 990 1,485 1,980 1,386 2,079 2,772 571 856 1,142 1,428 2,142 2,856 1,999 2,998 3,998 778 1,167 1,556 1,946 2,919 3,892 2,724 4,086 5,448 1,017 1,526 2,035 2,544 3,816 5,088 3,561 5,342 7,123 1,288 1,933 2,577 3,222 4,833 6,444

f 1,095 1,095 1,095 2,290 2,290 2,290 3,044 3,044 3,044 1,020 1,151 1,278 3,347 3,432 2,776 4,483 5,649 4,414 1,811 1,671 1,942 5,210 4,799 4,629 7,000 6,585 7,253 2,594 2,201 2,239 5,805 5,003 5,696 8,836 7,746 7,490 2,761 3,615 3,080 8,556 7,525 7,205

jCLj 22.00 22.00 22.00 42.00 42.00 42.00 69.00 69.00 69.00 22.00 22.67 21.00 56.33 55.00 51.67 80.67 86.00 73.33 25.00 28.00 42.00 67.00 64.33 72.00 97.00 93.33 99.00 34.00 27.00 28.00 91.00 71.33 92.00 200.00 103.67 99.67 34.33 41.00 36.00 92.00 85.67 81.00

SVNS Sol 48.00 54.33 57.33 76.33 86.67 92.00 110.00 116.67 123.67 41.33 49.00 61.00 97.33 106.67 105.00 128.33 162.67 148.00 58.33 63.33 70.67 126.33 125.33 144.33 179.33 178.33 199.33 68.33 63.67 71.33 144.67 130.33 168.33 209.00 193.67 191.00 74.67 94.67 93.33 174.00 167.00 172.67

Time 19.83 23.64 24.31 52.73 65.18 71.15 90.62 119.23 139.57 19.07 48.61 36.10 109.72 146.12 111.27 179.58 217.27 322.99 65.65 55.92 76.67 249.80 261.40 354.12 414.90 558.91 515.56 65.33 73.66 103.87 242.31 304.83 621.72 362.02 891.52 943.74 122.57 172.06 170.89 641.89 815.80 911.43 (continued)

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Table 2 (Continued) n 180 180 180 200 200 200 200 200 200 200 200 200

m 22,553 22,553 22,553 7,960 7,960 7,960 19,900 19,900 19,900 27,859 27,859 27,859

jLj 4,510 6,766 9,021 1,592 2,388 3,184 3,980 5,970 7,960 5,572 8,358 11,144

f 11,516 9,971 10,747 4,096 2,681 3,570 9,605 10,191 9,427 12,449 12,587 13,769

jCLj 125.67 115.33 129.33 37.00 29.00 37.00 104.33 95.00 96.00 135.67 135.67 136.67

SVNS Sol 233.67 217.33 244.67 83.67 69.33 98.00 183.00 191.00 198.33 257.67 233.67 253.33

Time 1,236.32 1,110.90 1,488.15 215.47 138.73 241.02 519.58 747.56 1,191.62 1,978.20 1,954.76 2,145.93

order to indicate whether 0, 1, or 2 instances were solved to optimality. The same notation is also used in a scenario for the relaxed model, where two instances have not been solved in the time limit; therefore, in this case, the average value reported in the table is not a certified average lower bound. The last three columns in Table 1 are related to the SVNS procedure. In particular, the average solution values are reported as well as running times and percentage gaps with respect to the optimal solution value (or the best upper bound available). Table 2 contains average results for the big instances. For these scenarios, the results obtained by CPLEX for RF, ILP1, and ILP2 are not reported; in fact, the solver was not able to find solutions within the time limit for most of the instances using these models. Therefore, the SVNS solutions are compared with the trivial lower bound given by jCLj. Looking at Table 1, it can be noticed that the RF formulation returns very good lower bounds. Indeed, in 6 out of 36 scenarios, it always finds certified optimal solutions, while the worst average upper bound is 13:6 % bigger than the optimum and the average gap is 3:79 %. Now, consider the two ILP formulations. It can be noticed that introducing necessary labels and nodes improves effectively the model, since in general the running times are reduced and, as a consequence, ILP2 is able to find five optimal solutions more than ILP1. In total, ILP1 fails to return a certified optimal value in 7 out of 108 instances, while ILP2 fails only twice. The SVNS also returns very good results in fast computational times. The instances with 20 and 40 nodes, it always returns optimal solutions in 52 out of 54 scenarios. In general, the maximum percentage gap is 4:92 %, and the average gap is under 1 %. The maximum average computational time is just 50:65 s. Regarding Table 2, it is possible to observe that the average gap among the very trivial lower bound jCLj and the SVNS solution values is 89:13 %. This might seem a weak result. However, it should be noticed that the average gap between jCLj and the optimal solution values in Table 1 is 63:5 %, and therefore a good performance

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of SVNS on these instances as well can be expected. Further research will be spent in finding nontrivial lower bounds for big instances.

5

Conclusion

In this chapter, the well-known maximum flow problem is reviewed, and an interesting variant on edge-labeled graphs is introduced. Since their introduction, flow problems defined on capacitated networks have been widely studied in the literature because of their great relevance. In fact, they can be used to effectively model and study many situations occurring in the real world. Furthermore, networkflow substructures are often used to model complex optimization problems. In the first part of this work, the most relevant contributions on the maximum flow problem are reviewed. In particular, the attention is focused on the description of augmenting path and preflow-push methods, which are the most commonly used classes of algorithms to face the problem. In the second part of the chapter, the maximum flow problem with the minimum number of Labels (MF-ML) is discussed, which aims at maximizing the amount of flow pushed from a given source to a given destination as well as the homogeneity of the solution on a capacitated network with logic attributes (usually known as colors or labels). This problem finds a practical application, for example, in the process of water purification and distribution. Our discussion starts with an overview of the class of optimization problems defined on labeled graphs, an area that has met a relevant interest in the last few years, in which many new problems have been defined and studied. Then the MF-ML problem is deeply explained by introducing some mathematical formulations and a skewed variable neighborhood search metaheuristic, whose efficiency is supported by a pool of preliminary computational results on randomly generated MF-ML instances.

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Modern Network Interdiction Problems and Algorithms J. Cole Smith, Mike Prince and Joseph Geunes

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Classical Interdiction Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 General Problem Descriptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Emerging Areas of Interdiction Research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fortification Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stochastic Interdiction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Asymmetric Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Multi-objective Interdiction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Competition, Interdiction, and Optimization Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Industrial Competition as an Interdiction Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Optimization Models for Competition with Interdiction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Connection to Robust Optimization and Survivable Network Design. . . . . . . . . . . . . . . . . . . . 5.1 Robust Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Survivable Network Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1950 1951 1952 1964 1970 1970 1972 1973 1974 1975 1975 1977 1979 1980 1983 1984 1984 1984

Abstract

A network interdiction problem usually involves two players who compete in a min–max or max–min game. One player, the network owner, tries to optimize its objective over the network, for example, as measured by a shortest path, maximum flow, or minimum cost flow. The opposing player, called the interdictor, alters the owner’s network to maximally impair the owner’s objective (e.g., by destroying arcs that maximize the owner’s shortest path). This chapter

J.C. Smith () • M. Prince • J. Geunes Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA e-mail: [email protected]; [email protected]; [email protected] 1949 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 61, © Springer Science+Business Media New York 2013

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summarizes the impressive recent development of this field. The first part of this chapter emphasizes interdiction foundations, applications, and emerging research areas. The links between this field and business competition models are then developed, followed by a comparison of interdiction research with parallel developments in robust optimization and survivable network design.

1

Introduction

This chapter discusses classical and emerging network interdiction problems, along with two- and three-stage mathematical programming models and solution techniques that are often employed to solve these problems. It is useful to think of these problems in terms of defenders and attackers of a network. A defender (sometimes called an “operator” or an “owner” of the network) may be a player who wishes to protect the network from damage or optimally push flows across the network using the infrastructure that exists. An attacker (“enemy,” “interdictor,” or “adversary” is also used) seeks to disrupt the defender’s objective to the maximum extent possible. Hence, the attacker may seek to optimally impair the defender’s infrastructure to limit the defender’s overall utility from the network. For instance, as discussed later in this chapter, a classical (though stylized) example may consist of a defender that builds a pipeline network with the intention of pushing as much flow as possible from an origin to a destination node. The attacker may attempt to destroy a strategic set of links (pipes connecting junction points), which would then minimize the defender’s maximum flow on the resulting network. As such, optimization models for this class of problems are sometimes referred to as minimax optimization models. Similarly, maximin models exist in which, for example, the attacker may wish to maximize the (optimal) cost that the defender must incur in operating its network. Most of the standard interdiction problems that have been proposed in the literature regard Stackelberg games [65], wherein the two entities (i.e., a leader and a follower) operate in turn with full knowledge of each other’s actions. In the example given above, the defender is presumably aware of which links have been attacked in the pipeline and adjusts flow accordingly. This assumption can be rather strong, though, and several emerging research areas have targeted problems that relax this assumption within a framework where both players act simultaneously. The role of “defender” and “attacker” is sometimes given in the context of security and antiterrorism applications, where the attacker plays the role of an adversary. However, this viewpoint is narrow with respect to the overall scope of interdiction research. For instance, it is common to model an attacker as a natural disaster such as a hurricane or earthquake, an accidental failure of infrastructure, or uncertainty about resource availability. None of these entities is, literally speaking, an intelligent and optimizing force that seeks to impede a defender. However, a defender wishing to protect itself in the worst case (restricted to whatever interdiction resources have been allocated to the attacker) must treat accidents as if they are malicious in nature. Hence, an analyst may wish to adopt the philosophy

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that the worst possible event will transpire in order to guarantee a minimum level of network performance. The defender may additionally be capable of combatting an attacker by fortifying its network infrastructure or when that is not practically achievable, installing enough redundancy in the network so that no small simultaneous collection of failures will substantially cripple the network’s functionality. One can thus consider a three-stage “defender–attacker–defender” game in which the players act sequentially (again with full knowledge of each other’s actions) in a Stackelberg fashion. These problems take the form of min–max–min (or max–min–max) games and are typically substantially more difficult than min–max or max–min games. The fortification actions that take place can be broadly conceived and may involve the fortification or design decisions mentioned above, a reduction of the attacker’s resources, deception regarding the defender’s infrastructure, and so on. There are several additional survey treatments of interdiction in the literature. An excellent starting point was provided by Brown et al. [16, 17] in survey and tutorial articles that explain the breadth of interdiction actions, particularly from the standpoint of security. See also [55] for an introductory treatment accessible to beginning optimization researchers and [57] for a recent discussion of the interdiction area. Section 2 presents some of the founding research principles behind network interdiction, along with key examples that demonstrate common mathematical programming approaches for solving these problems. Emerging areas of interdiction are discussed in Sect. 3. These areas include consideration of uncertain defense/attack strategies, uncertain or differing perceptions of network data, multiobjective interdiction, and fractional interdiction problems where defense/attack actions can be applied at intermediate levels of effectiveness. Applications of these problems pertaining to competition outside traditional interdiction application domains are presented in Sect. 4. The chapter then examines robust optimization theory and survivable network design techniques in Sect. 5 and compares their philosophies to that found in network interdiction studies. Finally, the chapter concludes in Sect. 6.

2

Classical Interdiction Study

This section describes four general types of interdiction problems: shortest path, maximum flow, facility interdiction, and minimum cost flow. These problems are defined on a graph G.N; A/, where N represents the vertex set and A represents a set of directed arcs. For each arc .i; j / 2 A, let cij represent a per-unit flow cost, and let uij represent the capacity of arc .i; j /. The forward star of node i is the set of arcs that leave node i . The reverse star of node i is the set of nodes that enter node i . Formally, the forward star F S.i / D fj 2 N W .i; j / 2 Ag, and the reverse star RS.i / D fj 2 N W .j; i / 2 Ag, for all i 2 N .

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In Sect. 2.1, each of the four problem types is described, formulated, and then illustrated with a numerical example. Section 2.2 includes various applications for interdiction problems.

2.1

General Problem Descriptions

First, the general formulation of a minimax network interdiction problem is given as follows: min max p.x/> y x2X

(1a)

s.t. Dy  r.x/

(1b)

y  0;

(1c)

where x and y are the leader and follower variables, respectively, and where p.x/ and r.x/ represent the follower’s flow profits and available resources, respectively, as a function of the first-stage decisions. The y-vector usually represents the flow, and p.x/ and r.x/ may or may not depend on the decisions made by the attacker. The constraints Dy  r.x/ usually consist of flow conservation and capacity constraints. The attacker’s feasible region, X, is typically of the form X D fx W fi .x/  bi for i D 1; : : : ; mI 0  x  eg where m is the number of constraints, fi is a function (usually linear) of x, and e is a vector of ones. For instance, X often consists of a single knapsack constraint, for example, an interdiction budget. The attacker is also commonly restricted to making binary decisions in which they must fully attack an arc, or not attack it at all.

2.1.1 Shortest Path Interdiction The shortest path problem seeks a path from a source node s to a destination node t such that the sum of the arc weights on the path is minimized. In the shortest path interdiction problem, the attacker seeks to maximize the defender’s shortest path cost on the network by increasing the cost of some arcs .i; j / 2 A from cij to cij C dij . Indeed, this construct permits the development of models in which either (a) the cost (or time) to traverse an arc is actually increased by dij or (b) arc .i; j / is completely destroyed by making dij a prohibitively large number that prevents traversal of .i; j / at optimality. Define N0 D N nfs; tg to be the set of intermediate nodes. The shortest path interdiction problem is formulated as follows: max min x2X

X

.cij C dij xij /yij

.i;j /2A

(2a)

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s.t.

X

yij 

j 2F S.i /

X

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8 ˆ ˆ dij g be the set of existing facilities that are farther than j from demand i . The decision variables include interdiction variables xj , 8j 2 F , and assignment variables yij , 8i 2 N; j 2 F . The variable xj equals 1 if facility j is interdicted, and 0 otherwise, and yij equals 1 if demand i is assigned to facility j , and 0 otherwise. The facility assignment interdiction formulation is given as follows: max

XX

ai dij yij

(11a)

i 2N j 2F

s.t.

X

yij D 1

8i 2 N

(11b)

j 2F

X

xj D r

(11c)

j 2F

X

yi k  xj

8i 2 N; j 2 F

(11d)

k2Tij

yij 2 f0; 1g 8i 2 N; j 2 F

(11e)

xj 2 f0; 1g 8j 2 F:

(11f)

The objective function (11a) is the sum of the weighted distances of the facilitydemand point assignments. Constraint (11b) requires each demand point to be assigned to a facility, and constraint (11c) requires r facilities be interdicted. Constraint (11d) ensures that demand i is assigned to the closest non-interdicted facility. Constraints (11e) and (11f) give the binary restrictions. It is important to note that if the x-variables are binary, then the y-variables are binary at optimality (given a binary x, there exists an optimal solution in which each demand is fully assigned to its closest non-interdicted facility). Therefore, it is possible to reduce the number of binary variables to p–one for each facility–while equivalently relaxing (11e) to yij  0; 8i 2 N; j 2 F (with the unitary upper bounds on yij implied by (11b)). Example 3 An army has five primary bases that must be assigned to cover nine remote outposts. For simplicity, suppose that each outpost is assigned to the closest primary base and that the effectiveness of the army’s operations is given by the sum of base-to-outpost distances. To assess the vulnerability of their coverages, the army would like to know which two large bases, if destroyed, would lead to the largest increase in minimum base-to-outpost distance assignments. Figure 3 shows the network representation along with the d -values and the optimal solution for the case in which no interdiction occurs (resulting an objective function value

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8 1 4 2

6 4

2

7

Outpost

3

5

9 1 3

5

1

2

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7

Fig. 3 Facility location example

of 24). For this particular case, ai D 1; 8i D 1; : : : ; 9. The following formulation corresponds to the illustration in Fig. 3, in which r D 2:

max

XX

dij yij

(12a)

i 2N j 2F

s.t. y11 C y12 C y13 C y14 C y15 D 1

(12b)

:: : y91 C y92 C y93 C y94 C y95 D 1

(12c)

x1 C x2 C x3 C x4 C x5 D 2

(12d)

0  x1

(12e)

y11 C y13 C y15  x2

(12f)

y11 C y15  x3

(12g)

y11 C y12 C y13 C y15  x4

(12h)

y11  x5

(12i)

:: : y92 C y93 C y94 C y95  x1

(12j)

y93 C y94 C y95  x2

(12k)

y94 C y95  x3

(12l)

0  x4

(12m)

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Fig. 4 Facility location solution

8 1 4 3

2

6 4

2

7

5

9 1 3

5

0  x5

(12n)

yij  0 8i D 1; : : : ; 9; j D 1; : : : ; 5

(12o)

xj 2 f0; 1g 8j D 1; : : : ; 5:

(12p)

The optimal solution has x4 D x5 D 1, resulting in an objective function value of 44. Figure 4 shows the revised assignments after the interdiction.

2.1.4 Minimum Cost Flow Interdiction The minimum cost flow problem seeks to satisfy all node demand requirements in a network at minimum cost and subsumes the cases discussed prior to now. In the minimum cost flow interdiction problem, the interdictor’s goal is to maximize the follower’s minimum flow cost by reducing or eliminating the capacity on some subset of arcs. Letting di be the supply (when positive) or demand (when negative) of node i 2 N , the minimum cost flow interdiction model is formulated as follows: max min x2X

X

cij yij

(13a)

.i;j /2A

s.t.

X j 2F S.i /

yij 

X

yij  uij .1  xij / y  0;

yj i D di

8i 2 N

(13b)

j 2RS.i /

8.i; j / 2 A

(13c) (13d)

P where i 2N di D 0. The goal for the follower is to minimize cost while satisfying the flow conservation constraints (13b) and capacity constraints (13c).

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Let ˛ and ˇ be the dual variables corresponding to (13b) and (13c), respectively. The dual formulation to (13) is max

X

di ˛i 

i 2N

X

uij .1  xij /ˇij

(14a)

.i;j /2A

s.t. ˛i  ˛j  ˇij  cij

8.i; j / 2 A

(14b)

ˇ0

(14c)

x 2 X:

(14d)

The form of the objective function, (14a), leads to a nonlinear optimization problem that seeks to optimally interdict a minimum cost flow. Remark 2 Unfortunately, it is not generally possible to restrict ˇ to be binary as in the maximum flow interdiction problem because the change in the objective function value as a function of capacity is dependent on the cost vector c. If x is binary, then linearization is still possible with suitable upper bounds on ˇ. Moreover, in many cases, X is given by a set of constraints fx W P x D q; 0  x  1g. If P is a totally unimodular matrix and q is integral (see e.g., [4]), then in fact an optimal solution exists in which the x-variables take binary values. This fact is due to the observation that, given values of ˛ and ˇ, problem (14) is a linear program, and an optimal solution would exist at an extreme point of X. Because the extreme points of X would be all binary (due to the total unimodularity of P and integrality of q), x is binary in some optimal solution.  Example 4 A company has three production facilities and three retail facilities. The company’s goal is to satisfy the demand at the retail facilities with the supply from the production facilities at the lowest cost. The company would like to know the worst-case cost associated with transportation in the event that up to two of the roads may be closed (or partially closed). In the network depicted in Fig. 5, the production facilities are labeled with their supply quantities (positive values) and the retail facilities are labeled with their demands (negative values). The table shows, for each arc .i; j / 2 A, the cost cij of sending one unit on the arc and the capacity uij on the arc. Without any road closures, an optimal solution is y16 D 4, y24 D 3, y34 D 1, and y36 D 2, with all other arcs having zero flow, and a total cost of 37. The interdiction problem is formulated as follows: max min 9y14 C 6y15 C 3y16 C 4y24 C 6y25 C 5y26 x2X

C5y34 C 8y35 C 4y36 C y54 C y56 s.t. y14 C y15 C y16 D 4 :: :

(15a) (15b) (15c)

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1

3

2

3

3

4

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-4

5

0

6

-6

(i,j)

cij

uij

(1,4) (1,5) (1,6) (2,4) (2,5) (2,6) (3,4) (3,5) (3,6) (5,4) (5,6)

9 6 3 4 6 5 5 8 4 1 1

4 5 6 3 2 3 7 4 3 5 7

Fig. 5 Minimum cost flow network

y16  y26  y36  y56 D 6

(15d)

y14  4.1  x14 /

(15e)

:: :

(15f)

y56  7.1  x56 /

(15g)

y  0;

(15h)

P where X D fx W .i;j /2A xij  2; 0  x  eg. After taking the dual of the minimization problem and combining it with the first-stage problem, the following maximization problem is obtained: max 4˛1 C 3˛2 C 3˛3  4˛4  6˛6  4.1  x14 /ˇ14 5.1  x15 /ˇ15  6.1  x16 /ˇ16  3.1  x24 /ˇ24  2.1  x25 /ˇ25 3.1  x26 /ˇ26  7.1  x34 /ˇ34  4.1  x35 /ˇ35  3.1  x36 /ˇ36 5.1  x54 /ˇ54  7.1  x56 /ˇ56 s.t. ˛1  ˛4  ˇ14  9

(16a) (16b)

:: :

(16c)

˛5  ˛6  ˇ56  1

(16d)

ˇ0

(16e)

x 2 X:

(16f)

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In this case, because the structural constraints of the set X consist only of the cardinality constraint enforcing the condition that only two interdictions are allowed, along with the unit upper bounds on the x-variables, the constraints defining X are totally unimodular. Hence, as explained in Remark 2, the x-variables are binary, and the nonlinear product terms can be linearized. The optimal solution in this case occurs when the roads corresponding to arcs .1; 5/ and .1; 6/ are fully closed, resulting in a cost of 63. 

2.2

Applications

This subsection presents several cases in which interdiction has been applied in real-world settings, as opposed to the simpler illustrative settings mentioned in the previous section.

2.2.1 Nuclear Smuggling (Shortest Path Interdiction) Morton et al. [40] give a shortest path interdiction model for a nuclear smuggling scenario. A joint effort between the United States and Russia, the goal was to determine the optimal locations for nuclear sensors to prevent the smuggling of nuclear material out of the former Soviet Union. Hence, in this case, the interdictors are the American/Russian governments, and the evaders are unknown criminals (as discussed in Remark 1). There is an underlying network on which the evaders travel, and there exists some probability of evasion on each arc. The authors developed a stochastic model in which the origin–destination pairs for evaders are unknown, but follow a known probability distribution. This results in a model where the objective of the interdictor is to minimize the maximum probability of evasion by the criminals. A directed graph G.N; A/ is used where F S.i / is the forward star of node i and RS.i / is the reverse star of node i . The interdictor has a total budget, b, with which it can install the sensors, each at a cost of cij , for all .i; j / 2 A. Associated with each arc .i; j / 2 A is a probability, pij , that the evader can traverse it undetected when no sensor is installed, as well as a probability, qij , that the evader can traverse it undetected with a sensor installed. The set of scenarios is denoted by , and each scenario, ! 2 , has a probability of p ! and an origin–destination realization of .s ! ; t ! /. The interdictor has one set of decision variables for which xij takes a value of 1 if a sensor is placed on arc .i; j /, and 0 otherwise. The evader has two sets of decision variables for which yij is positive only if the evader traverses arc .i; j / and a sensor is not installed, and zij is positive if the evader traverses arc .i; j / and a sensor is installed. This results in the following model: min x2X

X

p ! h.x; .s ! ; t ! //

(17)

!2

P where X D fx W .i;j /2AD cij xij  bI xij 2 f0; 1g; 8.i; j / 2 Ag, and h.x; .s ! ; t ! // is the optimal value of

Modern Network Interdiction Problems and Algorithms Fig. 6 Nuclear smuggling example

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1

s

t

2

max yt ! y;z

s.t.

(18a)

X

.ys ! j C zs ! j / D 1

(18b)

j 2F S.s ! /

X

.pj i yj i C qj i zj i / D

j 2RS.i /

yt ! D

X

X

.yij C zij /

8i 2 N nfs ! ; t ! g (18c)

j 2F S.i /

.pjt ! yjt ! C qjt ! zjt ! /

(18d)

j 2RS.t ! /

0  yij  1  xij ; zij  0

8.i; j / 2 A

8.i; j / 2 A:

(18e) (18f)

The objective function in (17) is the expected value of evasion, taken over all possible scenarios. The optimal value of (18) is the probability of evasion given the origin–destination pair .s ! ; t ! /. Constraint (18b) ensures that one unit of flow leaves the origin, constraint (18c) enforces flow conservation at the intermediate nodes, and constraint (18d) defines the probability of evasion, yt ! . Constraint (18e) ensures that yij must be zero if the arc is interdicted. If the evader uses an arc that is not interdicted, since pij > qij 8.i; j /, the corresponding yij -variable would be positive in order to maximize evasion probability. Therefore, given a path Ps ! ;t ! from s ! to t ! , this results in: yt ! D

Y



 pij .1  xij / C qij xij :

(19)

.i;j /2Ps! ;t !

To illustrate how yt ! is calculated, consider the following example illustrated in Fig. 6 in which there is no interdiction (the pij -values are located on the arcs.) Clearly, the optimal path is s  2  t, which has an evasion probability of .0:75/.0:5/ D 0:375. In satisfying constraint (18b), the values ys1 D 0 and ys2 D 1 are obtained. In satisfying constraint (18c) for node 2, the value y2t D 0:75 is

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obtained, and in satisfying constraint (18d), the value yt D 0:6y1t C 0:5y2t D 0 C .0:5/.0:75/ D 0:375 is obtained. The authors also developed a model in which information is asymmetric, that is, the interdictor and the smuggler have different perceptions of the network parameters. Reformulation and duality techniques were used as well as the introduction of step inequalities to improve and tighten the formulations. CPLEX was used to solve the resulting linear mixed-integer programs. See Sect. 3 for further discussion on handling challenges posed by stochasticity and information asymmetry.

2.2.2 Electrical Grids One highly researched area for applications of network interdiction models arises in electrical grid analysis (see, e.g., [23, 51, 52]). The goal is to determine the best way to operate a power system when the system is subject to disruptions. These disruptions may or may not be malicious. In the case of malicious disruptions, the interdictor attempts to maximize the amount of “load shedding” in the system. Load shedding occurs when the demand of the power grid exceeds the capacity, which results in significant costs to the electricity providers (as well as to those not receiving power). In [23], a two-stage model is introduced in which the interdictor (or terrorist in this case) attacks or destroys some of the power lines in the grid, and the operator attempts to minimize the amount of load shedding over the system. A simplified version of the formulation in the paper is as follows: max v

s.t.

X

n

(20a)

n2N

X .1  ve / D M

(20b)

l2L

ve 2 f0; 1g 8e 2 E;

(20c)

where E is the set of transmission lines; ve is a binary variable that takes a value of 0 if line e is destroyed, and 1 otherwise; M is the number of lines to be destroyed; N is the set of buses; and n is the load shed at bus n, which is obtained from the operator’s load-shedding minimization problem: X

n

(21a)

s.t. F .P; ı; w; v/ D 0

(21b)

min

;P;w;ı

n2N

>

A PD

(21c)

0    u l

P PP

u

(21d) (21e)

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ıl  ı  ıu

(21f)

we 2 f0; 1g 8e 2 E:

(21g)

Here P is the vector of variables corresponding to line power flows and generator power outputs; w is the vector of variables corresponding to the binary decision of disconnecting (we D 0) or not disconnecting (we D 1) line e 2 E; ı is the vector of variables corresponding to the phase angles at each of the buses; u , Pl , Pu , ı l , and ı u are vectors of lower and upper bounds for the variables; given v, F .P; ı; w; v/ is a set of nonlinear functions of P, ı, and w, which correspond to transmission line flows; and A> P D  is the set of linear constraints representing the power balance in each bus of the system. The authors develop a Benders’ decomposition technique for solving the model. Since the problem is nonconvex, the solution method incorporates multiple restarts in order to search a majority of the solution space. As a result, the algorithm is able to obtain optimal or near-optimal solutions in a reasonable amount of time. In [16], a defender–attacker–defender model is formulated in which the operator is able to prevent interdiction on some of the lines before the interdictor is able to act. Through duality techniques, they are able to turn the three-stage formulation into a single-level master problem that is conducive to Benders’ decomposition (see also the discussion in Sect. 3.1).

2.2.3 Drug Enforcement (Maximum Flow Interdiction) In [39], a maximum flow interdiction model is used to model city-level drug enforcement decisions. In the drug trade, there is usually some multilevel hierarchy of drug dealers and drug users. Assuming that the goal of the dealer, or “enemy,” at the highest level is to maximize the flow of drugs, the optimization problem from this dealer’s point of view is a maximum flow problem. A network is used to represent the hierarchy of dealers and users with source node ˛ that represents the “origin” of the drugs and a sink node ! that represents the sink of the flow. The origin is essentially the main dealer that supplies the city with drugs, and the sink captures all flow that reaches the users. Each dealer or user is represented in the network by a node. An arc .i; j / 2 A exists if the person represented by node i deals to the person represented by node j . Each arc has a capacity, and the formulation consists of the typical flow conservation and capacity constraints, with the goal of the “enemy” being to deliver as much flow from the source node to the sink node over a time horizon 1; : : : ; T . The goal of law enforcement is to minimize the maximum flow over the time horizon. One of the interesting ideas in this chapter is the way that law enforcement’s feasible region was characterized. The law enforcement officials are constrained not only by the resources of their departments but also by the real-life aspects of the drug trade. They discuss the idea that law enforcement must “climb the ladder” of the drug supply chain network in order to reach the dealers at the upper levels of the hierarchy. For example, arresting an individual may require that law enforcement officers monitor this

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individual for some period of time. Also, to be able to monitor a dealer that is high up in the hierarchy, it might be necessary to arrest several of the individuals that purchase from this dealer. The resource allocation of the department is constrained as follows: X .aij wijt C bij yijt /  B 8t D 1; : : : ; T; (22) .i;j /2A

where B is the number of available officers, aij is the number of officers required to arrest/remove arc .i; j /, bij is the number of officers required to monitor arc .i; j /, wijt is a binary variable representing the decision to arrest/remove arc .i; j / from the network in time period t, and yijt is a binary variable that represents the decision to monitor/target arc .i; j / in time period t. To restrict law enforcement to only be able to arrest an individual after they have been monitored for a set period of time, the following constraints are used: t 1 X

yijt 0  ij wijt

8t D 1; : : : ; T

(23a)

.1  zijt /  .1  zij;t 1 / C wijt

8t D 1; : : : ; T;

(23b)

t 0 D1

where ij is the number of time periods that .i; j / must be targeted before it is interdicted and zijt is a binary variable representing the availability of arc .i; j / in time period t. If it is necessary to monitor the target in consecutive time periods, then the summation in (23a) can be from t 0 D t  ij to t  1. In order to model the fact that law enforcement may have to arrest lower-level individuals in order to start to target a higher-level individual, each dealer is actually represented with two nodes, i and i 0 . The capacity of the arc between them, i i 0 , represents the number of arrests that are necessary in order to target the dealer. This is modeled with the following constraint set: X

.1  zjj 0 t /  i i 0 yi i 0 t

8t D 1; : : : ; T;

(24)

j 2A.i 0 /

where A.i 0 / is the adjacency list of node i 0 (or the set of buyers from dealer i .) The authors also detail further considerations that face law-enforcement agents in gaining cooperation from the criminals, and how these considerations affect the mathematical models they prescribe. Through the use of duality and linearization techniques, the authors develop a single mixed-integer linear program that can be solved with a commercial solver such as CPLEX.

2.2.4 Supply Chains (Facility Assignment Interdiction) Supply chains are vulnerable to a number of disruptions, many of them unavoidable such as hurricanes or other natural disasters. A great deal of research has focused on

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designing supply chains that are resilient to these types of disruptions. Snyder et al. [61] analyze the design problem for such supply chains, including a worst-case cost model. This model takes into account various scenarios and determines a design that minimizes the worst-case cost of operating the supply chain. Let I denote the set of customers, J the set of possible facility locations, and S a set of possible failure scenarios for the facilities. Not every possible scenario belongs to S –if so, the worst-case scenario would simply consist of the failure of all facilities–because the simultaneous failure of some combination of facilities may be sufficiently unlikely and hence not worthy of consideration. Instead, this set may, for instance, only include scenarios in which no more than four facilities are disrupted. Each customer i has an annual demand of hi units, and each unit shipped from facility j to customer i has a shipment cost of dij . There is a fixed cost fj for opening each facility j , and ajs represents whether facility j was disrupted (ajs D 1) in scenario s, or not (ajs D 0). The model uses two sets of binary decisions variables: xj which takes a value of 1 if facility j is opened, and 0 otherwise; and yijs which takes a value of 1 if customer i is assigned to facility j in scenario s, and 0 otherwise. The model is formulated as follows:

min U s.t. U  X

X j 2J

fj xj C

XX

(25a) hi dij yijs

8s 2 S

(25b)

i 2I j 2J

yijs D 1 8i 2 I; s 2 S

(25c)

j 2J

yijs  .1  ajs /xj

8i 2 I; j 2 J; s 2 S

(25d)

xj 2 f0; 1g 8j 2 J

(25e)

yijs 2 f0; 1g 8i 2 I; 8j 2 J; 8s 2 S:

(25f)

The objective minimizes U , where Constraint (25b) forces U to be greater than or equal to the minimum cost of operation in each of the scenarios (the right-hand side of the constraint is forced to be the minimum cost for each scenario, since this is a minimization problem.) Constraint (25c) ensures that each customer is assigned to a facility in every scenario. Constraint (25d) ensures that customer i can only be assigned to facility j in scenario s if the facility is opened and not disrupted in that scenario. It is important to note that there are clear differences between this model and the general facility assignment interdiction model presented earlier. Instead of an interdictor acting with a goal of maximizing the cost of operating the supply chain, there is a set of possible scenarios that must be considered. However, since the supply chain designer is preparing for the worst-case scenario, this results in the same solution as if there were an interdictor choosing the worst-case from the set of possible scenarios.

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Emerging Areas of Interdiction Research

This section describes a few research thrusts that have arisen primarily in the past decade. Fortification optimization problems are discussed in Sect. 3.1. Models and algorithms for stochastic interdiction problems are then presented in Sect. 3.2, followed by the consideration of asymmetric information in Sect. 3.3. Lastly, solution strategies are explored for multi-objective problems in Sect. 3.4.

3.1

Fortification Problems

Several fortification problems, such as those explored in [17, 19, 53, 59], seek to fortify a network in anticipation of an optimal attacker. These fortification actions are essentially taken in advance of the mathematical programming models described in Sect. 2. In many settings, it may be possible to integrate the presence of fortification actions directly within the mathematical programming interdiction model. However, for the examples mentioned here, a cutting-plane approach to solving these problems is evidently required. In the model described by (11), Church and Scaparra [19] prescribe a two-phased algorithm to optimally fortify facilities, where a fortified facility cannot be destroyed by the interdictor. In this approach, a master problem is solved first that prescribes fortification actions and computes a lower bound on the optimal objective function value. Fortification actions are constrained by w 2 W, where W D fw W e> w D q; w 2 f0; 1gjAj g restricts the fortification actions to be binary and enforces the condition that q fortifications are made. The approach taken is then to minimize some value function variable , subject to w 2 W, along with an exponentially sized set of affine inequalities that relate  to the fortification decisions w. Sets of these inequalities are generated one at a time, as needed, from a subproblem of the form (11), which has the additional restrictions that xi  .1 zi / for each i 2 N . To generate inequalities at iteration k of this process, consider any feasible interdiction h h solution P x to (11)h at iteration h of the algorithm, which induces a solution y . Define h di = j 2F dij yij as the distance from i to its closest non-interdicted facility in the solution given by .xh ; yh /. If dij < dih for some j 2 F , then by fortifying facility j , the total objective value may decrease by as much as .dih  dij /. Let Bih be the set of all closer facilities j 2 F such that dij < dih . One possible valid inequality is given by 0 1 X X  dih  dij wj A ; ai @dih   i 2N

j 2Bih

but this inequality is quite weak due to the fact thatP for some  i 2 N , fortifying multiple facilities in B 0  Bih promises a reduction of j 2B 0 dih  dij rather than   the actual (maximum possible) reduction of dih  minj 2B 0 fdij g . To strengthen this formulation and avoid the weakness presented by the foregoing inequality, Church

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and Scaparra [19] instead define additional variables vhij at iteration h, which equal 1 if facility j is the closest facility to i in Bih that is fortified, and 0 otherwise. This permits the inclusion of the stronger, though larger, inequality system for every iteration h enumerated by the algorithm:



X

0

X

ai @dih 

i 2N



1

dih  dij vhij A

(26a)

j 2Bih

vhij  wj 8i 2 N; j 2 Bih X vhij  1

(26b) (26c)

j 2Bih

vhij 2 f0; 1g 8i 2 N; j 2 Bih :

(26d)

Even though (26) increases the number of variables and constraints for each solution h, because they are added only as needed during the two-phase process (and not in general for every possible interdiction solution), the routine is able to solve realworld instances in acceptable computational limits. The process terminates when the predicted objective  found in the master problem matches the objective computed in the subproblem. The approach of Brown et al. [17] and Smith et al. [59] is tailored toward the more general problem of minimum cost network flow interdiction. Again letting W be the feasible region for fortification actions, the objective would be to minimize .w/ over w 2 W, where .w/ solves problem (14) with the additional objective term M.wij xij / for some large number M . This term serves as a penalty function to prohibit setting xij D 1 at optimality when wij D 1. Note that the feasible region of (14) is invariant with respect to w and that with w fixed, (14) is a disjointly constrained mixed-integer bilinear program, assuming that X contains only linear and (possibly) integrality constraints. (That is, ˛ and ˇ are constrained only by the polyhedron induced by (14b) and (14c), and x is only constrained to lie in X. If either set of variables is fixed, then assuming that X is defined by a set of linear constraints, (14) becomes a linear program.) Therefore, let ‚ denote the set of extreme points to (14b) and (14c), and let ƒ denote the set of extreme points to X (or to conv(X) if X includes binary restrictions). Because the problem is bilinear, an optimal solution lies at a combination of extreme points in  and ƒ. Then, the interdictor’s objective function can be rewritten as X

max

.˛ k ;ˇ k /2;

x2ƒ

i 2N

di ˛ik 

X

uij .1  xijk /ˇijk  M.wij xijk /:

.i;j /2A

The strategy for solving this class of problems constructs a master problem and subproblem as mentioned above, where the master problem contains a value function , and the subproblem solves (14) augmented with the fortification penalty

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term. If  does not match the subproblem objective term, an optimal solution .˛ ? ; ˇ ? ; x ? / is obtained from the subproblem and used to generate a (single) cut of the form 

X i 2N

3.2

di ˛i? 

X

uij .1  xij? /ˇij?  M.wij xij? /:

.i;j /2A

Stochastic Interdiction

Interdiction and fortification problems are further complicated in the presence of uncertainty. In practice, some of the data present in the model may be stochastic, and the success of interdiction actions may be uncertain. The former case is considered by Morton et al. [40] and Pan and Morton [44]. These papers consider an evader/interdictor, where the interdictor seeks to monitor network nodes and/or arcs with the purpose of minimizing the evader’s maximum probability of escape. The origin and destination of the evader are unknown in these models, which results in an NP-hard interdiction problem even if the underlying network is bipartite [41]. Cormican et al. [21] consider a stochastic maximum flow interdiction problem, which is subject to uncertainty of interdiction actions. In their base model, the network data is known with certainty, but each interdiction action is successful only with some probability pij , for each arc .i; j /. The authors extend their results for the case in which arc capacities are also uncertain, interdiction success is varied among several discrete levels, and multiple interdiction actions are possible. The approach taken in [21] is to consider a problem similar to the form of (6), where constraints (6c) are replaced with yij  uij .1  IQij xij /

8.i; j / 2 Anf.t; s/g;

(27)

and where IQij is a random variable that equals 1 if an interdiction on arc .i; j / would be successful (which occurs with probability pij ), and otherwise equals 0 if an interdiction would be unsuccessful (occurring with probability .1  pij /). Both successful and unsuccessful interdiction actions contribute to the interdictor’s overall budget. Hence, the interdictor is faced with the challenge of trading off the selection of arc interdictions that are maximally effective when all interdictions are successful, with those that correspond to large pij -values. The problem is now to identify the interdiction actions x 2 S that minimize the network owner’s maximum flow. Janjarassuk and Linderoth [34] approach this problem via sampling strategies as used in scenario-based stochastic problems. Cormican et al. [21] prescribe a sequential partitioning approach, where lower and upper bounds on the expected objective are gradually tightened over the partitioned space. The essence of the algorithm is to note that, given interdiction actions x 2 X, the maximum flow problem given by (6) (which includes (27) instead of (6c)) is a

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concave function over the set of possible outcomes of IQ. This problem is denoted here as SMF-1. An equivalent formulation, SMF-2, is given by min max yt s  x2X

X

.xij IQij /yij

(28a)

.i;j /2A

s.t. Constraints (6b)–(6d):

(28b)

Here, equivalence is claimed in the sense that an optimal solution to SMF-1 is also optimal to SMF-2, and although an optimal solution to SMF-2 need not be optimal to SMF-1, the optimal objective function values of the two problems match. Given a fixed interdiction action in X, the inner maximum flow problem of SMF-1 is a concave function of the random variables IQ, while the inner maximum flow problem of SMF-2 is a convex function of these random variables. (Note that stochasticity occurs in the constraints of the inner problem in SMF-1 and in the objective of the inner problem in SMF-2.) As a result, an initial lower bound (LB0 ) is given by replacing the random variables with their expectation in SMF-2, and an initial upper bound (UB0 ) is given by doing the same in SMF-1. These bounds are valid due to Jensen’s inequality (see, e.g., Kall and Wallace [35] for a discussion of this inequality’s use in stochastic programming). To refine this bound, one could partition the state space of IQ into K spaces I1 ; : : : ; IK , where the probability of IQ belonging to Ik is given by k , 8k D 1; : : : ; K. Then in the same manner above, one can compute LBk and UBk for each k D 1; : : : ; K and obtain new lower and upper bounds as LBK D

PK

UBK D

PK

kD1 k LBk

kD1

 LB0

k UBk  UB0 :

Moreover, as K becomes large, LBK and UBK both converge to the optimal expected interdiction objective. See [21] for methods that dynamically partition the original state space into subregions and terminate once UBK  LBK is sufficiently small.

3.3

Asymmetric Information

Bayrak and Bailey [3] and Morton et al. [40, 41] examine interdiction problems in which there exists so-called information asymmetry between the opposing players. In the context of interdiction problems, “asymmetric information” refers to the situation in which the opposing sides have different perceptions about the parameters of the model. Bayrak and Bailey [3] define cij as the length of arc .i; j / and dij as the amount that the arc length is increased if the arc is interdicted. The correct data values are known to the interdictor. The evader, on the other hand, only has estimates of these values: cNij and dNij . (Whether or not the interdictor in fact knows the correct data is immaterial, but in any case, the interdictor is optimizing

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with respect to c- and d -values, with knowledge that the evader optimizes with respect to cN and dN -values.) Variables xij , for each .i; j / 2 A, are the interdictor’s decision variables, which take a value of 1 if arc .i; j / is interdicted, and 0 otherwise. Variables yij are the evader’s decision variables, 8.i; j / 2 A, which take a value of 1 if arc .i; j / is on the shortest path, and 0 otherwise. The shortest path network interdiction problem with asymmetric information is formulated as follows: max

X

x

.cij C dij xij /yk

(29a)

.i;j /2A

s.t.

X

bij xij  B

(29b)

.i;j /2A

xij 2 f0; 1g 8.i; j / 2 A;

(29c)

where 9 8 = < X .cNij C dNij xij /yij y  2 argmin min ; : y .i;j /2A

s.t.

X j 2F S.i /

yij 

X

8 ˆ ˆ x s.t. Ax  b x  0;

(31a) (31b) (31c)

where A is an m  n matrix and where c and b have conforming dimensions, but where A is uncertain. In fact, the assumption that c and b are deterministic, with uncertainty limited to A, can be made without loss of generality by suitable variable substitutions. (Much of the robust optimization literature assumes a lower bound l on the x-variables, where l is not necessarily nonnegative; however, the brief exposition here is simplified with the assumption that x is nonnegative, and moreover, any linear programming problem can easily be transformed to the form (31).) Suppose that A belongs to the uncertainty set U. The robust counterpart formulation guarantees feasibility of x to (31b) for every A 2 U, that is, by changing (31b) to Ax  b; 8A 2 U. There is now potentially an infinite number of constraints to the problem, depending on U. However, this problem can be cast as a finite optimization problem by focusing only on certain forms of the set U and employing mathematical

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programming techniques reminiscent of those developed for interdiction models. It is first useful to state the following observations regarding robust optimization problems: • Let ai denote the i th row vector of A and define Ui as the set of all A-matrices whose i th row is ai . Because x must be feasible for any A 2 U, the following constraints are valid: a i x  bi

8i D 1; : : : ; m; ai 2 Ui :

(32)

Therefore, optimizing over the uncertainty set U is equivalent to optimizing over the possibly larger uncertainty set U1      Um . This is referred to as row-wise uncertainty, in which randomness is independent over the rows of A. • Similarly, optimization over U is equivalent to optimization over the convex hull of U and to the closure of this set (see [42]). • If U independently constrains the columns of A (referred to as column-wise uncertainty), then randomness is independent over both rows and columns. Randomness is thus independent over each element of A, and the robust counterpart replaces each element of A with its smallest possible value (in the case of  constraints). See [62] for an early exposition of this case. The remaining discussion in this section regards the more complex case of intercolumn dependence in U. For robust optimization problems, one can again employ minmax (or maxmin) optimization techniques as done in the context of Stackelberg games. Here, the first player (leader) selects a solution xO . The follower will, for each row i D 1; : : : ; m, minimize ai xO over all ai 2 Ui with the aim of forcing xO to be infeasible. Just as in the development of the interdiction models of Sect. 2, the inner problem can be dualized to uncover a single-level optimization problem. However, the order in which the defender (now the leader) and attacker (now the follower) play is reversed. The attacker’s problem is now quite simple, which admits a far simpler optimization model. Consider the case of budgeted uncertainty [12], where aij 2 Œaij  aO ij ; aij  for all elements of A and no more than some specified (nonnegative) value i of the coefficients ai1 ; : : : ; ai n are allowed to simultaneously deviate from aij , for all i D 1; : : : ; m. To minimize the left-hand side of the structural constraints, given the leader’s solution x, O the follower minimizes n X

aij xO j 

j D1

n X

aO ij xO j zij

8i D 1; : : : ; m;

(33)

j D1

which is equivalent to solving

max

n X j D1

aO ij xO j zij

(34a)

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s.t.

n X

zij  i

(34b)

j D1

0  zij  1

8j D 1; : : : ; n:

(34c)

Associating the dual variable i with (34b) and ij with the constraint zij  1, 8j D 1; : : : ; n, the dual formulation of (34) is min i i C

n X

ij

(35a)

j D1

s.t. i C ij  aO ij xO j i  0; ij  0

8j D 1; : : : ; n

(35b)

8j D 1; : : : ; n:

(35c)

P Because nj D1 aO ij xO j zij is given by the smallest possible value of (35a) and is constrained by (35b) and (35c), each constraint (32) is equivalent to 0 ai x  @ i i C

n X

1 ij A  bi

8i D 1; : : : ; m

(36a)

j D1

i C ij  aO ij xO j ;   0:

8j D 1; : : : ; n

(36b) (36c)

Note that the robust counterpart is also a linear program, as opposed to the models of Sect. 2, which become nonlinear (although they can be linearized when at least one of the bilinear terms is restricted to be binary). This class of robust models is therefore considerably easier to solve than the previously developed interdiction models. However, robust models are also far more pessimistic, as they permit the attacker to essentially act not only with full knowledge of any defender design decisions but also with full knowledge of the defender’s recourse decisions. By contrast, in the interdiction models, the defender has the capability of choosing its recourse based on the attacker’s actions. The most appropriate model for a given application should therefore take into consideration to what extent recourse decisions are allowed after an attack is made. (It is worth noting that some progress has also recently been made on robust optimization with recourse, see, e.g., [2, 13, 14].) Three other observations are relevant before continuing. First, the foregoing analysis can be applied when A belongs to a general polyhedral set. Second, much of the traditional robust optimization theory regards the case in which U is ellipsoidal, and for this case, the robust counterpart becomes a nonlinear (but still convex) program. See [11] for analogous derivations of robust counterparts with respect to more general uncertainty sets. Third, when the “true” range of possible A-values

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is restricted to lie in a constrained uncertainty set (e.g., budgeted or ellipsoidal uncertainty), it is possible to derive bounds on the probability that the constraint is in fact violated [6].

5.2

Survivable Network Design

Finally, it is worthwhile to note the relationship among the perspectives of other survivable network design research and interdiction studies. The survivable network design literature encompasses a broad study of network design and operations problems that remain functional in the face of failures. Hence, it is harder to put as precise a definition on this field (which some would consider to be encompassed by interdiction and/or robust optimization) than on the previously discussed lines of research. As a starting point, one may consider the case in which robustness is defined with respect to a collection of possible failure scenarios. Each failure scenario defines a simultaneous realization of uncertain data, and design decisions must be feasible with respect to every possible failure scenario. (The objective would thus be defined as the worst-case scenario in a robust modeling framework and perhaps as an expected cost from a stochastic programming perspective.) Kouvelis and Yu [37] show that with just two scenarios, the shortest path problem with nonnegative data becomes NP-hard, in contrast to the polynomial solvability of the deterministic version of this problem. Garg and Smith [27] provide another version of this problem, in which many failure scenarios exist, each of which represents a set of possible link failures. The authors employ Benders’ decomposition to mitigate the computational effort imposed by encountering a large set of failure scenarios. Several studies, for example, [22, 33, 63], assume that network failures are rare, especially when applied to accidental failure scenarios on equipment whose functionality is very reliable but will occasionally fail. Polyhedral integer programming studies by Gr¨otschel and Monma [30] and Gr¨otschel et al. [31] regard connectivity problems on networks. Other network systems implicitly guarantee survivability with respect to single points of failure. In the past two decades, synchronous optical networking (SONET) ring networks [66, 67] received much research attention from the combinatorial optimization community [5, 28, 29, 38, 54, 60, 64]. These network design problems can be seen within either the interdiction or robust viewpoint. From the interdiction perspective, the attacker destroys any link in an attempt to maximize damage (dropped data), while the recourse action of the defender reroutes data in response to the attack. From the robust perspective, the design decisions build the network with the implicit understanding that any interdiction action can be mitigated in the future (without explicitly stating variables and constraints that prove the existence of such recourse actions).

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Conclusion

Although network interdiction models have existed for some time, the past decade has seen an accelerated development within this field. This has resulted from a confluence of several factors. First, several major destructive events have occurred since the year 2000, including the terrorist attacks on the World Trade Center on September 11, 2001, and Hurricane Katrina in New Orleans in 2005. These events, and the state of the world in their aftermath, motivated researchers to intensify efforts on problems involving major catastrophic disruptions. Second, the operations research community and its available tools have seen advances in the development of models for handling singular events, uncertainties, and relatively large-scale network optimization problems in the past 20 years. Third, computing power and availability has increased to a degree that has permitted an ability to solve largescale problems on desktop computers. In a sense, however, the field of network interdiction modeling is in its early stages. The models discussed in this chapter remain somewhat limited in their abilities to handle dynamic changes in the data that drive these models. That is, the majority of the models this chapter has described take a snapshot of a problem in time and solve that snapshot or single-period version of the problem. Moreover, the limited action sets and information symmetry assumptions contained in the majority of existing models can limit their practical application. Finally, multiple interdictors with varying interdiction mechanisms serve to compound problem complexity. Accounting for a richer set of assumptions on the dynamic evolution of data, the potential actions of multiple interdictors, and various states of information will require greater modeling sophistication and computing power than currently available. Thus, while great advances have been made in both computing and operations research methods, a great deal of opportunity exists for developing models and methods for solving realistic versions of dynamic problems.

Cross-References Combinatorial Optimization in Transportation and Logistics Networks Maximum Flow Problems and an NP-Complete Variant on Edge-Labeled Graphs Network Optimization

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4. M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear Programming and Network Flows, 2nd edn. (Wiley, New York, 1990) 5. G. Belvaux, N. Boissin, A. Sutter, L.A. Wolsey, Optimal placement of add/drop multiplexers: static and dynamic models. Eur. J. Oper. Res. 108, 26–35 (1998) 6. A. Ben-Tal, L. El-Ghaoui, A. Nemirovski, Robust Optimization (Princeton University Press, Princeton, 2009) 7. A. Ben-Tal, A. Nemirovski, Stable truss topology design via semidefinite programming. SIAM J. Optim. 7, 991–1016 (1997) 8. A. Ben-Tal, A. Nemirovski, Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998) 9. A. Ben-Tal, A. Nemirovski, Robust solutions to uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999) 10. A. Ben-Tal, A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88(3), 411–424 (2000) 11. A. Ben-Tal, A. Nemirovski, Robust optimization – methodology and applications. Math. Program. Ser. B 92, 453–480 (2002) 12. D. Bertsimas, M. Sim, The price of robustness. Oper. Res. 52(1), 35–53 (2004) 13. D. Bertsimas, A. Thiele, Robust and data-driven optimization: Modern decision-making under uncertainty, in Tutorials in Operations Research: Models, Methods, and Applications for Innovative Decision Making, ed. by M.P. Johnson, B. Norman, N. Secomandi. INFORMS, 2006, pp. 95–122 14. D. Bertsimas, A. Thiele, A robust optimization approach to inventory theory. Oper. Res. 54(1), 150–168 (2006) 15. J.R. Birge, F.V. Louveaux, Introduction to Stochastic Programming (Springer, New York, 1997) 16. G.G. Brown, W.M. Carlyle, J. Salmer´on, K. Wood, Analyzing the vulnerability of critical infrastructure to attack and planning defenses, in Tutorials in Operations Research: Emerging Theory, Methods, and Applications, ed. by H.J. Greenberg, J.C. Smith. INFORMS, Hanover, 2005, pp. 102–123 17. G.G. Brown, W.M. Carlyle, J. Salmer´on, K. Wood, Defending critical infrastructure. Interfaces 36(6), 530–544 (2006) 18. S.D. Cartwright, Supply chain interdiction and corporate warfare. J. Bus. Strat. 21(2), 30–35 (2000) 19. R.L. Church, M.P. Scaparra, The r-interdiction median problem with fortification. Geogr. Anal. 39, 129–146 (2007) 20. R.L. Church, M.P. Scaparra, R. Middleton, Identifying critical infrastructure: The median and covering interdiction models. Ann. Assoc. Am. Geogr. 94(3), 491–502 (2004) 21. K.J. Cormican, D.P. Morton, R.K. Wood, Stochastic network interdiction. Oper. Res. 46(2), 184–197 (1998) 22. G. Dahl, M. Stoer, A cutting plane algorithm for multicommodity survivable network design problems. INFORMS J. Comput. 10(1), 1–11 (1998) 23. A. Delgadillo, J.M. Arroyo, N. Alguacil, Analysis of electric grid interdiction with line switching. IEEE Trans. Power Syst. 25(2), 633–641 (2010) 24. H.A. Eiselt, G. Laporte, J.-F. Thisse, Competitive location models: A framework and bibliography. Transport. Sci. 27(1), 44–54 (1993) 25. L. El-Ghaoui, H. Lebret, Robust solutions to least-square problems with uncertain data matrices. SIAM J. Matrix Anal. Appl. 18, 1035–1064 (1997) 26. L. El-Ghaoui, F. Oustry, H. Lebret, Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9, 33–52 (1998) 27. M. Garg, J.C. Smith, Models and algorithms for the design of survivable multicommodity flow networks with general failure scenarios. Omega 36(6), 1057–1071 (2008) 28. O. Goldschmidt, D.S. Hochbaum, A. Levin, E.V. Olinick, The SONET edge-partition problem. Networks 41(1), 13–23 (2003) 29. O. Goldschmidt, A. Laugier, E.V. Olinick, SONET/SDH ring assignment with capacity constraints. Discrete Appl. Math. 129(1), 99–128 (2003)

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30. M. Gr¨otschel, C.L. Monma, Integer polyhedra arising from certain network design problems with connectivity constraints. SIAM J. Discrete Math. 3(4), 502–523 (1990) 31. M. Gr¨otschel, C.L. Monma, M. Stoer, Polyhedral and computational investigations for designing communication networks with high survivability requirements. Oper. Res. 43(6), 1012–1024 (1995) 32. H. Hotelling, Stability in competition. Econ. J. 39(153), 41–57 (1929) 33. M. Iri, Extension of the maximum-flow minimum-cut theorem to multicommodity flows. J. Oper. Res. Soc. Jpn. 13(3), 129–135 (1971) 34. U. Janjarassuk, J. Linderoth, Reformulation and sampling to solve a stochastic network interdiction problem. Networks 52(3), 120–132 (2008) 35. P. Kall, S.W. Wallace, Stochastic Programming (Wiley, Chichester, 1994) 36. P. Kotler, R. Singh, Marketing warfare in the 1980s. J. Bus. Strat. 1(3), 30–41 (1980) 37. P. Kouvelis, G. Yu, Robust Discrete Optimization and Its Applications (Kluwer Academic, Norwell, 1997) 38. Y. Lee, H.D. Sherali, J. Han, S. Kim, A branch-and-cut algorithm for solving an intra-ring synchronous optical network design problem. Networks 35(3), 223–232 (2000) 39. A. Malaviya, C. Rainwater, T. Sharkey, Multi-period network interdiction models with applications to city-level drug enforcement. Working paper, Department of Industrial and Systems Engineering, Rensselaer Polytechnic Institute, Troy, 2011 40. D.P. Morton, F. Pan, K.J. Saeger, Models for nuclear smuggling interdiction. IIE Trans. 39(1), 3–14 (2007) 41. D.P. Morton, F. Pan, J.C. Smith, K. Sullivan, Inequalities for an asymmetric stochastic network interdiction problem. Technical report, Department of Industrial and Systems Engineering, University of Florida, Gainesville, 2011 42. A. Nemirovski, Lectures on robust convex optimization. http://www2.isye.gatech.edu/ nemirovs/ 43. F. Pan, W. Charlton, D.P. Morton, Interdicting smuggled nuclear material, in Network Interdiction and Stochastic Integer Programming, ed. by D.L. Woodruff (Kluwer Academic, Boston), 2003, pp. 1–20 44. F. Pan, D.P. Morton, Minimizing a stochastic maximum-reliability path. Networks 52, 111–119 (2008) 45. M. Parlar, Game theoretic analysis of the substitutable product inventory problem with random demands. Nav. Res. Logist. 35(3), 397–409 (1988) 46. A.E. Pearson, C.L. Irwin, Coca-Cola vs. Pepsi-Cola (A). Case study, Harvard Business School Case No. 9-387-108, Cambridge, 1988 47. F. Plastria, Static competitive facility location: An overview of optimisation approaches. Eur. J. Oper. Res. 129(3), 461–470 (2001) 48. M. Prince, J.C. Smith, J. Geunes, Optimizing exclusivity agreements in a three-stage procurement game. Working paper, Department of Industrial and Systems Engineering, The University of Florida, Gainesville, 2011 49. C.M. Rocco, J.E. Ramirez-Marquez, A bi-objective approach for shortest-path network interdiction. Comput. Ind. Eng. 59, 232–240 (2010) 50. J.O. Royset, R.K. Wood, Solving the bi-objective maximum-flow network-interdiction problem. INFORMS J. Comput. 19, 175–184 (2007) 51. J. Salmer´on, K. Wood, R. Baldick, Analysis of electric grid security under terrorist threat. IEEE Trans. Power Syst. 19, 905–912 (2004) 52. J. Salmer´on, K. Wood, R. Baldick, Worst-case interdiction analysis of large-scale electric power grids. IEEE Trans. Power Syst. 24, 96–104 (2009) 53. M.P. Scaparra, R.L. Church, A bilevel mixed-integer program for critical infrastructure protection planning. Comput. Oper. Res. 35(6), 1905–1923 (2008) 54. H.D. Sherali, J.C. Smith, Y. Lee, Enhanced model representations for an intra-ring synchronous optical network design problem allowing demand splitting. INFORMS J. Comput. 12(4), 284–298 (2000)

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55. J.C. Smith, Basic interdiction models, in Wiley Encyclopedia of Operations Research and Management Science, ed. by J. Cochran (Wiley, Hoboken, 2011) 56. J.C. Smith, S. Ahmed, Introduction to robust optimization, in Wiley Encyclopedia of Operations Research and Management Science, ed. by J. Cochran (Wiley, Hoboken, 2011) 57. J.C. Smith, C. Lim, Algorithms for network interdiction and fortification games, in Pareto Optimality, Game Theory and Equilibria, ed. by A. Migdalas, P.M. Pardalos, L. Pitsoulis, A. Chinchuluun, Nonconvex Optimization and Its Applications Series (Springer, New York, 2008), pp. 609–644 58. J.C. Smith, C. Lim, A. Alptekino˘glu, New product introduction against a predator: A bilevel mixed-integer programming approach. Nav. Res. Logist. 56, 714–729 (2009) 59. J.C. Smith, C. Lim, F. Sudargho, Survivable network design under optimal and heuristic interdiction scenarios. J. Global Optim. 38, 181–199 (2007) 60. J.C. Smith, A.J. Schaefer, J.W. Yen, A stochastic integer programming approach to solving a synchronous optical network ring design problem. Networks 44(1), 12–26 (2004) 61. L.V. Snyder, M.P. Scaparra, M.S. Daskin, R.L. Church, Planning for disruptions in supply chain networks, in Tutorials in Operations Research: Models, Methods, and Applications for Innovative Decision Making, ed. by M.P. Johnson, B. Norman, N. Secomandi. (INFORMS, 2006), pp. 234–257 62. A.L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973) 63. M. Stoer, G. Dahl, A polyhedral approach to multicommodity survivable network design. Numer. Math. 68(1), 149–167 (1994) 64. A. Sutter, F. Vanderbeck, L.A. Wolsey, Optimal placement of add/drop multiplexers: heuristic and exact algorithms. Oper. Res. 46(5), 719–728 (1998) 65. H. von Stackelberg, The Theory of the Market Economy (William Hodge, London, 1952) 66. T.-H. Wu, Fiber Network Service Survivability (Artech House, Boston, MA, 1992) 67. T.-H. Wu, M.E. Burrowes, Feasibility study of a high-speed SONET self-healing ring architecture in future interoffice networks. IEEE Comm. Mag. 28(11), 33–43 (1990)

Network Optimization Samir Khuller and Balaji Raghavachari

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Matching Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Matching Problem Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Matchings and Augmenting Paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Bipartite Matching Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Sample Execution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Matching Problem in General Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Assignment Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Applications of Matchings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Two Processor Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Weighted Edge-Cover Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Chinese Postman Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Network Flow Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Network Flow Problem Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Blocking Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Min-Cut Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Finding an s-t Min Cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Finding All-Pair Min Cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Applications of Network Flows (and Min Cuts). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Finding a Minimum-Weight Vertex Cover in Bipartite Graphs. . . . . . . . . . . . . . . . . . . 6.3 Maximum-Weight Closed Subset in a Partial Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Finding Densest Subgraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Algorithm for Densest Subgraph Without a Specified Subset. . . . . . . . . . . . . . . . . . . . 7.2 Algorithm for Densest Subgraph with a Specified Subset C . . . . . . . . . . . . . . . . . . . . .

1990 1991 1992 1992 1993 1994 1995 1995 1996 1998 1999 2000 2001 2002 2002 2006 2007 2008 2008 2008 2008 2009 2009 2010 2010 2011

S. Khuller () Department of Computer Science, University of Maryland, College Park, MD, USA e-mail: [email protected] B. Raghavachari Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA e-mail: [email protected] 1989 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 10, © Springer Science+Business Media New York 2013

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8 Minimum-Cost Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Min-Cost Flow Problem Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Multi-Commodity Flow Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Local Control Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Minimum Spanning and Other Light-Weight Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Minimum-Weight Branchings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Spanners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Light Approximate Shortest Path Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Coloring Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Vertex Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Edge Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Approximation Algorithms for Hard Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2012 2012 2014 2015 2016 2017 2018 2019 2020 2020 2020 2021 2022 2023 2023 2024 2025

Abstract

Many real-life problems can be modeled as optimization problems in networks. Examples include finding shortest paths, scheduling classes in classrooms, and finding the vulnerability of networks to disconnection because of link and node failures. In this chapter, a number of interesting problems and their algorithms are surveyed. Some of the problems studied are matching, weighted matching, maximum flow in networks, blocking flows, minimum cuts, densest subgraphs, minimum-cost flows, multi-commodity flows, minimum spanning trees, spanners, Light approximate shortest path trees, and graph coloring. Some interesting applications of matching, namely, two processor scheduling, edge covers, and the Chinese postman problem, are also discussed.

1

Introduction

The optimization of a given objective, while working with limited resources, is a fundamental problem that occurs in all walks of life, and therefore, optimization problems have been widely studied in Computer Science and Operations Research. Problems in discrete optimization vary widely in their complexity, and efficient solutions are derived for many of these problems by studying their combinatorial structure and understanding their fundamental properties. In this chapter, a number of problems arising in networks and advanced algorithmic techniques for solving them are studied. One of the basic topics in this field is network flow. The related optimization problems are fundamentally important. They have applications in various disciplines and provide a fundamental framework for solving problems. For example, the problem of efficiently moving entities, such as bits, people, or products, from one place to another in an underlying network, can be modeled as a network

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flow problem. Applications of flow include matching, scheduling, and connectivity problems. Because of its central role in the fields of Operations Research and Computer Science, considerable emphasis has been placed on the design of efficient algorithms for solving it. Many practical combinatorial problems such as the assignment problem, and the problem of finding the susceptibility of networks to failure due to faulty links or nodes, are special instances of the max-flow problem. There are several variations and generalizations of the max-flow problem including the vertex capacitated maxflow problem, the minimum-cost max-flow problem, the minimum-cost circulation problem, and the multi-commodity flow problem. In this chapter, a number of interesting and important optimization problems in networks are surveyed. Some of them such as network flow and matchings have been known for several decades, while others have recent results. The chapter starts with a discussion of matchings in Sect. 2. The single-commodity maximum flow problem is introduced in Sect. 4. The minimum-cut problem is discussed in Sect. 5. In Sect. 6 some applications of network flows are discussed. Section 7 discusses an interesting application of max flows to obtain the densest subgraph of a graph. Section 8 discusses the min-cost flow problem. The multi-commodity flow problem is discussed in Sect. 9. Section 10 introduces the problem of computing optimal branchings. Section 11 discusses the problem of coloring the edges and vertices of a graph. Section 12 discusses approximation algorithms for NP-hard problems. At the end, references to current research in graph algorithms are provided.

2

The Matching Problem

First, the matching problem, which is a special case of the max-flow problem, is introduced. Then, the assignment problem, a generalization of the matching problem, is studied. A matching is a set of edges such that no two of them are incident to the same vertex. Finding a matching of maximum cardinality within a given graph is called the matching problem. A matching is perfect if every vertex of the graph is matched. An entire book [35] has been devoted to the study of various aspects of the matching problem, ranging from necessary and sufficient conditions for the existence of perfect matchings to algorithms for solving the matching problem. Many of the basic algorithms such as breadth-first search, depth-first search, and shortest paths play important roles in developing efficient implementations for network flow and matching algorithms. The maximum matching problem is discussed in detail only for bipartite graphs. The same principles are used to design efficient algorithms to solve the matching problem in arbitrary graphs. The algorithms for general graphs are complex due to the presence of odd-length cycles called blossoms, and the reader is referred to [38, Chap. 10] or [43, Chap. 9] for detailed treatments on how blossoms are handled.

1992

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S. Khuller and B. Raghavachari

Matching Problem Definitions

Given a graph G D .V; E/, a matching M is a subset of the edges such that no two edges in M share a common vertex. In other words, the problem is that of finding a set of independent edges that have no incident vertices in common. The cardinality of M is usually referred to as its size. The following terms are defined with respect to a matching M . The edges in M are called matched edges, and edges not in M are called free edges. Likewise, a vertex is a matched vertex if it is incident to a matched edge. A free vertex is one that is not matched. The mate of a matched vertex v is its neighbor w that is at the other end of the matched edge incident to v. A matching is called perfect if all vertices of the graph are matched in it. When the number of vertices is odd, one vertex is permitted to remain unmatched. The objective of the maximum matching problem is to maximize jM j, the size of the matching. If the edges of the graph have weights, then the weight of a matching is defined to be the sum of the weights of its edges. A path p D Œv1 ; v2 ; : : : ; vk  is called an alternating path if the edges .v2j 1 ; v2j /, j D 1; 2; : : : are free and the edges .v2j ; v2j C1 /, j D 1; 2; : : : are matched. An augmenting path p D Œv1 ; v2 ; : : : ; vk  is an alternating path in which both v1 and vk are free vertices. Observe that an augmenting path is defined with respect to a specific matching. The symmetric difference of a matching M and an augmenting path P , M ˚ P , is defined to be .M  P / [ .P  M /. It can be shown that M ˚ P is also a matching. Figure 1 shows an augmenting path p D Œa; b; c; d; g; h with respect to the given matching. The symmetric difference operation can also be used as above between two matchings. In this case, the resulting graph consists of a collection of paths and cycles with alternate edges from each matching.

2.2

Matchings and Augmenting Paths

The following theorem gives necessary and sufficient conditions for the existence of a perfect matching in a bipartite graph.

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Theorem 1 (Hall’s Theorem) A bipartite graph G D .X; Y; E/ with jX j D jY j has a perfect matching if and only if 8S  X; jN.S /j  jS j, where N.S /  Y is the set of vertices that are neighbors of some vertex in S . Although the above theorem captures exactly the conditions under which a given bipartite graph has a perfect matching, it does not lead to an algorithm for finding perfect matchings directly. The following lemma shows how an augmenting path with respect to a given matching can be used to increase the size of a matching. An efficient algorithm will be described later that uses augmenting paths to construct a maximum matching incrementally. Lemma 1 Let P be the edges on an augmenting path p D Œv1 ; : : : ; vk  with respect to a matching M . Then, M 0 D M ˚ P is a matching of cardinality jM j C 1. Proof Since P is an augmenting path, both v1 and vk are free vertices in M . The number of free edges in P is one more than the number of matched edges in it. The symmetric difference operator replaces the matched edges of M in P by the free edges in P . Hence the size of the resulting matching, jM 0 j, is one more than jM j. The following theorem provides a necessary and sufficient condition for a given matching M to be a maximum matching. Theorem 2 A matching M in a graph G is a maximum matching if and only if there is no augmenting path in G with respect to M . Proof If there is an augmenting path with respect to M , then M cannot be a maximum matching, since by Lemma 1 there is a matching whose size is larger than that of M . To prove the converse, it is shown that if there is no augmenting path with respect to M , then M is a maximum matching. Suppose that there is a matching M 0 such that jM 0 j > jM j. Consider the subgraph of G induced by the edges M ˚ M 0 . Each vertex in this subgraph has degree at most two, since each node has at most one edge from each matching incident to it. Hence each connected component of this subgraph is either a path or a simple cycle. For each cycle, the number of edges of M is the same as the number of edges of M 0 . Since jM 0 j > jM j, one of the paths must have more edges from M 0 than from M . This path is an augmenting path in G with respect to the matching M , contradicting the assumption that there were no augmenting paths with respect to M .

2.3

Bipartite Matching Algorithm

High-Level Description: The algorithm starts with the empty matching M D ; and augments the matching in phases. In each phase, an augmenting path with respect to the current matching M is found, and it is used to increase the size of the matching.

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An augmenting path, if one exists, can be found in O.jEj/ time, using a procedure similar to breadth-first search. The search for an augmenting path proceeds from the free vertices. At each step, when a vertex in X is processed, all its unvisited neighbors are also searched. When a matched vertex in Y is considered, only its matched neighbor is searched. This search proceeds along a subgraph referred to as the Hungarian tree. The algorithm uses a queue to hold vertices that are yet to be processed. Initially, all free vertices in X are placed in the queue. The vertices are removed one by one from the queue and processed as follows. In turn, when vertex v is removed from the queue, all edges incident to it are scanned. If it has a neighbor in the vertex set Y that is free, then the search for an augmenting path is successful; update the matching, and the algorithm proceeds to its next phase. Otherwise, add the mates of all the matched neighbors of v to the queue if they have never been added to the queue, and continue the search for an augmenting path. If the algorithm empties the queue without finding an augmenting path, its current matching is a maximum matching, and it terminates.

2.4

Sample Execution

Figure 2 shows a sample execution of the matching algorithm. It depicts a partial matching and the structure of the resulting Hungarian tree. In this example, the search starts from the free vertex b. Its neighbors are v and z, and both are matched.

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Fig. 2 Sample execution of matching algorithm

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Their mates, c and e, are added to a queue Q. When c is explored, d is added to Q and then f and a. Since a has a free neighbor u, an augmenting path from vertex b to vertex u is found by the algorithm.

2.5

Analysis

If there are augmenting paths with respect to the current matching, the algorithm will find at least one of them. Hence, when the algorithm terminates, the graph has no augmenting paths with respect to the current matching, and the current matching is optimal. Each iteration of the main while loop of the algorithm runs in O.jEj/ time. The construction of the auxiliary graph A and computation of the array f ree also take O.jEj/ time. In each iteration, the size of the matching increases by one, and thus, there are at most min.jX j; jY j/ iterations of the while loop. Therefore, the algorithm solves the matching problem for bipartite graphs in time O.min.jX j; jY j/jEj/. Hopcroft and Karp (see [38]) showed how to improve the running time by finding a maximal set of disjoint augmenting paths in ap single phase in O.jEj/ time. They also proved that thep algorithm runs in only O. jV j/ phases, yielding a worst-case running time of O. jV jjEj/.

2.6

The Matching Problem in General Graphs

The techniques used to solve the matching problem in bipartite graphs do not extend directly to non-bipartite graphs. The notion of augmenting paths and their relation to maximum matchings (Theorem 2) remain the same. Therefore, it is still possible to use the natural algorithm of starting with an empty matching and increasing its size repeatedly with augmenting paths until no augmenting path exists in the graph. But the problem of finding augmenting paths in non-bipartite graphs is harder. The main trouble is due to odd-length cycles known as blossoms that appear along alternating paths explored by the algorithm when it is looking for augmenting paths. This difficulty is illustrated by the example in Fig. 3. The search for an augmenting path from an unmatched vertex, such as e, could go through the following sequence of vertices Œe; b; g; d; h; g; b; a. Even though the augmenting path satisfies the “local” conditions for being an augmenting path, it is not a valid augmenting path since it is not simple. The reason for this is that the odd-length cycle .g; d; h; g/ causes the path to “fold” on itself – a problem that does not arise in the bipartite case. In fact, the matching does contain a valid augmenting path Œe; f; c; d; h; g; b; a. Not all odd cycles cause this problem, but odd cycles that are as dense in matched edges as possible, that is, it depends on the current matching. By “shrinking” blossoms to single nodes, the algorithm can get rid of them [38]. Subsequent work focussed on efficient implementation of this method. Edmonds (see [38]) gave the first polynomial-time algorithm for solving the maximum matching problem in general graphs. The current fastest algorithm for this

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Fig. 3 Difficulty in dealing with blossoms

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h

d

p problem is due to Micali and Vazirani [36] and their, algorithm runs in O.jEj jV j/ steps, which are the same bound obtained by the Hopcroft-Karp algorithm for finding a maximum matching in bipartite graphs.

2.7

Assignment Problem

In this section, the assignment problem – that of finding a maximum-weight matching in a given bipartite graph in which edges are given nonnegative weights – is studied. There is no loss of generality in assuming that the graph is a complete bipartite graph, since zero-weight edges may be added between pairs of vertices that are nonadjacent in the original graph without affecting the weight of a maximumweight matching. The minimization version of the weighted version is the problem of finding a minimum-weight perfect matching in a complete bipartite graph. Both versions of the weighted matching problem are equivalent, and it is shown below how to reduce the minimum-weight perfect matching to maximum-weight matching. Choose a constant W that is larger than the weight of any edge, and assign each edge a new weight of w0 .e/ D W  w.e/. Observe that maximum-weight matchings with the new weight function are minimum-weight perfect matchings with the original weights. In this section, attention is restricted to the study of the maximum-weight matching problem for bipartite graphs. Similar techniques have been used to solve the maximum-weight matching problem in arbitrary graphs (see [33, 38]). The input is a complete bipartite graph G D .X; Y; X  Y /, and each edge e has a nonnegative weight of w.e/. The following algorithm is known as the Hungarian method (see [1, 35, 38]). The method can be viewed as a primal-dual algorithm in the framework of linear programming [38]. No knowledge of linear programming is assumed here. A feasible vertex-labeling ` is defined to be a mapping from the set of vertices in G to the real numbers such that for each edge .xi ; yj /, the following condition holds: `.xi / C `.yj /  w.xi ; yj /:

Network Optimization

1997

The following can be verified to be a feasible vertex labeling. For each vertex yj 2 Y , set `.yj / to be 0, and for each vertex xi 2 X , set `.xi / to be the maximum weight of an edge incident to xi : `.yj / D 0; `.xi / D max w.xi ; yj /: j

The equality subgraph, G` , is defined to be the spanning subgraph of G which includes all vertices of G but only those edges .xi ; yj / which have weights such that `.xi / C `.yj / D w.xi ; yj /: The connection between equality subgraphs and maximum-weighted matchings is established by the following theorem. Theorem 3 If the equality subgraph, G` , has a perfect matching, M  , then M  is a maximum-weight matching in G. Proof Let M  be a perfect matching in G` . By definition, w.M  / D

X

w.e/ D

e2M 

X

`.v/:

v2X [Y

Let M be any perfect matching in G. Then, w.M / D

X e2M

w.e/ 

X

`.v/ D w.M  /:

v2X [Y

Hence, M  is a maximum-weight perfect matching. High-Level Description: The above theorem is the basis of the following algorithm for finding a maximum-weight matching in a complete bipartite graph. The algorithm starts with a feasible labeling, then computes the equality subgraph and a maximum cardinality matching in this subgraph. If the matching found is perfect, by Theorem 3, the matching must be a maximum-weight matching, and the algorithm returns it as its output. Otherwise the matching is not perfect, and more edges need to be added to the equality subgraph by revising the vertex labels. The revision should ensure that edges from the current matching do not leave the equality subgraph. After more edges are added to the equality subgraph, the algorithm grows the Hungarian trees further. Either the size of the matching increases because an augmenting path is found, or a new vertex is added to the Hungarian tree. In the former case, the current phase terminates, and the algorithm starts a new phase since the matching size has increased. In the latter case, new nodes are added to the Hungarian tree. In jX j phases, the tree includes all the nodes, and therefore, there are at most jX j phases before the size of the matching increases.

1998

S. Khuller and B. Raghavachari

Fig. 4 Sets S and T as maintained by the algorithm

S

T

−δ

+δ

S

T

It is now described in more detail how the labels are updated and which edges are added to the equality subgraph. Suppose M is a maximum matching found by the algorithm. Hungarian trees are grown from all the free vertices in X . Vertices of X (including the free vertices) that are encountered in the search are added to a set S , and vertices of Y that are encountered in the search are added to a set T . Let S D X  S and T D Y  T . Figure 4 illustrates the structure of the sets S and T . Matched edges are shown in bold; the other edges are the edges in G` . Observe that there are no edges in the equality subgraph from S to T , even though there may be edges from T to S . The algorithm now revises the labels as follows. Decrease all the labels of vertices in S by a quantity ı (to be determined later), and increase the labels of the vertices in T by ı. This ensures that edges in the matching continue to stay in the equality subgraph. Edges in G (not in G` ) that go from vertices in S to vertices in T are candidate edges to enter the equality subgraph, since one label is decreasing and the other is unchanged. The algorithm chooses ı to be the smallest value such that some edge of G  G` enters the equality subgraph. Suppose this edge goes from x 2 S to y 2 T . If y is free, then an augmenting path has been found. On the other hand, if y is matched, the Hungarian tree is grown by moving y to T and its matched neighbor to S , and the process of revising labels is continued.

3

Applications of Matchings

Matchings lie at the heart of many optimization problems, and the problem has many applications. The name comes from the marriage problem, where the objective is to find a maximum number of compatible pairs of men and women for marriage,

Network Optimization

1999

given their pairwise compatibility information. Applications of matching include assigning workers to jobs, assigning a collection of jobs with precedence constraints to two processors such that the total execution time is minimized, determining the structure of chemical bonds in Chemistry, matching moving objects based on a sequence of snapshots, and localization of objects in space after obtaining information from multiple sensors (see [1]). A few applications of matching are discussed.

3.1

Two Processor Scheduling

First, the problem of scheduling jobs on two processors is studied. The original paper is by Fujii, Kasami, and Ninomiya [17]. There are two identical processors and a collection of n jobs that need to be scheduled. Each job requires unit time. There is a precedence graph associated with the jobs (also called the DAG). If there is an edge from i to j , then i must be finished before j is started by either processor. How should the jobs be scheduled on the two processors so that all the jobs are completed as quickly as possible. The jobs could represent courses that need to be taken (courses have prerequisites), and if a student is taking at most two courses in each semester, the question really is: How quickly can he graduate? This is a good time to note that even though the two processor scheduling problem can be solved in polynomial time, the three processor scheduling problem is not known to be solvable in polynomial time and is known to be NP-complete. In fact, the complexity of the k processor scheduling problem when k is fixed is not known. From the acyclic graph G, a compatibility graph G  is constructed as follows.  G has the same nodes as G, and there is an (undirected) edge .i; j / if there is no directed path from i to j or from j to i in G. In other words, i and j are jobs that can be done together. A maximum matching in G  indicates the maximum number of pairs of jobs that can be processed simultaneously. Clearly, a solution to the scheduling problem can be used to obtain a matching in G  . More interestingly, a solution to the matching problem can be used to obtain a solution to the scheduling problem! Suppose a maximum matching M in G  is found. An unmatched vertex is executed on a single processor while the other processor is idle. If the maximum matching has size m , then 2m vertices are matched and n  2m vertices are left unmatched. The time to finish all the jobs will be m C n  2m D n  m . Hence a maximum matching will minimize the time required to schedule all the jobs. It is now shown that, given a maximum matching M  , a schedule of size n  m can be extracted from it. The key idea is that each matched job is scheduled in a slot together with another job (which may be different from the job it was matched to). This ensures that no more than n  m slots are used. Let S be the subset of vertices that have indegree 0. The following cases show how a schedule can be constructed from G and M  . Basically, it must be ensured

2000

S. Khuller and B. Raghavachari

a

b

J1

J1

J1

J1

J2

J2

J2

J2

J3

J3

nodes in S

matched edge in M ∗ new matched edge

nodes in S matched edge in M ∗

Fig. 5 (a) Changing the schedule (b) Continuing the proof

that for all jobs that are paired up in M  , they are also paired up in the schedule. The following rules can be applied repeatedly. 1. If there is an unmatched vertex in S , schedule it and remove it from G. 2. If there is a pair of jobs in S that are matched in M  , then schedule the pair and delete both the jobs from G. If none of the above rules are applicable, then all jobs in S are matched; moreover each job in S is matched with a job not in S . Let J1 2 S be matched to J10 … S . Let J2 be a job in S that has a path to J10 . Let J20 be the mate of J2 . If there is path from J10 to J20 , then J2 and J20 could not be matched to each other in G  (this cannot be an edge in the compatibility graph). If there is no path from J20 to J10 , then the matching is changed to match .J1 ; J2 / and .J10 ; J20 / since .J10 ; J20 / is an edge in G  (see Fig. 5a). The only remaining case is when J20 has a path to J10 . Recall that J2 also has a path to J10 . The above argument is repeated with J20 in place of J10 . This will yield a new pair .J3 ; J30 /, etc (see Fig. 5b). Since the graph G has no cycles at each step, a distinct pair of jobs is generated; at some point of time this process will stop (since the graph is finite). At that time a pair of jobs in S have been found that can be scheduled together.

3.2

Weighted Edge-Cover Problem

Another application of matching is the problem of finding a minimum-weight edge cover of a graph. Given a graph G D .V; E/ with weights on the edges, a set of edges C  E form an edge cover of G if every vertex in V is incident to at least one edge in C . The weight of the cover C is the sum of the weights of its edges. The objective is to find an edge cover C of minimum weight. The problem can be reduced to the minimum-weight perfect matching problem as follows: Create an identical copy G 0 D .V 0 ; E 0 / of graph G, except the weights of edges in E 0 are all 0. Add an edge from each v 2 V to v0 2 V 0 with weight wmin .v/ D minx2N.v/ w.v; x/. The final graph H is the union of G and G 0 , together with the edges connecting the

Network Optimization

2001

two copies of each vertex; H contains 2jV j vertices and 2jEj C jV j edges. There exist other reductions from minimum-weight edge cover to minimum-weight perfect matching; the reduction outlined above has the advantage that it creates a graph with O.jV j/ vertices and O.jEj/ edges. The minimum-weight perfect matching problem may be solved using the techniques described earlier for the bipartite case, but the algorithm is more complex [33, 38]. Theorem 4 The weight of a minimum-weight perfect matching in H is equal to the weight of a minimum-weight edge cover in G. Proof Consider a minimum-weight edge cover C in G. There is no loss of generality in assuming that a minimum-weight edge cover has no path of three edges, since the middle edge can be removed from the cover, thus reducing its weight further. Hence the edge cover C is a union of “stars” (trees of height 1). For each star that has a vertex of degree more than one, one of its leaf nodes can be made to match to its copy in G 0 , with weight at most the weight of the edge incident on the leaf vertex, thus reducing the degree of the star. This is repeated until each vertex has degree at most one in the edge cover, that is, it is a matching. In H , select this matching, once in each copy of G. Observe that the cost of the matching within the second copy G 0 is 0. Thus, given C , a perfect matching in H can be found, whose weight is no larger. To argue the converse, it is now shown how to construct a cover C from a given perfect matching M in H . For each edge .v; v0 / 2 M , add to C a least weight edge incident to v in G. Also include in C any edge of G that was matched in M . It can be verified that C is an edge cover whose weight equals the weight of M .

3.3

Chinese Postman Problem

One of the first problems to be studied in graph theory was the Euler tour problem. It asks whether a given graph has a tour that traverses each edge of the graph exactly once. Graphs that have Euler tours are called Eulerian graphs. It is well known that a graph is Eulerian if and only if it is connected and the degree of every vertex is even. Similarly, a necessary and sufficient condition for a directed graph to have an Euler tour is that it is strongly connected and the number of incoming edges at each vertex is equal to the number of outgoing edges from it. Given an Eulerian graph, whether undirected or directed, an Euler tour can be found in linear time. A natural optimization problem, known as the Chinese postman problem (CPP), arises in non-Eulerian graphs, which are that of finding a shortest tour that traverses each edge of a given graph at least once. This problem and its generalizations were studied by Edmonds and Johnson [13]. CPP can be solved by using algorithms for weighted matching as follows. It is assumed that the underlying graph is connected, since otherwise, the problem is not feasible. CPP in undirected graphs: Here, the given graph G needs to be augmented by adding additional edges such that the degree of every vertex becomes even. The subgraph that is added, known as a T-join, has odd degree at odd-degree nodes

2002

S. Khuller and B. Raghavachari

of G and has even degree (possibly zero) at other nodes. It is easy to show that any graph has an even number of odd-degree nodes and that a minimum-weight T-join can be decomposed into a collection of edge-disjoint paths connecting odddegree nodes. This leads us to the following algorithm for finding a shortest CPP in undirected graphs. Construct a new graph containing just the odd-degree nodes of G. Add edges connecting each pair of these nodes, with the weight of an edge being the length of a shortest path in G connecting those two nodes. Now, find a minimum-weight perfect matching in this graph. For each edge in this matching, a shortest path in G is added between its end vertices. The resulting multigraph has even degree at all nodes, and an Euler tour can be found in it. CPP in directed graphs: In this case, it must be ensured that every node is balanced, that is, the number of outgoing edges is equal to the number of its incoming edges. For a vertex v, its imbalance, b.v/, is defined to be its outdegree minus its indegree. Construct an auxiliary graph with just the vertices that have b.v/ ¤ 0. In this graph, it is necessary to add as many copies of a vertex as its imbalance, that is, there are jb.v/j copies of vertex v. The vertices are partitioned into those with positive imbalance (Vp ) and vertices with negative imbalance (Vn ). Take the complete bipartite graph between Vp and Vn , where the weight of an edge .p; n/ is equal to the length of a shortest path in G from p 2 Vp to n 2 Vn . Now, find a minimum-weight perfect matching M in this bipartite graph, and when G is augmented by adding the paths in G corresponding to M , the resulting multigraph is Eulerian. Note that it is also possible to solve the problem by using a solution for min-cost flow (see Sect. 8), by defining demands to be b.v/, by setting the weight of each edge to be its cost, and by finding a min-cost flow (without capacities) that satisfies all the demands.

4

The Network Flow Problem

A number of polynomial-time flow algorithms have been developed over the last two decades. The reader is referred to the books by Ahuja, Magnanti, and Orlin [1] and Tarjan [43] for a detailed account of the historical development of the various flow methods. An excellent survey on network flow algorithms has been written by Goldberg, Tardos, and Tarjan [23]. The book by Cormen, Leiserson, Stein, and Rivest [11] describes the preflow-push method, and to complement their coverage, an implementation of the “blocking flow” technique of Malhotra, Kumar, and Maheshwari (see [38]) is discussed here.

4.1

Network Flow Problem Definitions

Flow Network: A flow network G D .V; E/ is a directed graph, with two specially marked nodes, namely, the source s and the sink t. A capacity function u W E 7! 0. Figure 6 shows a network on the left with a given flow function. Each edge is labeled with two values: the flow through that edge and its capacity. The figure on the right depicts the residual network corresponding to this flow. An augmenting path for f is a path P from s to t in GR .f /. The residual capacity of P , denoted by u0 .P /, is the minimum value of u0 .v; w/ over all edges .v; w/ in the path P . The flow can be increased by u0 .P /, by increasing the flow on each edge of P by this amount. Whenever f .v; w/ is changed, f .w; v/ is also correspondingly changed to maintain skew symmetry. Most flow algorithms are based on the concept of augmenting paths pioneered by Ford and Fulkerson [16]. They start with an initial zero flow and augment the flow in stages. In each stage, a residual graph GR .f / with respect to the current flow function f is constructed, and an augmenting path in GR .f / is found, thus increasing the value of the flow. Flow is increased along this path until an edge in

Network Optimization

2005

this path is saturated. The algorithms iteratively keep increasing the flow until there are no more augmenting paths in GR .f / and return the final flow f as their output. Edmonds and Karp [14] suggested two possible improvements to this algorithm to make it run in polynomial time. The first was to choose shortest possible augmenting paths, where the length of the path is simply the number of edges on the path. This method can be improved using the approach due to Dinitz (see [43]) in which a sequence of blocking flows is found. The second strategy was to select a path which can be used to push the maximum amount of flow. The number of iterations of this method is bounded by O.jEj log C / where C is the largest edge capacity. In each iteration, a path is found through which a maximum amount of flow can be pushed. This step is implemented by a suitable modification of Dijkstra’s shortest path algorithm. The following lemma is fundamental in understanding the basic strategy behind these algorithms. Lemma 2 Let f be a flow and f  be a maximum flow in G. The value of a maximum flow in the residual graph GR .f / is jf  j  jf j. Proof Let f 0 be any flow in GR .f /. Define f Cf 0 to be the flow f .v; w/Cf 0 .v; w/ for each edge .v; w/. Observe that f C f 0 is a feasible flow in G of value jf j C jf 0 j. Since f  is the maximum flow possible in G, jf 0 j  jf  j  jf j. Similarly define f   f to be a flow in GR .f / defined by f  .v; w/  f .v; w/ in each edge .v; w/, and this is a feasible flow in GR .f / of value jf  j  jf j, and it is a maximum flow in GR .f /. Blocking Flow: A flow f is a blocking flow if every path from s to t in G contains a saturated edge. Blocking flows are also known as maximal flows. Figure 7 depicts a blocking flow. Any path from s to t contains at least one saturated edge. For example, in the path Œs; a; t, the edge .a; t/ is saturated. It is important to note that a blocking flow is not necessarily a maximum flow. There may be augmenting paths that increase the flow on some edges and decrease the flow on other edges (by increasing the flow in the reverse direction). The example in Figure 7 contains an augmenting path Œs; a; b; t, which can be used to increase the flow by 3 units. Layered Networks: Let GR .f / be the residual graph with respect to a flow f . The level of a vertex v is the length of a shortest path from s to v in GR .f /. The level graph L for f is the subgraph of GR .f / containing vertices reachable from s and only the edges .v; w/ such that level.w/ D 1 C level.v/. L contains all shortest length augmenting paths and can be constructed in O.jEj/ time. The maximum flow algorithm proposed by Dinitz starts with the zero flow and iteratively increases the flow by augmenting it with a blocking flow in GR .f / until t is not reachable from s in GR .f /. At each step, the current flow is replaced by the sum of the current flow and the blocking flow. This algorithm terminates in jV j  1 iterations, since in each iteration the shortest distance from s to t in the residual graph increases. The shortest path from s to t is at most jV j  1, and this gives an upper bound on the number of iterations of the algorithm.

2006

S. Khuller and B. Raghavachari a

Fig. 7 Example of blocking flow – all paths from s to t are saturated. A label of .f; c/ for an edge indicates a flow of f and capacity of c

(6,6)

(3,6)

t

(3,4)

s

(3,3)

(0,3) b

An algorithm to find a blocking flow that runs in O.jV j2 / time is described here (see [38] for details), and this yields an O.jV j3 / max-flow algorithm. There are a number of O.jV j2 / blocking flow algorithms available, some of which are described in [43].

4.2

Blocking Flows

Dinitz’s algorithm finds a blocking flow as follows. The main step is to find paths from the source to the sink and push as much flow as possible on each path and thus saturate an edge in each path. It takes O.jV j/ time to compute the amount of flow that can be pushed on a path. Since there are jEj edges, this yields an upper bound of O.jV jjEj/ steps on the running time of the algorithm. The following algorithm shows how to find a blocking flow more efficiently. Malhotra-Kumar-Maheshwari Blocking Flow Algorithm: The algorithm has a current flow function f and its corresponding residual graph GR .f /. Define for each node v 2 GR .f / a quantity tpŒv that specifies its maximum throughput, that is, either the sum of the capacities of the incoming arcs or the sum of the capacities of the outgoing arcs, whichever is smaller. The quantity tpŒv represents the maximum flow that could pass through v in any feasible blocking flow in the residual graph. Vertices with zero throughput are deleted from GR .f /. The algorithm selects a vertex x with least throughput. It then greedily pushes a flow of tpŒx from x towards t, level by level in the layered residual graph. This can be done by creating a queue which initially contains x, which is assigned the task of pushing tpŒx out of it. In each step, the vertex v at the front of the queue is removed, and the arcs going out of v are scanned one at a time, and as much flow as possible is pushed out of them until v’s allocated flow has been pushed out. For each arc .v; w/ that the algorithm pushed flow through, it updates the residual capacity of the arc .v; w/ and places w on a queue (if it is not already there) and increments the net incoming flow into w. Also tpŒv is reduced by the amount of flow that was sent through it now. The flow finally reaches t, and the algorithm never comes across a vertex that has incoming flow that exceeds its outgoing capacity since x was chosen

Network Optimization

2007

as a vertex with least throughput. The above idea is repeated to pull a flow of tpŒx from the source s to x. Combining the two steps yields a flow of tpŒx from s to t in the residual network that goes through x. The flow f is augmented by this amount. Vertex x is deleted from the residual graph, along with any other vertices that have zero throughput. The above procedure is repeated until all vertices are deleted from the residual graph. The algorithm has a blocking flow at this stage since at least one vertex is saturated in every path from s to t. In the above algorithm, whenever an edge is saturated, it may be deleted from the residual graph. Since the algorithm uses a greedy strategy to pump flows, at most O.jEj/ time is spent when an edge is saturated. When finding flow paths to push tpŒx, there are at most jV j times (once for each vertex) when the algorithm pushes a flow that does not saturate the corresponding edge. After this step, x is deleted from the residual graph. Hence the above algorithm to compute blocking flows terminates in O.jEj C jV j2 / D O.jV j2 / steps. Karzanov (see [43]) used the concept of preflows to develop an O.jV j2 / time algorithm for developing blocking flows as well. This method was then adapted by Goldberg and Tarjan (see [11]), who proposed a preflow-push method for computing a maximum flow in a network. A preflow is a flow that satisfies the capacity constraints, but not flow conservation. Any node in the network is allowed to have more flow coming into it than there is flowing out. The algorithm tries to move the excess flows towards the sinks without violating capacity constraints. The algorithm finally terminates  with a feasible flow that is a max flow. It finds a max jV j2 flow in O jV jjEj log jEj time.

5

The Min-Cut Problem

Two problems are studied in this section. The first is the s-t min-cut problem: Given a directed graph with capacities on the edges and two special vertices s and t, the problem is to find a cut of minimum capacity that separates s from t. The value of a min-cut’s capacity can be computed by finding the value of the maximum flow from s to t in the network. Recall that the max-flow min-cut theorem shows that the two quantities are equal. It is now shown how to find the sets S and T that provides an s-t min cut from a given s-t max flow. The second problem is to find a smallest cut in a given graph G. In this problem on undirected graphs, the vertex set must be partitioned into two sets such that the total weight of the edges crossing the cut is minimized. This problem can be solved by enumerating over all fs; tg pairs of vertices and finding a s-t min cut for each pair. This procedure is rather inefficient. However, Hao and Orlin [24] showed that this problem can be solved in the same order of running time as taken by a single maxflow computation. Stoer and Wagner [41] have given a simple and elegant algorithm for computing minimum cuts based on a technique due to Nagamochi and Ibaraki [37]. Recently, Karger [27] has given a randomized algorithm that runs in almost linear time for computing minimum cuts without using network flow methods.

2008

5.1

S. Khuller and B. Raghavachari

Finding an s-t Min Cut

Let f  be a maximum flow from s to t in the graph. Recall that GR .f  / is the residual graph corresponding to f  . Let S be the set of all vertices that are reachable from s in GR .f  /, that is, vertices to which s has a directed path. Let T be all the remaining vertices of G. By definition s 2 S . Since f  is a max flow from s to t, t 2 T . Otherwise the flow can be increased by pushing flow along this path from s to t in GR .f  /. All the edges from vertices in S to vertices in T are saturated and form a minimum cut.

5.2

Finding All-Pair Min Cuts

For an undirected graph G with jV j vertices, Gomory and Hu (see [1]) showed that the flow values between each of the jV j.jV j  1/=2 pairs of vertices of G can be computed by solving only jV j  1 max-flow computations. Furthermore, they showed that the flow values can be represented by a weighted tree T on jV j nodes. Each node in the tree represents one of the vertices of the given graph. For any pair of nodes .x; y/, the maximum flow value from x to y (and hence the x-y min cut) in G is equal to the weight of the minimum-weight edge on the unique path in T between x and y.

6

Applications of Network Flows (and Min Cuts)

There are numerous applications of the maximum flow algorithm in scheduling problems of various kinds. It is used in open-pit mining, vehicle routing, etc. A few interesting applications of the flow problem are discussed. See [1] for further details.

6.1

Connectivity

Given a graph G, its connectivity is a measure of its susceptibility to disconnection upon the removal of some of its edges or nodes. It is an important concept in the area of telecommunication networks, where failure of nodes (called hubs) and edges (called links) can lead to network separation, resulting in millions of dollars in losses. For two nodes v and w, define their edge connectivity to be the maximum number of edge-disjoint paths between v and w in G. Similarly, their vertex connectivity or simply connectivity is the maximum number of vertex-disjoint (also known as openly disjoint) paths connecting them. One can also define the terms using the minimum number of edges/vertices that have to be removed to disconnect v from w. Menger showed that these two definitions are equivalent: The maximum number of edge-disjoint and openly disjoint paths from v to w are, respectively, equal to the

Network Optimization

2009

minimum number of edges and vertices that must be removed to separate v from w in G. Given a graph G, the edge connectivity of its vertices v and w is computed as follows. Replace each edge of G by two (antiparallel) directed edges between its end points. Assign the capacity of each edge to be 1. In the resulting flow network, find a maximum flow from v to w. The value of the maximum flow is equal to the edge connectivity of v and w in G. A min cut separating v and w is a minimum set of edges of G, whose removal separates v from w in G. For vertex connectivity, openly disjoint paths are required. This can be done by setting capacities of vertices also to be 1, which can be accomplished by replacing each vertex u by two vertices ui n and uout , adding the edge from ui n to uout of capacity 1. All incoming edges to u are sent to ui n . All outgoing edges from u are sent out of uout . Now, a maximum flow from vout to wi n gives us the maximum number of openly disjoint paths from v to w in G.

6.2

Finding a Minimum-Weight Vertex Cover in Bipartite Graphs

A vertex cover of a graph is a set of vertices that is incident to all its edges. The weight of a vertex cover is the sum of the weights of its vertices. Finding a vertex cover of minimum weight is NP-hard for arbitrary graphs (see Sect. 12 for more details). For bipartite graphs, the problem is solvable efficiently using the maximum flow algorithm. Given a bipartite graph G D .X; Y; E/ with weights on the vertices, the goal is to find a subset C of vertices of minimum total weight, so that for each edge, at least one of its end vertices is in C . The weight of a vertex v is denoted by w.v/. Construct a flow network N from G as follows: Add a new source vertex s, and for each x 2 X , add a directed edge .s; x/, with capacity w.x/. Add a new sink vertex t, and for each y 2 Y , add a directed edge .y; t/, with capacity w.y/. The edges in E are given infinite capacity, and they are oriented from X to Y . Let C be a subset of vertices in G. In the following discussion, XC denotes X \ C , and YC denotes Y \ C . Let XC D X  XC and YC D Y  YC . If C is a vertex cover of G, then there are no edges in G connecting XC and YC , since such edges would not be incident to C . A cut in the flow network N , whose capacity is the same as the weight of the vertex cover, can be constructed as follows: S D fsg [ XC [ YC and T D ftg [ XC [ YC . Each vertex cover of G gives rise to a cut whose capacity is equal to the weight of the cover. Similarly, any cut of finite capacity in N corresponds to a vertex cover of G whose weight is equal to the capacity of the cut. Hence the minimum-weight s-t cut of N corresponds to a minimum-weight vertex cover of G.

6.3

Maximum-Weight Closed Subset in a Partial Order

Consider a directed acyclic graph (DAG) G D .V; E/. Each vertex v of G is given a weight w.v/ that may be positive or negative. A subset of vertices S  V is said to be closed in G if for every s 2 S , the predecessors of s are also in S .

2010

S. Khuller and B. Raghavachari

The predecessors of a vertex s are vertices that have directed paths to s. Consider the objective of computing a closed subset S of maximum weight. This problem occurs, for example, in open-pit mining, and it can be solved efficiently by reducing it to computing the minimum cut in the following network N : • The vertex set of N D V [ fs; tg, where s and t are two new vertices. • For each vertex v of negative weight, add the edge .s; v/ with capacity jw.v/j to N . • For each vertex v of positive weight, add the edge .v; t/ with capacity w.v/ to N . • All edges of G are added to N , and each of these edges has infinite capacity. Consider a minimum s-t cut in N . Let S be the positive weight vertices whose edges to the sink t are not in the min cut. It can be shown that the union of the set S and its predecessors is a maximum-weight closed subset.

7

Finding Densest Subgraphs

The density of a subgraph is defined as the ratio of the number of edges in the induced graph to the number of vertices in the subgraph. The problem can be extended to the weighted case when edges and vertices have positive integer weights defined by functions w0 and w, respectively, by defining the density of a subgraph as the ratio of the total weight of its edges to the total weight of its vertices. Given a graph G, the objective is to find a subgraph of G, whose density is the largest among all its subgraphs. Despite the fact that there are exponentially many subgraphs, it is possible to find a maximum density subgraph in polynomial time using a procedure for the max-flow (min-cut) problem. An interesting extension of the problem is when a subset C of vertices is given, and the goal is to find a densest subgraph that must include all nodes of C .

7.1

Algorithm for Densest Subgraph Without a Specified Subset

First, the basic algorithm for finding a densest subgraph by a series of max-flow (min-cut) computations [33] is discussed. Guess ˛, the density of the maximum density subgraph, and then refine the guess by doing a network flow computation. Suppose a maximum density subgraph S  has density ˛  . Let the current guess be ˛. By appropriately defining a flow network as shown below and examining its min-cut structure, it is possible to determine if ˛ D ˛  or ˛ < ˛  or ˛ > ˛  . It can be verified that ˛  lies between 0 and w0 .E/ (total weight of all edges), and therefore, ˛  can be found by using binary search. Since densities are rational numbers, once the interval size drops below jV1j2 , the algorithm can stop. The construction of the flow network for a given guess ˛ is described below. Create a flow network G 0 with a source s and sink t. Each edge in G has a node in G 0 , and let E 0 be the set of these nodes of G 0 . In addition, each node in G also has its own node in G 0 , and let V 0 be the set of these nodes in G 0 . Add edges from s to

Network Optimization

2011

e 2 E 0 of capacity w0 .e/ and an edge from v 2 V 0 to t with capacity ˛w.v/. Add edges from e D .x; y/ 2 E 0 to both x 2 V 0 and y 2 V 0 with capacity 1. In fact, this method works even in the case of hyper-graphs, and in this case, E 0 is a set of hyper-edges, and such edges are added from e to all x 2 V 0 such that x 2 e. Since infinite capacity edges were added from e to x and y, any finite capacity .s; t/ cut must have all of e, x, and y on the same side of the cut whenever e is on the same side as s. Therefore, all such finite cuts correspond to subgraphs of G. If the problem is given without a specified subset (C D ;), then the algorithm proceeds as follows. First, note that there is a s-t cut of value w0 .E/. This is obtained by setting S D fsg and T D ftg [ E 0 [ V 0 . Suppose the max density subset 0 .S  /  has density ˛  . Suppose the guess ˛ < ˛  D ww.S  / , then it follows that ˛w.S / 0  < w .S /. Now consider an s-t cut .s [ V1 ; t [ V2 / in the flow network G 0 , then let S D V1 \ V 0 . The cut includes all the edges from nodes in S to t of capacity ˛w.S / as well as edges from s to nodes in E 0 that are not in V1 . All edges e D .x; y/ 2 E with one end in V n S must be in V2 since otherwise there will be an edge of 1 capacity across the cut. If all the edges in the induced graph formed by S are not in V1 , a smaller cut can be found by moving such edges from V2 to V1 . Therefore, s-t min cuts have all edges of the induced subgraph of S in V1 . The capacity of such a cut is exactly w0 .E nE.S //C˛w.S /. Note that w0 .E nE.S // includes the weight of all edges that are incident on some node in E n S , and so, w0 .E n E.S // C ˛w.S / D w0 .E/  w0 .S / C ˛w.S / D w0 .E/  .w0 .S /  ˛w.S //. But for the optimal subset S  , w0 .S  /  ˛w.S  / > 0, and thus, there is a cut whose capacity is smaller than w0 .E/. Therefore, if the capacity of a min cut happens to be smaller than w0 .E/, then it implies that the guess for ˛ is smaller than ˛  . Similarly, if the guess for ˛ is greater than ˛  , then the (unique) min cut has value w0 .E/. When ˛ D ˛  , then there could be multiple min cuts of value w0 .E/. Any min cut other than the trivial cut gives a correct solution.

7.2

Algorithm for Densest Subgraph with a Specified Subset C

It is now shown (see [40]) how to modify the above construction when C ¤ ;. In the following discussion, w.C / denotes the sum of the weights of the vertices in C , and w0 .C / denotes the sum of the weights of edges in the induced subgraph of C . Create a new source s 0 and add an edge to s with capacity w0 .E/  ˛w.C /. In addition, remove C from V 0 . Again suppose that ˛ < ˛  . In this case, a subset S  exists 0  //Cw0 .C /Cw0 .S  ;C / such that w .E.S w.S > ˛. Thus, w0 .E.S  // C w0 .C / C w0 .S  ; C /   /Cw.C / ˛.w.S  / C w.C // > 0. Replace w0 .E/  w0 .E.V n .S  [ C /// for the first term. Note that w0 .E.V n .S  [ C /// includes all edges incident on nodes in V n .S  [ C / and not only the edges induced by those nodes. Hence, w0 .E/  w0 .E.V n .S  [ C ///  ˛.w.S  / C w.C // > 0:

2012

S. Khuller and B. Raghavachari

.w0 .E/  ˛w.C //  .w0 .E.V n .S  [ C /// C ˛w.S  // > 0: .w0 .E/  ˛w.C // > w0 .E.V n .S  [ C /// C ˛w.S  /: This means that a min cut exists (e.g.,defined by the subset S  ) that is smaller than w0 .E/  ˛w.C /. So again, by looking at the min-cut structure, it is possible to know that ˛ < ˛  . If ˛ > ˛  , then the trivial min cut separating s 0 from the rest of the graph is unique. Again a binary search for ˛ can be done. Note: A simple method that does not work is to snap the edges to C as self loops and to then compute the densest subgraph in G with C removed. If the density of the densest subgraph found is lower than w0 .C /=w.C /, then the algorithm just returns C as the answer. Otherwise, it returns S [C . The main problem is that the density of S could get lowered when C is added back. The level of dilution depends on the size of the densest subgraph in G with C removed; hence a subgraph with slightly lower density than the optimal solution but of much larger size could be a better choice.

8

Minimum-Cost Flows

In this section, one of the most important problems in combinatorial optimization, namely, the minimum-cost network flow problem, is studied. Some of the basic ideas behind the problem are outlined, and the reader can find a wealth of information in [1] about efficient algorithms for this problem. The min-cost flow problem can be viewed as a generalization of the max-flow problem, the shortest path problem, and the minimum-weight perfect matching problem. To reduce the shortest path problem to the min-cost flow problem, notice that if just one unit of flow needs to be shipped, it is best to send it on a shortest path. The minimum-weight perfect matching problem on a bipartite graph G D .X; Y; X  Y / is reduced to min-cost flow as follows. Introduce a special source s and sink t, and add unit capacity edges from s to vertices in X and from vertices in Y to t. Compute a flow of value jX j to obtain a minimum-weight perfect matching.

8.1

Min-Cost Flow Problem Definitions

The flow network G D .V; E/ is defined as before. Other than the capacity function u, a cost function c W E ! R is also given. The cost function specifies the cost of shipping one unit of flow through an edge. Associated with each vertex v, there is a supply b.v/. If b.v/ > 0, thenPv is a supply node, and if b.v/ < 0, then it is a demand node. It is assumed that v2V b.v/ D 0; if the supply exceeds the demand, an artificial sink can be added to absorb the excess flow. As before, a flow function satisfies the capacity and skew symmetry constraints. The flow conservation is rewritten to indicate that each vertex v has an outgoing flow of b.v/: P • (Flow Conservation) For all vertices v 2 V , w2V f .v; w/ D b.v/.

Network Optimization

2013

The objective is to find a flow function that minimizes the cost of the flow, min z.f / D

X

c.e/  f .e/:

e2E

Before studying any specific algorithm to solve the problem, it is noted that a feasible solution (not necessarily optimal) can be found, if one exists, by finding a max flow and ignoring costs. To see this, add two new vertices, a source vertex s and a sink vertex t. Add edges from s to all vertices v with b.v/ > 0 of capacity b.v/ and edges from P all vertices w with b.w/ < 0 to t of capacity jb.w/j. Find a max flow of value b.v/>0 b.v/. If such a flow does not exist, then the flow problem is not feasible. The algorithm for the min-cost flow problem also uses the concept of residual networks. As before, the residual network GR .f / is defined with respect to a specific flow. Edges in the residual graph have both capacity and cost. If there is a flow f .v; w/ on edge e D .v; w/, then its capacity in the residual network is u.e/  f .e/ and its residual cost is c.e/. The reverse edge .w; v/ has residual capacity f .e/, but its residual cost is c.e/. Note that sending a unit of flow on the reverse edge actually reduces the original amount of flow that was sent, thereby decreasing the total cost, and hence its residual cost is negative. Only edges with strictly positive residual capacity are retained in the residual network. The following theorem characterizes optimal solutions. Theorem 5 A feasible solution f is an optimal solution if and only if the residual network GR .f / has no directed cycles of negative cost. The above theorem can be used to develop an algorithm for finding a min-cost flow. As indicated earlier, a feasible flow can be found by the techniques developed in the previous section. To improve the cost of the solution, the algorithm identifies negative-cost cycles in the residual graph and pushes as much flow around them as possible and reduces the cost of the flow. This is repeated until there are no negative cycles left, and a min-cost flow has been found. A negative-cost cycle may be found by using an algorithm like the Bellman-Ford algorithm, which takes O.jEjjV j/ time. An important issue that affects the running time of the algorithm is the choice of a negative-cost cycle. For fast convergence, one should select a cycle that decreases the cost as much as possible but finding such a cycle in NP-hard. Goldberg and Tarjan [22] showed that selecting a cycle with minimum mean cost (the ratio of the cost of cycle to the number of arcs on the cycle) yields a polynomial-time algorithm. There is a parametric search algorithm [1] to find a min-mean cycle that works as follows. Let  be the average edge weight of a min-mean cycle. Then  is the largest value such that reducing the weight of every edge by  does not introduce a negative-weight cycle into the graph. One can search for the value of  using binary search: Given a guess , decrease the weight of each edge by  and test if

2014

S. Khuller and B. Raghavachari

the graph has no negative-weight cycles. If the smallest cycle has zero weight, it is a min-mean cycle. If all cycles are positive, the guess is increased by . If the graph has a negative-weight cycle, the guess is decreased by . Karp (see [1]) has given an algorithm based on dynamic programming for this problem that runs in O.jEjjV j/ time. The first polynomial-time algorithm for the min-cost flow problem was given by Edmonds and Karp [14]. Their idea is based on scaling capacities, and the running time of their algorithm is O.jEj log U.jEj C jV j log jV j//, where U is the largest capacity. The first strongly polynomial algorithm (whose running time is only a function of jEj and jV j) for the problem was given by Tardos [42].

9

The Multi-Commodity Flow Problem

A natural generalization of the max-flow problem is the multi-commodity flow problem. In this problem, there are a number of commodities 1; 2; : : : ; k that have to be shipped through a flow network from source vertices s1 ; s2 ; : : : ; sk to sink vertices t1 ; t2 ; : : : ; tk , respectively. The amount of demand for commodity i is specified by di . Each edge and each vertex has a nonnegative capacity that specifies the total maximum flow of all commodities flowing through that edge or vertex. A flow is feasible if all the demands are routed from their respective sources to their respective sinks while satisfying flow conservation constraints at the nodes and capacity constraints on the edges and vertices. A flow is an -feasible flow for a di positive real number  if for each demand i , it satisfies at least 1C of that demand. There are several variations of this problem as noted below: In the simplest version of the multi-commodity flow problem, the goal is to decide if there exists a feasible flow that routes all the demands of the k commodities through the network. The flow corresponding to each commodity is allowed to be split arbitrarily (i.e., even fractionally) and routed through multiple paths. The concurrent flow problem is identical to the multi-commodity flow problem when the input instance has a feasible flow that satisfies all the demands. If the input instance is not feasible, then the objective is to find the smallest fraction  for which there is an -feasible flow. In the minimum-cost multi-commodity flow problem, each edge has an associated cost for each unit of flow through it. The objective is to find a minimum-cost solution for routing all the demands through the network. All of the above problems can be formulated as linear programs and, therefore, have polynomial-time algorithms using either the ellipsoid algorithm or the interior point method [28]. A lot of research has been done on finding approximately optimal multicommodity flows using more efficient algorithms. There are a number of papers that provide approximation algorithms for the multi-commodity flow problem. For a detailed survey of these results, see [12, Chap. 5]. In this section, a paper that introduced a new algorithm for solving multicommodity flow problems is discussed. Awerbuch and Leighton [3] gave a simple and elegant algorithm for the concurrent flow problem. Their algorithm is based

Network Optimization

2015

on the “liquid-flow paradigm.” Flow is pushed from the sources on edges based on the “pressure difference” between the end vertices. The algorithm does not use any global properties of the graph (such as shortest paths) and uses only local changes based on the current flow. Due to its nature, the algorithm can be implemented to run in a distributed network since all decisions made by the algorithm are local in nature. They extended their algorithms to dynamic networks, where edge capacities are allowed to vary [4]. The best implementation of this algorithm runs in O.kjV j2 jEj 3 ln3 .jEj=// time. In the integer multi-commodity flow problem, the capacities and flows are restricted to be integers. Unlike the single-commodity flow problem, for problems with integral capacities and demands, the existence of a feasible fractional solution to the multi-commodity flow problem does not guarantee a feasible integral solution. An extra constraint that may be imposed on this problem is to restrict each commodity to be sent along a single path. In other words, this constraint does not allow one to split the demand of a commodity into smaller parts and route them along independent paths (see the recent paper by Kleinberg [31] for approximation algorithms for this problem). Such constraints are common in problems that arise in telecommunication systems. All variations of the integer multi-commodity flow problem are NP-hard.

9.1

Local Control Algorithm

In this section, Awerbuch and Leighton’s local control algorithm is discussed. It finds an approximate solution for the concurrent flow problem. First, the problem is converted to the continuous flow problem. In this version of the problem, di units of commodity i is added to the source vertex si in each phase. Each vertex has queues to store the commodities that arrive at that node. The node tries to push the commodities stored in the queues towards the sink nodes. As the commodities arrive at the sinks, they are removed from the network. A continuous flow problem is stable if the total amount of all the commodities in the network at any point in time is bounded (i.e., independent of the number of phases completed). The following theorem establishes a tight relationship between the multi-commodity flow problem (also known as the static problem) and its continuous version. Theorem 6 A stable algorithm for the continuous flow problem can be used to generate an -feasible solution for the static concurrent flow problem. Proof Let R be the number of phases of the stable algorithm for the continuous 1 flow problem at which 1C fraction of each of the commodities that have been injected into the network have been routed to their sinks. It is known that R exists since the algorithm is stable. The average behavior of the continuous flow problem in these R phases generates an -feasible flow for the static problem. In other words,

2016

S. Khuller and B. Raghavachari

the flow of a commodity through a given edge is the average flow of that commodity through that edge over R iterations, and this flow is -feasible. In order to get an approximation algorithm using the fluid-flow paradigm, there are two important challenges. First, it must be shown that the algorithm is stable, that is, that the queue sizes are bounded. Second, the number of phases that it takes the algorithm to reach “steady state” (specified by R in the above theorem) must be minimized. Note that the running time of the approximation algorithm is R times the time it takes to run one phase of the continuous flow algorithm. Each vertex v maintains one queue per commodity for each edge to which it is incident. Initially all the queues are empty. In each phase of the algorithm, commodities are added at the source nodes. The algorithm works in four steps: 1. Add .1 C /di units of commodity i at si . The commodity is spread equally to all the queues at that vertex. 2. Push the flow across each edge so as to balance the queues as much as possible. If i is the discrepancy of commodity i in the queues at either end of edge e, then the flow fi crossing the edge in that step is chosen so as to maximize P 2 i fi .i  fi /=di without exceeding the capacity constraints. 3. Remove the commodities that have reached their sinks. 4. Balance the amount of each commodity in each of the queues at each vertex. 1 The above 4 steps are repeated until 1C fraction of the commodities that have been injected into the system have reached their destination. It can be shown that the algorithm is stable and the number of phases is O.jEj2 k 1:5 L 3 /, where jEj is the number of edges, k is the number of commodities, and L  jV j is the maximum path length of any flow. The proof of stability and the bound on the number of rounds are not included here. The actual algorithm has a few other technical details that are not mentioned here, and the reader is referred to the original paper for further details [3].

10

Minimum Spanning and Other Light-Weight Trees

Efficient algorithms are well known for finding minimum-weight spanning trees of undirected graphs. There are several greedy algorithms known for solving the problem. The problem turns out to be more difficult to solve in directed graphs, and the greedy algorithm fails since not all maximal, acyclic subgraphs are trees in directed graphs. One solution for finding a minimum-weight branching, as minimum spanning trees are called in directed graphs, is by using an algorithm for matroid intersection, which is similar to the algorithm for weighted matching. A simpler and more efficient algorithm for finding optimal branchings is discussed. Another well-studied problem in networks is that of finding shortest paths. One may be interested in finding shortest paths from a single-source node or

Network Optimization

2017

between all pairs of nodes. For the single-source case, if all edges have nonnegative weights, then the problem of finding shortest paths from the source to all nodes of the graph can be solved efficiently using Dijkstra’s algorithm. The algorithm also outputs a “shortest path tree” such that for any vertex, the shortest path to it from the root is the path in the tree from the root to that vertex. In general, shortest path trees can be much heavier than minimum spanning trees. It is useful in practice if it is possible to find light subgraphs in which shortest path distances are approximately preserved with respect to the given graph. Spanners are light subgraphs in which the distance between every pair of vertices is approximately preserved. Light approximate shortest path trees (LAST) do the same for singlesource shortest paths.

10.1

Minimum-Weight Branchings

A natural analog of a spanning tree in a directed graph is a branching (also called an arborescence). For a directed graph G and a vertex r, a branching rooted at r is an acyclic subgraph of G in which each vertex but r has exactly one outgoing edge, and there is a directed path from any vertex to r. This is also sometimes called an in-branching. Replacing “outgoing” in the above definition of a branching to “incoming” defines an out-branching. An optimal branching of an edge-weighted graph is a branching of minimum total weight. Unlike the minimum spanning tree problem, an optimal branching cannot be computed using a greedy algorithm. Edmonds gave the first polynomial-time algorithm to find optimal branchings (see [20]). Let G D .V; E/ be an arbitrary graph, let r be the root of G, and let w.e/ be the weight of edge e. Consider the problem of computing an optimal branching rooted at r. In the following discussion, assume that all edges of the graph have nonnegative weights. Two key ideas are discussed below that can be converted into a polynomialtime algorithm for computing optimal branchings. First, for any vertex v ¤ r in G, suppose that all outgoing edges of v are of positive weight. Let  > 0 be a number that is less than or equal to the weight of any outgoing edge from v. Suppose the weight of every outgoing edge from v is decreased by . Observe that since any branching has exactly one edge out of v, its weight decreases by exactly . Therefore, an optimal branching of G with the original weights is also an optimal branching of G with the new weights. In other words, decreasing the weight of the outgoing edges of a vertex uniformly leaves optimal branchings invariant. Second, suppose there is a cycle C in G consisting only of edges of zero weight. Suppose the vertices of C are combined into a single vertex and the resulting multigraph is made into a simple graph by replacing each multiple edge by a single edge whose weight is the smallest among them and by discarding self loops. Let GC be the resulting graph. It can be shown that the weights of optimal branchings of G and GC are the same. An optimal branching of GC can also be converted into an optimal branching

2018

S. Khuller and B. Raghavachari

of G by adding sufficiently many edges from C without introducing cycles. The above two ideas can be used to design a recursive algorithm for finding an optimal branching. The fastest implementation of a branching algorithm is due to Gabow, Galil, Spencer, and Tarjan [18].

10.2

Spanners

Given a graph G D .V; E/ and a positive number t  1, a t-spanner is a spanning subgraph H D .V; E 0 / of G with the property that for all pairs of vertices v and w, distH .v; w/  t  distG .v; w/: In other words, a spanner is a subgraph that approximates shortest paths between every pair of vertices. The factor t is called the stretch factor. The goal is to minimize the size and weight of H for a given value of t. The size of H refers to the number of edges in H , and the weight of H refers to the total weight of the edges in H . The subject of graph spanners has received much attention recently [2, 9, 39]. The following elegant algorithm computes a t-spanner [2]. SPANNER .G; t/ 1 H .V; ;/. 2 Sort all edges in increasing weight, w.e1 /  w.e2 /  : : :  w.em /. 3 for i D 1 to m do 4 let ei D .v; w/. 5 let P .v; w/ be the shortest path in H from v to w. 6 if w.P .v; w// > t  w.ei /, then 7 H H [ ei . 8 end-if 9 end-for end-proc It is not difficult to show that this algorithm outputs a t-spanner. The following property is used to establish the size of the spanner output by the algorithm. Lemma 3 Any cycle in the graph H output by algorithm SPANNER has at least t C 2 edges. Proof Suppose for contradiction that H contains a cycle C with at most t C1 edges. Let emax D .v; w/ be a heaviest weight edge on the cycle C . When the algorithm considers emax , all the other edges on C have already been added. This implies that there is a path of length at most t  w.emax / from v to w, since the path contains t edges each of weight at most w.emax /. Thus, the edge emax is not added to H , a contradiction.

Network Optimization

2019 2

Using this property, it can be shown that H has at most O.jV j1C t 1 / edges by applying a theorem from extremal graph theory relating the number of edges in a graph and its girth (length of a shortest cycle) [8]. In a subsequent chapter, Chandra et al. [9] proved the following theorem. Theorem 7 Let G be a connected edge-weighted graph. For all  > 0, algorithm 2C SPANNER constructs a t-spanner of G with weight O.jV j t 1 /  w.M S T /. The constant depends only on t and .

10.3

Light Approximate Shortest Path Trees

In this section, the problem of computing light-weight spanning trees that approximate shortest paths to every vertex from a single fixed root vertex is considered. Let G be a graph and let r be the root vertex in G. Let Tmin be a minimum spanning tree of G. For any vertex v, let d.r; v/ be the length  of a shortestpath from r to v in G. An .˛; ˇ/-light approximate shortest path tree .˛; ˇ/-LAST of G is a spanning tree T of G with the property that the distance from the root to any vertex v in T is at most ˛  d.r; v/, and the weight of T is at most ˇ times the weight of Tmin . Awerbuch, Baratz, and Peleg [5], motivated by applications in broadcast-network design, made a fundamental contribution by showing that every graph has a shallowlight tree – a tree whose diameter is at most a constant times the diameter of G and whose total weight is at most a constant times the weight of a minimum spanning tree. Cong et al. [10] studied the same problem and showed that the problem has applications in VLSI-circuit design; they improved the approximation ratios obtained in [5] and also studied variations of the problem such as bounding the radius of the tree instead of the diameter. Khuller, Raghavachari, and Young [29] modified the algorithm in [5] and obtained a stronger result; they showed that the distance from the root to each vertex can be approximated within a constant factor. Their algorithm also runs in linear time if a minimum spanning   tree and a shortest path tree are provided. The algorithm 2 computes an ˛; 1 C ˛1 -LAST. Independently, Awerbuch, Baratz, and Peleg [6] modified their algorithm from [5]. They obtained the same algorithm  as in 4[10]  but gave a stronger analysis, proving that the algorithm computes an ˛; 1 C ˛1 LAST. The basic idea is as follows: Initialize a subgraph H to be a minimum spanning tree Tmin . The vertices are processed in a preorder traversal of Tmin . When a vertex v is processed, its distance from r in H is compared to ˛  d.r; v/. If the distance exceeds the required threshold, then the algorithm adds to H a shortest path in G from r to v. When all the vertices have been processed, the distance in H from r to any vertex v meets its distance requirement. A shortest path tree in H is returned by the algorithm as the required LAST. An analysis shows that the entire subgraph H satisfies the weight guarantee.

2020

S. Khuller and B. Raghavachari

11

Coloring Problems

11.1

Vertex Coloring

A vertex coloring of a graph G is a coloring of the vertices of G such that no two adjacent vertices of G receive the same color. The objective of the problem is to find a coloring that uses as few colors as possible. The minimum number of colors needed to color the vertices of a graph is known as its chromatic number. The register allocation problem in compiler design and the map coloring problem are instances of the vertex coloring problem. The problem of deciding if the chromatic number of a given graph is at most a given integer k is NP-complete. The problem is NP-complete even for fixed k  3. For k D 2, the problem is to decide if the given graph is bipartite, and this can be solved in linear time using depth-first or breadth-first search. The vertex coloring problem in general graphs is a notoriously hard problem, and it has been shown to be intractable even for approximating within a factor of n for some constant  > 0 [12, Chap. 10]. A greedy coloring of a graph yields a coloring that uses at most  C 1 colors, where  is the maximal degree of the graph. Unless the graph is an odd cycle or the complete graph, it can be colored with  colors (known as Brooks’ theorem) [20]. A celebrated result on graph coloring is that every planar graph is 4colorable. However, checking if a planar graph is 3-colorable is NP-complete [19]. For more information on recent results on approximating the chromatic number, see [26]. In the equitable coloring problem, the objective is to find a coloring of the nodes of a graph with the additional constraint that each color class should have roughly the same cardinality. Chapter A Unified Approach for Domination Problems on Different Network Topologies surveys results in this area. Coloring problems are also discussed in Chap. Resource Allocation Problems.

11.2

Edge Coloring

The edge coloring problem is similar to the vertex coloring problem. In this problem, the goal is to color the edges of a given graph using the fewest colors such that no two edges incident to a common vertex are assigned the same color. The problem finds applications in assigning classroom to courses and in scheduling problems. The minimum number of colors needed to color a graph is known as its chromatic index. Since any two incident edges must receive distinct colors, the chromatic index of a graph with maximal degree  is at least . In a remarkable theorem, Vizing (see [20]) has shown that every graph can be edge colored using at most  C 1 colors. Deciding whether the edges of a given graph can be colored using  colors is NP-complete (see [12]). Special classes of graphs such as bipartite graphs and planar graphs of large maximal degree are known to be -edgecolorable.

Network Optimization

12

2021

Approximation Algorithms for Hard Problems

Many graph problems are known to be NP-complete (see Sects. A.1 and A.2 of [19]). The area of approximation algorithms explores intractable (NP-hard) optimization problems and tries to obtain polynomial-time algorithms that generate feasible solutions that are close to optimal. In this section, a few fundamental NP-hard optimization problems in graphs that can be approximated well are discussed. For more information about approximating these and other problems, see the chapter on approximation algorithms (Algorithmic Aspects of Domination in Graphs). We also mention several books on this topic under “Further Information and Recommended Reading” in Sect. 13. In the following discussion, G D .V; E/ is an arbitrary undirected graph, and k is a positive integer. Vertex cover: A set of vertices S  V such that every edge in E is incident to at least one vertex of S . The minimum vertex cover problem is that of computing a vertex cover of minimum cardinality. If the vertices of G have weights associated with them, a minimum-weight vertex cover is a vertex cover of minimum total weight. Using the primal-dual method of linear programming, one can obtain a 2-approximation. Dominating set: A set of vertices S  V such that every vertex in the graph is either in S or adjacent to some vertex in S . There are several versions of this problem such as the total dominating set (every vertex in G must have a neighbor in S , irrespective of whether it is in S or not), the connected dominating set (induced graph of S must be connected) and the independent dominating set (induced graph of S must be empty). The minimum dominating set problem is to compute a minimum cardinality dominating set. The problems can be generalized to the weighted case suitably. All but the independent dominating set problem can be approximated within a factor of O.log n/. For more information on domination problems, see Chaps. Combinatorial Optimization Techniques for Network-Based Data Mining and Fuzzy Combinatorial Optimization Problems. Steiner tree problem: A tree of minimum total weight that connects a set of terminal vertices S in a graph G D .V; E/. There is a 2-approximation algorithm for the general problem. There are better algorithms when the graph is defined in Euclidean space. For more information, see [12, 25]. Gradient constrained Steiner trees are discussed in Chap. Online Frequency Allocation and Mechanism Design for Cognitive Radio Wireless Networks and Steiner minimal trees in Chap. Neural Network Models in Combinatorial Optimization. Minimum k-connected graph: A graph G is k-vertex-connected, or simply k-connected, if the removal of up to k  1 vertices along with their incident edges leaves the remaining graph connected. It is k-edge-connected if the removal of up to k  1 edges does not disconnect the graph. In the minimum k-connected graph problem, the objective is to compute a k-connected spanning subgraph of G of minimum cardinality. The problem can be posed in the context of vertex or edge connectivities, and with edge weights. The edge-connectivity problems can be approximated within a factor of 2 from optimal, and a factor smaller than 2 when the

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edges do not have weights. The unweighted k-vertex-connected subgraph problem can be approximated within a factor of 1 C 2=k, and the corresponding weighted problem can be approximated within a factor of O.log k/. When the edges satisfy the triangle inequality, the vertex-connectivity problem can be approximated within a factor of about 2 C 2=k. These problems find applications in fault-tolerant network design. For more information on approximation algorithms for connectivity problems, see [12, Chap. 6]. Traveling salesman problem (TSP): Finding a shortest tour that visits all the vertices in a given graph with weights on the edges. It has received considerable attention in the literature [34]. The problem is known to be computationally intractable (NP-hard). Several heuristics are known to solve practical instances. Considerable progress has also been made for finding optimal solutions for graphs with a few thousand vertices. Degree constrained spanning tree: A spanning tree of maximal degree k. This problem is a generalization of the traveling salesman path problem. There is a polynomial-time approximation algorithm that returns a spanning tree of maximal degree at most k C 1 if G has a spanning tree of degree k (see [12, Chap. 7]). Max cut: A partition of the vertex set into .V1 ; V2 / such that the number of edges in E \ .V1  V2 / (edges between a vertex in V1 and a vertex in V2 ) is maximized. It is easy to compute a partition in which at least half the edges of the graph cross the cut. This is a 2-approximation. Semi-definite programming techniques can be used to derive an approximation algorithm with a performance guarantee of about 1.15 (see [12, Chap. 11]).

13

Conclusion

Some of the recent research efforts in graph algorithms have been in the areas of dynamic algorithms, graph layout and drawing, and approximation algorithms and randomized algorithms. The methods illustrated in this chapter find use in algorithms for many optimization problems. For approximation algorithms see the books by Hochbaum [12], Vazirani [44], and Williamson and Shmoys [45]. More information on graph layout and drawing can be found in an annotated bibliography by DiBattista et al. [7]. An exciting result of Karger [27] is a randomized, near-linear time algorithm for computing minimum cuts. Finding a deterministic algorithm with similar bounds is a major open problem. Iterative rounding [32] has emerged and has a very useful paradigm to design approximation algorithms for a collection of NP-hard optimization problems dealing with networks. Computing a maximum flow in O.jEjjV j/ time in general graphs is still open. Several recent algorithms achieve these bounds for certain graph densities. A recent result by Goldberg and Rao [21] breaks this barrier, by paying an extra log U factor

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in the running time, when the edge capacitiespare in the range Œ1 : : : U . Finding a maximum matching in time better than O.jEj jV j/ and obtaining nontrivial lower bounds are open problems.

Further Reading The area of graph algorithms continues to be a very active field of research. There are several journals and conferences that discuss advances in the field. Some of the important meetings are “ACM Symposium on Theory of Computing (STOC)”; “IEEE Conference on Foundations of Computer Science (FOCS)”; “ACM-SIAM Symposium on Discrete Algorithms (SODA)”; “International Colloquium on Automata, Languages and Programming (ICALP)”; and the “European Symposium on Algorithms (ESA).” There are many other regional algorithms/theory conferences that carry research papers on graph algorithms. The primary journals that carry articles on current research in graph algorithms are “Journal of the ACM,” “SIAM Journal on Computing,” “SIAM Journal on Discrete Mathematics,” “ACM Transactions on Algorithms,” “Algorithmica,” “Journal of Computer and System Sciences,” “Information and Computation,” “Information Processing Letters,” “Networks,” “Internet Mathematics,” “Journal of Graph Algorithms and Applications,” and “Theoretical Computer Science.” To find more details about some of the graph algorithms described in this chapter, the reader is referred to the books by Cormen, Leiserson, Stein, and Rivest [11]; Even [15]; Gibbons [20]; Kleinberg and Tardos [30]; and Tarjan [43]. Ahuja, Magnanti, and Orlin [1] have provided a comprehensive book on the network flow problem. The survey chapter by Goldberg, Tardos, and Tarjan [23] provide an excellent survey of various min-cost flow and generalized flow algorithms. A detailed survey describing various approaches for solving the matching and flow problems can be found in Tarjan’s book [43]. Papadimitriou and Steiglitz [38] discuss the solution of many combinatorial optimization problems using a primal-dual framework. For understanding recent methods on iterative rounding, the book by Lau, Ravi, and Singh [32] is suggested. For the area of approximation algorithms with an extensive coverage of graph algorithms, books by Hocbaum [12], Vazirani [44], and Williamson and Shmoys [45] are available.

Glossary Assignment problem: the problem of finding a matching of maximum (or minimum) weight in an edge-weighted graph. Augmenting path: a path used to augment (increase) the size of a matching or a flow. Blocking flow: a flow function in which any directed path from s to t contains a saturated edge.

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Blossoms: odd-length cycles that appear during the course of the matching algorithm on general graphs. Branching: a spanning tree in a rooted graph, such that each vertex has a path to the root (also known as in-branching). An out-branching is a rooted spanning tree in which the root has a path to every vertex in the graph. Capacity: the maximum amount of flow that is allowed to be sent through an edge or a vertex. Chromatic index: the minimum number of colors with which the edges of a graph can be colored. Chromatic number: the minimum number of colors needed to color the vertices of a graph. Chinese postman problem: the problem of finding a minimum length tour that traverses each edge at least once. Concurrent flow: a multi-commodity flow in which the same fraction of the demand of each commodity is satisfied. Edge coloring: an assignment of colors to the edges of a graph such that no two edges incident to a common vertex receive the same color. Euler tour: a tour of the graph that visits each of its edges exactly once. Eulerian graph: a graph that has an Euler tour. Integer multi-commodity flow: a multi-commodity flow in which the flow through each edge of each commodity is an integral value. The term is also used to capture the multi-commodity flow problem in which each demand is routed along a single path. Matching: a subgraph in which every vertex has degree at most one. Maximum flow: the maximum amount of flow that can be sent from a source vertex to a sink vertex in a given network. Multi-commodity flow: a network flow problem involving multiple commodities, in which each commodity has an associated demand and source-sink pairs. Network flow: an assignment of flow values to the edges of a graph that satisfies flow conservation, skew symmetry, and capacity constraints. Traveling salesman problem: the problem of computing a minimum length tour that visits all vertices exactly once. s-t cut: a partition of the vertex set into S and T such that s 2 S and t 2 T . Vertex coloring: an assignment of colors to the vertices of a graph such that no two adjacent vertices receive the same color.

Cross-References A Unified Approach for Domination Problems on Different Network Topologies Combinatorial Optimization Techniques for Network-Based Data Mining Fuzzy Combinatorial Optimization Problems Neural Network Models in Combinatorial Optimization Online Frequency Allocation and Mechanism Design for Cognitive Radio

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Optimal Partitions Quadratic Assignment Problems Resource Allocation Problems Binary Unconstrained Quadratic Optimization Problem

Recommended Reading 1. R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network flows (Prentice Hall, Englewood Cliffs, 1993) 2. I. Alth¨ofer, G. Das, D. Dobkin, D. Joseph, J. Soares, On sparse spanners of weighted graphs. Discret. Comput. Geom. 9(1), 81–100 (1993) 3. B. Awerbuch, T. Leighton, A simple local-control approximation algorithm for multicommodity flow, in Proceedings of the 34th Annual Symposium on Foundations of Computer Science, Palo Alto, California, 3–5 November 1993, pp. 459–468 4. B. Awerbuch, T. Leighton, Improved approximation algorithms for the multi-commodity flow problem and local competitive routing in dynamic networks, in Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, Montr´eal, 23–25 May 1994, pp. 487–496 5. B. Awerbuch, A. Baratz, D. Peleg, Cost-sensitive analysis of communication protocols, in Proceedings of the Ninth Annual ACM Symposium on Principles of Distributed Computing, Quebec City, 22–24 Aug 1990, pp. 177–187 6. B. Awerbuch, A. Baratz, D. Peleg, Efficient broadcast and light-weight spanners. Manuscript (1991) 7. G.D. Battista, P. Eades, R. Tamassia, I.G. Tollis, Annotated bibliography on graph drawing algorithms. Comput. Geom. 4, 235–282 (1994) 8. B. Bollob´as, Extremal Graph Theory (Academic, New York, 1978) 9. B. Chandra, G. Das, G. Narasimhan, J. Soares, New sparseness results on graph spanners, in Proceedings of the 8th Annual ACM Symposium on Computational Geometry (ACM, Berlin, Germany, 1992), pp. 192–201 10. J. Cong, A.B. Kahng, G. Robins, M. Sarrafzadeh, C.K. Wong. Provably good performancedriven global routing. IEEE Trans. CAD 11(6), 739–752 (1992) 11. T.H. Cormen, C.E. Leiserson, C. Stein, R.L. Rivest, Introduction to Algorithms (MIT, Cambridge, 1989) 12. D.S. Hochbaum (ed.) Approximation Algorithms for NP-Hard Problems (PWS Publishing, Boston, 1996) 13. J. Edmonds, E.L. Johnson, Matching, Euler tours and the Chinese postman. Math. Program. 5, 88–124 (1973) 14. J. Edmonds, R.M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems. J. Assoc. Comput. Mach. 19, 248–264 (1972) 15. S. Even, Graph Algorithms (Computer Science Press, Rockville, 1979) 16. L.R. Ford, D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1962) 17. M. Fujii, T. Kasami, K. Ninomiya, Optimal sequencing of two equivalent processors. SIAM J. Appl. Math. 17(4), 784–789 (1969) 18. H.N. Gabow, Z. Galil, T. Spencer, R.E. Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6, 109–122 (1986) 19. M.R. Garey, D.S. Johnson, Computers and intractability: A guide to the theory of NP-completeness (Freeman, San Francisco, 1979) 20. A.M. Gibbons, Algorithmic Graph Theory (Cambridge University Press, New York, 1985) 21. A. Goldberg, S. Rao, Beyond the flow decomposition barrier, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Miami Beach, Florida, 20–22 October 1997, pp. 2–11 22. A.V. Goldberg, R.E. Tarjan, Finding minimum-cost circulations by canceling negative cycles. J. Assoc. Comput. Mach. 36(4), 873–886 (1989)

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´ Tardos, R.E. Tarjan, Network flow algorithms, in Algorithms and 23. A.V. Goldberg, E. Combinatorics Volume 9: Flows, Paths and VLSI Layout, ed. by B. Korte, L. Lov´asz, H. J. Pr¨omel, A. Schrijver (Springer, Berlin, 1990), pp. 101–164 24. J. Hao, J.B. Orlin, A faster algorithm for finding the minimum cut in a directed graph. J. Algorithm. 17(3), 424–446 (1994) 25. F. Hwang, D.S. Richards, P. Winter, The Steiner Tree Problem (North-Holland, Amsterdam, 1992) 26. D. Karger, R. Motwani, M. Sudan, Approximate graph coloring by semidefinite programming, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science Santa Fe, New Mexico, 20–22 November 1994, pp. 2–13 27. D.R. Karger, Minimum cuts in near-linear time, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, 22–24 May 1996, pp. 56–63 28. H. Karloff, Linear Programming (Birkh¨auser, Boston, 1991) 29. S. Khuller, B. Raghavachari, N. Young, Balancing minimum spanning trees and shortest-path trees. Algorithmica 14(4), 305–321 (1995) ´ Tardos, Algorithm Design (Pearson Education, Boston, 2006) 30. J. Kleinberg, E. 31. J.M. Kleinberg, Single-source unsplittable flow, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, Burlington, Vermont, 14–16 October 1996, pp. 68–77 32. L-C. Lau, R. Ravi, M. Singh, Iterative Methods in Combinatorial Optimization (Cambridge University Press, Cambridge, 2011) 33. E. L. Lawler, Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976) 34. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization (Wiley, New York, 1985) 35. L. Lov´asz, M.D. Plummer, Matching Theory (Elsevier, New York, 1986) p 36. S. Micali, V.V. Vazirani, An O. jV jjEj/ algorithm for finding maximum matching in general graphs, in Proceedings of the 21st Annual Symposium on Foundations of Computer Science, Syracuse, New York, 13–15 October 1980, pp. 17–27 37. H. Nagamochi, T. Ibaraki, Computing edge-connectivity in multi-graphs and capacitated graphs. SIAM J. Disc. Math. 5(1), 54–66, 1992. 38. C.H. Papadimitriou, K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity (Prentice Hall, Englewood Cliffs, 1982) 39. D. Peleg, A. Sch¨affer, Graph spanners. J. Graph Theory 13, 99–116 (1989) 40. B. Saha, A. Hoch, S. Khuller, L. Raschid, X-N. Zhang, Dense subgraphs with restrictions and applications to gene annotation graphs, in RECOMB, ed. by B. Berger. Volume 6044 of Lecture Notes in Computer Science (Springer, Berlin, 2010) 41. M. Stoer, F. Wagner, A simple min-cut algorithm. J. Assoc. Comput. Mach., 44(4), 585–590 (1997) ´ Tardos, A strongly polynomial minimum cost circulation algorithm. Combinatorica 5, 42. E. 247–255 (1985) 43. R.E. Tarjan, Data and Network Algorithms (Society for Industrial and Applied Mathematics, Philadelphia, 1983) 44. V.V. Vazirani, Approximation Algorithms (Springer, Berlin, 2001) 45. D.P. Williamson, D.S. Shmoys, Design of Approximation Algorithms (Cambridge University Press, New York, 2011)

Neural Network Models in Combinatorial Optimization Mujahid N. Syed and Panos M. Pardalos

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Objective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Artificial Neural Networks (ANNs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Example: Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hopfield and Tank (H-T) Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Durbin-Willshaw (D-W)’s Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Kohonen Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Lagrangian Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 General Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Example: Mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Optimality Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 System Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lyapunov Energy Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Penalty-Based Energy Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Lagrange Energy Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Escaping Local Minima. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Stochastic Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Chaotic Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Convexification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Hybridiziation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 General Optimization Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Linear Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M.N. Syed () Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA e-mail: [email protected]; [email protected] P.M. Pardalos Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA e-mail: [email protected]

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5.2 Convex Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Quadratic Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Nonlinear Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Complementarity Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Mixed Integer Programming Problems (MIPs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Discrete Optimization Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Graph Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Shortest Path Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Number Partitioning Problems (NPP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Assignment Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Sorting Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Traveling Salesman Problems (TSP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Criticism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter reviews the theory and application of artificial neural network (ANN) models with the intention of solving combinatorial optimization problems (COPs). Brief introductions to the theory of ANNs and to the classical models of ANNs applied to COPs are presented at the beginning of this chapter. Since the classical ANN models follow gradient-based search, they usually converge to a local optimal solution. To overcome this, several methods that extend the capability of ANNs to avoid the local minima have been reviewed in this chapter. Apart from that, not all the ANNs converge to a local minimum; thus, stability and/or convergence criteria of various ANNs have been addressed. The thin wafer that divides continuous and discrete optimization problems while applying ANNs to solve the COPs is highlighted. Applications of ANNs to solve the general optimization problems and to solve the discrete optimization problems have been surveyed. To conclude, issues regarding the performance behavior of the ANNs are discussed at the end of this chapter.

1

Introduction

Combinatorial optimization problems (COPs) are special problems in the field of operations research, where a real-valued mathematical function is optimized over a given domain (feasible set). Either some or all the variables of these problems are discrete in nature. The main criterion in proposing a solution method to solve the COPs is to provide a solution strategy better than the complete enumeration method. The conventional methods that are used to find the solution of a COP include branching, bounding, cutting planes, etc. Usually, COPs are NP-Hard in nature; the solution time of the conventional solution methods grows exponentially with the problem size. A curiosity to improve the solution time of finding an optimal solution of a COP directed the research toward unconventional methods. The primary unconventional methods for solving the COPs are the heuristics, and the most

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prominent ones among them are the metaheuristics, such as simulated annealing, genetic and evolutionary algorithms, tabu search, and greedy randomized adaptive search procedure [15, 38, 44, 46, 53, 87, 94, 98, 115]. On the other hand, there were simultaneous research [32,112] to use the analog circuits instead of the conventional computers for solving COPs. A method for solving the linear and the quadratic programs using resistors, diodes, transformers, current sources, and voltage sources (the basic building blocks of the analog circuit) was proposed in [32]. However, this method was impractical due to the unavailability of an ideal diode. Thus, this method remained in the theory until improved methods were proposed in [23, 62]. The first known analog circuit implementations which were capable of solving linear and quadratic programming problems were presented in [24, 131]. However, these methods were not ideal due to the use of negative resistors [131]. Later in 1985, Hopfield and Tank [121] proposed an analog circuit consisting of neurons,1 called artificial neural networks (ANNs), to solve the COPs. This method caught the attention of many researchers, when Hopfield and Tank [121] solved some sufficiently large instances of the symmetric traveling salesman problem (TSP). The proposed method illustrated an alternate approach to solve the COPs, which is independent of the digital computers. After this illustration, many researchers were curious to utilize ANNs as a tool to solve various COPs.

1.1

Objective

The main objective of this chapter is to present different ideas of applying ANNs in solving the COPs. Furthermore, the issues related to the optimality and/or the stability analysis of various ANNs will be dealt. In addition to that, the implementation and the mapping of ANNs to solve the COPs will be illustrated by examples. Nevertheless, a brief history about the ANNs will be reviewed. In addition to that, usage of ANNs in solving the general optimization problems will be addressed.

1.2

Outline

In this chapter, ANN models which are applied in solving the COPs are described. This chapter is organized as follows: an introduction of ANNs in general and a review of the classical models in ANNs that are suitable for solving the COPs in particular are presented in Sect. 2. A systematic method, which illustrates the stability and convergence analysis of ANNs, is described in Sect. 3. Subsequently, the extensions and the improvements that can be incorporated to the classical ANN models are presented in Sect. 4. In addition to that, a methodology of mapping and

1

A term similar to the smallest processing unit in the brain, in ANNs it consist of a small independent circuit with resistors, diodes, capacitors etc.

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solving some of the general optimization problems is discussed in Sect. 5. Moreover, the applications of ANNs for solving various discrete programming problems are surveyed in Sect. 6. Nonetheless, in Sect. 7, the criticism of using ANNs to solve the COPs is presented. Finally, this chapter ends with the concluding remarks in Sect. 8.

2

Review

Human brain and its functionality has been the topic of research for many years, and until now, there has been no model that could aptly imitate the human brain. Curiosity of studying human brain lead to the development of ANNs. Henceforth, ANNs are the mathematical models mimicking the interactions among the neurons2 within the human brain. Initially, ANNs were developed to provide a novel solution approach to solve the problems which do not have any algorithmic solution approach. The first model that depicted a working ANN used the concept of perceptron3 [82, 101]. After the invention of perceptron, ANNs were the area of interest for many researchers for almost 10 years. However, this area of research was crestfallen due to the results shown by Minsky and Papert in [85]. Their results showed that the perceptron is incapable to model the simple logical operations. However, their work also indirectly suggested the usage of multilayer ANNs. After a decade, as a requital, this area of research became popular with Hopfield’s feedback4 ANN [56, 57]. A pioneer practical approach to solve the COPs was presented in [58, 121]. The mathematical analysis illustrated by Hopfield demonstrated that ANNs can be used as a gradient descent method in solving optimization problems. Moreover, some instances of traveling salesman problem (TSP) were mapped and solved using ANNs. This approach kindled enthusiasm among many researchers to provide an efficient solution methodology for solving NP-Hard problems. Since then, ANNs were modified and applied in numerous ways to solve the COPs.

2.1

Artificial Neural Networks (ANNs)

Neurons are the primary elements of any ANN. They posses some memory represented by their state. The state of a neuron is represented either by a single state or a dual state (internal and external state). The state of a neuron can take any real value between the interval Œ0; 1.5 Furthermore, the neurons interact with each other via links to share the available information. In general, the external state of the

2

Biological neurons are the basic building blocks of the nervous system. A simple single-layered artificial neural network invented by Frank Rosenblatt. 4 Feedback and feed-forward are two main classes of neural networks and will be explained in the following subsections. 5 Some authors prefer to use the values between [1,1]; however, both the definition are interchangeable and have the same convergence behavior. Hence, in this work [0,1], representation is followed. 3

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neuron takes part in the outer interactions, whereas the internal state of the neuron takes part in the firing6 decision. The intensity and sense of interactions between any two connecting neurons are represented by the weight (or synaptic7 weight) of the links. A single neuron itself is incapable to process any complex information. However, it is the collective behavior of the neurons that results in the intelligent information processing. Usually, neurons update their states based on the weighted cumulative effect of the states of the surrounding neurons. The update rule for the states is Ii D

n X

wij xi  Ui

(1)

j D1;j ¤i

xi D  .Ii /

(2)

where xi and Ii represent the external and internal states of an i th neuron, respectively, wij represents the weight between the i th neuron and the j th neuron, and Ui represents the threshold level of the i th neuron. Function P  ./ is calledthe n is transfer function.8 If the collective input to the i th neuron j D1;j ¤i wij xi greater than the threshold (Ui ), then the neuron will fire (value of xi is set as 1). This is the basic mechanism of any ANN. With appropriate network structure and link weights, ANNs are capable of making any logical operations. Example 1 Consider the following neural networks: In Fig. 1, a simple neural network, which performs logical AND and logical OR operations, is depicted. The number over the links represents the weight of the link. The number inside the neurons represents the threshold value. Each neural network has two binary input signals and one binary output signal. Similarly, in Fig. 2, another elementary logical operator XOR is presented (where XOR is logical exclusive OR operator). From this figure, it can be seen that for a specific operation, neural network may not have a unique representation. From Figs. 1 and 2, it can be seen that although a single neuron does not possess the computation ability, collectively, they can perform any logical operations very easily. Moreover, these are just the elementary form of logical operations that can be performed using simple neural networks. ANNs can be used for the complex logical operations as well. In order to use ANNs for complex logical operations, feedback method is used. Based on the feed mechanism, ANNs can be classified as feed-forward neural networks and Feedback neural networks as shown in Fig. 3.

6

Whether to set the value of the corresponding external state to 0 or 1 The term synaptic is related to the nervous system and is used in ANNs to indicate the weight between any two neurons. 8 This function may be continuous or discontinuous based on the type of the neuron. That is, for a discrete case, this function is similar to a signum function, whereas for continuous case, this function is similar to a sigmoidal function. 7

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a

b 1

1

1

1 0.5

1

3.5

1

1

1

Fig. 1 Example:1 simple logical operations in ANNs. (a) OR. (b) AND

a

b 1

1 1 1.5

−2

0.5

1 1

−1 1

1

1

−1

1 1

1

1

1 1

1

Fig. 2 Example:2 simple logical operations in ANNs. (a) XOR. (b) XOR Fig. 3 Major categories of ANNs. (a) Feed-forward ANN. (b) Feedback ANN

a

b i

j

i

j

In the feed-forward neural networks, information is processed only in one direction. Whereas in the case of feedback neural networks, information is processed in both the directions. From the literature, typically, there are three main types of applications of ANNs in information processing. They are categorized as: Type 1: Pattern Recognition • This is the most explored application of the ANNs [14, 19, 88, 89]. In pattern recognition, two sets of data, namely, training data, and testing data are given. Generally, the data sets consists of input parameters which are believed to be the cause of the output features. The aim of pattern recognition is to find a relation that maps the input parameters with the output features. This is obtained by learning, where the weights of the neural network are updated by error backpropagation. When the error between the generated features and the required output features falls below some acceptable limit, the weights are frozen or set to the current value. These weight are used to validate the performance of the obtained neural network using the testing data. This approach is used in function approximation, data prediction, curve fitting, etc. Type 2: Associative Memory • Advancement in the theory of ANNs from feed-forward neural networks to feedback neural networks led to this application [18, 84, 93]. In associative memory application, a neural network is set to the known target memory pattern. Later, if an incomplete or partly erroneous pattern is presented to the trained

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ANN, then the ANN is capable to generate or repair the input pattern to the most resembling target pattern. This approach is used in handwriting recognition, image processing, etc. Type 3: Optimization • Further extensions of associative memory approach led to the application of ANNs in solving the optimization problems. This application is fundamentally different from the pattern recognition application. In this approach, weights of the neural network are fixed, and the inputs are randomly (synchronously or asynchronously) updated such that the total energy of the network is minimized. This minimization is achieved if the network is stable. Based on the different optimality conditions, the system may converge to the local (or global) optimal solution. ANNs have been applied not only for solving the COPs [3, 43, 64, 97, 110, 111] but also for solving the optimization problems in general [39, 86, 90, 104] In the following subsection, an example implementation of ANNs for solving a general linear programming (LP) problem is presented.

2.2

Example: Implementation

One of the conventional methods of solving any COP is to relax the problem into a series of continuous problems, and solve them until the desired discrete solution is obtained [47]. Although simplex is the most widely used method to solve the linear relaxation of COPs, it has been shown in the literature that other methods like dual-simplex, barrier methods, and interior point methods perform better than simplex on certain occasions [66, 99, 133]. Apart from that, the convergence time of simplex method is exponential in time [69], yet in practice, it performs better than polynomial time interior point method. Thus, finding a polynomial time algorithm (polynomial both in theory and in practice) for solving linear programming problems is still an attractive area of research. Therefore, it becomes an attractive choice to study the method of solving an LP problem using an ANN. Apart from that, the purpose of selecting an LP model to show the implementation of ANNs is due to the simple nature of LP models. Thus, a reader familiar with the theory of LP will find it easy to understand the analog circuit of the LP problem. Consider a standard form of a linear programming problem described as minimize W cT x subject to W Ax D b 0  xi  xmax

8i D 1; 2; : : : ; n

(3)

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where c 2 Rn , b 2 Rm , A 2 Rmn , and x 2 Rn . In order to apply an ANN to solve this problem, the above stated problem has to be transformed into an unconstrained optimization problem using either a penalty function or a Lagrange function method. For the sake of simple illustration, a penalty function approach will be presented. Let x.t/ represent the instantaneous or dynamic value of the solution vector at time t. Let E.x.t/; t/ represent the modified (penalized) objective function, which is given as E.x.t/; t/ D

C1 .Ax.t/  b/T .Ax.t/  b/ C C2 cT .x C x.t/e C3 t / 2

(4)

where C1 ; C2 , and C3 represent the positive scaling parameters of the system. Let the system dynamics be defined as rt I D rx.t / E.x.t/; t/

(5)

where I represents the internal energy of the neuron. Based on the system dynamics defined in Eq. (5), the system can be described as rt I D C1 AT Ax.t/ C C1 AT b  C2 c e C3 t

(6)

and

xmax 1 C e .C4 I.t // where C4 is a positive scaling parameter, xi .t/ D

 rt I D and

 rx.t / E.x.t/; t/ D

d In d I1 d I2 ; ;:::; dt dt dt

T ;

(7)

I 2 Rn

@E.x.t/; t/ @E.x.t/; t/ @E.x.t/; t/ ; ;:::; @x1 @x2 @xn

T

Equations (6) and (7) are the main equations which are used to design the ANN. The first thing in designing the neural network is to know the number of neurons. Since each variable is represented by a neuron, there will be altogether n neurons required for the construction of ANN. Let Ii represent the internal state of an i th neuron and xi represent the external state of an i th neuron. At any given time, both the states of an i th neuron can be represented by ŒIi .t/; xi .t/. The next step will be to connect the network using links. These links will have resistors, which represent the weights (synaptic weights) of the link. Based on the coefficients in Eq. (6), the weights are taken as the corresponding coefficients of the first-order term. The constant term in Eq. (6) is taken as the input bias to the neurons. From the Eq. (6), the following weight matrix and bias vector is obtained: W D C1 AT A and .t/ D C1 AT b  C2 c e C3 t

(8)

Neural Network Models in Combinatorial Optimization

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or wij D C1

m X

aki akj

and i .t/ D C1

X

k D 1m aki bk  C2 ci e C3 t

(9)

kD1

where aij is the i th row j th column element of A and wij is the i th row j th column element of W . Now, in order to set the appropriate weights, we use resistors. The absolute value of synaptic weight wij (between i th and j th neuron) is obtained as the ratio of the resistors Rf and Rij , that is, Rf jwij j D (10) Rij The next step is to assign input bias to each neuron. This can be done using the voltage Ei and the resistor Ri , given as i D

Rf Ei Ri

(11)

The other part of the input bias, that is, C2 ci e C3 t is obtained by providing a discharging loop, with an additional set of capacitor Cd , and resistor Rd . The configuration is shown in Fig. 4. The values of Rd and Rf are taken arbitrarily such that Rd  Rf . The initial value of Cd is assigned as C2 ci 8 i . Once these values are set, the other values are given as Rij D

Rf wij

C3  1=.Rd Cd /

(12) (13)

The positive (or negative) value of weight is obtained by using the xi th terminal (or xi th terminal) of the neuron. For the sake of simplicity, the neuron in Fig. 4 can be represented as depicted in Fig. 5. Once the structure of a single neuron is formed, it is connected to the other neurons to form the ANN which can be represented as in Fig. 6. As the time proceeds, the neurons update their internal and external energies based on the update rules and based on the dynamics of selecting the neurons for the update. If the network is stable and the dynamics is appropriate, the network will converge to a limit point(local minimum), which will give the optimal solution of the LP [39]. Applying ANNs to solve the LP problems has been studied and analyzed in [25–28, 35, 67, 100, 136]. The objective of depicting the method of developing analog circuit for solving LP was to illustrate the reader various complexities involved in the deployment of an analog ANN for solving any COP. In the following subsections, some of the classical models of the ANNs that are applied in solving COPs will be described.

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− +

Rni

−Xn

− g

+

Xn

Neuron i

Ri

Cd

Re

Rn Rd −

+

En

Fig. 4 Circuit representation of a neuron Input

Connections from other neurons

Output

Connection to other neurons Neuron i

Input bias Feedback to itself

Fig. 5 Block representation of a neuron

These are the basic models proposed in 1980s–1990s as an alternate analog method to solve COPs. These models can be broadly classified as: • Gradient-based methods – Hopfield and Tank’s discrete and continuous models • Self-organizing methods – Durbin-Willshaw’s and Kohonen’s models • Projective gradient methods – Lagrange-based models

Neural Network Models in Combinatorial Optimization Fig. 6 Interconnected neurons

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1

2

i

Input bias

Synaptic Weights

n

2.3

Hopfield and Tank (H-T) Models

In 1982, Hopfield [56] proposed a novel feedback type neural network, which is capable of performing computational tasks. Although feedback type models were proposed earlier by Anderson and Kohonen [10, 71], however, Hopfield [57] provided a complete mathematical analysis of feedback type neural networks. Thus, these feedback type neural networks were named as Hopfield networks. There are two different models proposed by Hopfield and Tank [58] for solving COPs. They are described below: (a) H-T’s Discrete Model The discrete model consists of n interconnected neurons. The strength of the connection between the i th neuron and the j th neuron is represented by the weight wij . A presence of connection between any two neurons is indicated by a nonzero weight. Weights maybe positive or negative, representing the sense of connection (excitatory or inhibitory). In this model, each neuron has been assigned with the two states (external and internal states, like that in McCullough and Pitts [82] model) representing the embedded memory at each neuron. Moreover, the internal state of i th neuron Ii is continuous valued, whereas the external state xi is binary valued. The internal and external state of a neuron are related as

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Ii .t C 1/ D

n X

wij xj .t/ C i

j D1

xi .t C 1/ D ‰d .Ii /

(

1

if Ii > Ui

0

if Ii  Ui

(14)

(15)

where i is a constant representing an external input bias to the i th neuron. ‰d ./ represents a transfer function, which assign binary values to xi . Ui represents the threshold value of the i th neuron. If the aggregate sum of the internal energies supplied to the i th neuron is greater than Ui , then this neuron will “fire,” that is, xi will be set to 1. On the other hand, if the aggregate sum of the internal energies is less than the threshold, the neuron will be in a dormant or “notfiring” state. For solving the COP, wij ’s are calculated based on the objective function and the constraints of a given COP. The only decision variables of this model are xi and Ii . Unless or otherwise specified, the values of Ui ’s are taken as 0. Initially, a random value is assigned to each external state, and the neurons update their states based on the updating rules. Usually, the neurons are selected randomly for the update (asynchronous updating). This updating rule is proven to converge to one of the stable states [83], which minimizes the energy function, provided the energy function is given by Eq. (16): X 1 XX wij xi xj  i xi 2 i D1 j D1 i D1 n

Ed D 

n

n

(16)

Since one of the states is binary and the update sequence of neurons is random, this model is known as a discrete model with stochastic updates. With the above details, Algorithm 1 summarizes the mechanism of this model. This model can be used to compute the local minimum of any quadratic function. Hopfield and Tank [58, 121] showed that the way energy function is constructed will always result in decrease of energy and the algorithm leads to one of the stable states. The algorithm proceeds as the gradient descent method, converging to a local minimum. The main step in solving a COP using H-T’s discrete model is to convert the constrained COP into an unconstrained COP using a penalty function method (or Lagrange method). Once the unconstrained penalty objective function is obtained, it is compared with the energy function given in Eq. (16). (b) H-T’s Continuous Model In the subsequent research [57], H-T proposed another model in which both the states of the neuron are continuous. In addition, in this model, two additional properties have been assigned to the neurons, namely, the input capacitance Ci and the transmembrane resistance Ri . Moreover, a finite impedance Rij between the

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Algorithm 1 Hopfield Discrete Model Randomly assign initial values to xi termination = false while (termination) do Select one neuron randomly Update the states Pn Ii and xi : Ii .t C 1/ D j D1 wij xj .t / C i if Ii > 0 then xi .t C 1/ D ‰d .Ii / D 1 else xi .t C 1/ D ‰d .Ii / D 0 end if Calculate the d: Pnenergy Pnof the system EP n Ed D  12 iD1 j D1 wij xi xj  iD1 i xi if termination criteria is met then termination = true end if end while

i th neuron and the output xj from j th neuron is assigned on the link ij . The main difference between the continuous model and the discrete model is in the continuous model, the external states of the neurons are continuous valued between 0 and 1. The equations of motion for this model are given as X d Ii Ii D C i wij xj .t/  dt  j D1

(17)

xi D ‰c .Ii /

(18)

n

where  D Ri Ci and usually,  is assigned a value of 1, if the time step of any discrete time simulation is less than unity. A widely used transfer function is continuous sigmoidal function which is of the form, ‰c .Ii / D 12 .1 C tanh. ITi //. The slope of the transfer function is controlled by parameter T . Similar to the previous model, this model will converge to a stable state which will minimize the energy function, given by Eq. (19): X 1 XX wij xi xj  i xi C 2 i D1 j D1 i D1 n

Ec D 

n

n

Z

xi 0

‰c1 .a/ da

(19)

The only decision variables in this model are xi and Ii . Algorithm 2 describes the mechanism of this model. From this illustration, it can be seen that a continuous ANN can be applied to solve discrete optimization problems, just by tuning the parameters of the transfer function. In specific, if the value of the slope, T , in the transfer function is set to a very high value, then this model will approximately behave as the discrete model.

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Algorithm 2 Hopfield Continuous Model Randomly assign initial states xi termination = false while (termination) do Select randomly one neuron Update the states Ii and xi : Pn d Ii D j D1 wij xj .t /  Ii C i dt xi .t C 1/ D ‰c .Ii / D 12 .1 C t anh. ITi // Calculate the Rx Pd n: Pnenergy Pn of the system E Ec D  12 iD1 j D1 wij xi xj  iD1 i xi C 0 i ‰c1 .a/ da if termination criteria is met then termination = true end if end while

The first attempt to solve the COP using ANN was demonstrated by Hopfield and Tank [58] in 1985. They illustrated the usage of ANN by solving sufficiently large instances of the famous traveling salesman problem (TSP). After this illustration by Hopfield and Tank, numerous approaches and extensions were proposed to solve various COPs using ANNs. Most of the proposed extensions were in the direction of improving H-T’s ANN model. In the following subsection, the alternative approaches which are significantly different from H-T’s ANN model are presented.

2.4

Durbin-Willshaw (D-W)’s Model

In 1987, Durbin and Willshaw [34] proposed a fundamentally different approach to the H-T’s approach for solving the geometric COPs. The proposed method was named as elastic nets and often referred as deformable templates. The origin of this method is the seminal work of Willshaw and Malsburg [130]. In this model, the neurons have no memory or states. Moreover, the network is not arbitrarily connected based on the values of the weights. Instead, it is connected in the form of a ring (elastic ring), where the neurons are placed on the circumference of the ring. Since this model is used to solve the geometric COPs, we will describe the model with respect to TSP. In this model, there are M neurons on the circumference of the ring, where M is greater than N, the number of cities. The first step is to find the center of gravity of the network. This point is taken as the center for the ring, and iteratively neurons on the rings are pulled apart. In every iteration, neurons are pulled in such a way that the distance between the neurons and the closest city is minimized. However, there is another force from the ring that tries to keep the length of the ring as small as possible. Thus, in the end, when each city has been close enough to a corresponding neuron, a Hamiltonian tour is obtained. The equation describing the movement of j th neuron is given as

Neural Network Models in Combinatorial Optimization

yj D ˛

N X

2041

wij .ai  yi / C ˇK.yj C1 C yj 1  2yj /

(20)

i D1

where ai is the coordinate of the i th vertex of the TSP; ˛; ˇ; and K are the scaling parameters: .dai yj ; K/ wij D PM 8i; j (21) kD1 .dai yk ; K/ dai yj is the Euclidean distance between the i th city and the j th neuron; and .dai yj ; K/ D e dai yj

2

=2K 2

(22)

˛ represents the scaling factor that drives the neurons toward the cities whereas ˇ represents the scaling factor for the force that keeps the neighboring neurons on the ring closer. Parameter K is gradually decreased, so that the neurons get closer to the cities. This parameter acts like temperature in simulated annealing and requires a cooling schedule. The energy of this system is given by Eq. (23): Ed w D ˛K

N X i D1

ln

M X j D1

.dai yj ; K/ C

M ˇX 2 d 2 j D1 yj yj C1

(23)

The mechanism of this model can be explained by Algorithm 3 .

2.5

Kohonen Model

Another fundamentally different approach to the H-T’s approach was proposed by Kohonen [72] for solving geometric COPs. This model utilizes self-organizing maps (SOMs) [70, 71] which fall under the category of the competitive neural networks. In this model, there are two layers of neurons (input and output). The input layer of the neurons are fully connected to the output layer of the neurons. In general, the neurons at the inner layer represent the input from a high-dimensional space.

Algorithm 3 Durbin-Willshaw model Find the center of gravity of the system, and assign it as the center of the circle with M equispaced ring points. termination = false while (termination) do Update the coordinates yi of ring points using equation Eq. (20) if (min1;:::;M fdai ;yi g  ") or (K < Kmi n ) then termination = true end if K D ˛K ˛ 2 .0; 1/ end while

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However, the neurons at the output layer represent output to a low-dimensional space. Therefore, the whole model act as a mapping from a higher dimensional space to a lower dimensional space. The property of self-organizing maps is to map the neurons close in the input space to the neurons that are close in the output space. The output layer of the neurons is arranged according to a topological structure, which is based on the geometry of the problem under consideration. For example: for the TSP, the output layer is arranged in a ring structure. Let T ai D .a1i ; a2i ; : : : ; ani / 8i D 1; : : : ; N be the coordinates of i th city to be j visited. Let w D .w1j ; w2j ; : : : ; wNj /T ; 8j D 1; : : : ; M; M  N represent the weight vector of the j th output neuron. Unlike the previous models, in this model, the weights are the decision variables. They are changed over the time using the following equation: wj D wj C fj;j  .ai  wj /;

8j D 1; 2 : : : ; M;

0 0 ) xi .t C 1/ D 1 ) .4EDD1 .t C 1//i  0 8 i 2 N: • If Ii .t/  0 ) xi .t C 1/ D 0 ) .4EDD1 .t C 1//i  0 8 i 2 N: From the above implication, for every i th neuron, .4EDD1 .t C 1//i  0. Thus, the system is converging. At the converging limit, .4EDD1 .t C 1//i D 0. This implies the following cases: • Ii .t/ D 0 ) xi .t/ D xi .t C 1/ D 0 8 i 2 N , a trivial case and can be discarded • xi .t  1/  xi .t C 1/ D 0 8 i 2 N: – ) xi .t  1/ D xi .t C 1/ ¤ xi .t/ 8 i 2 N ) two-edge cycles. – ) xi .t  1/ D xi .t C 1/ D xi .t/ 8 i 2 N ) local minima. Thus, the system will converge to a local minimum or to a two-edge cycle.  The next case to consider will be DSDD system with parallel dynamics. If a similar energy function is used, then following a similar argument, it is observed for the parallel dynamics that if i 2 S.t  1/ and i … S.t/, then the sign of Eq. (3.2) is undetermined. Thus, there could be an increase in the energy of the system, if

Neural Network Models in Combinatorial Optimization

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parallel dynamics is followed. However, this case will not arise, if the set of neurons for parallel update is selected in a specified way and if a different energy function is used. The following theorem will give the necessary and sufficient condition for convergence of parallel dynamics. Theorem 2 If W is a symmetric matrix with zero diagonals and if the update rule is parallel, then a discrete system defined by Eq. (47) will converge to a local minima if and only if S.t/ contains an independent set of neurons for each iteration t. Where S.t/ is said to contain independent set of neurons, if i; j 2 S.t/, then wi;j D 0 Proof Consider a generalized Lyapunov energy function given by Eq. (47). Let S.t/ be the independent set of neurons. The change in energy of the system is given by 4EDD2 .t C 1/ D .EDD2 .t C 1//  .EDD2 .t// 1 0 n n n X 1 @ XX D wij xi .t C 1/xj .t C 1/ C i .xi .t C 1//A  2 i D1 j D1 i D1 1 0 n n n X 1 @ XX  wij xi .t/xj .t/ C i .xi .t//A  2 i D1 j D1 i D1 Splitting the above equation for the cases i; j property of W , we get 0 4EDD2 .t C 1/ D

1@  2

X X i 2S.t / j …S.t /

(54)

2 S.t/ and using symmetric

wij xi .t C 1/xj .t C 1/ C

n X

1 i .xi .t C 1//A

i D1

1 0 n n n X 1 @ XX  wij xi .t/xj .t/ C i .xi .t//A  2 i D1 j D1 i D1

(55)

Now, based on the above equation, the following two scenarios may happen: • If Ik .t/ > 0 ) xk .t C 1/ D 1 ) 4EDD2 .t C 1/  0 • If Ik .t/  0 ) xk .t C 1/ D 0 ) 4EDD2 .t C 1/  0 Thus, the above system is converging. At the converging limit, 4EDD2.t C1/ D 0, which implies the following cases: • Ik .t/ D 0 ) xk .t/ D xk .t C 1/ D 0 8 k 2 N , a trivial case and can be discarded. • xk .t/  xk .t C 1/ D 0 8 k 2 N ) local minima. Thus, the system will converge to a local minimum. For the proof of the converse, see [83]. 

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Theorem 3 If W is symmetric, the update rule is consecutive, then a discrete system defined by Eq. (47) will converge to a local minimum. Proof Consider the Lyapunov energy function given by Eq. (47). The change in energy is given by 4EDDC .t C 1/ D .EDDC .t C 1//  .EDDC .t// 1 0 n n n X 1 @ XX wij xi .t C 1/xj .t C 1/ C i .xi .t C 1//A D  2 i D1 j D1 i D1 1 0 n X n n X X 1  @ wij xi .t/xj .t/ C i .xi .t//A 2 i D1 j D1 i D1

(56)

Splitting the above equation for cases when i D K and when i ¤ K and using the property that W is symmetric, we get 4EDDC .t C 1/ D .xk .t/  xk .t C 1// Ik .t/

(57)

The following two scenarios may happen: • If Ik .t/ > 0 ) xk .t C 1/ D 1 ) 4EDDC .t C 1/  0 • If Ik .t/  0 ) xk .t C 1/ D 0 ) 4EDDC .t C 1/  0 Thus, the above system is converging. At the converging limit, 4EDDC .t C1/ D 0, which implies the following cases: • Ik .t/ D 0 ) xk .t/ D xk .t C 1/ D 0 8 k 2 N , a trivial case and can be discarded. • xk .t/  xk .t C 1/ D 0 8 k 2 N ) local minimum. Thus, the system will converge to a local minimum.  Theorem 4 If W is symmetric positive semidefinite, then a discrete system defined by Eq. (47) will converge to a local minimum. 

Proof See [83].

The above analysis is performed for DSDD system. Another important class of system is CSDD system. Similar to the discrete state, three main types of dynamics can be applied for the continuous state system; they can be described as • Synchronous Dynamics 0 xi .t C 1/ D ‰c @

n X

j D1

1 wij xj .t/ C i A

8 i D 1; : : : ; n

(58)

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• Parallel Dynamics

xi .t C 1/ D

8  < ‰ Pn c

:xi .t/

 8 i 2 S.t/

j D1 wij xj .t/ C i

(59)

otherwise

• Consecutive Dynamics

xi .t C 1/ D

8  < ‰ Pn c

:xi .t/



j D1 wij xj .t/

C i

i DK

(60)

otherwise

where S.t/ and K have the same meaning as in the DSDD case. The main difference between the above equations and the discrete state equations is the definition of the transfer function. Here the transfer function is not similar to sign function. However, here the transfer can be a general real valued monotonically increasing continuous function with the following properties: if a ! 1; ‰c .a/ ! 1

8a 2 R

if a ! 1; ‰c .a/ ! 1

8a 2 R

(61)

Apart from the above properties, the following property is also incorporated to maintain an easy shift from a discrete state to a continuous state neuron. This is one of the important property of the transfer function, which adds the flexibility of using a continuous state model for solving a discrete COP. ‰c .a/ ! ‰d .a/

as  ! 1

(62)

Even the monotonic increasing criteria is not necessary. To be specific, any monotonic nondecreasing continuous function with the above properties may be used as the transfer function. However, the monotonic increasing continuous function is used only for the sake of simplicity and clearness [83]. There are many standard mathematical functions that appropriately fit the above restriction for the transfer function. The following are some of the widely used transfer functions: 1. ‰c .a/ D tan h.a/ 2. ‰c .a/ D 2 tan1 .a/ 1ea 3. ‰c .a/ D 1Ce a 4. ‰c .a/ D cot h.a/  a1 In the following part of this section, some more theorems without proof for the CSDD systems are presented. Interested readers are directed to [83] for the detailed proofs of the following theorems. Theorem 5 If W is symmetric, wi i  0, and the update rule is synchronous, then a discrete system will converge to a local minima or a two-edge cycle, if the following energy function is used.

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ECDS .t C 1/ D 

n n X X

wij xi .t C 1/xj .t/ C

i D1 j D1 n X

i xi .t C 1/ C xi .t/ C

i D1

Z n X

xi .t C1/ 0

i D1

‰c 1 i .a/ d.a/

Z

xi .t /

C 0

!! ‰c 1 i .a/ da (63)

Theorem 6 If W is symmetric and positive semidefinite, then a discrete system will converge to a local minima (irrespective of the type of dynamic used), if the following energy function is used. 1 XX wij xi .t C 1/xj .t/ 2 i D1 j D1 n

n

ECD .t C 1/ D 

C

n X

i .xi .t C 1// C

i D1

n Z X i D1

xi .t C1/ 0

‰c 1 i .a/ da

(64)

The restriction of positive semidefinite W can be relaxed if ‰c and ‰c 1 are differentiable. Let i be a scalar, defined as ‰c 0i  i

1 i

and Œ‰c 1 0 

(65)

If such i ’s exists, then the following theorem holds: Theorem 7 Let b wij D wij C function be defined as

C

, where ıi;j is Kronecker delta. Let the energy

1 XX wij xi .t C 1/xj .t/ 2 i D1 j D1 n

ECD .t C 1/ D 

ıij ij

n X i D1

n

i .xi .t C 1// C

n Z X i D1

xi .t C1/ 0

‰c 1 i .a/ da

(66)

b is positive definite, then a discrete system will converge to a local minimum If W (irrespective of the type of dynamic used). Furthermore, for the specific case of consecutive dynamics, a more relaxed optimality condition can be used. Theorem 8 Let the energy function be defined as

Neural Network Models in Combinatorial Optimization

1 XX wij xi .t C 1/xj .t/ 2 i D1 j D1 n

ECD .t C 1/ D 

C

2057

n X

n

i .xi .t C 1// C

i D1

n Z X i D1

xi .t C1/ 0

‰c 1 i .a/ da

(67)

If wi i  0, wij C 1i > 0, the update rule is consecutive, and ‰c & ‰c 1 are differentiable, then a discrete system will converge to a local minimum. A similar analysis can be done for continuous dynamics. Interested readers are referred to [83] for detailed analysis. The above theorems can be easily extended to f0; 1g systems. Continuous dynamics are usually hard to simulate on the digital computers and are mostly used in designing the analog circuits. The typical way of implementing continuous dynamics on the digital computer is to discretize the time into small steps. Nevertheless, the objective of presenting the proofs for discrete state systems is to highlight the reader that the convergence of ANNs is a critical issue. Moreover, the convergence of ANNs depends not only on the energy function but also on the systems dynamics. However, it can also be seen from the above proofs that a careful design of energy function (based on various properties of the weight matrix, W ) will converge the system to a local minimum, irrespective of the dynamics. Although Lyapunov function analysis is enough to understand that only for specific scenarios ANNs will converge to local optima. Lyapunov analysis is useful for those optimization problems whose constraints and objective function can be converted into a quadratic form. Moreover, the constraints are to be incorporated into the Lyapunov function using an appropriate penalty function. In general, it is hard to obtain a Lyapunov energy function for a given system. This does not mean that ANNs cannot be applied to these systems. Therefore, for such systems, different types of energy functions are designed. The primary approach that replaced the usage of Lyapunov energy function analysis is the usage of a general penalty function analysis (by incorporating constraints of a given optimization problem [77]). In the following subsections, the convergence and stability criteria for penalty- and Lagrange-based ANNs will be presented.

3.3

Penalty-Based Energy Functions

Consider the following constrained optimization problem: minimize W f .x/

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subject to W gi .x/  0 8i D 1; : : : ; m x 2 Rn

(68)

where, f; gi W Rn 7! R. It is assumed that both f; gi are continuously differentiable functions. The above problem can be converted to an unconstrained optimization problem as minimize W Epen .x/ subject to W x 2 Rn where Epen .x/ D f .x/ C

m X

(69)

ci Pi .x/

(70)

i D1

and Pi .x/ is the penalty function. Now, the convergence of ANNs to solve the above problem depends upon the structure of the penalty function. In this work, convergence analysis based on the penalty function of the form Pi .x/ D 1 2 2 Œminf0; gi .x/g 8 i D 1; : : : ; m will be presented. One of the properties of this penalty function is that it can be converted to a continuous differentiable penalty function [77]. Consider the following transformation: Let ( ci gi .x/ if gi .x/ < 0

i D 8 i D 1; : : : ; m (71) 0 if gi .x/  0 then,

ci 1 Œminf0; gi .x/g2 D i gi .x/ 8 i D 1; : : : ; m (72) 2 2 since gi ; 8 i D 1; : : : ; m; is assumed to be differentiable, the above penalty function is differentiable. The dynamics of this system12 is defined as ci Pi .x/ D

1 @E.x/ dxi D dt Ci @xi

8 i D 1; : : : ; n

(73)

Thus, these systems are also called as gradient-based systems. These systems [77] have some of the useful properties that can be used to solve the COPs. These properties are described in the following theorems:

12

Using Kirchoffs law, see [77]

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Theorem 9 Let the system be described by Eq. (73). The isolated equilibrium point of the system is asymptotically stable if and only if it is a strict local minimizer of function Epen . Theorem 10 Let the system be described by Eq. (73). If the isolated equilibrium point of the system is stable, then it is a local minimizer of function Epen . For the gradient-based systems, assuming Lipschitz continuity, Han et al. [51] proposed many other convergence properties. One of the difficulties in applying the penalty based method in the ANNs, is the selection of various penalty parameters. Due to the physical limits on the analog circuits, ci cannot be set to a large value. In order to overcome the difficulties of the penalty method, and to generalize the use of ANNs, Lagrange based energy function was proposed [139]. Following subsection discuss some of the results for the Lagrange-based ANNs.

3.4

Lagrange Energy Functions

Lagrange based neural networks are inspired by the saddle point theory and the duality analysis. The focus of these networks is to search for a point that satisfies K.K.T’s first order necessary conditions of optimality [12]. Therefore, analogous to the gradient descent systems, these network are also known as projective gradient systems. In the following subsections, necessary and sufficient conditions related to Lagrange type energy functions will be presented. Consider the following optimization problem: minimize W f .x/ subject to W hi .x/ D 0

8i D 1; : : : ; m

(74)

x 2 Rn where, f; hi W Rn 7! R. It is assumed that both, f; hi are continuously differentiable functions. The above problem can be converted to an unconstrained optimization problem using the Lagrange function, as: L.x; / D f .x/ C T h.x/

(75)

In this subsection, some of the basic optimality condition in the form of theorems are stated without proof. Interested readers may refer to [12] for the complete discussion, with proofs. Theorem 11 Let us assume that x is a regular point (one that satisfies constraint qualification). If x is the local minimum of problem Eq. (74), then there exists  2 Rm such that:

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rx L.x ;  / D 0 and

2 dT rxx L.x ;  /d  0;

(76)

8 d 2 fd W rx h.x /T d D 0g





(77)

Theorem 12 Let us assume that h.x / D 0. If 9  2 R such that: m

rx L.x ;  / D 0 and

2 dT rxx L.x ;  /d > 0;

8 d 2 fd W d ¤ 0I rx h.x /T d D 0g

(78)

(79)



then x is the local minimum of problem Eq. (74). Similar to the case of gradient-based dynamics, Lagrange systems has the following dynamic equations: rt x D rx L.x; /

(80)

rt  D r L.x; /

(81)

Although, the above equations are similar to the gradient-based systems, due to the difference in type of variables (ordinary variables and Lagrange variables), the system is named as projective gradient systems. In the presence of different types of the variables, the Lagrange will decrease w.r.t. one type of the variables and increase w.r.t. the other. This leads to the saddle point theory. For example, it is very easy to see that  n n  X X dL.x; / ˇˇ @L.x; / dxi dxi 2 D D 0 ˇ Dconstant dt @xi dt dt i D1 i D1  m m  X X dL.x; / ˇˇ @L.x; / d i d i 2 D D 0 ˇ xDconstant dt @ i dt dt i D1 i D1

(82)

(83)

Thus, .x ;  / act as a saddle point of the system. 2 Theorem 13 If rxx L.x; / is positive definite 8 x 2 Rn and 8  2 Rm , then the proposed ANN is Lyapunov stable.

The proof of this theorem is presented by Zhang and Constantinides in [139]. From the above theorem, following corollaries can be obtained: Corollary 1 x tends to a limit x at which rx L.x ; / D 0. Corollary 2 If rhi .x / 8 i converges to  .

D 1; : : : ; m are linearly independent, then 

Theorem 14 Let the network be described by Eqs. (75), (80), and (81). If the network is physically stable, then the ANN converges to a local minimum. The point of convergence is a Lagrange solution to problem Eq. (74).

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The above restriction of the global convexity can be relaxed to a local convexity, and the following theorem can be stated: Theorem 15 Let the network be described by Eqs. (75), (80), and (81). If following conditions hold: 1. 9.x ;  / such that .x ;  / is a stationary point 2 2. rxx L.x ;  / > 0 3. x is a regular point of problem Eq. (74) the initial point is in proximity of .x ;  /, then the ANN will asymptotically converge to .x ;  /. In this section, different ways to study convergence and stability performance of ANNs were illustrated. Various theorems were presented to understand the critical role of the energy functions in the convergence behavior of ANNs. However, the convergence will guarantee local optimality provided suitable energy function, and dynamics are used to update the states of neurons. For obtaining global optimality, there are numerous extensions proposed in the literature. In the following section, a discussion on the methods that can be used to escape from local minima, in the hope of finding the global minimum, is presented.

4

Escaping Local Minima

The very thought of the gradient descent method will kindle the question regarding global optimality. Classical ANNs perform gradient descent method and will hardly converge to the global minimum. In order to overcome such difficulties, various extensions have been proposed. The extension methods that may converge ANNs to global minima by escaping local minima can be broadly classified as: • Stochastic extensions • Chaotic extensions • Convexification • Hybridization Clearly, any extension that can be applied to H-T models can be easily applied to the general penalty or the Lagrange-based ANNs. Thus, in most of the cases, the extensions for the H-T’s model will be reviewed. In the following part of this section, a brief discussion of the above extension methods will be presented.

4.1

Stochastic Extensions

There are basically four ways to include stochasticity in the H-T’s model [107]. They are: 1. Introducing stochasticity in the transfer function 2. Introducing noise in the synaptic weights

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M.N. Syed and P.M. Pardalos

3. Introducing noise in the input bias 4. Any combination of the above three The prominent stochastic extensions that can be applied to the classical ANNs are discussed in the following parts of this subsection.

4.1.1 Boltzmann Machines This is the first extension of H-T’s model, which proposes a way to escape from the local minima [1, 2, 54, 55]. This is done by replacing the deterministic transfer function with a probabilistic transfer function. Let the change in energy of the i th neuron of the H-T’s model be represented by rEi . Then, irrespective of the previous state, the i th neuron will fire with probability pi , which is defined as pi D

1 .1 C exp rEi =T / (

and xi D ‰Blz .Ii / D

1

with probability pi

0

with probability .1  pi /

(84)

(85)

where T is the parameter similar to the concept of temperature in simulated annealing.

4.1.2 Gaussian Machines These are the extensions to the Boltzmann machines, where noise is introduced in the inputs [7]. The updating dynamics of an i th neuron in the H-T’s model will be modified as n X Ii D wij xj C i C "i (86) j D1

where "i is the term representing the noise, which follows Gaussian distribution with mean zero and variance  2 . The deviation of  depends upon the parameter T , the temperature of the Gaussian network. In addition to the noise, activation level ai is being assigned to every neuron, which has the following dynamics: ai dai D  C Ii (87) dt  The external state of the i th neuron is determined by the following equation:

xi D ‰Gss .ai / D

    ai 1 tanh C1 2 a0

where a0 is the additional parameter added to the Gaussian network.

(88)

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4.1.3 Cauchy Machines This system is an extension of the Gaussian machines, where instead of a Gaussian noise, a Cauchy noise is added to the system [114, 116]. The reason of replacing the Gaussian noise with the Cauchy noise is based on the observation that the Cauchy noise produces both global random flights and local random flights, whereas Gaussian noise produces only local random noise. Thus, the system has a better chance of convergence to the global minimum. The probability that a neuron will fire is given as   1 Ii 1 arctan. / (89) pi D C 2  T (

and xi D ‰Cau .Ii / D

1 with probability pi 0 with probability .1  pi /

(90)

where T is the temperature of the system which has the similar meaning as in the Gaussian or Boltzmann machines. In [63], comparison of lower bounds for Boltzmann and Cauchy machines is presented.

4.2

Chaotic Extensions

Let EHop represent the general H-T’s energy function. Then chaotic extensions for the H-T’s model can be obtained by adding an additional function to EHop [73,111], which can be described as ECNN D EHop C H.x; W; /

(91)

Different definitions for the function H./ lead to different chaotic extensions. Some of the chaotic extension methods for the H-T’s model will be presented in the following subsection.

4.2.1 Chen and Ahira’s Model Chen and Ahira [20, 21] proposed a chaotic addition to the energy function, which is given as .t/ X xi .xi  1/ (92) HCA D 2 i where .t/ is the decay parameter. If this term is added to H-T’s energy function, then the system’s dynamics is defined by the following equation: @ECNN d Ii D dt @xi The update rule for the internal energy of this model is given as

(93)

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M.N. Syed and P.M. Pardalos

0 Ii .t C 1/ D kIi .t/ C ˛ @

n X

1 wij xj .t/ C i A  z.t/.xi .t/  0 /

(94)

j D1;j ¤i

z.t C 1/ D .1  ˇ/z.t/

(95)

where ˛ D rt, z.t/ is the self-feedback connection weight z.t/ D .t/rt, ˇ is the damping factor, 0 D 1=2, and k D 1  rt=. Initially, z is very high to create strong self-coupling, thereby leading to the chaotic dynamics. Chaotic dynamics acts like an exploration stage in the simulated annealing, searching for a global minima. In the later stages, z is decayed so that the system starts to converge to a stable fixed point (similar to exploitation stage in the simulated annealing).

4.2.2 Wang and Smith’s Model Wand and Smith [124] proposed the following chaotic addition to the H-T’s energy function, which is given as HWS D .ˇ T  1/Ec

(96)

where Ec is the original H-T’s energy function and ˇ is the parameter which takes the values between Œ0; 1. If this term is added to H-T’s energy function, then the system is defined as @ECNN d Ii D (97) dt @xi The update rule of the internal energy for this model is given as 1 0   n X ˇ T rt.0/ Ii .t C 1/ D 1  Ii .t/ C ˇ T rt.0/ @ wij xj .t/ C i A 

(98)

j D1;j ¤i

ˇ T rt.0/ D rt.t/

(99)

Initially, rt is sufficiently large to trigger chaotic search. As t ! 1, rt gradually decreases.

4.2.3 Chaotic Noise Model Recently, Wang et al. [125] proposed the following chaotic addition to the H-T’s energy function, which is given as H D 

X

i .t/xi .t/

(100)

i

where i is the noise term. In general, i is a normalized time series. Initially, the neuron is set with the chaotic time series. If this term is added to H-T’s energy function, then the system is defined as

Neural Network Models in Combinatorial Optimization

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@ECNN d Ii D dt @xi

(101)

The update rule of the internal state for this model is given as 0 1   n X rt Ii .t/ C rt @ Ii .t C 1/ D 1  wij xj .t/ C i C i .t/A  j D1

(102)

This model has a linear form of function H./, when compared to the quadratic forms presented in the earlier models. This may be the reason for its superior optimization performance among the chaotic extension models.

4.3

Convexification

One of the important properties of a convex optimization problem is that the local minimum is the global minimum. Thus, convex reformulation of the problem and finding a local minimum of the reformulation will give the global minimum. If the actual model is nonconvex, then convexification leads to different approximations (or relaxations), which in turn may be useful in studying various bounds (lower and/or upper bound) of a given COP. Both penalty-based ANNs and Lagrangebased ANNs can be convexified. Convexification of COPs using penalty function depends upon the type of penalty function. Thus, it will not be discussed in this subsection. Nevertheless, convexification of Lagrange-based ANNs will be discussed. From the Theorem 15, it can be seen that the local convexity of Lagrange systems is hard to obtain in practice. However, an augmented Lagrange method can be used to convexify any Lagrange-based ANN. In this subsection, one of the convexification methods that can be used for Lagrange-based ANN is presented. Let us assume that problem Eq. (74) is nonconvex, then its augmented Lagrange can be written as 1 LAug .x; / D f .x/ C T h.x/ C cjh.x/j2 2

(103)

Using Eq. (103), different ANNs can be designed which are known as Augmented Lagrange Neural Networks (ALNN). The state equations can be obtained as X @hj dxi @f D  . j C chj / dt @xi j D1 @xi m

d j D hj ; dt

8 i D 1; : : : ; n

8 j D 1; : : : ; m

(104)

(105)

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By selecting a sufficiently large value of c, the local convexity for any Lagrange neural networks can be obtained under some restrictions (see [13]). However, due to analog circuit and implementing restrictions, the selection of c will become a critical issue [139].

4.4

Hybridiziation

ANNs have been coupled with metaheuristics like simulated annealing [103, 123– 125], tabu search [43], and evolutionary algorithm [102, 126] for solving COPs. The main purpose of these hybridizations is to find the global optimal solution, by adding flexibility to the classical H-T’s model. The usual method of coupling H-T’s model with a metaheuristic is to incorporate the constraint handling technique of the ANNs in the global search technique of the metaheuristic. Moreover, there has been research to hybridize H-T’s model with SONNs (see [108]). One of the successful implementations is illustrated by Smith et al. [109]. The experimental result from the illustration indicates that a careful and appropriate hybridization of H-T’s model with SONNs will result in a better solution methodology, which can be competitive to the state-of-the art optimization solvers.

5

General Optimization Problems

In this section, a discussion on applying ANNs to some of general optimization problems will be presented. For the sake of simplicity and convenience, following general optimization problem will be considered: • Linear programming problems • Convex programming problems • Quadratic programming problems • Nonlinear programming problems • Complementarity problems • Mixed integer programming problem The objective of selecting mixed integer programming problem is to highlight the reader, the method of transformation between discrete and continuous variables, when applying ANNs.

5.1

Linear Programming

Linear programming (LP) is the most widely used optimization technique for solving numerous optimization problems. Dantzig [30], the father of linear programming, proposed the famous and widely used simplex method to solve LP problems. However, simplex method has an exponential complexity. A polynomial time algorithm (ellipsoid method) for solving LP problems was presented by Khachiyan [68]. Although the ellipsoid method is polynomial time algorithm, in practice simplex

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outperforms ellipsoid method. Later, in 1984, Karmarkar [66] proposed a more efficient algorithm than ellipsoid method, and it outperforms simplex method in few complicated problems like scheduling, routing, and planning. Based on the solution methodology, the simplex method is categorized as exterior point method, whereas Karmarkar’s algorithm is categorized as interior point method. Further studies [122] on the classification of these two fundamentally different ideas lead to a conclusion that LP problems are finite in nature. Moreover, simplex method is a finite algorithm suitable to solve LP problems. However, it was also noted that sometimes, infinite algorithms like interior point method may outperform exterior point algorithms. Thus, there has been always a curiosity among researchers to investigate the infinite algorithms. ANNs approach in solving COPs can be seen as an infinite approach. From the demonstration of Hopfield and Tank [58], ANNs were seen as an alternative solution approaches to solve the COPs. Many studies has been conducted to solve LP problems using ANNs [25–28, 35, 67, 100, 136]. Most of these models were extensions of H-T’s continuous model. As noted in Sect. 2.7, these models will have an energy function-based model for minimization. Before presenting the models, consider the following notations for LP problem in canonical LPc and standard LPs form: LPs

LPc minimize W

minimize W e cT x

cT x subject to W gi .x/  0

subject to W e g i .x/ D 0 8i D 1; : : : ; m

8i D 1; : : : ; m

Following are the prominent models from the literature.

5.1.1 Kennedy and Chua’s Model Kennedy and Chua’s [67] model has neurons similar to the H-T’s continuous model. The external states of this model are represented by x. The equations of updates for states and energy function are given as rx D C 1 .c  rg.x/ .g.x/// m Z gj .x/ X E.x.t// D cT x C j .s/ ds j D1 0

where C is a matrix of capacitance and is usually taken as I for most of the cases. Kennedy and Chua showed that Tank and Hopfield’s model is a special case of the proposed model. Furthermore, they proved that their model will converge to a steady state, under the assumption that E is bounded from below. Selection of ./ is similar to that of a penalty function. In [80], it was shown that ./ is indeed

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M.N. Syed and P.M. Pardalos

an implicit form of the penalty function. The most widely used representation of ./ is s gi .x/, where s > 0 is the penalty weight. Moreover, gi .x/ is same as minfgi .x/; 0g. Based on the idea of applying penalty function, there were many extension of neural networks for LP. Some of the prominent extensions can be seen in [100, 138]. These prominent extensions will be presented in the following part of this section.

5.1.2 Rodriquez-Vazquez et al. Model Rodriquez-Vazquez et al. [100] proposed another model for LP. This model is described as rx D u.x/c  s.1  u.x//rg.x/ v.x/ where v.x/ D Œv1 ; : : : vm T , ( u.x/ D ( vj .x/ D

1 if gi .x/  0

8j D 1; : : : ; m;

0 otherwise 1 if gj .x/  0 0 otherwise

and s > 0. This model can be viewed as a dynamic system, which can start from any arbitrary point. Initially, the model will move from an infeasible point to a feasible point. Once a feasible point is obtained, the model will try to move to an optimal point. In [45], the convergence analysis of Rodriquez-Vazquez et al. [100] model was performed. It was shown that if the problem is convex, network will converge. However, Zak et al. [138] proposed that discrete time implementation of Rodriquez-Vazquez et al. model with an adaptive step size may have problem in reaching feasible region. In the following paragraph, the model proposed by Zak et al. will be presented.

5.1.3 Zak et al. Model This model differs from the earlier ANNs for the LP, as Zak et al. used the exact penalty function (non-differentiable) to formulate the energy function. This selection of penalty function eliminates the selection of very high penalty weight. Apart from that, all the earlier models were proposed for the canonical representation of LP. However, this model is applicable for both standard and canonical representation of LP. Although, in the theory, there is no difference in solving the problem in any representation. But in the case of neural networks, changing from one representation to other will increase in the number of neurons (due to slack or artificial variables) and may increase in the complexity of the ANN model. The proposed model can be described as rx D rEp;; .x/

Neural Network Models in Combinatorial Optimization

rEp;; .x/ D e cT x C rke g.x/kp C  r

2069 m X

! xi

i D1

where xi D minfxi ; 0g;  > 0;  > 0; 1  p  1. Although the model uses exact penalty function, this penalty function is also non-differentiable and hence has few drawbacks. In [137], convergence analysis of Zak et al.’s model is performed. It was shown that the system trajectory will converge to the solution of LP under some positivity restriction on ; ; p. In fact, it is shown that equilibrium points of the ANN are solutions to the corresponding LP problem [137]. These are some of the prominent models of ANNs applied to solve LP problems. Similar to the duality theory in LP, there were attempts to solve LP using dual ANNs. Interested reader may refer to [29, 81].

5.2

Convex Programming

Consider the following convex problem: minimize W f .x/ subject to W gi .x/ D 0 8i D 1 : : : m x2X

(106)

where f .x/; gi .x/ W Rn 7! R are any convex functions of x. In order to apply ANNs to solve this problem, the problem is reformulated into unconstrained optimization problem using penalty function. Let ˆ./ be the penalty function which has the following properties: ( D 0 if x is feasible (107) ˆ.x/ > 0 otherwise Then, consider the following transformed convex problem: minimize W E.x; s/ subject to W x2X where E.x; s/ D f .x/ C s The update equation is

Pm

i D1

ˆ.gi .x//.

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rx D  rE.x; s/ rE.x; s/ D rf .x/ C s

m X

(108)

.gi .x// rgi .x/

i D1

The stability and convergence analysis of this model has been performed in [22]. It has been shown that dynamics given by Eq. (108) leads to an equilibrium point.

5.3

Quadratic Programming

Similar to the linear programming, quadratic programming problems can be modeled using ANNs. Consider the following quadratic programming problem: minimize W xT H x C c T x subject to W Ax D b x2X

(109)

where x 2 Rn A; H 2 Rnn . This program can be converted to an unconstrained problem using a penalty function method or a Lagrange method. A Lagrange method will be presented in the following part of this section. The Lagrange of problem Eq. (109) is written as L.x; / D xT H x C c T x C T .Ax  b/

(110)

The equations of motion that describe this model are rt x D rx L.x; /

(111)

rt  D r L.x; /

(112)

where  rt x D  rT  D  rx L.x; / D

dxn dx1 dx2 ; ;:::; dt dt dt

T

d m d 1 d 2 ; ;:::; dt dt dt

x 2 Rn

; T ;

2 Rm

@L.x; / @L.x; / @L.x; / ; ;:::; @x1 @x2 @xn

T

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and  r L.x; / D

@L.x; / @L.x; / @L.x; / ; ;:::; @ 1 @ 2 @ m

T

The above model is iterative Lagrangian neural network. The convergence analysis of this model is presented in Sect. 3 and in [139]. Apart from that, quadratic programming problems with ANNs have been exhaustively studied in [39, 80, 134, 135].

5.4

Nonlinear Programming

Similar to the quadratic and convex programming, general nonlinear programming problems can be solved using ANNs. One of the ways is to use a penalty method approach [67]. Another approach is based on the Lagrange method, presented by Zhang and Constantinides [139]. The main differences with penalty function method and the advantages of Lagrange methods are discussed in Sect. 3 of this chapter. In the following part of this subsection, the method of writing a Lagrange for any generalized nonlinear problem will be presented. minimize W f .x/ subject to W hi .x/ D 0 8i D 1 : : : m x 2 Rn

(113)

where f .x/; hi .x/ W Rn 7! R are any continuous and twice differentiable functions of x. The Lagrange of the above COP is written as L.x; / D f .x/ C T h.x/

(114)

The equations of motion that describes this model are rt x D rx L.x; /

(115)

rt  D r L.x; /

(116)

where  rt x D

dxn dx1 dx2 ; ;:::; dt dt dt

T

; x 2 Rn

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 d m T d 1 d 2 ; ;:::; ; 2 Rm dt dt dt   @L.x; / @L.x; / @L.x; / T rx L.x; / D ; ;:::; @x1 @x2 @xn 

rt  D

and

 r L.x; / D

@L.x; / @L.x; / @L.x; / ; ;:::; @ 1 @ 2 @ m

T

The above model is iterative, similar to that of H-T’s continuous model. Both the types of neuron are decision variables. Lagrangian variables tend to take the solution to a feasible space, whereas the ordinary variables will take the solution to an optimal value. The mathematical properties and proofs related to convergence of this model have been discussed in Sect. 3.

5.5

Complementarity Problem

5.5.1 Linear Complementarity Problem (LCP) Given a matrix M 2 RnN and a vector q 2 Rn , the LCP can be stated as: find W x subject to W Mx C q  0 T

x .M x C q/ D 0 x0

(117)

where x 2 Rn . This problem can be solved by ANNs, by converting it to a nonlinear problem byp using Fischer function. Let .a; b/ be the Fischer function, defined as .a; b/ D a2 C b 2  a  b. The purpose of using this function is that it has an important property, which can be stated as .a; b/ D 0 , a  0; b  0 and ab D 0

(118)

Thus, this property is exploited while solving LCP using ANN. The above property mimics the LCP, which can be represented as i .xi ; Fi .x// D 0 8 i

(119)

where Fi .x/ D xi .M x C q/i . Let ˆ.x/ represent a vector, where the i th element is i .xi ; Fi .x//. Then, solving LCP is same as solving the following problem:

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minimize W F .x/

(120)

where

1 jjˆ.x/jj2 (121) 2 Now, the above problem can be solved by ANNs as a quadratic programming problem, using either a H-T’s model or Lagrange model. F .x/ D

5.5.2 Nonlinear Complementarity Problem (NCP) Given a set of functions Fi W Rn 7! Rn ; 8 i D 1; : : : ; n, the NCP can be stated as find W x subject to W Fi .x/  0 xi F .x/ D 0 x0

(122)

where x 2 Rn and Fi is assumed to be continuously differentiable function. ANNs can be used to solve NCP by using the idea of Fischer function [75]. Similar to the LCP case, define a vector ˆ.x/, where each i th element is represented by i .xi ; Fi .x// D 0 8 i

(123)

Thus, solving NCP is same as solving the following problem: minimize W E.x/

(124)

where

1 jjˆ.x/jj2 (125) 2 Now, this above problem can be solved by ANNs using a Lagrange model. E.x/ D

5.6

Mixed Integer Programming Problems (MIPs)

The difficulty in solving any optimization problems is not due to the nonlinear (or discrete) representation of the problem. The criteria that separate

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difficult and easy problems are the convexity of the problem. A convex problem in a linear or nonlinear representation is tractable, whereas a nonconvex problem in any representation is difficult to solve. By presenting the example of a general MIPs, we would like to highlight the thin line that separates a continuous optimization problem from a discrete optimization problem while applying ANNs. Consider the following MIPs defined as minimize W f .x; y/ subject to W gi .x; y/  0 8i D 1; : : : ; m hj .x; y/ D 0 8j D 1; : : : ; p lk  xk  uk

8k D 1; : : : ; n

yr 2 f0; 1g 8r D 1; : : : ; q

(126)

Similar to any previous approaches, this problem can be converted to unconstrained optimization problem using either a penalty function or a Lagrange function. For the sake of completeness, both approaches are illustrated in the following part of this subsection.

5.6.1 Penalty Function Approach The modified (penalized) energy function for problem Eq. (126) can be written as Emip .x; y/ D C1 f .x; y/ C C2

m X

ˆŒgi .x; y/

i D1

CC3

p X

h2j .x; y/ C C4

j D1

q X

yr .1  yr /

(127)

rD1

where C1 ; C2 ; C3 ; and C4 are scaling constants, used as the penalty parameters. ˆ is the penalty function. The dynamics of the network defined by Eq. (127) can be described as d Ikx @Emip D x dt @xk

8k D 1; : : : ; n

(128)

8r D 1; : : : ; q

(129)

y

@Emip d Ir D y dt @yr xk D ‰c .Ikx /

8k D 1; : : : ; n

(130)

yr D ‰d .Iry / 8r D 1; : : : ; q

(131)

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where x and y are positive scaling coefficients for the dynamics of the system. In addition to that, ‰d can be replaced by the tuned ‰c that results in the discrete values of yr .

5.6.2 Lagrange Function Approach The Lagrange of problem Eq. (126) can be written as Lmip .x; y; ; 1 ; 2 / D f .x; y/ C

m X

i gi .x; y/

i D1

C

p X

j1 hj .x; y/ C

j D1

q X

r2 yr .1  yr /

(132)

rD1

where i  0 8 i D 1 : : : m and j1 ; r2 2 R 8 j D 1 : : : p and 8 r D 1 : : : q. Similar to penalty function approach, the system dynamics can be described as rt x D rx Lmip .x; y; ;  1 ;  2 /

(133)

rt y D ry Lmip .x; y; ;  1 ;  2 /

(134)

rt 1 D r 1 Lmip .x; y; ;  1 ;  2 /

(135)

rt 2 D r 2 Lmip .x; y; ;  1 ;  2 /

(136)

8 2 D d 2 e C 1. Theorem 103 Let G be a planar graph with girth g: 1. If g  6, then lc.G/  d 2 e C 4. 2. If g  5, then lc.G/  d 2 e C 14. More recently, Theorem 102 was extended by Wang and Li [357] by showing the following result. Theorem 104 Every graph G embeddable in a surface of nonnegative characteristic has lc.G/ D d 2 e C 1 if there is a pair .k; g/ 2 f.13; 7/; .9; 8/; .5; 10/; .3; 13/g such that G satisfies   k and G has girth at least g. Dong, Xu, and Zhang [132] improved partially the results in Theorem 103 by showing:

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Theorem 105 Let G be a planar graph with girth g: 1. If g  8, then lc.G/  d 2 e C 2. 2. If g  6, then lc.G/  d 2 e C 3. 3. If g  5, then lc.G/  d 2 e C 10. The latest results on linear chromatic number were established by Li et al. [247]. Theorem 106 Let G be a graph: 1. lc.G/  12 .2 C /. 2. If   4, then lc.G/  8. 3. If   5, then lc.G/  14. 4. If G is a planar graph with   52, then lc.G/  b0:9c C 5.

3.4.4 Generalized Acyclic Colorings A generalization of the acyclic edge chromatic number was introduced by Gerke et al. in [163]. Let r  3 be a positive integer. Given a graph G, let 0aWr .G/ be the minimum number of colors required to color the edges of the graph such that every k-cycle has at least minfk; rg colors, for all k  3. So 0a .G/ D 0aW3 .G/. The r-acyclic edge chromatic number 0aWr .G/ is the minimum number of colors required to color the edges of the graph G such that every cycle C of G has at least minfjC j; rg colors. Replacing the word edges with vertices gives the definition of the r-acyclic chromatic number aWr .G/ of G. A proper vertex (respectively, edge) coloring is r-acyclic if each k-cycle has at least minfk; rg colors, for k  3. Obviously, a proper r-acyclic coloring of the line graph L.G/ of a graph G gives a proper racyclic edge coloring of G. Hence, 0aWr .G/  aWr .L.G//. Note that the complete bipartite graph Kd;d satisfies aW4 .Kd;d / D d 2 , since every edge must receive a distinct color to avoid a 4-cycle that is colored with less than four colors. This is an example of a class of graphs where the 4-acyclic edge chromatic number of any member is quadratic in its maximum degree. For the case r  6, Greenhill and Pikhurko [167] considered the generalized acyclic chromatic number and generalized acyclic edge chromatic number of graphs with maximum degree d and constructed -regular graphs G with 0aWr .G/  cr br=2c , where cr depends on r but is a constant with respect to . They also proved upper bounds on 0aWr .G/ of graphs with order O./br=2c . Exactly, they proved the following theorem by using the Local Lemma while considering the Hamming cube graphs. The upper bounds and the lower bounds were given in [167]. Before stating the following theorem, some notation is first introduced. Let d be a positive integer. Let aWr ./ and 0aWr ./, respectively, be the maximum of aWr .G/ and 0aWr .G/, over all graphs G with maximum degree . The following theorem was obtained in [167]. Theorem 107 Let r  4 be fixed. There exist positive constants c, C , c 0 , C 0 such that cbr=2c  aWr ./  Cbr=2c, c 0 br=2c  0aWr ./  C 0 br=2c for   3.

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In addition, the authors in [167] showed that some well-known algebraic constructions produce families of graphs which, unlike the Hamming cubes, have girth at least r but still provide fairly good lower bounds for the r-acyclic chromatic numbers when r D 6; 8; 12. Theorem 108 Suppose that d D q C 1 where q is a prime or prime power. Then there is a graph G which is d -regular, has girth 6, and satisfies aW6 .G/ D 2.d 2  d C 1/, 0aW6 .G/ D d.d 2  d C 1/. Theorem 109 Suppose that d D q C 1 where q is a prime or prime power. Then there is a graph G which is d -regular, has girth 6, and satisfies aW8 .G/ D 2d.d 2  2d C 2/, 0aW8 .G/ D d 2 .d 2  2d C 2/. Theorem 110 Suppose that d D q C 1 where q is a prime or prime power. Then there is a graph G which is d -regular, has girth 12, and satisfies aW12 .G/ D 2d.d 4  4d 3 C 7d 2  6d C 3/, 0aW12 .G/ D d 2 .d 4  4d 3 C 7d 2  6d C 3/. In [163], it was shown that typical -regular graphs have much smaller r-acyclic edge chromatic number. More precisely, it was shown that a random -regular graph on n vertices has, with probability tending to 1 as n tends to infinity, an racyclic edge chromatic number at most .r  2/. A property holds asymptotically almost surely (a.a.s.) if the probability that it holds tends to 1 as n tends to infinity. Theorem 111 ([163]) Let   2 and r  4 be fixed. The r-acyclic edge chromatic number of a uniformly chosen -regular graph on n vertices is a.a.s. at most .r  2/. Therefore, a typical regular graph has an r-acyclic edge chromatic number that is linear in its maximum degree. A natural question is thus to ask which structures force a class of graphs to have an r-acyclic edge chromatic number that is quadratic in . In 2006, Gerke and Raemy [164] showed that such a class has to contain graphs with small girth. More specifically, they showed the following theorem: Theorem 112 Let r  4 be an integer. For any graph G with maximum degree  and girth g  3.r  1/, it holds that 0aWr .G/  6.r  1/. Theorem 112 showed that for the class of graphs with large girth, the r-acyclic edge chromatic number is linear in  if r is fixed. The upper bound is at most a factor 12 away from the optimal bound as it is known [163] that every d -regular graph with girth r  3 satisfies 0aWr .G/ > .r  1/d=2.

4

Vertex-Distinguishing Coloring

This section is concerned with vertex-distinguish coloring problems. To start with this, a new concept is needed, namely, edge-weighting. Let G be a graph. A k-edge weighting of G is a mapping W E.G/ ! f1; 2; : : : ; kg. Let .e/

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denote the weight of edge e. Usually, it is needed to seek a much smaller size of f1; 2; : : : ; kg such that there exists a mapping satisfying several given constraints. Given a graph G and an edge-weighting of G, for v 2 V .G/, let S .v/ D f .e/jv 2 eg be the set of edge-weights appearing on edges incident to v under the edge-weighting . Based on the Ph.D. dissertation [293], the class of edge-weighting problems can be roughly split into two parts. The one is the proper edge-weightings (also named as edge-coloring), that is, edge weightings of G where any two adjacent edges must be mapped to different weights by . The other is the improper edge-weightings of G, that is, edge-weightings of G where two adjacent edges may be mapped to the same weight by .

4.1

Proper Edge-Weighting

Firstly, proper edge weighting (namely, edge-coloring) is considered. It is evident that  is the lower bound for all the considered edge-coloring problems. Moreover, if G has two adjacent vertices of maximum degree, then  C 1 is the corresponding lower bound.

4.1.1 Vertex-Distinguishing Edge-Colorings An edge-coloring of G is called vertex-distinguishing (or vde-coloring for short), if any two vertices u; v have different weight sets, i.e., S .u/ ¤ S .v/. A vde-coloring is also called a strong edge coloring, or point-distinguishing edgecoloring. The vertex-distinguishing chromatic index of G, denoted by 0vd .G/ (or vdi.G/), is the smallest integer k such that there is a k-vde-coloring of G. This graph parameter is introduced by Burris and Schelp [88, 89] and, indeˇ y et al. [96], and Horˇna´ k and pendently (as observability of a graph), by Cern´ Sot´ak [195]. If G has an isolated edge uv, then S.u/ D S.v/ always and G cannot have a vde-coloring. Similarly, if G has two isolated vertices u; v, then S.u/ D S.v/ D ;. However, every graph without isolated edges and with at most one isolated vertex must have a vde-coloring: simply assign different colors to all edges of G. Such graphs are called as nice graphs. In [88], the author conjectured two different upper bounds. The first one has been confirmed by Bazgan et al. [42]. Theorem 113 If G is a nice graph on n vertices, then 0vd .G/  n C 1. The sharpness of Theorem 113 can be confirmed by considering the complete graph on n vertices with n even. Given a graph G, let nd denote the number of vertices of degree d . It is easy to see that if there exists a k-vde-coloring of G, then .kd /  nd for all 1  d  . The second conjecture of Burris and Schelp [88, 89] remains open, which is analogous to the Vizing’s Theorem on edge coloring:

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Conjecture 15 If G is a nice graph and k is the minimum integer such that .kd /  nd for all ı  d  , then k  0vd .G/  k C 1. In the same papers [88, 89], Burris and Schelp gave an upper bound of 0vd .G/ for a graph G. Theorem 114 For a graph G; m1  0vd .G/  . C 1/b2m2 C 5c, where  m1 D max

1k

.kŠnk /1=k C

k1 1=k ; and m2 D max nk : 1k 2

Corollary 7 For an r-regular graph G of order n, 0vd .G/  .r C 1/b2n1=r C 5c. Recently, there has been significant progress on this conjecture. Balister [28] considered the vde-coloring of random graphs by giving a strong bound below: Theorem 115 If G is a random graph on n vertices with edge probability p D pn .1p/n 0 p.n/, and if log n ; log n ! 1 as n ! 1, then the probability that vd .G/ D 4 goes to 1 as n ! 1. Balister et al. [31] proved that Conjecture 15 is true for graphs with large maximum degree. p Theorem 116 If G is a graph with n vertices,   2n C 4; ı  5, and k is the smallest positive integer such that nd  .kd / for all ı  d  , then k  0vd .G/  k C 1. Balister et al. [29] confirmed Conjecture 15 if G consists of just paths or of just cycles. Specially, they proved the following result. Theorem 117 1. Let G be a vertex-disjoint union of cycles, and let n2 D jGj  .k2 /, with k as small as possible. Then 0vd .G/ D k or k C 1. 2. Let G be a vertex-disjoint union of paths of length at least 2. Let n1  k and n2  .k2 /, with k as small as possible. Then 0vd .G/ D k or k C 1. 3. Let G be any nice graph with  D 2. Let n1  k and n2  .k2 /, with k as small as possible. Then k  0vd .G/  k C 5. Theorem 117 implies that Conjecture 15 holds for 2-regular graphs. Since the conjecture holds for graphs with large maximum degree [31], it is natural to ask whether the conjecture holds for small maximum degree, for example, 3-regular graphs. Many people tried to think about the bounds of 0vd .G/ for some regular graphs G, such as Fibonacci and Lucas cubes [124]; Qn [197]; nG, where G is a 3-regular graph with 1-factor on at most 12 vertices or G 2 fK4;4 ; K5;5 ; K6;6 ; K7;7 ; K6 g [307]; ladders and pK4 [325]; and extended Fibonacci cubes [380]. The following result has been testified for d -regular graphs with small components by Balister et al. [32].

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Theorem 118 Let d  3 and assume that G is a d -regular graph which contains s S d  2 disjoint 1-factors. Let G D Gi , where G1 ; G2 ; : : : ; Gs are components of G. If jGj  .kd / and jGi j 

3.k1/ 4.d 1/

i D1

for all i , then 0vd .G/  k C 1.

Balister, L Riordan, and Schelp [32] defined a few parameters for nice graphs. Here, is the symmetric difference operator, which produces the set of items which appear in exactly an odd number of its operands. Suppose that G is a nice graph. Let k.G/ be the minimum integer k such that .kj /  nj for all j with ı  j  . Recall for v 2 V .G/, S.v/ denotes the set of colors used to color the edges incident to v. If a vde-coloring with L k colors exists for G, then each color meets an even number of vertices. Thus, v S.v/ D ;. Let k 0 .G/ be the minimum k such that there exist distinct sets S.v/  f1; 2; : : : ; kg with jS.v/j D d.v/ for all v 2 V .G/, such that L S.v/ D ;. Therefore, k 0 .G/  k.G/. Let k 00 .G/ be the smallest k such that for v any set of vertices X  V .G/,Lthere exist distinct sets S.v/  f1; 2; : : : ; kg, v 2 X , such that jS.v/j D d.v/ and j v2X S.v/j  jE.X; X c /j, where E.X; X c / is the set of edges between X and X c D V .G/nX . Thus, 0vd .G/  k 00 .G/  k 0 .G/  k.G/. Also, in [32], it is shown that k 00 .G/  k.G/ C 1. Hence, they presented the following conjecture. Conjecture 16 For all nice graphs G, 0vd .G/ D k 00 .G/. Observe that this conjecture is consistent with the inequalities mentioned above. Also, this conjecture implies clearly Conjecture 15. It is not even known whether there exists an absolute constant c such that 0vd .G/  k.G/ C c, when G is a nice graph. Yet, for graphs with large minimum degree, Bazgan, Harkat-Benhamdine, Li, and Wo´zniak [43] obtained the following result. Theorem 119 Let G be a nice graph of order n  3. If ı > n3 , then 0vd .G/  C5. For an integer k with k  ˛.G/, where ˛.G/ stands for the independence number of G, k .G/ is defined by

k .G/ D min

k ˚P i D1

d.vi /jfv1 ; v2 ; : : : ; vk g is an independent set of G :

If ˛.G/ D 1, then G is complete and the exact value of 0vd .G/ is computed in [89], that is, 0vd .Kn / D n if n is odd and 0vd .G/ D n C 1 if n is even for n  4. For ˛.G/  2, Liu and Liu [254] presented the following result. Theorem 120 Let G be a connected graph of order n  3. If 2 .G/  0vd .G/   C 5. It is easy to see that if ı > n3 , then 2 .G/  Theorem 119.

2n . 3

2n 3 ,

then

Thus, Theorem 120 generalizes

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4.1.2 Adjacent Vertex-Distinguishing Edge-Coloring Instead of requiring that any two vertices have different weight sets, it can be required that only adjacent vertices have different weight sets. An edge-coloring satisfying this condition is adjacent vertex-distinguishing (also called an adjacent strong edge-coloring, 1-strong edge-coloring, or neighbor-distinguishing edgecoloring and denoted as avde-coloring for short). A k-avde-coloring is an avdecoloring using k colors. The smallest integer k such that there is a k-avde-coloring of G is called the adjacent vertex-distinguishing chromatic index of G, denoted by 0avd .G/. Obviously, when G contains an isolated edge, there is no any k-avde-coloring of G for any k  1. Thus, only simple graphs without isolated edges are considered in the rest of this section. This graph parameter was introduced by Zhang et al. [396]. They completely determined the adjacent vertex-distinguishing chromatic indices for paths, cycles, complete graphs, and complete bipartite graphs. In light of these examples, they proposed the following conjecture. Conjecture 17 If G is a connected graph and G ¤ C5 , then 0avd .G/   C 2: Note that 0a .C5 / D 5 D  C 3. It is unknown if C5 is the unique exception graph. The conjecture has been confirmed by Balister et al. [30] for bipartite graphs and for graphs of maximum degree 3. They also gave a general upper bound on 0avd .G/, depending on the maximum degree  and chromatic number .G/: Theorem 121 If G is a graph, then 0avd .G/   C O.log .G//: Akbari et al. [10] obtained an upper bound of 0avd .G/, which was improved by Dai and Bu [123] by one. Theorem 122 ([10]) For any graph G, 0avd .G/  3. Theorem 123 ([123]) For any graph G, 0avd .G/  3  1. Later, Ghandehari and Hatami [165] improved these bounds by restricting the maximum degree of the graph: p Theorem 124 If G is a graph with   106 , then 0avd .G/   C 27  ln . Hatami [179] gave an asymptotically stronger upper bound on 0avd .G/ by using a random edge-weighting technique. Theorem 125 If G is a graph with  > 1020 , then 0avd .G/   C 300: Despite the large maximum degree requirement, this result is of importance as it shows that a simple additive constant is enough for an upper bound on 0avd .G/. For graphs with small maximum average degree, Wang and Wang [371] verified Conjecture 17:

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Theorem 126 Let G be a graph: 1. If mad.G/ < 3 and   3, then 0avd .G/   C 2. 2. If mad.G/ < 52 and   4, or mad.G/ < 73 and  D 3, then 0avd .G/   C 1. 3. If mad.G/ < 52 and   5, then 0avd .G/ D  C 1 if and only if G contains adjacent vertices of maximum degree. Recall that if G is a planar graph with girth g, then mad.G/  together with Theorem 126 leads to the following corollary:

2g g2 .

This fact

Corollary 8 Let G be a planar graph with girth g: 1. If g  6 and   3, then 0avd .G/   C 2. 2. If g  10 and   4, or g  4 and  D 3, then 0avd .G/   C 1. 3. If g  10 and   5, then 0avd .G/ D  C 1 if and only if G contains adjacent vertices of maximum degree. It should be pointed out that Bu et al. [86] also proved that Conjecture 17 holds for planar graphs of girth at least six, that is, if G is a planar graph with girth g  6, then 0avd .G/   C 2. Liu and Liu [255] considered the adjacent vertex-distinguishing chromatic index for graphs with dominating subgraphs. A subgraph H of a graph G is called a dominating subgraph of G if V .G/  V .H / is an independent set. The following three theorems appeared in [255]: Theorem 127 If G is a Hamiltonian graph with .G/  3, then 0avd .G/   C 3. Theorem 128 If G is a graph with .G/  3 and G has a Hamiltonian path, then 0avd .G/   C 4. Theorem 129 If a graph G has a dominating cycle or a dominating path H such that .GŒV .H //  3, then 0avd .G/   C 5. For regular graphs, Greenhill and Ruc´nski [168] used a probabilistic method to obtain the following better bound: Theorem 130 Let r  4. Then a random r-regular graph G a.a.s. has 0avd .G/  d3r=2e. When r D 4, it can be deduced from the above theorem that almost always all 4-regular graphs satisfy Conjecture 17. Edwards et al. [139] proved that 0avd .G/   C 1 if G is bipartite, planar, and   12. Chen and Guo [103] completely determined the adjacent vertexdistinguishing chromatic index of hypercubes. In precise, they showed that if Qd is a hypercube of dimension d  3, then 0avd .Qd / D d C 1. Liu et al. [256] proved that if G is a 2-connected outerplanar graph with   5, then   0avd .G/   C 1, and 0avd .G/ D  C 1 if and only if G contains adjacent vertices of maximum degree. Wang and Wang [372] extended this result to the following:

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Theorem 131 Let G be a K4 -minor-free graph with   4. Then the following statements hold: 1.   0a .G/   C 1. 2. If   5, then 0a .G/ D  C 1 if and only if G contains adjacent vertices of maximum degree. It is somewhat surprising that there exists an interesting relationship between 0avd .G/ and 00 .G/, the total chromatic number of a graph G. Zhang et al. [397] proposed the following conjectures: Conjecture 18 For a graph G with no K2 or C5 component, 0avd .G/  00 .G/. Conjecture 19 For an r-regular graph G with no C5 component (r  2), 0avd .G/ D 00 .G/. In [397], Conjecture 19 was shown to be true for many classes of graphs, such as graphs with 00 .G/ D  C 1; 2-regular, 3-regular, and .jGj  2/-regular graphs; bipartite regular graphs; balanced complete multipartite graphs; hypercubes; and joins of two matchings or cycles.

4.1.3 r-Strong Chromatic Index Akbari et al. [10] introduced another graph parameter, called as r-strong edge coloring. A proper edge coloring of a graph G is called an r-strong edge-coloring if for any two distinct vertices u; v 2 V .G/ with the distance d.u; v/  r, S.u/ ¤ S.v/. The r-strong chromatic index 0si .G; r/ is the minimum number of colors required for an r-strong edge-coloring of the graph G. Thus, 0si .G; 1/ D 0avd .G/, and for any connected graph G with n vertices, 0si .G; n/ D 0vd .G/. Akbari et al. [10] generalized a result in [396] about the parameter 0avd .T /, where T is a tree as follows: Theorem 132 Let T be a tree with at least three vertices: 1. 0si .T; 2/   C 1. Furthermore, if for any two vertices u and v of maximum degree, d.u; v/  3, then 0si .T; 2/ D . 2. If   3, then 0si .T; 3/  2  1. 3. For any integers k  4 and   1, there exists a tree T with maximum degree  such that 0si .T; k/  .  1/bk=2c1 : Ghandehari and Hatami [165] presented an upper bound of 0si .G; k/ for regular graphs: Theorem 133 For each r-regular graph G with r>100, 0si .G; k/  r C .k C 2/ log2 r. As a direct corollary of Theorem 133, the following holds: Corollary 9 For an r-regular graph G with r > 100, 0avd .G/  r C 3 log2 r.

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Non-proper Edge-Weighting

In this section, non-proper edge-weighting is concerned. Unlike the case of proper edge-weighting, there is no lower bounds compared to 0 .G/.

4.2.1 Point-Distinguishing Chromatic Index Similar to the vertex-distinguishing edge-coloring problem, an edge-weighting (not needed to be proper) such that all vertices have different weight sets is considered here. This graph parameter is introduced by Harary and Plantholt [177] (as pointdistinguishing chromatic index) and is denoted by 0 .G/. Note that if a graph G has at least two isolated vertices or an isolated edge, then point-distinguishing chromatic index cannot be defined very well. This means that once G admits the point-distinguishing chromatic index, then G must be a nice graph. Harary and Plantholt [177] determined the value of this parameter for paths, cycles, complete graphs, and hypercubes. Furthermore, they presented a lower bound for complete bipartite graphs. Even for complete bipartite graphs, it seems that the problem of determining 0 .Km;n / is not easy. In fact, it was shown in [177] that 0 .Km;n /  dlog2 .m C n/e and 0 .Km;n /  dlog2 ne C 2 if dlog2 ne C 1  m  n. Zagaglia Salvi [393] proved that the sequence f0 .Kn;n /g1 nD2 is non-decreasing with jumps of value 1. As a consequence, there is an increasing sequence fnk g1 kD2 of integers in which n2 D 1 and for any k  3, 0 .Kn;n / D k if and only if nk1 C 1  n  nk . Only few exact values of nk are known, namely, n3 D 2; n4 D 5; n5 D 11; n6 D 22 [392], and n7 D 46 [196]. According to Zagaglia Salvi [393], nk nk nkC1  2nk and so f 2k1 g is convergent and 46  lim 2k1  1. Horˇna´ k and 64 p nk Sot´ak [198] succeeded in showing that lim 2k1  3  5. Horˇna´ k and Zagaglia Salvi [200] asked the following question. nk < 1? Problem 4 Is it true that lim 2k1

They also determined 0 .Km;n / for all n sufficiently large with respect to m in [200]. Let k; l; m; n be integers with 2  m  n, l D dlog2 me, k  minfl C1; 3g.

dm.k/

.k/ bm

8 < 2m; D 2m  1; : m; D

m X k  i

if k  mI if k D m C 1I if k  m C 2:

 dm.k/ ;

i D0

Im.k/

.k1/ .k/ D 7Œbm C 1; bm ;

.k1/ C 1; 2k1  m  l  1; L.k/ m D Œbm .k/ .k/ Rm D Œ2.k1/  m  l; bm ;

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where Œp; q denotes the interval of all integers z with p  z  q for fixed integers p; q. The following several results and one problem appeared in [200]: .k/

Theorem 134 If n 2 Im , then 0 .Km;n /  k. Theorem 135 0 .Km;n /  k C 1 follows from either of the following conditions: 1. k  m C 1 and n  2k  m  l. .k/ 2. k  m C 2 and n  bm C l. .k/

Theorem 136 The equality 0 .Km;n / D k , n 2 Im follows from any of the following conditions: 1. k  2l C 2. 2. m 2 f2; 3g. Corollary 10 If m  7 and n 2 Œ2  4l  2m C 1; 2mC1  2m, then 0 .Km;n / D dlog2 .n C 2m/e. Theorem 137 The equality 0 .Km;n / D k follows from any of the following conditions: .k/ 1. k 2 Œl C 2; 2l C 1 and n 2 Rm . .k/ 2. k D l C 1 and n 2 Im . .k/

Theorem 138 Suppose that m 2 Œ4; 10 and n 2 Lm . Then 0 .Km;n / D k  1 if and only if .k; m; n/ D .6; 10; 13/. Problem 5 Decide whether there is a triple of integers .k; m; n/ such that m  11; .k/ k 2 Œl C 3; 2l C 1, n 2 Lm , and 0 .Km;n / D k  1.

4.2.2 General Neighbor-Distinguishing Index Also, a generalized version of adjacent vertex-distinguishing edge-coloring may be considered, in which edge weighting may be not proper. The corresponding invariant is the general neighbor-distinguishing index of a graph G, denoted by gndi.G/. This graph parameter was first introduced and investigated by Gy˝ori et al. [174]. Clearly, G is a nice graph when considering this parameter. First, four authors in [174] established an interesting relationship between gndi.G/ and .G/: Theorem 139 For any graph G, the following statements are equivalent: 1. gndi.G/ D 2. 2. G is bipartite and there is a bipartition fX1 [X2 ; Y g of V .G/ such that X1 \X2 D ; and any vertex of Y has at least one neighbor in both X1 and X2 . Theorem 140 If G is a connected bipartite graph with jE.G/j  2, then gndi.G/  3. Theorem 141 gndi.G/  2dlog2 .G/e C 1 for any nice graph G.

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Thus, when .G/  2, a good characterization on gndi.G/ has been established. If .G/  3, Horˇna´ k and Sot´sk [199] improved the above bound: Theorem 142 If G is a connected graph with .G/  3, then dlog2 .G/e C 1  gndi.G/  blog2 .G/c C 2 . Therefore, when log2 .G/ is not an integer, then gndi.G/ D dlog2 .G/e C 1. A natural question is as follows: what about the case where log2 .G/ is an integer? This problem is recently solved by Gyo˝ ri and Palmer [175]: Theorem 143 If G is a connected graph with .G/  3, then gndiG D dlog2 .G/e C 1 .

4.2.3 Irregularity Strength Instead of distinguishing the color set, the problem of distinguishing the total sum of weights can be considered. Chartrand et al. [100] defined the irregularity strength of a graph G, denoted by s.G/, to be the smallest integer k such that there is a k-edge-weighting of G and for any two vertices u; v, X e3u

.e/ ¤

X

.e/:

e3v

It is easy to see that for every nice graph G, there exists an irregular strength. If G is not a nice graph, set s.G/ D 1. Most of the upper bounds which have been determined for special classes of graphs are based on the minimum and maximum degree. The proofs rely on two techniques, either using the congruence method to find special spanning graphs and then determining a labeling of this spanning graph explicitly or through employing the probabilistic method. In [100], it was shown that for any graph G with n  3 vertices, s.G/  2n  3. This bound was later improved by Nierhoff [290] by refining the idea of Aigner and Triesch [8] to s.G/  n  1 for any graph G with n  4 vertices. For regular graphs, stronger upper bounds on s.G/ are known. Faudree and Lehel [145] showed that if G is d -regular .d  2/, then d nCdd 1 e  s.G/  d n2 eC9. They also proposed the following conjecture. Conjecture 20 If G is a d -regular graph with d  2, then s.G/  d dn e C c for some constant c. The question was motivated by the following facts: (i) For every d -regular graph of order n, s.G/  d.n  1/=d e C 1 (see [100]). (ii) In all those cases where the irregularity strength of a regular graph is known, its value is very close to d.n  1/=d e C 1. For example, Amar [23] proved that if G is an .n  3/- or .n  4/-regular graph of order n, then s.G/ is 3 (except if G D K3;3 ). In light of this result, Amar [23] proposed the following conjecture:

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Conjecture 21 The irregular strength of an r-regular graph of order n  2r is 3, except if G is Kl;l with l odd. Frieze, Gould, Karo´nski, and Pfender [157] used a probabilistic analysis to show that for a graph G with minimum degree ı and maximum degree , s.G/  c1 n=ı if   n1=2 and s.G/  c2 .log n/n=ı if  > n1=2 , where c1 and c2 are explicit constants. For regular graphs, they obtained the following results in the same paper. Theorem 144 Let G be a d -regular graph of order n with no isolated vertices or edges: 1. If d  b.n= ln n/1=4 c, then s.G/  10n=d C 1. 2. If b.n= ln n/1=4 c C 1  d  bn1=2 c, then s.G/  48n=d C 1. 3. If d  bn1=2 c C 1, then s.G/  240.log n/n=d C 1. Cuckler and Lazebnik [122] further obtained the following result: Theorem 145 Let G be a graph of order n: 1. If ı  10n3=4 log1=4 n, then s.G/  48.n=ı/ C 6. 2. If G is d -regular with d  104=3 n2=3 log1=3 n, then s.G/  48.n=d / C 6. However, these results do not confirm even a weaker form of Conjecture 20, namely, s.G/  c1 n=d C c2 holds for all d -regular graphs of order n, with c1 and c2 being absolute positive constants. A big step toward this direction has been achieved by Przybyło, who shows that actually constants c1 and c2 exist, both for regular graphs in [295] and general graphs in [296]. Theorem 146 Let G be a d -regular graph of order n with no isolated vertices or edges. Then s.G/ < 16n=d C 6. Theorem 147 Let G be a graph of order n with no isolated vertices or edges. Then s.G/ < 112n=ı C 28. Recently, Kalkowski et al. [218] obtained a much better upper bound. Theorem 148 Let G be a graph of order n and of minimum degree ı. Then s.G/  6dn=ıe. Recall that ni figures the number of vertices of degree i in a graph G. Then a simple counting argument shows that &

' k 1X s.G/  ˇ.G/ D max ni : k k i D1 Lehel [245] felt that ˇ.G/ with addition of an absolute constant was enough for s.G/. Namely, Conjecture 22 If G is a connected graph, then s.G/  ˇ.G/Cc for some absolute constant c.

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Note that if G is r-regular with r  2, then ˇ.G/ D dn=re. Therefore, Conjecture 20 is a special case of Conjecture 22. Baril et al. [33] studied the irregularity strength of circulant graphs of degree 4. Let Cn .1; k/ denote the circulant graph which consists of a cycle of length n plus chords joining vertices at distance k on the cycle. Theorem 149 For n  4k C 1 and k  2,  s.Cn .1; k// D

 nC3 : 4

For grids, Dinitz et al. [127] proved that for every positive integer d , s.Xm;m / D ˇ.Xm;n / or ˇ.Xm;n / C 1 for all but a finite number of m  n grids Xm;n where nm D d . This finite number was computed in [127] to at most d C13. Togni [336] proved that the irregularity strength of the toroidal grid Cm  Cn is b.mn C 3/=4c. Amar and Tongi [24] showed that s.T / D ˇ.T / D n1 for all trees T with n2 D 0 and n1  3. For general trees, it is not even the case that s.T / is within an additive constant of n1 . Bohman and Kravitz [52] p presented an infinite sequence of trees 11 5 2 with irregularity strength converging to 8 n1 > n1 > n1 Cn 2 . Ferrara et al. [146] presented an iterative algorithm to show that s.T / D ˇ.T / for another class of trees, 2 but this time, n1 < n2 , that is, s.T / D d n1 Cn 2 e. If the point-distinguishing chromatic index problem is modified so that for any two vertices u; v 2 V .G/, the multiset of all edge weights appearing on edges incident to u is different from the multiset of all edge weights appearing on edges incident to v (where the multiplicity of an element is exactly how many times that edge weight appears on edges incident to the corresponding vertex). For a given graph G, the smallest k such that a mapping satisfying these conditions exists is denoted by c.G/. It is easy to see that c.G/  s.G/. Aigner, Triesch, and Tuza introduced this problem in [9]. They proved the following result for regular graphs: Theorem 150 If G is a regular graph, then there exist constants c1 ; c2 , such that c1 n1=r  c.G/  c2 n1=r . For other known results of parameter s.G/, see a nice survey paper of Lehel [245].

4.2.4 On 1; 2 ; 3-Conjecture Motivated by the results on irregularity strength, Karo´nski et al. [221] studied the improper edge-weighting problem where it is only required that for adjacent vertices u; v, X X .e/ ¤ .e/: e3u

e3v

The smallest integer k such that there is a mapping satisfying the above condition is denoted by .G/ and is called the vertex-coloring k-edge-weighting. Notice that .G/  s.G/ for every nontrivial graph G. Karo´nski et al. [221] proved that .G/  3 for graphs G without an edge component and having chromatic

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number .G/  3 and conjectured that .G/  3 for all graphs G without an edge component. This is sometimes referred to as 1,2,3-conjecture. Conjecture 23 If G is a graph without isolated edges, then .G/  3. Conjecture 23 has been attacked primarily in two ways. The first is to show it being true for smaller classes of graphs and the second is to find general upper bound on .G/. Karo´nski et al. [221] first proved that for all graphs G without an edge component, .G/  213. Addario-Berry et al. [3] improved this upper bound to 30, which was improved later by Addario-Berry et al. [4] to 16 and by Wang and Yu [376] to 13. More recently, Kalkowski et al. [217] improved the bound to 5, based on a method developed in [216]. Note that 2 is not sufficient as seen for instance in complete graphs. Even if it is still far from providing a positive answer to the conjecture, actually .G/  2 for many graphs. In fact, Addario-Berry et al. [4] showed that .G/  2 for almost all graphs. Theorem 151 Let G be a random graph chosen from Gn;p for constant 0 < p < 1. Then there exists, a.a.s., a vertex-coloring 2-edge-weighting for G. In fact, there exists a 2-edge-weighting such that the colors of two adjacent vertices are distinct mod 2.G/. In [99], Chang et al. gave some sufficient conditions for bipartite graphs with .G/  2. Theorem 152 Every connected bipartite graph G D .A; BI E/ without isolated edges admits .G/  2 if one of following conditions holds: 1. jAj or jBj is even. 2. ı.G/ D 1. 3. bd.u/=2c C 1 ¤ d.v/ for any edge of G. Also, they proposed the following problem. Problem 6 Characterize 2-colorable graphs G (i.e., bipartite graphs) with .G/ D 2. Lu et al. [260] continued to consider this problem. They also obtained several sufficient conditions for graphs with .G/  2. In particular, they showed that 3-connected bipartite graphs admit .G/  2, and every r-regular nice bipartite graph (r  3) has .G/  2. There exist infinitely many bipartite graphs (e.g., generalized -graphs) which are 2-connected and have a vertex-coloring 3-edge-weighting but not a vertex-coloring 2-edge-weighting. Bartnicki et al. [34] considered the choosability version of edge weighting. A graph is said to be k-edge-weight choosable if the following holds: for any list assignment L which assigns to each edge e a set L.e/ of k real numbers, G has a proper edge-weighting f such that f .e/ 2 L.e/ for each edge e. They conjectured

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that every graph without isolated edges is 3-edge weight choosable and verified the conjecture for complete graphs, complete bipartite graphs, and some other graphs. Similar to the problem posed in [8], it can be considered a multiset version of vertex coloring k-edge-weighting. In this case, for any edge uv 2 E.G/, the multiset of weights on edges incident to u is different from the multiset of weights on edges incident to v. A k-edge-weighting satisfying the above condition is called as vertexcoloring k-edge partition. Karo´nski et al. [221] proved that if .G/  3, then three colors are enough for a graph. Formally, they proved the following result. Theorem 153 Let G be a connected graph with at least three vertices. If .G/  3, then G permits a vertex-coloring 3-edge partition. Addario-Berry et al. [2] proved the following two theorems concerning this version of edge-weighting. Theorem 154 If G is a graph without isolated edges, then G permits a vertexcoloring 4-edge partition. Theorem 155 If G is a graph without isolated edges and with minimum degree 1,000, then G permits a vertex-coloring 3-edge partition. Recently, Bian et al. [49] improved their results. Theorem 156 If G is a graph without isolated edges t and with minimum degree 188, then G permits a vertex-coloring 3-edge partition. It is unknown if there exists a graph G permits a vertex-coloring 4-edge partition. If such a graph exists, then by [49], .G/  4 and ı.G/ < 188.

4.3

Some Related Topics

All of the previous vertex-distinguishing edge-weighting problems can be generalized to the vertex-distinguishing total-weighting problems. Given a graph G, a k-total-weighting is a mapping W V .G/ [ E.G/ ! f1; 2; : : : ; kg. Given a graph G and a total-weighting of G, for v 2 V .G/, let S .v/ D f .e/jv 2 eg [ f .v/g be the set of weights appearing on the vertex v and the edges incident to v under the total-weighting . A k-total-weighting is proper, if .e/ ¤ .f / for any pair of adjacent edges e; f ; .v/ ¤ .e/ for any vertex v and incident edge e; and .u/ ¤ .v/ for any two adjacent vertices u; v.

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4.3.1 Adjacent Vertex-Distinguishing Total Coloring In [395], Zhang et al. defined the adjacent vertex-distinguishing total chromatic number of a graph G, denoted by 00avd .G/, which is the smallest integer k such that there exists a proper total k-coloring of G such that for any edge uv 2 E.G/, S .u/ ¤ S .v/. The authors [395] determined the adjacent vertex-distinguishing total chromatic numbers for some special classes of graphs including cycles and complete graphs. Based on these results, they made the following conjecture: Conjecture 24 If G is a connected graph with n  2 vertices, 00avd .G/   C 3. If Conjecture 24 is true, then the upper bound  C 3 is tight, for instance, a complete graph on odd order can attain this bound. Chen [102] further constructed a class of graphs, that is, the joint graphs sP3 _ Kt , attaining the upper bound. Chen [102], and independently Wang [347], confirmed Conjecture 24 for graphs G with   3. Hulgan [207] presented a more concise proof for this result and proposed the following question: Question 8 For a graph G with  D 3, is the bound 00avd .G/  6 sharp? The following result follows immediately from the definitions of parameters under consideration. Proposition 3 For any graph G, 00avd .G/  .G/ C 0 .G/. By Proposition 3, Conjecture 24 holds for all bipartite graphs. For planar graphs, by the Four-Color Theorem and the Vizing’s Theorem on edge coloring, 00avd .G/   C 5. Moreover, if G is of Class 1, that is, 0 .G/ D , then 00avd .G/   C 4. Coker and Johannson [116] used a probabilistic method to give an upper bound for 00avd .G/. Theorem 157 There exists a constant c > 0 such that for every graph G, 00avd .G/   C c. For graphs of larger maximum degree, Liu et al. [252] showed the following result. p Theorem 158 If G is a graph with  sufficiently large and ı  32  ln , then p 00avd .G/   C 1026 C 2  ln . For graphs with the smaller maximum average degree, Wang and Wang [370] verified Conjecture 24. More precisely, the following theorem and corollary were showed: Theorem 159 Let G be a graph: 1. If mad.G/ < 3 and   5, then C1  00avd .G/  C2; and 00avd .G/ D C2 if and only if G contains adjacent vertices of maximum degree. 2. If mad.G/ < 3 and  D 4, then 00avd .G/  6. 3. If mad.G/ < 83 and  D 3, then 00avd .G/  5.

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As an immediate consequence of Theorem 159, the following corollary holds: Corollary 11 Let G be a planar graph with girth g: 1. If g  6 and   5, then  C 1  00avd .G/   C 2; and 00avd .G/ D  C 2 if and only if G contains adjacent vertices of maximum degree. 2. If g  6 and  D 4, then 00avd .G/  6. 3. If g  8 and  D 3, then 00avd .G/  5. Wang and Wang [373] also completely characterized the adjacent vertexdistinguishing total chromatic number of outerplanar graphs by showing the following: Theorem 160 Let G be an outerplanar graph with   3. Then  C 1  00avd .G/   C 2; and 00avd .G/ D  C 2 if and only if G contains adjacent vertices of maximum degree. In [368], Theorem 160 was generalized to the following form: Theorem 161 Let G be a K4 -minor-free graph with   3. Then  C 1  00avd .G/   C 2; and 00avd .G/ D  C 2 if and only if G contains adjacent vertices of maximum degree. When studying the adjacent vertex-distinguishing edge-coloring of hypercubes, Chen and Guo [103] considered also the adjacent vertex-distinguishing total coloring of this kind of graphs. Their result is stated below: Theorem 162 If Qd is a hypercube of dimension d  2, then 00avd .Qd / D d C 2.

4.3.2 Total Vertex Irregularity Strength Przybyło and Wo´zniak [297] introduced an improper k-total-weighting question in the light of the problem posed by Karo´nski et al. [221]. A k-total-weighting of a graph G is called neighbor-distinguishing (or vertex-coloring, see [4]) if for every edge uv, X X .u/ C .e/ ¤ .v/ C .e/: e3u

e3v

If it exists, then G permits a neighbor-distinguishing k-total-weighting. The smallest integer k for which G permits a neighbor-distinguishing k-total-weighting is called total vertex irregularity strength, denoted by .G/. Note that if a graph permits a vertex-coloring k-edge-weighting, then it also permits a neighbor-distinguishing k-total-weighting (it is enough to put ones at all vertices). Thus .G/  5 for all graphs. Przybyło and Wo´zniak [297] made the following conjecture. Conjecture 25 Every simple graph permits a neighbor-distinguishing 2-totalweighting. In [297], the authors gave a positive answer to Conjecture 25 whenever G is 3-colorable, complete, or 4-regular. They also presented upper bounds 11 or

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b .G/ 2 c C 1 for .G/. Przybyło [294] showed that .G/  7 for all regular graphs. Later, a big breakthrough is due to Kalkowski [216], who proved that .G/  3 for all graphs. Anholcer et al. [26] presented an upper bound of .G/  3dn=ıe C 1 for each graph G of order n and with ı > 0. The values of .G/ for some special graphs are computed. Kristiana and Slamin [239] computed .G/ for wheels Wn , fans Fn , suns Sn , and friendship graphs nC2 nC1 fn , namely, .Wn / D d nC3 4 e for n  3, .Fn / D d 4 e for n > 3, .Sn / D d 2 e 2nC2 for n  3, and .fn / D d 3 e for all n. The choosability version of total weighting was studied in [383] and in [298]. For positive integers k; k 0 , a .k; k 0 /-total list assignment is a mapping L which assigns to each vertex v a set L.v/ of k real numbers as permissible weights and assigns to each edge e a set L.e/ of k 0 real numbers as permissible weights. A graph is called .k; k 0 /-totally weight choosable if for any .k; k 0 /-total list assignment L, G has a proper total-weighting f such that f .y/ 2 L.y/ for all y 2 V .G/ [ E.G/. It was known in [383] that a graph is .k; 1/-totally weight choosable if and only if it is k-choosable. The following conjectures were proposed in [383] by Wong and Zhu. Conjecture 26 Every graph is .2; 2/-totally weight choosable. Conjecture 27 Every graph with no isolated edges is .1; 3/-totally weight choosable. However, it is unknown if there are constants k; k 0 such that every graph is .k; k 0 /-totally weight choosable. It was shown in [383] that complete graphs, trees, cycles, and generalized -graphs are (2,2)-totally weight choosable, K2;n are (1,2)-totally weight choosable, and K3;n are (2,2)-totally weight choosable. Wong et al. [382] introduced a method, the max-min weighting method, for finding proper L-total weightings of graphs. Using this method, they proved that Kn;m;1;1;:::;1 are (2,2)-totally weight choosable, and complete bipartite graphs other than K2 are (1,2)-totally weight choosable.

5

L.p; q/-Labeling of Graphs

Motivated by the frequency channel assignment problem, Griggs and Yeh [169] introduced the L.2; 1/-labeling of graphs. This notion was subsequently generalized to the L.p; q/-labeling problem of graphs. Let p and q be two nonnegative integers. An L.p; q/-labeling of a graph G is a function f from its vertex set V .G/ to the label set f0; 1; : : : ; kg such that jf .x/  f .y/j  p if x and y are adjacent and jf .x/  f .y/j  q if x and y are at distance 2. The L.p; q/-labeling number

p;q .G/ of G is the smallest integer k such that G has an L.p; q/-labeling f with maxff .v/jv 2 V .G/g D k.

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The L.p; q/-labeling of graphs has been studied rather extensively in recent years [98, 219, 276, 313, 348]. In particular, the case .p; q/ D .p; 1/ with p 2 f0; 1; 2g has received much more attention.

5.1

L.2 ; 1/-Labeling

In 1991, Roberts [304] proposed a variation of the frequency assignment problem in which close transmitters must receive different frequencies and very close transmitters must receive frequencies that are at least two units apart. To translate the problem into the language of graph theory, the transmitters are represented by the vertices of a graph; two vertices are very close if they are adjacent and close if they are at distance 2 in the graph. This is indeed the case L.2; 1/-labeling problem of graphs. Along this direction, Griggs and Yeh [169] investigated the L.2; 1/-labeling problem. For a latest survey on this field, see [388]. There are a considerable number of papers studying L.2; 1/-labelings (see [97, 98, 169, 214, 308]). Most of the papers considered the values of 2;1 .G/ on particular classes of graphs G; however, Griggs and Yeh [169] provided an upper bound 2 C 2 for general graphs with maximum degree . Later, Chang and Kuo ˇ [98] improved it to 2 C . Recently, Kr´al and Skrekovski [237] reduced the bound to 2 C   1. In general, Griggs and Yeh [169] suggested the following conjecture: Conjecture 28 (L.2; 1/-Labeling Conjecture) For any graph G with   2,

2;1 .G/  2 . It is obvious that Conjecture 28 cannot hold for  D 1, as .K2 / D 1 but

2;1 .K2 / D 2. However, if G is a diameter 2 graph, then 2;1 .G/  2 . Besides, the upper bound is attainable by Moore graphs (diameter 2 graph with order 2 C1). Moreover, in [169], the authors proved that determining 2;1 .G/ of any graph G is NP-complete, and Bodlaender et al. [51] showed that the problem of finding the minimum number 2;1 for planar graphs, bipartite graphs, chordal graphs, and split graphs is also NP-complete. Fiala and Kratochvil [150] even proved that for every integer p  3, it is still NP-complete to decide whether a p-regular graph admits an L.2; 1/-labeling of span (at most) p C 2. For general graphs, in [166], Gonc¸alves constructed a labelling algorithm to get a little bit better upper bound: Theorem 163 For any graph G with   2, 2;1 .G/  2 C   2. Recently, Havet et al. [182] showed that Conjecture 28 is true for any graph G with sufficiently large , for example,   1069 . This is the first paper addressing the solution of Conjecture 28 for general graphs, although it does not completely solve it. More generally, they showed the following:

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Theorem 164 For any positive integer d , there exists a constant d such that every graph with   d has an L.d; 1/-labeling with span at most 2 C 1. This yields that, for each positive integer d , there is an integer Cd such that every graph has an L.d; 1/-labeling with span at most 2 C Cd . In the following, it is concerned with the L.2; 1/-labeling of some special graphs such as planar graphs, outerplanar graphs, K4 -minor-free graphs, trees, and chordal graphs. Some recent results will be displayed.

5.1.1 Planar Graphs For planar graphs, Bella et al. [47] showed that Conjecture 28 holds for all planar graphs G with   4. Kang [219] showed for every subcubic Hamiltonian graph G, 2;1 .G/  9, that is, Conjecture 28 holds for this type of graphs. Heuvel and McGuinness [189] proved that p;q .G/  .4q  2/ C 10p C 38q  23 for every planar graph G. It implies that 2;1 .G/  2 C 35 for a planar graph G. Molloy and Salavatipour [276] improved this result to p;q .G/  qd 35 e C 18p C 77q  18 for every planar graph G. It implies that 2;1 .G/  d 53 e C 95 for a planar graph G. By considering planar graphs without short cycles, Wang and Cai [351] showed that every planar graph G without 4-cycles has p;q .G/  minf.8q  4/ C 8p  6q 1; .2q 1/C10p C84q 47g. It means that 2;1 .G/  minf4C9; C57g for such a planar graph G. 5.1.2 Outerplanar Graphs For any outerplanar graph G, Bodlaender et al. [50] observed that at most  C 9 colors are sufficient to give an L.2; 1/-labeling for outerplanar graph and they suspected this bound is not tight. Later, Calamoneri and Petreschi [93] proved that

2;1 .G/  maxf10;  C 2g for any outerplanar graph G and 2;1 .G/  8 if  D 3. Their results imply that 2;1 .G/   C 6 for every outerplanar graph G and

2;1 .G/   C 2 if   8. By investigating the circular distance two labeling of graphs, Liu and Zhu [253] also proved independently that 2;1 .G/   C 6 for an outerplanar graph G and 2;1 .G/  C2 if   15. Wang and Luo [363] improved these results by showing the following: (1) 2;1 .G/   C 5 for any outerplanar graph G and (2) 2;1 .G/  7 for an outerplanar graph G with  D 3. In [50], the authors conjectured that every outerplanar graph G has 2;1 .G/   C 2. Calamoneri and Petreschi [93] disproved this conjecture by constructing an outerplanar graph G with  D 3 and 2;1 .G/ D 6 and further conjectured that every outerplanar graph G with   4 has 2;1 .G/   C 2. Unfortunately, Wang and Luo [363] constructed another example of outerplanar graph G with  D 4 and

2;1 .G/ D 7 to disprove their new conjecture. However, it is yet unknown if every outerplanar graph G with  2 f5; 6; 7g has 2;1 .G/   C 2. 5.1.3 K4 -Minor Free Graphs Wang and Wang [369] first studied the L.p; q/-labeling of K4 -minor-free graphs. Their result is as follows:

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Theorem 165 Let G be a K4 -minor-free graph. If p C q  3, then p;q .G/  2.2p  1/ C .2q  1/b 3 2 c  2. c C 4. Corollary 12 If G is a K4 -minor-free graph, then 2;1 .G/  b 3 2 It should be pointed out that Theorem 165 is a generalization of results in [251]. Corollary 12 implies that Conjecture 28 holds for all K4 -minor-free graphs. More recently, Kr´al and Nejedly [236] showed that there exists 0 such that for every K4 -minor-free graph G with   0 , p;q .G/  qb 3 c, which is best possible. 2

5.1.4 Trees and Chordal Graphs Griggs and Yeh [169] proved that every tree T satisfies  C 1  2;1 .T /   C 2. Chang and Kuo [98] presented a polynomial algorithm for determining the precise value of 2;1 .T / for any tree T . Wang [348] provided a sufficient condition for a tree T to have 2;1 .T / D  C 1. More precisely, he showed the following: Theorem 166 If a tree T contains no two vertices of maximum degree at distance either 1, 2, or 4, then 2;1 .T / D  C 1. Theorem 166 is the best possible in the sense that for each   3, there exists a tree T of maximum degree  such that 2;1 .T / D  C 2 and the distance between any two vertices of maximum degree is at least 4. Such examples of trees were constructed in [348]. To attack Conjecture 28, Sakai [308] considered the class of chordal graphs. She showed that 2;1 .G/  . C 3/2 =4 for any chordal graph G. For a unit interval graph G, which is a very special chordal graph, she also proved that 2.G/  2  3=2

2;1 .G/  2.G/. In [235], Kr´al showed that 2;1 .G/  p6 C O./ if G is a chordal graph. Besides, he conjectured the following: Conjecture 29 For a chordal graph G, 2;1 .G/ 

3=2 2 p 3 3

C O./  1.

For a detailed survey on L.2; 1/-labeling problem of graphs, see [92].

5.2

Coloring the Square of Graphs

The square G 2 of a graph G is the graph defined on the vertex set V .G/ such that two vertices u and v are adjacent in G 2 if and only if 1  d.u; v/  2 in G. A mapping is a k-coloring of G 2 if and only if .u/ ¤ .v/ whenever 1  d.u; v/  2 in G. Since G 2 contains a clique of order at least .G/ C 1, .G 2 /  .G/ C 1 for any graph G. If G is a tree, then .G 2 / D .G/ C 1. On the other hand, .G 2 /  2 .G/. Hence, .G 2 /  2 .G/ C 1 for any graph G. The 5-cycle and Petersen graph G satisfy .G 2 / D 2 .G/C1. By the definition of L.p; q/-labeling,

1;1 .G/ D .G 2 /  1. In 1977, Wegner [378] first investigated the chromatic number of the square of a planar graph. He proved that .G 2 /  8 for every planar graph G with  D 3 and

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Fig. 3 A planar graph with  D 2k that needs at least 3k C 1 D 32  C 1 colors

k vertices

k – 1 vertices

y

z

k vertices

conjectured that the upper bound could be reduced to 7, which has been confirmed by Thomassen [332]. Moreover, Montassier and Raspaud [281] proved that 7 colors are necessary by giving an example. For planar graphs of maximum degree   4, Wegner [378] proposed the following conjecture. Conjecture 30 Let G be a planar graph with maximum degree . Then  2

.G / 

 C 5; if 4    7; b 32 c C 1; if   8.

Wegner also gave some bounds for the case   7 and constructed planar graphs to attain the conjectured bounds. For   8 even, these examples are sketched below. The graph in Fig. 3 has maximum degree 2k, and yet all vertices except z are adjacent in its square. Hence, to color these 3k C1 vertices, at least 3k C1 D 32 C1 colors is required. Many upper bounds on .G 2 / for planar graphs in terms of  have been obtained in the past 15 years. In [6], Agnarsson and Halldorsson showed the following: Theorem 167 For every planar graph G with maximum degree   749, .G 2 /  b 95 c C 2. A similar result was obtained by Borodin et al. [60]. Theorem 168 For every planar graph G with maximum degree   47, .G 2 /  d 95 e C 1. Nearly at the same time, Heuvel and McGuinness [189] gave a general upper bound without maximum degree restriction.

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Theorem 169 For every planar graph G with maximum degree , .G 2 /  2 C 25. The currently best known upper bound on this parameter for planar graphs was obtained by Molloy and Salavatipour [276]. Theorem 170 For every planar graph G with maximum degree , .G 2 /  d 53 e C 78. Also, Molloy and Salavatipour [276] showed that if G is a planar graph with maximum degree   241, then .G 2 /  d 53 e C 25. Lih, Wang, and Zhu [251] studied the coloring of the squares of K4 -minor-free graphs. More precisely, they proved the following: Theorem 171 Let G be a K4 -minor-free graph with maximum degree . Then   C 3; if 2    3; 2 .G /  b 32 c C 1; if   4. The upper bound in Theorem 171 is tight. For  D 2, it is easy to obtained that .C5 / D 2 and .C52 / D 5. For  D 3, let G be the graph consisting of three internally disjoint paths joining two vertices x and y, where two of the paths are of length 2 and the third one is of length 3. Then  D 3 and .G 2 / D .K6 / D 6. For  D 2k  4, let G2k be the graph consisting of k internally disjoint paths joining x and y, k internally disjoint paths joining x and z, and k internally disjoint paths joining y and z. All these paths are of length 2, except one path joining x and y is of length 1, and one path joining x and z is of length 1. Then .G2k / D 2k and 2 .G2k / D .K3kC1 / D 3k C 1. For  D 2k C 1  5, let G2kC1 be obtained from G2k by adding a new path of length 2 joining y and z. Then .G2kC1 / D 2k C 1 2 and .G2kC1 / D .G3kC2 / D 3k C 2. In 2001, Kostochka and Woodall [234] investigated the list coloring of square of graphs and raised the following conjecture: Conjecture 31 For every graph G, .G 2 / D l .G 2 /. If true, this conjecture (together with Thomassen’s result [332]) implies that every planar graph G with  D 3 satisfies l .G 2 /  7. Recently, Cranston and Kim [119] proved that every connected graph (not necessarily planar) with  D 3 other than the Petersen graph satisfies l .G 2 /  8 (and this is best possible by the Wagner graph). In addition, Cranston and Kim [119] showed that if G is a planar graph with ˇ  D 3 and girth g  7, then l .G 2 /  7. Dvoˇra´ k, Skrekovski, and Tancer showed that if G is a planar graph with  D 3 and g  10, then l .G 2 /  6, which has been improved to g  9 in [119]. For planar graphs G with   8, Conjecture 31 leads directly to the following conjecture:

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Conjecture 32 For every planar graph G with   8, l .G 2 /  b 3 2 c C 1. Havet et al. [181] announced the following theorem. Theorem 172 The square of every planar graph G of maximum degree  has list chromatic number at most .1 C o.1// 23 . Moreover, given lists of this size, there is a proper coloring in which the colors on every pair of adjacent vertices of G differ 1 by at least  4 . The o.1/ term here is as  ! 1. The first-order term 32  is essentially best 1 possible; see the example described above. On the other hand, the term  4 is probably far from best possible; it was chosen to keep the proof simple. In what follows, it is going to study the coloring of squares of some special graphs such as hypercubes, outerplanar graphs, chordal graphs, and planar graphs with high girth.

5.2.1 Planar Graphs with High Girth By investigating the L.p; q/-labeling of planar graphs, Wang and Lih [360] obtained a general result of planar graphs with high girth. Theorem 173 Let G be a planar graph with maximum degree  and girth g. Then 8 < .2q  1/ C 4p C 4q  4; if g  7I

p;q .G/  .2q  1/ C 6p C 12q  9; if g  6I : .2q  1/ C 6p C 24q  15; if g  5: As it is mentioned at the beginning of this section, every proper coloring of the square of a graph is exactly an L.1; 1/-labeling of the graph. So Theorem 173 implies the following: Corollary 13 Let G be a planar graph with maximum degree  and girth g. Then 8 <  C 5; if g  7I .G 2 /   C 10; if g  6I :  C 16; if g  5: 8 <  C 8; if g  7I

2;1 .G/   C 15; if g  6I :  C 21; if g  5: c C 1   C 5 when   8, b 3 c C 1   C 10 when   18, Note that b 3 2 2 3 and b 2 c C 1   C 16 when   30. Thus, the following result is an immediate consequence of Corollary 13. Corollary 14 Conjecture 30 holds for planar graphs G with g  7, or g D 6 and   18, or g D 5 and   30.

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Borodin et al. [63] proved that if G is a planar graph with maximum degree  and girth at least 7, then .G 2 / D  C 1 whenever   30. On the other hand, there are planar graphs G with girth at most 6, arbitrarily large maximum degree , and .G 2 / >  C 1. Recently, Dvoˇra´ k et al. [134] discussed the coloring of the square of a planar graph with girth 6. Their result is as follows: Theorem 174 Let G be a planar graph with maximum degree   8821 and girth g  6. Then .G 2 /   C 2. This result has been improved and extended by Borodin and Ivanova [69, 70]. Theorem 175 ([69]) Let G be a planar graph with maximum degree   18 and girth g  6. Then .G 2 /   C 2. Theorem 176 ([70]) Let G be a planar graph with maximum degree   36 and girth g  6. Then l .G 2 /   C 2. Borodin et al. [73] gave a sufficient condition for planar graphs with given maximum degree and girth with respect to the list coloring. Theorem 177 For every planar graph G with maximum degree  and girth g, l .G 2 / D  C 1 in each of the following cases: .1/  D 3 and g  24:

.2/  D 4 and g  15: .3/  D 5 and g  13:

.4/  D 6 and g  12:

.5/   7 and g  11: .6/   9 and g D 10:

.7/   15 and g D 8:

.8/   30 and g D 7:

5.2.2 Outerplanar Graphs For the class of outerplanar graphs, Lih and Wang [250] proved the following result: Theorem 178 Let G be an outerplanar graph with maximum degree . Then 1. .G 2 /   C 2 if   3. 2. .G 2 / D  C 1 if   7. An alternative proof of Theorem 178 was given in [361]. Moreover, Calamoneri and Petreschi [93] also obtained the same result through an algorithmic aspect. Wang and Luo [363] extended the result of Theorem 178 to the case  D 6. That is, they proved the following: Theorem 179 If G is an outerplanar graph with  D 6, then .G 2 / D 7. Theorem 179 was dependently obtained by Aganarsson and Halld´orsson [7]. Moreover, it was shown in [7] that there exist infinitely many biconnected outerplanar graphs G with maximum degree  D 5 and .G 2 / D 7 D  C 2. This shows that Theorems 178 and 179 are the best possible with respect to the requirement of maximum degree.

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An outerplanar graph G is said to be maximal if G C xy is not outerplanar for any two nonadjacent vertices x and y. Let Q denote a maximal outerplanar graph obtained by adding three chords y1 y3 ; y3 y5 ; y5 y1 to a 6-cycle y1 y2 : : : y6 y1 . Luo [262] characterized completely the chromatic number of the square of a maximal outerplanar graph. More precisely, she showed the following: Theorem 180 Let G be a maximal outerplanar graph with maximum degree . Then  C 1  .G 2 /   C 2; and .G 2 / D  C 2 if and only if G is isomorphic to Q.

5.2.3 Other Graphs The coloring of the square of hypercube Qd has been studied by Wan in [346]. Such coloring problem is known to be related to the conflict-free channel set assignment for the hypercube cluster interconnection topology. According to the definition of hypercube, the vertices with at most one coordinate equal to form a clique of size d C1 in Qd2 ; hence, .Qd2 /  d C2. An independent set in Qd2 is precisely an errorcorrecting code. So, a proper coloring of this graph may be viewed as a covering of the hypercube with codes. Wan [346] constructed a coloring of Qd2 using 2blog2 d cC1 colors. Furthermore, he conjectured that this is optimal. Note that Qd2 contains a subgraph isomorphic to Qd2 1 . Thus, it suffices to prove the conjecture when d is a power of 2. The cases d D 1; 2; 4 are rather easy to verify. However, this conjecture is false, since it was proved by Royle that 13  .Q82 /  14, see [212]. For a chordal graph G with p maximum degree , Kr´al [235] showed that the kth power G k of G is O. k.kC1/=2 /-degenerate for even values of k and O..kC1/=2 /-degenerate for odd values. In particular, this bounds the chromatic number .G k / of the kth power of G. The bound proven for odd values of k is the best possible. More generally, the author in [235] showed that p;q .G/  3=2 3=2 p p c.2q  1/ C .2p  1/, which implies that .G 2 /  b .C1/ c C  C 1. b .C1/ 6 6 3=2 On the other hand, a construction of such graphs with p;q .G/  . q C p/ was found.

5.3

Backbone Coloring

Let G be a graph and H a spanning subgraph of G. For an integer  2, a backbone-k-coloring of .G; H / is a mapping f : V .G/ ! f1; 2; : : : ; kg such that jf .u/  f .v/j  if uv 2 E.H / and jf .u/  f .v/j  1 if uv 2 E.G/nE.H /. The

-backbone chromatic number, denoted by  .G; H / or BBC .G; H /, of .G; H / is the smallest integer k such that .G; H / has a -backbone-k-coloring. In particular, when D 2, it is called, for short, the backbone coloring and the backbone chromatic number, denoted by b .G; H / or BBC.G; H /. The backbone coloring of graphs was introduced in 2003 by Broersma et al. [81] and later was investigated by some authors; see [82, 83, 270, 271, 350]. The following result is an easy, useful observation:

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Theorem 181 If H is a spanning subgraph of G, then b .G; H /  2.G/  1. Some better upper bounds for b .G; H / can be derived from Theorem 181: (i) If G is a planar graph, then .G/  4 by the Four-Color Theorem, and hence, b .G; H /  7. (ii) If G is a bipartite graph, then .G/  2, and hence, b .G; H /  3. (iii) If G is a triangle-free planar graph, then .G/  3 by [170], and hence, b .G; H /  5. (iv) If G is a k-degenerate graph, then .G/  kC1, and thus, b .G; H /  2kC1. In particular, if G is a K4 -minor-free graph (especially, outerplanar graph and series-parallel graph), then G is 2-degenerate, and therefore, b .G; H /  5. It is easy to verify that if G is a connected non-bipartite graph and H is a connected spanning subgraph of G, then b .G; H /  4. Moreover, if G is a connected graph with at least two vertices and H is a connected spanning subgraph of G, then b .G; H / D 3 if and only if G is a bipartite graph. Now, consider the relation between the backbone chromatic number and ordinary chromatic number of a graph. How far away from .G/ to  .G; H / in the worst case? To answer this question, some symbols are needed. Suppose that k is a positive integer. Set T .k/ D maxf .G; T / W G is a graph with a spanning tree T , and .G/ D kg. P .k/ D maxf .G; P / W G is a graph with a Hamiltonian path P , and .G/ D kg. M .k/ D maxf .G; M / W G is a graph with a perfect matching M , and .G/ D kg. S .k/ D maxf .G; S / W G is a graph with disjoint spanning stars S , and .G/ D kg. If D 2, simply write T .k/ for T2 .k/, P .k/ for P2 .k/, M.k/ for M2 .k/, and S.k/ for S2 .k/. Broersma et al. [81, 82] determined the exact values of the first three parameters T .k/, P .k/, and M.k/. More precisely, their results are described in Theorems 182–184. Theorem 182 T .k/ D 2k  1 for all k  1. Theorem 183 For k  1, P .k/ takes the following values: (a) If 1  k  4, then P .k/ D 2k  1. (b) P .5/ D 8 and P .6/ D 10. (c) If k  7 and k D 4t, then P .4t/ D 6t. (d) If k  7 and k D 4t C 1, then P .4t C 1/ D 6t C 1. (e) If k  7 and k D 4t C 2, then P .4t C 2/ D 6t C 3. ( f) If k  7 and k D 4t C 3, then P .4t C 3/ D 6t C 5. Theorem 184 For k  1, M.k/ takes the following values: .a/ M.4/ D 6. .b/ If k D 3t, then M.3t/ D 4t.

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.c/ If k D 4 and k D 3t C 1, then M.3t C 1/ D 4t C 1. .d / If k D 3t C 2, then M.3t C 2/ D 4t C 3. For a general problem, that is, may be equal to or greater than 2, Broersma et al. [83] established the following two results: Theorem 185 For  2, M .k/ takes the following values: 8 ˆ if ˆ C k  1; ˆ ˆ ˆ 2k  2; if ˆ ˆ < 2k  3; if M .k/ D ˆ 2t ; if ˆ ˆ ˆ ˆ 2t C 2c  1; if ˆ ˆ : 2t C 2c  2; if

2  k  I

C 1  k  2 I k D 2 C 1I k D t. C 1/ with t  2I k D t. C 1/ C c with t  2 and 1  c  C3 2 I  c 

: k D t. C 1/ C c with t  2 and C3 2

Theorem 186 For  2, S .k/ takes the following values: .a/ S .2/ D C 1, S2 .3/ D 5, and S2 .4/ D 6. .b/ If 3  k  2  3, then S .k/ D d 3k 2 e C  2. .c/ If 2  2  k  2  1 with  3, then S .k/ D k C 2  2. .d / If k D 2 with  3, then S .k/ D 2k  1. .e/ If k  2 C 1, then S .k/ D 2k  d k e. Concerning the computational complexity of backbone coloring, Broersma et al. [81, 83] gave the following two theorems: Theorem 187 (a) The following problem is polynomially solvable for any l  4: Given a graph G and a spanning tree T , decide whether b .G; T /  l. (b) The following problem is NP-complete for all l  5: Given a graph G and a Hamiltonian path P , decide whether b .G; T /  l. Theorem 188 .a/ The following problem is polynomially solvable for any l  C 1: Given a graph G and a star backbone S , decide whether  .G; S /  l. .b/ The following problem is NP-complete for all l  C 2: Given a graph G and a star backbone S , decide whether  .G; S /  l.

5.3.1 Split Graphs A split graph is a graph whose vertex set can be partitioned into a clique and an independent set, with possibly edges in between. Recall that the size of a largest clique in G is denoted by !.G/. It is known that a split graph G is perfect, and hence, .G/ D !.G/. In [81], the authors proved the following: Theorem 189 Let G be a split graph: .a/ For every spanning tree T in G, b .G; T /  .G/ C 2. .b/ If !.G/ ¤ 3, then for every Hamiltonian path P in G, b .G; P /  .G/ C 1. Furthermore, they showed that both bounds in Theorem 189 are tight. Explain here why for split graphs with clique number 3 the statement in Theorem 189 does

On Coloring Problems Fig. 4 A split graph G with a Hamiltonian path P such that !.G/ D 3 and b .G; P / D 5

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w

b

u

v

a

not work. Consider the split graph G on five vertices from Fig. 4. Vertices v; u; w form a clique, vertex a is adjacent to v, u, and vertex b is adjacent to u, w. The clique number (and hence the chromatic number) of this graph is equal to 3. Let P D avuwb be a Hamiltonian path. Then b .G; P / > .G/ C 1 D 4. Assume to the contrary that .G; P / has a backbone coloring with colors 1; 2; 3, and 4. It is easy to see that u cannot be colored with color 2 or 3; otherwise, v and w are forced to be colored with the same color, an obvious contradiction. Now suppose that u is colored with color 1 (the case when u is colored with color 4 is similar). Then one of its neighbors in P must have color 3 and the other one color 4. Without loss of generality, assume that v has color 3. Vertex a is adjacent in P to v, so the colors 2,3, and 4 are forbidden for a and the only valid color for a is 1. But a is adjacent in G to u which has color 1 as well. This contradiction completes the proof of the claim. In 2006, Salman [309] investigated the -backbone colorings of split graphs with a spanning tree. He obtained the following three theorems, where all the upper bounds are tight. Theorem 190 Let  2 and let G be a split graph. For every tree backbone T of G, 8 if .G/ D 1I < 1;  .G; T /  1 C ; if .G/ D 2I : .G/ C ; if .G/  3: Theorem 191 Let  2 and let G be a split graph with .G/ D k  2. For every matching backbone M of G, 8 ˆ

C 1; if k ˆ ˆ ˆ ˆ if k < k C 1;  .G; M /  k C 2; if k ˆ k ˆ ˆ e C

; if k d ˆ ˆ : k2 d 2 e C C 1; if k

D 2I  4 and  minf k2 ; kC5 3 gI   d k2 eI D 9 or k  11 and kC6 3 k D 3; 5; 7 and  d 2 eI D 4; 6 or k  8 and  d k2 e C 1:

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Theorem 192 Let  2 and let G be a split graph with .G/ D k  2. For every star backbone S of G,   .G; S / 

k C ; if either k D 3 and  2 or k  4 and D 2I k C C 1; otherwise:

5.3.2 Halin Graphs Suppose that T is a tree with no vertex of degree 2 and at least one vertex of degree 3 or more. A Halin graph is a plane graph G D T [ C , where C is a cycle connecting the leaves, the vertices of degree 1, of T in the cyclic order determined by the drawing of T . Vertices of C are called outer vertices of G, and vertices in V .G/ n V .C / are called inner vertices of G. A Halin graph G is called a wheel if G contains only one inner vertex. When  D 3, G is 3-regular. In particular, K4 is a 3-regular Halin graph. It is easy to see that every Halin graph is 3-connected. It is easy to show that for a Halin graph G, .G/ D 4 if G is a wheel of order even and .G/ D 3 otherwise. Wang et al. [350] discussed the backbone coloring of Halin graphs and got the following result. Theorem 193 Let G D T [ C be a Halin graph with X [ Y as a bipartition of T . Then 1. 4  b .G; T /  5. 2. b .G; T / D 5 if and only if jM.T /j is odd, and M.T /  X or M.T /  Y , where M.T / denotes the set of all leaves of T . An almost binary tree is a tree T with  D 3 and jT j  4 that contains no 2 vertices. Let  denote the set of almost binary trees T with the properties: (1) jM.T /j is odd and (2) M.T /  X or M.T /  Y , where X [ Y is a bipartition of T . Let A denote the graph obtained from a path uxrys by gluing a leaf v at x and a leaf t at y. The vertex r is called the root of A. Given an almost binary tree T with a leaf z, A is attached to T at z if z is identical to the root r of A. And A is an end subgraph of T if A  T and u; v; s; t are leaves of T and r; x; y are of degree 3 in T . Removing an end subgraph A from T means that six vertices x; y; u; v; s; t are removed from T . Let ƒ0 denote the set of graphs G0 ; G1 ; G2 ; : : :, where G0 D K1;3 , and Gi is obtained by consecutively attaching two copies of A to the graph Gi 1 for i D 1; 2; : : :. In [350], the authors first showed the equality that ƒ0 D ƒ and afterward gave the following interesting characterization: Theorem 194 Let G D T [C be a 3-regular Halin graph. Then 4  b .G; T /  5; and b .G; T / D 5 if and only if T 2 ƒ0 , that is, T can be obtained by consecutively attaching 2k, k  0, copies of the graph A to the graph K1;3 . Notice that the backbone spanning tree T taken in Theorems 193 and 194 is just the tree in the definition of Halin graph G D C [ T . What happens when picking

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up other spanning tree as backbone in a Halin graph? The authors in [350] posed the following problem: Problem 7 Given a spanning tree T  , different from T , in a Halin graph G D T [ C , determine b .G; T  /. It seems very difficult to settle Problem 7. The following is a partial solution by considering a class of special spanning trees [350]. For a Halin graph G D T [ C , take e 0 D u0 v0 2 E.T / with u0 2 M.T / and 00 e D x 00 y 00 2 E.C /. So, x 00 ; y 00 2 M.T /. Let T 0 D T  M.T /. Define a spanning tree T  of G as follows: T  D T 0 [ .C nfe 00 g/ [ fe 0 g: Theorem 195 Let G D T [ C be a Halin graph. Let T  be a spanning tree of G defined as above. If G satisfies one of the following conditions, then b .G; T  / D 4: 1. jM.T /j is even. 2. e 0 is adjacent to e 00 in G.

5.3.3 Other Related Results In [81], the authors asked the following: Is there a constant c such that b .G; T /  .G/ C c for every triangle-free graph G with T as a backbone tree? Miˇskuf et al. [270] answered negatively this question. More precisely, they proved the following: Theorem 196 For every n 2 N , there exist a (finite) triangle-free graph Rn and its spanning tree Tn such that b .Rn ; Tn / D 2.Rn /  1 D 2n  1. Wang et al. [350] considered the backbone coloring of graphs with small maximum average degree. Their result is stated as follows: Theorem 197 If G is a connected graph with mad.G/ < spanning tree T of G such that b .G; T /  4.

5 , 2

then there exists a

Corollary 15 If G is a connected non-bipartite graph with mad.G/ < 52 , then there is a spanning tree T such that b .G; T / D 4. Note that every planar graph G with girth g  10 satisfies mad.G/ < Proposition 2. This fact, together with Corollary 15, gives the following fact:

5 2

by

Corollary 16 If G is a connected non-bipartite planar graph with girth at least 10, then there is a spanning tree T such that b .G; T / D 4. Let  denote the smallest integer k such that for every non-bipartite planar graph G with girth at least k, there exists a spanning tree T of G such that b .G; T / D 4. By virtue of Corollary 16,   10. What is the exact value of ? Bu and Zhang [87] partially answered this question:

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Fig. 5 A planar graph G  with a spanning tree T  such that b .G  ; T  / D 6

Theorem 198 If G is a connected non-bipartite planar graph without 4-cycles, then there exists a spanning tree T of G such that b .G; T / D 4. The following result was given in [350]: Theorem 199 There exist infinitely many graphs G with   3 that contain a Hamiltonian path P such that b .G; P / D 5.

5.3.4 Some Open Problems The Four-Color Theorem implies that b .G; T /  7 for any planar graph G with an arbitrary spanning tree T . However, this bound 7 is probably not best possible. Can it be improved to 6? The planar graph G  in Fig. 5, see [82], demonstrates that this bound cannot be improved to 5. Note that the graph G  consists of four copies of K4 , each having a K1;3 as a spanning tree. In any backbone coloring of such a K4 with only five colors, its central vertex must receive color 1 or 5. With this observation, it is easy to see that b .G  ; T  /  6. Problem 8 Is b .G; T /  6 for any planar graph G with a spanning tree T ? How about the Hamiltonian path case? In this case, it is natural to ask whether the bound 7 is best possible. It can be shown that one cannot improve this bound to 5. Let G2 be the planar graph from Fig. 6 in which the Hamiltonian path P is indicated by the bold edges. To prove that b .G2 ; P /  6, the following claim is needed. Let G1 be the subgraph of G2 with the indicated Hamiltonian path P1 as shown in Fig. 6.

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v1

v2

v3

v4

v5

G1

u1

u2

u3

u4

u5

G2

Fig. 6 A planar graph G2 with a Hamiltonian path P such that b .G2 ; P / D 6

Claim 1 In any backbone coloring of G1 with colors f1; 2; 3; 4; 5g, at least one vertex has color 3. Proof Suppose that there exists a backbone coloring c for .G1 ; P1 / with colors 1, 2, 4, and 5. By symmetry, assume that c.v1 / 2 f1; 2g. Then, it is obvious that c.v2 / 2 f4; 5g, c.v3 / 2 f1; 2g, and c.v4 / 2 f4; 5g. Hence, it follows that c.v4 /, c.v5 / 2 f4; 5g, contradicting the fact that v1 , v3 , and v5 induce a K3 in G1 .  Now suppose that there exists a backbone coloring for .G2 ; P / with colors f1; 2; 3; 4; 5g. Then Claim 1 implies that there is a vertex with color 3 among the neighbors of u1 , hence, u1 cannot receive color 3. Similarly, Claim 1 implies that u2 , u3 , u4 , and u5 cannot receive color 3. But the graph induced by fu1 ; u2 ; u3 ; u4 ; u5 g is isomorphic to G1 , contradicting Claim 1. Problem 9 Is b .G; P /  6 for any planar graph G with a Hamiltonian path P ? What about the perfect matching case? It was shown in [50] that b .G; M /  6 for any planar graph G with a perfect matching M . This bound 6 seems to be not best possible. In order to show that this bound cannot be reduced to 4, the authors in [50] constructed an example of planar graph G3 , as shown in Fig. 7, with an 0 0 0 0 indicated perfect matching M D fab ; bc ; cd ; da g. The detailed argument is here omitted. Problem 10 Is b .G; M /  5 for any planar graph G with a perfect matching M ?

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a⬘

Fig. 7 A graph G3 with a perfect matching M such that b .G3 ; M / D 5

d

c⬘

b⬘

a d⬘ b

5.4

c

Injective Coloring

The injective coloring of a graph was first introduced by Hahn et al. in [188]. A vertex coloring of a graph G is called injective if any two vertices having a common neighbor get different colors. Let i .G/ be the minimum number of colors in injective colorings of G. The injective coloring is originated in complexity theory and is used in coding theory [188]. Note that an injective coloring is not necessarily a proper coloring and vice versa. It is easy to see that if G is a triangle-free graph, then the injective coloring is just the L.0; 1/-labeling of the graph G. The common neighbor graph Gcn of a graph G is the graph defined by V .Gcn / D V .G/ and E.Gcn / D fuvj there is a path of length 2 in G joining u and vg: By the definition of L.p; q/-labeling, the following relations among several parameters, that is, i .G/, .G 2 /, and .Gcn /, can be easily deduced: (i) i .G/ D .Gcn /. (ii) If G is a triangle-free graph, then 0;1 .G/ D i .G/  1. (iii)   i .G/  .G 2 /  jGj. Let H denote the prism, which is a graph obtained by joining three chords u1 u5 ; u2 u4 ; u3 u6 to a 6-cycle u1 u2 : : : u6 u1 . It is easy to inspect that H 2 is K6 , and Hcn is K6  M , where M is a perfect matching of H .

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By (iii), i .H /   D 3. On the other hand, an injective 3-coloring c is easily constructed as follows: c.u1 / D c.u2 / D 1, c.u3 / D c.u6 / D 2, and c.u4 / D c.u5 / D 3. Thus, i .H / D 3. Moreover, it is easy to derive that i .H / D .Hcn / D 3 < 6 D .H 2 /. This example shows that there exists a graph G such that .G 2 / D 2i .G/. In fact, it was proved in [227] that .G 2 /  2i .G/ for any graph G. The authors in [188] considered the injective chromatic number of some special graphs such as complete graphs, paths, cycles, and stars. Moreover, they gave the following results: Proposition 4 Let G be a graph with maximum degree : 1. If G is d -regular and if i .G/ D d , then d divides jGj. 2. If G is connected and not K2 , then .G/  i .G/. 3. If G has the diameter 2 and the independence number ˛, then i .G/  ˛. 4. i .G/  .  1/ C 1. Theorem 200 Let Qn be the hypercube of dimension n: 1. i .Qn / D n if and only if n is a power of 2. 2. i .Qn /  2n  2. 3. i .Q2nC1 /  2i .QnC1 /. 4. i .Q2m j / D 2m for 0  j  3. The algorithmic aspect of the injective coloring for some special graphs was also considered in [188]: Theorem 201 It is NP-complete for a given split (and hence chordal) graph G and an integer k, to decide whether the injective chromatic number of G is at most k. Theorem 202 It is NP-complete to decide, for a given split (and hence chordal) graph G, whether i .G/ D .G 2 /  1. Theorem 203 The injective chromatic number of a power chordal graph can be computed in polynomial time. Bu et al. [84] studied the injective chromatic number of planar graphs having large maximum degree and girth. Theorem 204 Let G be a planar graph with maximum degree  and girth g: 1. i .G/ D  if either g  20 and   3 or g  7 and   71. 2. i .G/   C 1 if g  11. 3. i .G/   C 2 if g  8. Luˇzar et al. [264] improved some results in [84] by showing the following: Theorem 205 Let G be a planar graph with maximum degree  and girth g: 1. i .G/   if g  19 and   3. 2. i .G/   C 1 if g  10. 3. i .G/   C 4 if g  5 and   3.

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Instead of studying planar graphs with high girth, Doyon et al. [133] considered the injective chromatic number of graphs with small maximum average degree. Their result is the following theorem: Theorem 206 Let G be a graph with maximum degree : 1. If mad.G/ < 14 5 , then i .G/   C 3. 2. If mad.G/ < 3, then i .G/   C 4. 3. If mad.G/ < 10 3 , then i .G/   C 8. 2g Again, applying the relation mad.G/ < g2 for a planar graph G with girth g, the following results hold: If G is a planar graph, then i .G/   C 3 if g  7, i .G/   C 4 if g  6, and i .G/   C 8 if g  5. In [120, 121], Cranston et al. improved the upper bounds given in [133, 264] in certain cases. More specifically, they studied sufficient conditions to imply i .G/ D  and i .G/   C 1.

Theorem 207 ([121]) Let G be a graph with maximum degree : 1. If mad.G/ < 14 5 and   4, then i .G/   C 2. 2. If mad.G/ < 36 and  D 3, then i .G/  5. 13 Furthermore, an example to show that the maximum average degree in the second conclusion of Theorem 207 is sharp was constructed in [121]. Chen et al. [104] obtained some upper bounds of the injective chromatic numbers for some graphs, that is, planar graphs, K4 -minor-free graphs, etc. The following three theorems were established in [104]. Theorem 208 Let G be a K4 -minor-free graph with   1. Then i .G/  d 32 e: Theorem 208 is best possible in the sense that there exist K4 -minor-free graphs G such that i .G/ D d 32 e, where  is even. Such an example was constructed in [104]. However, it is unknown if, for each given odd integer k  3, there exists a K4 -minor-free graph G with  D k and i .G/ D 3kC1 . 2 Theorem 209 If G is a planar graph with   3, then i .G/  2  . It follows from Theorem 209 that every planar graph G with   3 has i .G/  6. Some examples of planar subcubic graphs G with i .G/ D 5 have been constructed by different authors (in private communications). Figure 8 is such a planar cubic graph example, which has ten vertices. Theorem 210 Let G be a connected graph with   3. Then i .G/ D 2   C 1 if and only if Gcn D 2K.1/C1. By Theorem 210, the following consequences hold automatically: (i) If G is a graph with   4, then i .G/  13. Moreover, i .G/ D 13 if and only if Gcn D 2K13 . (ii) If G is a graph with   3, then i .G/  7. Moreover, i .G/ D 7 if and only if G is isomorphic to the Heawood graph.

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Fig. 8 A planar graph G with  D 3 and i .G/ D 5

Fig. 9 Graph G 0 with g.G 0 / D 6 and i .G 0 / D  C 1 for   2

In the end of the paper [265], the authors put forward the challenging problem: Conjecture 33 Let G be a planar graph with maximum degree . Then (a) i .G/  5, if  D 3. (b) i .G/   C 5, if 4    7. (c) i .G/  b 32 c C 1, if   8. In 2011, Borodin and Ivanova [72] first investigated the list injective colorings of planar graphs. A graph G is said to be injectively k-choosable if any list L of size k allows an injective coloring ' such that '.v/ 2 L.v/ whenever v 2 V .G/. The injective choice number of G, denoted by li .G/, is the least integer k such that G is injectively k-choosable. Clearly, li .G/  i .G/   for every graph G. Theorem 211 ([72]) If G is a planar graph with maximum degree  and girth g, then li .G/ D i .G/ D  in each of the following cases: 1.   16 and g D 7. 2.   10 and 8  g  9. 3.   6 and 10  g  11. 4.  D 5 and g  12. Observe that a cycle C4n1 has i .C4n1 / D .C4n1 / C 1 D 3, whereas the graph G 0 in Fig. 9 has girth g D 6 and i .G/ D  C 1 for arbitrarily large . It means that the assumption on g D 7 of Theorem 211 (1) is tight. Theorem 212 ([72]) Every planar graph G with   24 and g  6 has li .G/   C 1. Furthermore, Borodin and Ivanova [72] posed the following problem. Problem 11 Find a precise upper bound for the injective choice number of planar graphs with given girth and maximum degree.

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.d; 1/-Total Coloring

This section is devoted to the .d; 1/-total coloring problem of graphs. Recall that a total k-coloring of a graph G is a coloring of V .G/ [ E.G/ using k colors such that no two adjacent or incident elements in V .G/ [ E.G/ receive the same color. The total chromatic number 00 .G/ of G is the smallest integer k such that G has a total k-coloring. By the definition, 00 .G/   C 1 for any graph G. Behzad [46] and Vizing [340] posed independently the famous conjecture, named as the Total Coloring Conjecture: Conjecture 34 (Total Coloring Conjecture) For any simple graph G, 00 .G/   C 2. Conjecture 34 remains still open. For the case  D 3, it was verified by Rosenfeld [306] and independently by Vijayaditya [338]. Later, Kostochka [230–232] gradually settled the case 4    5. For planar graphs G, Conjecture 34 was confirmed by Borodin [56] for   9, then by Yap [387] for  D 8, and later by Sanders and Zhao for  D 7 [311]. However, it remains open for planar graphs with  D 6. Recently, the total coloring of planar graphs with certain restrictions, that is, without some special short cycles or with given maximum degree, attracts much attention and has been deeply investigated in literatures [77,204,315,316,349,374]. Whittlesey et al. [381] investigated the L.2; 1/-labeling of incidence graphs. The incidence graph, denoted by GI , of a graph G is the graph obtained from G by replacing each edge by a path of length 2. The L.2; 1/-labeling of the incidence graph of G is equivalent to an assignment of integers to each element of V .G/ [ E.G/ such that adjacent vertices have different labels, adjacent edges have different labels, and incident vertex and edge have the difference of labels by at least 2. Let d be a positive integer and G be a simple graph. A .d; 1/-total labeling of G is an integer-valued function f defined on the set V .G/ [ E.G/ such that 8 < 1; if verticse x and y are adjacent in G; jf .x/  f .y/j  1; if edges x and y are adjacent in G; : d; if vertex x and edge y are incident. Notice that jf .x/  f .y/j for adjacent elements x and y may be required to be greater than or equal to p, instead of 1, in the above, defining inequality for some given positive integer p to obtain a more general notion of .d; p/-total labeling. A .d; 1/-total labeling taking values in the set f0; 1; : : : ; kg is called a .d; 1/-k-total labeling. The span of a .d; 1/-total labeling is the maximum difference between two labels. The minimum span, that is, the minimum k, among all .d; 1/-k-total labelings of G, denoted by Td .G/, is called the .d; 1/-total number of G.

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It is easy to observe that a .1; 1/-total labeling is a total coloring of G and that

T1 .G/ D 00 .G/  1. Moreover, Td .G/ D d;1 .GI /, where GI is the incidence graph of G. The concept of .d; 1/-total labeling was firstly introduced and investigated by Havet and Yu [184, 185]. They proposed the following conjecture: Conjecture 35 (.d; 1/-Total Labeling Conjecture) For any graph G and any integer d  1, Td .G/  minf C 2d  1; 2 C d  1g. Notice that when d D 1, Conjecture 35 is equivalent to Conjecture 34. By generalizing a result of [191], Havet and Yu in [185] showed that Td .G/  2  2 log. C 2/ C 2 log.16d  8/ C d  1 for any graph G. Moreover, Havet and Thomass´e [183] showed that determining .d; 1/-total number of graphs is very difficult, even for bipartite graphs. For instance, if   2d  4, or if 2d  1    d C 2  5, then the -bipartite .d; 1/-total labeling problem is NP-complete; if d  3, then the .d C 1/-bipartite .d; 1/-total labeling problem is NP-complete.

5.5.1 General Bounds Observing the label of a vertex with maximum degree and its incident edges, it is easy to see that Td .G/   C d  1. This lower bound may be increased in some cases. In [185], the authors proved that if G is -regular, or if d  , then

Td .G/  Cd . For upper bound, the following obvious fact was indicated in [185]: Proposition 5 Let G be a graph. Then 1. Td .G/  .G/ C 0 .G/ C d  2. 2. Td .G/  2 C d  1. Since 0 .G/ D  for a bipartite graph G (see [228]), it follows from Proposition 5(1) that every bipartite graph G satisfies Cd 1  Td .G/  Cd . In particular, if d D 2, then  C 1  Td .G/   C 2. In [267], McDiarmid and Reed proved that if G is a graph with n vertices and k is an integer such that kŠ > n, then 00 .G/  0 .G/ C k C 1. A slight modification of the proof can be applied to the following extension [185]: Theorem 213 If G is a graph with n vertices and k is an integer with

kŠ .2p1/k 0

> n,

then Td .G/  0 .G/ C k C 3d  3. Hence, as n ! 1, Td .G/   .G/ C log n O. log.log n/ /. It was proved in [191] that if  is large enough, then 00 .G/   C O.log10 /. This result was also extended to the .d; 1/-total labeling situation [185]: Theorem 214 There exists a 0 such that for each graph G with   0 , Td .G/   C 2 log10  C 3d  2.

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Molloy and Reed [274] affirmed that there is a constant c so that the total chromatic number of a graph G is at most  C c as long as  is sufficiently large, where c  1026 . It is possible that a similar proof leads to an analogous theorem for .d; 1/-total labeling but with a larger constant. The following bound, which slightly improves the known upper bound 2 C d  1, also appeared in [185]. Theorem 215 For any graph G and an integer d  1, Td .G/  2  2log . C 2/ C 2log.16d  8/ C d  1: Employing a probabilistic discussion, Havet and Yu [185] showed the following result: Theorem 216 If G is a graph with n vertices and k is an integer with then

Td .G/

0

kŠ .2d 1/k

> n,

  C k C 3d  3.

It was shown in [185] that T2 .G/  2 for any graph G with   2, and  2  1 if   5 is odd. This implies that T2 .G/  6 if   3. Also it was conjectured [185] that T2 .G/  5 if G is a graph with   3 and G ¤ K4 . Moreover, it was affirmed that T3 .G/  7 if   3. Lih et al. [248] also discussed the general upper bounds of .d; 1/-total number of graphs. They gave the following two results and one problem:

T2 .G/

Theorem 217 For any graph G with the list chromatic index k, Td .G/  k C 4d  3. Theorem 218 For any graph G, Td .G/  b 32 c C 4d  3: Conjecture 36 Let G be a graph with .G/ > maxf2; d g. Then Td .G/  .G/ C 0 .G/ C d  3:

5.5.2 Special Graphs The .d; 1/-total labeling for some kind of special graphs have been studied, for example, complete graphs [185], complete bipartite graphs [248], planar graphs [44], graphs with a given maximum average degree [280], trees for d D 2 [205,352], and outerplanar graphs for d D 2 [110]. Complete graphs In their papers, Havet and Yu [185, 186] occupied nearly five pages to study the .d; 1/-total labeling of the complete graphs Kn , according to the various values of n and d . In particular, they determined completely the exact value of T2 .Kn / for any n  1. Here, only several main results are listed on this topic. For more details, readers may refer to [185]. Theorem 219 Let n; d  1 be integers: 1. If d  n, then Td .Kn / D 2n C d  3. 2. If d  n  1, then Td .Kn /  n C 2d  2. 3. If n  d , then Td .Kn /  n C 2d  3.

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4. If n is odd, then Td .Kn /  n C 2d  3. 5. If n is even and n > 6d 2  10d C 4, then Td .Kn / D n C 2d  2. 6. If n  4 is even and d  n  3, then Td .Kn /  n C 2p  3. Lih et al. [248] studied the .d; 1/-total number of complete bipartite graphs Km:n. In particular, they achieved the exact values of Td .Km;n / for d D 1; 2; 3. Three main results are stated as follows: Theorem 220 If m < n C d , then Td .Km;n / D m C d . Theorem 221 Let 1  n  m. Then 

T2 .Km;n / D

m C 2; if m  n C 1, or m D n C 2 and n  3; m C 1; otherwise.

Theorem 222 Let 1  n  m. Then 8 < m C 3; if m  n C 2, or m D n C 3 and n  4,

T3 .Km;n / D or m D n C 4 and n D 5; 9; 10; 13; 14; 15; : m C 2; otherwise. Planar graphs Suppose that G is a planar graph with maximum degree . By the Four-Color Theorem, .G/  4. By the Vizing’s Theorem [339], 0 .G/   C 1. Thus, by Proposition 5(1), it is immediate to conclude that Td .G/   C d C 3. Further, if   7, then 0 .G/ D  by [312] or [394], and therefore, Td .G/   C d C 2. If G is triangle-free and   5, then .G/  3 [170] and 0 .G/ D  [246]; therefore, Td .G/   C d C 1. In [44], Bazzaro et al. proved that Td .G/   C 2d  2 for a planar graph G with high girth and large maximum degree . Precisely, their main result is stated below: Theorem 223 Let G be a planar graph with maximum degree  and girth g. Then

Td .G/   C 2d  2 with d  2 in each of the following cases: 1.   2d C 1 and g  11. 2.   2d C 2 and g  6. 3.   2d C 3 and g  5. 4.   8d C 2. As pointed out above, the case d D 1 corresponds just to the total coloring of graphs. So, Theorem 223 extends actually the partial results obtained by Borodin et al. [77] regarding the total coloring of planar graphs with large girth. It should be emphasized that Bazzaro et al. [44] constructed a planar graph G which needs at least C2d 1 colors for a .d; 1/-total labeling when d D 2. Furthermore, they posed the following risky conjecture: Conjecture 37 For any triangle-free planar graph G with   3, Td .G/  Cd .

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In [280], Montassier and Raspaud proved the following: Theorem 224 Let G be a graph with maximum degree . Then Td .G/C2d 2 with d  2 in each of the following cases: 1.   2d C 1 and mad.G/ < 52 . 2.   2d C 2 and mad.G/ < 3. 3.   2d C 3 and mad.G/ < 10 . 3 Moreover, Montassier and Raspaud [280] gave examples to show that Theorem 224 is optimal for d D 2. Let G be an outerplanar graph with   3. It is known that .G/  3 and 0 .G/ D . Thus, it follows from Proposition 5(1) that Td .G/   C d C 1. In particular, when d D 2, then Td .G/   C 3. The upper bound  C 3 is not the best possible. In 2007, Chen and Wang [110] obtained the following result: Theorem 225 Let G be an outerplanar graph: 1. If   5, then T2 .G/   C 2. 2. If G is 2-connected and  D 3, then T2 .G/  5. 3. If  D 4 and G does not contain intersecting triangles, then T2 .G/  6. Recently, Hasunumaa et al. [178] improved the results (2) and (3) of Theorem 225. That is, they showed that every outerplanar graph G with 3    4 also satisfies T2 .G/   C 2. Other graphs For a tree T , it is easy to prove that  C 1  T2 .T /   C 2. In [352], Wang and Chen gave a complete characterization for a tree T with  D 3 to have T2 .T / D 4 or T2 .T / D 5. Suppose that T is a tree. A path PkC2 D x0 x1 : : : xk xkC1 of length k C 1 of T is called a k-chain if d.x0 / > 2, d.xkC1 / > 2, and d.xi / D 2 for all i D 1; 2;    ; k. Note that a 0-chain is exactly an edge of T with two ends of degree more than 2. Let Vi .T / denote the set of vertices of degree i in T . Assume that .T / D 3. A subtree T  of T is called bad if it satisfies the following conditions (1)–(4): 1. V3 .T  / ¤ ;. 2. V1 .T  / [ V3 .T  /  V3 .T / and V2 .T  /  V2 .T /. 3. Every vertex x 2 V1 .T  / is connected to some vertex y 2 V3 .T  / by a 1-chain. 4. Every two closest vertices x; y 2 V3 .T  / are connected by a 2-chain or 1-chain. A tree T is called good if it contains neither bad subtrees nor 3 vertices adjacent to two other 3 vertices. Theorem 226 ([352]) If T is a tree with  D 3, then T2 .T / D 4 if and only if T is good. Wang and Chen [352] also proposed the following interesting problem: Problem 12 Give a complete classification for trees T with   4 according to their .2; 1/-total numbers.

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It seems hard to settle the above problem. Huang et al. [205] gave a partial solution. For a tree T , let D .T / denote the set of integers k for which there exist two distinct vertices of maximum degree of distance at k in T . Based on this definition, the authors in [205] established the following sufficient condition: Theorem 227 Let T be a tree with   4. If 1 62 D .T / or 2 62 D .T /, then

T2 .T / D  C 1. This result is best possible in the sense that for any fixed integer k  3, there exist infinitely many trees T with   4 and k … D .T / such that T2 .T / D  C 2. Chen and Wang [109] determined completely the .2; 1/-total number of the Cartesian product of cycles or paths. Theorem 228 If m; n  3, then T2 .Cm  Cn / D 6. Theorem 229 Let m  n  2. Then 8 < 4; if m D 2 and n D 2; 3I

T2 .Pm  Pn / D 5; if m D 2 and n  4, or m D 3 and 3  n  6, or m D n D 4; : 6; otherwise. Recall that the join G _ H of two vertex-disjoint graphs G and H is the graph obtained by joining each vertex of G to each vertex of H . Wang et al. [356] provided a complete characterization for the .2; 1/-total number of the join of cycles or paths. Theorem 230 Let n  m  3. Then

T2 .Cm _ Cn / D

8 < n C 3; :

n C 4;

if either n  m C 2 and m is even, or n D m C 1 and m  2; 4 (mod 12); otherwise.

Theorem 231 Let n  m  1. Then 8 n C 1; ˆ ˆ < n C 2;

T2 .Pm _ Pn / D ˆ n C 3; ˆ : n C 4;

6

if m D 1 and n  4; if m D 1 and 1  n  3, or m D 2 and n  4; if m D 2 and n D 3, or m  3 and n  m C 1; if m D n  2.

Conclusion

Graph coloring is an important branch in graph theory. From the viewpoint of historical development, graph theory came first from the graph coloring problems, for example, the Four-Coloring Conjecture. Since the famous Four-Coloring Conjecture

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was proposed, many variants, generalizations, techniques, and applications on graph coloring problem have emerged. There have existed a large amount of literature regarding this area. In this chapter, some subjects of graph coloring problems are selected to investigate, including classical chromatic number, choice number, acyclic chromatic number, acyclic chromatic index, vertex-distinguishing edge-weighting (especially, adjacent vertex-distinguishing edge coloring and total coloring), and L.p; q/labeling problems. So far there have been many well-known conjectures and open problems on these selected topics, such as Haj´os Conjecture, Hadwiger Conjecture, Steinberg’s Conjecture, Acyclic Edge-Coloring Conjecture, Griggs and Yeh’s Conjecture, Wegner’s Conjecture, and Havet and Yu’s Conjecture. Some main results and progresses on each of these problems are surveyed, without proofs provided. Of course, methods and tools to solve the related problems are rarely mentioned, because of the limited pages. For the interested reader, they can refer to a mass of literatures listed below. Acknowledgements The authors deeply thank their postgraduate students M. Chen, D. Huang, Q. Shu, S. Zhang, W. Gao, and Y. Wang for their careful search for a number of literatures and patience inspection for this manuscript.

Cross-References Equitable Coloring of Graphs Graph Searching and Related Problems Graph Theoretic Clique Relaxations and Applications Max-Coloring Maximum Flow Problems and an NP-Complete Variant on Edge-Labeled Graphs Online and Semi-online Scheduling Optimal Partitions Partition in High Dimensional Spaces

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Online and Semi-online Scheduling Zhiyi Tan and An Zhang

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Online Problems and Competitive Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Structure and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Online Over List. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Non-preemptive Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preemptive Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Randomized Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Other Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Semi-online Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Motivation and Taxonomy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Results on Basic Semi-online Models of Type I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results on Basic Semi-online Models of Type II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Results on Combined Semi-online Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Results on Disturbed Semi-online Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Online Over Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Single Machine Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Results on Parallel Machine Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Supported by the National Natural Science Foundation of China (10971191, 11271324), Zhejiang Provincial Natural Science Foundation of China (LR12A01001), and Fundamental Research Funds for the Central Universities. Z. Tan () Department of Mathematics, Zhejiang University, Hangzhou, People’s Republic of China e-mail: [email protected] A. Zhang Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou, People’s Republic of China e-mail: [email protected] 2191 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 2, © Springer Science+Business Media New York 2013

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5 Other Variants of Online Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Online Shop Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Batch Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Online Scheduling with Machine Eligibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

During the last two decades, online and semi-online scheduling problems have received considerable research interest. This chapter provides definitions of several online paradigms, including online over list, online over time, and semi-online. This chapter presents details of the basic online algorithms and referencing other significant results.

1

Introduction

1.1

Scheduling

Scheduling is the earliest studied branch in combinatorial optimization and also one of the problems with the most fruitful results achieved. Early scheduling problems are motivated from manufacturing systems. Later, more and more scheduling problems are formalized in the areas of information science, supply chain management, public management, and so on. In the following, we summarize the basic concepts and notations which will be used later. For more detailed introduction on scheduling theory, please refer to [21, 36, 117, 143]. Scheduling problems can be generally described as follows. There is a sequence J of n jobs J1 ; J2 ;    ; Jn , which need to be processed on a set M of m machines M1 ; M2 ;    ; Mm . A schedule is an assignment for each job to one or more time intervals of one or more machines. Schedules that satisfy various requirements of the problem are called feasible. Scheduling problems are specified by the machine environment, the job characteristics, and an optimality criterion. There are two basic types of machine systems: single-stage system and multistage system. In a single-stage system, each job requires one operation, while in multiple-stage system the jobs may require several operations at different stages. There are also two classical single-stage systems: single machine and parallel machines. In a single machine system, job Jj must be processed on the unique machine M1 with a processing time pj ; j D 1; 2;    ; n. In a parallel machines system, each job can be processed by one of the m machines. The time needed for job Jj to be processed on Mi is pj i ; j D 1;    ; n; i D 1;    ; m. If pj i D pj for any i D 1;    ; m and j D 1;    ; n, then the machine system is called identical. p s If there exist si , i D 1;    ; m such that pjj ii1 D sii2 for any j D 1;    ; n, then the 2 1 machine system is called uniform, and si is called the speed of Mi , i D 1;    ; m. W.l.o.g., we assume s1  s2      sm D 1 and let pj D pj m, which is the

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normalized processing time of job Jj . In the case of two uniform machines, it is of more convenience for discussion to replace the speed parameters s1 and s2 .D 1/ by the speed ratio s D ss12 , and both are equivalent. Parallel machines system that does not fall into above categories is called unrelated. In a multi-stage system, each operation of a job has a separate processing time, and different operations of the same job cannot be scheduled at the same time. The multi-stage system is usually distinguished into open shop, when different operations of the same job may be scheduled in any order; flow shop, if the order of the operations is fixed and same for all the jobs; and job shop, if the order of the operations is fixed and possibly different for each job. Apart from the processing time of the jobs, the processing mode can also be classified into the job characteristics. Some scheduling models allow preemption of jobs, that is, the processing of a job may be interrupted and resumed later on possibly a different machine. If each job can be arbitrarily split between the machines and parts of the same job can run on different machines in parallel, this model is called fractional assignment. There is another possibility allowing jobs to restart, which means that a processing job can be stopped and restarted later to finish the full processing time. Given a schedule , let Cj ./ (or Cj if there is no confusing) be the completion time of Jj , j D 1; 2;    ; n. Some commonly used objective functions include: The makespan: max1j n CP j ./. The total completion time: nj D1 Cj ./, P The total weighted completion time: nj D1 wj Cj ./, where wj is the weight of Jj . All these objectives are for minimization problems. There is another criteria which maximizes the continuous period of time when all machines are busy. Such criteria have applications in the sequencing of maintenance actions for modular gas turbine aircraft engines [49] and fairly allocation of indivisible goods [15]. Problems with this objective are often called machine covering. The period when one machine is not assigned to any job during the scheduling process is called idle time. For most non-preemptive problems, there is no benefit to introduce idle times. But idle time is of great help if preemption is allowed, especially for uniform machines scheduling. Let Li be the completion time of machine Mi in schedule . The makespan of  equals to maxi D 1; ;m Li . However, mini D 1; ;m Li is not always identical with the objective value of the machine covering problem if idle times exist (Fig. 1). In general, scheduling problems are specified in terms of a three-field notation [88]. Here, we adopt the notation from it. In this chapter, we will use 1; P; Q; R to denote single, identical, uniform, unrelated machine system and F; O; J to denote flow shop, open shop, and job shop system. Problems allowing preemption, fractional assignment, and restart will be denoted by pmtn, frac, and res, respectively. The objectives of minimizing makespan, minimizing the total completion time, and P minimizingP the total weighted completion time will be shortly denoted by Cmax ; Cj , and wj Cj , respectively. The objective of machine covering is denoted Cmi n .

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Fig. 1 Machine completion times and objective value of a schedule

Cmax

M1 M2

idle

L1

1.2

L2

Online Problems and Competitive Analysis

Online problems began to draw attention in the middle period of the eighties of the twentieth century. The basic character of online models is “lack of information,” while the previously studied problems having the access to full information are called offline. For more detailed introduction of online theories and variants, please refer to [20, 75]. For scheduling problems, there are a variety of online models, among which the following two are the most common. Online over list: Jobs are ordered in a list and arrived one by one (at time 0). The information of a job is known to the scheduler only when all the jobs before it have been assigned. And the assignment of jobs is irrevocable. Online over time: Each job has a release time after which the information of the job is known and can be processed. The release time of Jj is denoted by rj , j D 1; 2;    ; n. Since almost all problems considered in this chapter are online, we will ignore online over list in the second field of the three-field notation. On the other hand, rj will be included in the second field of the three-field notation for the online over time model. The competitive analysis is the most common method in analysis of online algorithms, which is formally introduced by Sleator and Tarjan [161]. The performance of an online algorithm is measured by its competitive ratio, which has some similarity as worst-case ratio for offline problems. For a job sequence J and an online algorithm A, if A produces a schedule with objective value C A .J / for the minimization (or maximization) problem and the optimal offline objective value is C  .J /, then the competitive ratio of A is defined as rA D inffrjC A .J /  rC  .J /g .or rA D inffrjC  .J /  rC A .J /g/: J

J

If there is an online algorithm A for the problem with a competitive ratio r, then A is called an r-competitive algorithm. On the contrary, if no online algorithm has

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a competitive ratio smaller than , then the problem must have a lower bound . Therefore, an online algorithm is called optimal or best possible for the problem if its competitive ratio just matches the lower bound. Here, optimal does not mean that it always produces an optimal schedule. It only indicates the fact that no online algorithm can be better. An online algorithm is stated to possess competitive ratio 1 if it always produces an optimal schedule, but such algorithm seldom exists. In this chapter, we use optimal bound to denote the competitive ratio of the optimal algorithm of the problem for simplicity. Symmetrically, a problem has upper bound r if there exists an online algorithm for the problem with competitive ratio r. A randomized algorithm is one which somehow bases its decision making on the outcome of random coin flips. For a randomized algorithm, the objective value C A .J / is a random variable. Thus, the competitive ratio of a randomized algorithm (for a minimization problem) is defined as rA D inffrjE.C A /.J /  rC  .J /g; J

where the expectation is taken over the random choices of the algorithm. In addition, concepts of randomized upper and lower bound can also be defined accordingly. Algorithms which do not use randomization are called deterministic algorithm. Since deterministic algorithm can be viewed as a special randomized algorithm, the randomized lower bound cannot be larger than deterministic lower bound. In this chapter, without special mention, all algorithms (competitive ratio, lower bound) are deterministic. For a general introduction on randomized algorithms, please refer to [134].

1.3

Structure and Notation

Online scheduling boasts of extensive results; thus, it is impossible to cover all the problems in one chapter, and only part of paradigms together with the according main results is included. There have been two excellent surveys [144,156] concerning online scheduling, and some paradigms discussed elaborately there will be omitted. We will mainly introduce paradigms approaching classical scheduling problems and withal active in recent research. There are several interesting and important problems not covered due to limited space and number of references, and we will sketch some of them in the following. Firstly, scheduling problems with deadline constraints, including interval scheduling, are not included in this chapter. Secondly, it does not cover problems concerning other objective functions, for example, the (weighted) total flow time and the Lp norm of machine completion times, together with objectives deriving from bicriteria situations, for example, scheduling with rejection and scheduling with machine cost. Thirdly, performance measures other than competitive analysis, including resource argumentation, will not be contained in this chapter. Finally, we skip problems which have specific

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applications, such as scheduling problems in power management, broadcasting, and supply chain management. The chapter is organized as follows. In Sect. 2, we survey main results for the online over list model. Section 3 is devoted to semi-online variants of online over list model. In Sect. 4, we introduce online and semi-online problems of the online over time model. Other variants of online scheduling are contained in Sect. 5. The following notations as well as those defined above willP be used frequently in the reminder of this chapter. For any i D 1;    ; m, let Si D ilD1 sl be the total speeds of the first i machines in uniform machine system, and we simplify Sm as S . For any j D 1;    ; n, let Jj be the subset of J containing the first j jobs and Pj be the total processing times of the first j jobs. We simplify Pn as P . Let p.j / be the processing time of the job which has the j th largest processing time, that is, j p.1/  p.2/      p.n/ . For any i D 1;    ; m and j D 1;    ; n, let Li be the j completion time of machine Mi after the job Jj is assigned. Li is also called the load of Mi if no idle time exists. Clearly, Lni equals to Li .

2

Online Over List

This section summarizes main results for online over list model on parallel machines. In this paradigm, the makespan, which is the most common objective function with no doubt, will be considered throughout this section except the last subsection.

2.1

Non-preemptive Scheduling

2.1.1 Identical Machines P mjjCmax is the most studied and historical problem for online scheduling. In his epoch making paper, Graham [86] designed the first approximation algorithm in combinatorial optimization called List Scheduling (LS for short). LS uses greedy idea and has a very simple structure. It always assigns the current job to the machine where it can be started to process the earliest, that is, the machine with the smallest current load, ties are broken arbitrary. Such principle of assigning jobs is sometimes called LS rule in the literature. Graham proved the following result. Theorem 1 ([86]) The competitive ratio of LS for P mjjCmax is 2 

1 m.

Proof (Sketch) Without loss of generalization, the last job Jn determines the makespan. Let sn be the start time of Jn . By LS rule, no machine will idle during n Œ0; sn . Hence, sn  P p m . Clearly, C 

P m

(1)

Online and Semi-online Scheduling

and

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C   max pj :

(2)

1j n

Hence, C LS D sn Cpn 

  P  pn P m1 m1  1 Cpn D C pn  C  C C D 2 C : m m m m m t u

LS is a typical online algorithm. When assigning a new job, it does not use any information of jobs that have not arrived yet. Not only the rule of LS but also the ideas behind the proof are frequently used in design and analysis of other online algorithms. Twenty years have passed since the appearance of LS , and online problems are drawing more and more attentions. Feigle et al. [74] proved that LS is the optimal algorithm for P mjjCmax when m D 2; 3. It should be emphasized that no other algorithm has been proved to be optimal for any m  4 up till now. Though LS is irreplaceable in online scheduling, it is gradually recognized that LS cannot be the optimal algorithm for P mjjCmax when m  4. Hence, algorithms and lower bounds have been refined bit by bit, and brief results and history are summarized in Table 1. In these new algorithms, jobs are not always assigned to the least loaded machine. Sometimes, the current job may be assigned to the second smallest or the machine whose load lies approximately in middle of all machines. Almost all these algorithms contain some parameters which have been carefully selected, and the analysis of the algorithm is much more involved. The instances used to prove the lower bound are likewise very sophisticated. The current best lower bound is obtained even by exhaustive search using computers. It seems that improvement of the algorithm could be carried on since there is still a gap 0:066 between the current best upper and lower bounds. However, there is a turning point when Albers [2] proposed her important results. When proving the competitive ratio of an online algorithm, it is in fact that some lower bounds instead of the exact value of the optimum itself are used. For example, lower bounds

Table 1 Improvement of lower and upper bounds of P mjjCmax Reference

Competitive ratio

Reference

Graham [86]

2

Falgle et al. [74]

p 1C 2 .m 2

Galambos and Woeginger [82]

2

Bartal et al. [17]

1:837.m  3454/

Bartal et al. [18]

2

Albers [1]

1:852.m  80/

Karger et al. [110] Albers [1] Fleischer and Wahl [77]

1:945 1:923 1:9201c

Gormley et al. [85]

1:85358d

1 m 1 m 1 70

 m

a

 1:986b

m ! 0.m ! 1/, the first algorithm which beats LS for m  4 The first algorithm with competitive ratio strictly less than 2 for any m c The best current known algorithm d The best current known lower bound a

b

Lower bound  4/

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(1) and (2) are used in the proof of Theorem 1. Another useful lower bound on the optimum is C   2p.mC1/: (3) Theorem 2 ([2]) If only using (1), (2) and (3) as lower bounds on the optimum, it is impossible to prove an online algorithm A has a competitive ratio less than 1:919. Note that the competitive ratio of the algorithm could be smaller than 1:919, but it cannot be proved using existing techniques. The bound 1:919 in Theorem 2 is very close to the current best upper bound 1:9201. Hence, it seems more urgent to find new lower bounds on the optimum than to designing more delicate algorithms. Though Theorem 2 does not give new lower bounds, it does point out the direction of future research. Such ideas can be adopted to other hard online scheduling problems as well. When the number of machines is small, better lower bounds and algorithms can be obtained. For example, Chen et al. [31] presented an algorithm with competitive ratio at most ( ) p 4m2  3m 2.m  1/2 C 1 C 2m.m  1/  1 max ; p 2m2  2 .m  1/2 C 1 C 2m.m  1/  1 for P mjjCmax , m  4. Rudin p III and Chandrasekaran [147] proved the lower bound of P 4jjCmax is at leastp 3, which is in line with the conjecture that the optimal bound of P 4jjCmax is 3. Numerical bounds are summarized in Table 2.

2.1.2 Uniform and Unrelated Machines The competitive analysis for QmjjCmax is even more difficult than that for P mjjCmax . Both the competitive ratios of online algorithms and lower bounds of the problems are functions of machine speeds si , i D 1;    ; m. This might be the reason why optimal or tight bounds can only obtained for small values of m. For larger m or arbitrary number of machines, it is of more significance and practical that either to consider some special combinations of si or to obtain overall upper and lower bounds, that is, the maximal value among all possible choice of si , i D 1;    ; m for both. Since the machines have different speeds, the machine where a job can be started to process the earliest may not be the machine where it can be completed the earliest. Therefore, it is necessary to distinguish two variants of LS , namely, LSc and LS s. LSc and LS s assign the arriving job to the machine which the job can be completed or started the earliest, respectively. It is evident to see that for QmjjCmax , LSc is better than LS s. The competitive ratio of LSc for QmjjCmax is at most (P min

1km

m i D1 si

C .k  1/s1 Pk i D1 si

)

1:74625 [31]

1:77301 [31]

1:79103 [31]

1:85358 [85]

5

6

7

1

4

3

2

m

Non-preemptive Deterministic Lower bounds 3 D 1:5000 [74] 2 5  1:6667 [74] 3 p 3  1:732 [147]

D 1:5000 [86]

5 3

1:9201 [77]

26  1:7333 [31] 15 85  1:77083 [31] 48 9 D 1:8000 [31] 5 175  1:82292 [31] 96

D 1:5000 [86]

3 2

Upper bounds

Randomized Lower bounds 4  1:3333 [18] 3 27 > [174] 19 256  1:46286 [32, 155] 175 3;125  1:48739 [32, 155] 2;101 46;656  1:50353 [32, 155] 31;031 823;543  1:51496 [32, 155] 543;607 e1  1:582 [32, 155] e

Table 2 Current best upper and lower bounds for online over list model on identical machines

 1:3333 [18]

1:916 [2]

1:81681 [151]

1:78295 [151]

1:73376 [151]

1:65669 [153]

1:53727 [153]

4 3

Upper bounds

Preemptive Optimal bounds 4  1:3333 [33] 3 27  1:42105 [33] 19 256  1:46286 [33] 175 3;125  1:48739 [33] 2;101 46;656  1:50353 [33] 31;031 823;543  1:51496 [33] 543;607 e1  1:582 [33] e

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[90]. The overall competitive ratio is at most (

p 1C 5 2p ; 2C 2m2 ; 2

m D 2, m  3,

and the bound is tight in the overall sense when m  6 [44]. For large m, the bound O.log m/ obtained by using an alternative method different from those in Theorem 1 is better [10]. Specifically, for m D 2, the bound is (

2sC1 sC1 ; sC1 ; s

1s s>

p 1C 5 2p ; 1C 5 ; 2

and LSc is the optimal algorithm [72]. For m D 3, the bound is 8 s1 C s2 C s3 ˆ ˆ ; s12  s22 C s1 s2 C s2 s3 ; ˆ ˆ s ˆ 1 < 2s C s C s 1 2 3 ; s12 < s22 C s1 s2 C s2 s3 and s12 C s1 s2  s32 C s1 s3 C 2s2 s3 ; ˆ s C s 1 2 ˆ ˆ ˆ 3s1 C s2 C s3 2 ˆ : ; s C s1 s2 < s32 C s1 s3 C 2s2 s3 ; s1 C s2 C s3 1 and it is tight for any combinations of machine speeds [90]. However, p it has been shown that LSc is optimal only for the case when s1 D s2 D 1  . 2  1/s3 and s12  s22 C s1 s2 C s2 p[90, 136], and there does exist another online algorithm which

beats LSc when 1C8 97 < ss12 D ss13 < 2 [90]. It implies that the optimality of LS for P 3jjCmax cannot be generalized to the problem with different machine speeds. Since the competitive ratio of LSc tends to infinite when m becomes very large [44], it is natural to raise the question that whether an algorithm with finite competitive ratio exists. The first such algorithm with competitive ratio 8 was obtained by using a so-called doubling strategy [10]. Their results were later improved p by Berman [19], where a new algorithm with overall competitive ratio 3 C 8  5:828 and a overall lower bound 2:4380 are given. The lower bound was recently improved to  by Ebenlendr and Sgall [60], where   2:564 is the R 1 ln  solution of the equation 0 ln.1 x / dx D 1. For the most studied special case of s1 > s2 D s3 D    D sm D 1, the overall competitive ratio of LSc for QmjjCmax is at most 3m1 , where the maximum value mC1 achieves at s1 D 2 [44]. The bound matches the overall lower bound 2 when m D 3 [118]. For m  4, Li and Shi [118] designed a new algorithm which has a smaller competitive ratio around s1 D 2. Thus, the overall competitive ratio can be decrease to 2:8795. Cheng et al. [42] further raised an algorithm which has a competitive ratio of 2:45 if the speed of the fastest machine is restricted to 1  s1  2. For RmjjCmax , Aspnes et al. [10] proved that the competitive ratio of LSc is at most m, and thus, LSc is optimal when m D 2. They also designed a new algorithm

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with competitive ratio O.log m/. Since the lower bound of the problem is at least dlog2 .m C 1/e [13], their algorithm is optimal up to a constant factor.

2.2

Preemptive Scheduling

Results of preemptive online scheduling are more complete than those of nonpreemptive problems. The main reason may be that the optimal offline makespan can be obtained in polynomial time. The following lemma plays an important role in designing online algorithms for PmjpmtnjCmax and QmjpmtnjCmax . Lemma 1 ([84, 101]) For QmjpmtnjCmax , ( Pn 

C D min

j D1 pj Pm i D1 si

Pl

p.i / ; max Pi D1 l 1lm1 i D1 si

) :

Chen et al. [33] designed an optimal algorithm preemptive for PmjpmtnjCmax . mm e1 Let m D mm .m1/  1:582. Recall that for any j , m . Note that limm!1 m D e  C .Jj / can be calculated by Lemma 1. Algorithm Preemptive j 1 j 1 Let the current job be Jj . Reindex the machines such that L1  L2 j 1 j 1      Lm . If Lm C pj  m C  .Jj /, then assign Jj to Mm . Otherwise, j 1 C pj  m C  .Jj /g. Assign part of Jj of processing let l D minfi jLi j 1 to Ml and the remaining part of Jj of processing time time m C  .Jj /  Ll j 1  Ll C pj  m C .Jj / to Ml1 . Theorem 3 ([33]) The competitive ratio of Preemptive for P mjP mtnjCmax is m . Proof (Sketch) Prove by induction that for any j , the following two inequalities hold Ljm  m C  .Jj /; k X i D1

j

Li 

m k . m1 / 1 Pj ; k D 1;    ; m: m m . m1 /  1

Then, it can be confirmed that the schedule produced by Preemptive is feasible, and the competitive ratio holds.  Several attempts have been made to modify and generalize Preemptive to handle uniform machines, which resulted in an optimal algorithm for 2 Q2jpmtnjCmax with competitive ratio ss 2C2sC1 [72, 180] and an optimal algorithm CsC1 i for QmjpmtnjCmax with nondecreasing speed ratios, that is, sis1  sisC1 ; i 2  i  m  1 [64]. But it seems difficult to make further generalizations using similar ideas.

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Recently, Ebenlendr et al. [62] introduced a novel technique which leads to an optimal algorithm of QmjpmtnjCmax for any combinations of machine speeds. The competitive ratio .s1 ;    ; sm / of the algorithm is the optimal value of the following linear programming with decision variables q1 ;    ; qm ; O1 ;    ; Om and parameters s1 ;    ; sm . max q1 C q2 C    C qm s:t: q1 C    C qk  .s1 C s2 C    C sm /Ok ; k D 1;    ; m; qj C qj C1 C    C qk  .s1 C s2 C    C skj C1 /Ok ; 2  j  k  m; (4) 1 D s1 Om C s2 Om1 C    C sm O1 ; qj  qj C1 ; j D 2;    ; m  1; q1  0; q2  0: However, it can only be proved that the algorithm is optimal for any m and any combinations of machine speeds, but the concrete expression of the optimal bound .s1 ;    ; sm / is not known yet, even the overall bounds are hard to get. The currently known best lower and upper bound for arbitrary m is 2:054 (achieves at m D 100 and is obtained by numerical calculation) and e  2:718, respectively [62]. When the number of machines is small or some assumptions are made on the values of si , (4) can be solved theoretically [62]. For example, the optimal bounds for Q3jpmtnjCmax are ( .s1 ; s2 ; s3 / D

. sS1 C .1 

s1 s2 S /S

S2

s12 Cs22 Cs32 Cs1 s2 Cs1

C .1  ; s Cs s 3

2 3

s1 2 s3 1 S/ S/ ;

if if

s1 s2 s1 s2

 

s2 s3 s2 s3

C 1; C 1.

Another question requiring answer is that whether the optimal bounds will increase if idle time is prohibited. Note that algorithms in [33, 64, 72, 180] all do not introduce idle time, but no algorithm which allows idle time can have a smaller competitive ratio under certain machine environment mentioned there. However, the algorithm given in [62] does use idle time. It is not known yet that whether an algorithm which does not use idle time will necessarily have a strict larger competitive ratio than .s1 ;    ; sm / for general QmjpmtnjCmax .

2.3

Randomized Algorithm

As far as ever known results are concerned, for case on parallel machine scheduling problem with objective to minimize makespan, any lower bound for deterministic preemptive online algorithm (allowing idle time) is also a lower bound for randomized preemptive or non-preemptive online algorithms. For example, the m randomized lower bound for P mjjCmax and QmjjCmax is mm.m1/ [32, 155] and m .m1/ .s1 ;    ; sm / [62], respectively. But the converse is not true. Tich´y [174] showed that there does not exist a randomized algorithm for P 3jjCmax with competitive

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ratio 27 19 . However, his proof does not yield a new randomized lower bound with a strict larger value. The first randomized algorithm for parallel machine scheduling is due to [18], where an optimal randomized algorithm for P 2jjCmax is given. The algorithm was restated in a simpler version in [155] as follows. Algorithm Random 1. If possible, assign the current job Jj randomly so that afterwards the expected makespan equals 23 Pj . 2. Otherwise, assign job Jj always on the less loaded machine. Theorem 4 ([18, 155]) The competitive ratio of Random for P 2jjCmax is 43 . Proof (Sketch) Prove by induction that for any j , the following two invariants hold E.C Rand om .Jj //  23 Pj ; If E.C Rand om .Jj // > 23 Pj ; then max1lj pl  34 E.C Rand om .Jj //:

(5)

Clearly, (5) is also valid for j D n. Then, by (1) and (2), the theorem is thus proved.  Design effective randomized algorithm for P mjjCmax is rather a challenging work. When the number of machines is small, the competitive ratios of algorithms proposed by Seiden [151, 153] can be less than the best known deterministic algorithms (See Table 2). For arbitrary m, Albers [2] designed a randomized algorithm with competitive ratio 1:916, which is a combination of two deterministic algorithms with equal probability, and it is better than any known deterministic algorithm. There is even less study on randomized algorithm for uniform machines. For Q2jjCmax , Epstein et al. [72] showed that randomization does not help when s  2. They also presented numerical randomized lower bounds for 1  s  2 by solving 2 .sC2/ linear programmings and a smaller analytical randomized lower bound .sC1/ 3s 2 C5sC1 p for 1  s  2. Three randomized algorithms are proposed, and thus, an overall upper bound 1:52778 can be achieved, but the gap between the lower and upper bounds for any s  1 is still relatively large. For Q3jjCmax with restricted machine speeds s1 D s2  s3 D 1, Musitelli and Nicoletti [136] presented a randomized algorithm with competitive ratio 8 p 4s2 C 2 ˆ < ; 1  s2 < 10C2 ; 3 3s2 p 6s C 2 ˆ : 2 ; s2  10C2 : 3 3s2 C 2 It is only a little smaller than the competitive ratio of LSc, and no randomized lower bound has been reported.

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Other Objectives

2.4.1 Total Completion Time The property of the objective of minimizing the total completion time is different from that of minimizing makespan. The objective value of the schedule will be affected by not only the set of jobs assigned to each machine but also the process sequence of it. As a result, there exists no algorithm with constant competitive ratio, even for the single machine case. Theorem 5 ([76]) For every function f W N ! RC that fulfills conditions: 1. P f .n/ is nondecreasing, 1 2. 1 nD1 nf .n/ converges, P there exists an online algorithm for 1jj Cj with competitive ratio at most O.f .n//, where n is the number of jobs. On the other hand, let g W N ! RC be a function that fulfills condition: 1. g.n/ P is nondecreasing, 1 2. 1 nD1 ng.n/ diverges, 3. g.n/ O.log2 n/,  D 4. g logn2 n D .g.n//, P then there does not exist an online algorithm for 1jj Cj with competitive ratio o.f .n//. By selecting specific functions which satisfy the respective conditions, it can be found that the gap between the lower and upper bounds is rather small. Corollary 1 ([76]) For every  > 0, there exists an online algorithm with competitive ratio at most .log n/1C , but no online algorithm with competitive ratio log n can exist.

2.4.2 Machine Covering The online version of non-preemptive machine covering problem is relatively easy. It should be mentioned that for machine covering problems, LS s is better than LSc in most cases. Woeginger [181] and Epstein [66] proved that LS s is the optimal algorithm for P mjjCmi n and Q2jjCmi n, respectively. In fact, their analysis can be generalized to QmjjCmi n. LS s remains the optimal algorithm with P competitive ratio m a randomized algorithm i D1 si . Azar and Epstein [12] designed p for P mjjCmi n with an overall competitive ratio O. m log m/ and proved an overall p m randomized lower bound 4 . Surprisingly, machine covering problem becomes much more difficult if preemption is allowed, even though the optimal offline objective value Pn min

0j m1

lDj C1 p.l/

Pm

lDj C1 sl

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for QmjpmtnjCmi n canPbe calculated similarly as QmjpmtnjCmax [108]. Jiang 1 et al. [108] proved that m i D1 i is a lower bound of P mjpmtnjCmi n and m is an overall lower bound of QmjpmtnjCmi n, but no algorithm with matched competitive ratio has been given. For Q2jpmtnjCmi n, they designed an optimal algorithm with competitive ratio 2sC1 . However, their algorithm may introduce idle time, even sC1 when it assigns the first job. If idle time is not allowed, then no deterministic algorithm can have a smaller competitive ratio than 8 p 2s C 1 ˆ < ; 1  s < 1C2 5 ; sC1 p 1C 5 ˆ : s; : s 2 A corresponding algorithm prohibiting idle time with matched competitive ratio was also given in [108]. Therefore, whether idle time is allowed or not does affect the optimal bounds for machine covering problems. It should be noted that different from algorithms allowing idle time, for non-idle time situation, there is no evidence yet that the lower bound also holds for randomized algorithm. Such phenomenon also appears in similar situations in the following.

3

Semi-online Scheduling

3.1

Motivation and Taxonomy

There are two basic characteristics for the online over list model. The first is that jobs arrive one by one and it is required to assign each job without any knowledge of the jobs that arrive later. The second is that the assignment of the job should be determined immediately upon its arrival and cannot be changed later. The reason of the first characteristic is “lack of information,” while the essence of the second is “restriction of scheduling.” There are often voices of criticism on the model for the sake of both theoretical study and practical application. In practice, most problems are not pure offline or online but somehow in between. Theoretically, the competitive ratio of an online algorithm tends to be over large in contract with the actual performance, since offline algorithms are more powerful than online algorithms. Moreover, the lower bound of the problem can also be ineffective, namely, too large, as the adversary can arbitrarily determine the instances without any restriction. The reasons aforementioned bring semi-online scheduling about an active research area. Semi-online can be looked upon as relaxation of online or an intermediate state of offline and online. The primary significance of study on semionline settings is as follows. First and foremost, without reasonable understanding of semi-online problems, there will be no more than online algorithm to implement faced with semi-online cases, which will give rise to inevitable loss definitely, just as LS algorithm performs not satisfactorily on offline problems. Furthermore, deep

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study on semi-online problems will lead to initiative search for crucial factors in algorithm design and improving, which will certainly save the cost and energy on many pointless factors. In theory, research on semi-online scheduling is of help to clarify influence of various restrictions of both information accessed to and algorithm design, and to reveal some deep and dominant properties. For example, it is interesting to consider the question that whether it is possible or under what kind of circumstance that an online approximation scheme, that is, a class of semionline algorithms with competitive ratio arbitrarily close to 1 can exist. In contrast to P TAS of offline problems, whose time complexity increases as the worst-case ratio of the algorithm decreases, more partial information or more freedom of scheduling is required in online cases when pursuing a better competitive ratio. In fact, one such approximation scheme has already been obtained [160]. Last but not least, it will be meaningful for algorithm design and analysis on online problems when researching on semi-online models, as will see later. Though semi-online model can be viewed as an intermediate state between online and offline, its properties and research approach are almost the same as the online model. Competitive analysis is still the primary technique adopted, and lower bound of the problem, together with competitive ratio of the algorithm, can be defined accordingly. The value of a semi-online model, denoted by …s , is evaluated by comparing its upper (or lower) bound with that of a corresponding online model reacting to the same machine environment and objective function, which is represented by …o . The semi-online model is called valuable if there exists an online algorithm for …s whose competitive ratio is smaller than the optimal bound (or lower bound) of the …o . On the other hand, if the lower bound of …s is no smaller than the competitive ratio of an algorithm for …o , it is called valueless. Of course, a semionline model is valuable or may not be determined by certain machine environment or objective functions. In this section, mainly identical and uniform machines, as well as minimizing makespan and maximizing minimum machine load, are considered. It is also possible to compare variants of semi-online models qualitatively, according to the extent to which the optimal bound of pure online problem can be improved. However, the outcomes may be quite sensitive to the machine environment or objective function. Various semi-online paradigms have been studied in the literature, which can be divided into several categories. The taxonomy and their notations, which will be included in the middle field of the three-field notation, are summarized in the Table 3. Their specific implications will be introduced in the rest of the subsection.

3.1.1 Basic Semi-online Models of Type I Basic semi-online models of Type I modify the first assumption of the online over list model, that is, some partial information about the future jobs are known in advance. The following four kinds of partial information are the most heated. decr [154]: The jobs are arriving in nonincreasing order of their processing times, that is, p1  p2      pn .

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Table 3 Semi-online models Basic

Type I

Combined

Type II Type I and Type I

Disturbed

Type I and Type II inexact information

Valuable: sum, opt, max, decr Valueless: num, LB, UB, min, incr, lookahead buffer, parallel, reassignment UB & LB, UB & sum, LB & max, max & sum, max & opt, decr & sum, decr & opt, end of sequence sum & buffer, sum & parallel, sum & reassignment inexact sum, inexact opt, inexact max

P sum [111]: The total processing time of all jobs P D nj D1 pj is known before the first job is arrived. opt [14]: The optimal offline makespan C  is known before the first job is arrived. max [111]: The maximal processing time of all jobs pmax D maxj D1; ;n pj is known before the first job is arrived. The motivation of decr, sum, and max is obvious, whereas it may seem to be strange at first glance that the optimal offline makespan can be known in advance. In fact, it can be interpreted from the realistic scenario of remote file transfer [14]. Besides, online problems with C  is known in advance have already been studied before semi-online began to draw attention, since the following lemma is useful in competitive analysis of difficult online problems [10]. Lemma 2 For some scheduling problem of online over list model with objective to minimize makespan, if there exists an algorithm for the opt variant with competitive ratio r, then there exists an algorithm for the pure online model with competitive ratio at most 4r. Some variants reacting to other partial knowledge of input are also reasonable, which is quite common in practice. For example, num: The number of all jobs n is known before the first job is arrived. incr: The jobs are arriving in nondecreasing order of their processing times, that is, p1  p2      pn . UB(LB): The upper (lower) bound on the processing time of all jobs pUB  maxj D1; ;n pj (pLB  minj D1; ;n pj ) is known before the first job is arrived. min: The minimal processing time of all jobs pmi n D minj D1; ;n pj is known before the first job is arrived. However, such paradigms, which seem worthless under certain machine environment and objective functions discussed so far, have not drawn much attention. Another kind of partial information which is common in practice is called lookahead [111]. When assigning the current job, the information of the next k jobs is known, where k is a constant number. Lookahead tends to be of benefit for scheduling according to practical experience, whereas it turns out to be still valueless as far as competitive ratio is concerned.

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3.1.2 Basic Semi-online Models of Type II There are other variants which lie between online and offline. However, they have nothing to do with the partial information of the input but more freedom acquired in scheduling. These variants belong to Type II semi-online models. It is a much more potential research area compared with Type I semi-online models, which is fruitful of results. In the buffer [111] paradigm, there is a buffer of size K which can store at most K jobs. The incoming job can either be scheduled immediately or be stored in the buffer, which enables it to be scheduled later. The root cause of the improved performance attributed to buffer lies in the fact that it can reorder the job sequence. Pure online and offline models are equivalent to problems with buffer size 0 and 1, respectively. In another related class of models reassignment [160, 167], it is allowed to reassign some of the jobs during or after the process. Recall that in the pure online model, no job can be reassigned. A technique frequently used in designing algorithm for the offline problems is to run several procedures in parallel and then select the best result as output. However, it does not fit pure online setting. Semi-online variant which allows to use such technique is called parallel. A real-world system which resembles such situation was claimed to exist in [111]. Kellerer et al. [111] proved that the optimal bound of P 2jparal lel.2/jCmax is 43 ; here, paral lel.2/ means that two procedures are run in parallel.

3.1.3 Combined Semi-online Models Since some semi-online variants are indeed of value, it is natural to raise the question whether a more efficient algorithm can be designed for a problem possessing characteristics of several semionline variants in the meantime. A combination is useful, if there exists an online algorithm for the combined semi-online problem whose competitive ratio is smaller than the optimal bound (or lower bound) of any one of the basic semi-online model. Otherwise, the combination is useless. The combination could be of two basic semi-online variants of Type I, as well as that of Type I and II, respectively. In spite of the fact that combination consisting of more than three variants can certainly be concerned with, such cases are always quite complicated and seldom studied.

3.1.4 Disturbed Semi-online Models The disturbed semi-online models were first studied in [165]. In some cases, it is allowed to use some additional information or slightly more resources than the offline algorithm, whereas the information or resources might be unreliable. For example, it is known in advance that the total processing time lies in the interval ŒP0 ; ˛P0 , but the exact value of P is still unknown. It can be believed that the value of inexact partial information lies largely on the value of ˛, which is called

Online and Semi-online Scheduling

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disturbance parameter. For problem with inexact partial information, special stress is laid on determining the value of ˛ which insures the information to be useful and further designing algorithm making use of the beneficial information acquired.

3.2

Results on Basic Semi-online Models of Type I

3.2.1 Identical Machines It has been widely known for a long time that in combinatorial optimization, the performance of an algorithm can be improved after appropriate preprocessing for the input. Graham [87] proved that the worst-case ratio of algorithm Longest Processing Time first (LP T for short), where LS is implemented after a preprocessing step, namely, reordering the jobs in a nonincreasing sequence of their processing time, 1 is 43  3m . However, it is not allowed to reorder the sequence in the pure online setting. But if the jobs are arrived in nonincreasing order of their processing times, the performance of LS will be just the same as that of LP T . Above situation was reformulated as a semi-online variant by Seiden et al. [154]. They proved that LS is the optimal algorithm for P 2jdecrjCmax . However, LS is no longer optimal when m  3, since there exists an algorithm with competitive ratio 54 for m  4 and an p optimal algorithm with competitive ratio 1C6 37 for m D 3 [43]. If the total processing time of all jobs P is known in advance, a lower bound Pm on the optimal makespan is readily available, which will be of great help in designing algorithm with a better competitive ratio. Take the optimal algorithm reacting to two machines [111] for example. Jobs are successively assigned to M1 until there exists a crucial job, such that the load of M1 would exceed a threshold 43  P2 if the crucial job is still assigned to M1 ; here, 43 is the expected competitive ratio of the algorithm. Then, the crucial job is assigned to M2 , and the remaining jobs will no longer have any negative effect on the competitive ratio. The core idea of the algorithm is to keep one machine lightly loaded so as to serve the long jobs which might present later. The significance of sum lies in the fact that it is impossible to determine the threshold without access to P . ForpP mjsumjCmax , Angelelli et al. [6] proposed an algorithm with competitive ratio 6C1  1:725 and proved that the lower bound of the problem is 1:565 for 2 sufficiently large m. The lower bound was recently improved to 1:585 by Albers and Hellwig [3]. By more careful classification according to the processing time of the jobs as well as the current load of the machines, Cheng et al. [41] designed an improved algorithm with competitive ratio 85 . They also showed that 32 is a lower bound of the problem for any m  6. For the case of m D 3, the current best lower p 1293 and upper bounds are  1:393 and 27 6 19  1:421, respectively [9]. The semi-online variant opt is closely associated with the variant sum. In fact, many instances used to show the tightness of the competitive ratio of some algorithms satisfy C  D Pm . However, the instance with C  > Pm does exist. Therefore, it is not easy to answer whether corresponding two problems regarding

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sum and opt variants have the same optimal bound. Nevertheless, D´osa et al. [54] proved the following result. Lemma 3 ([54]) (i) If  is a lower bound of QmjoptjCmax , then the lower bound of QmjsumjCmax is at least . (ii) If there is an algorithm for QmjsumjCmax with competitive ratio r, then there also exists an algorithm for QmjsumjCopt with competitive ratio at most r. By Lemma 3, the algorithm for P mjsumjCmax given in [41] can be modified to solve P mjoptjCmax with at most the same competitive ratio 85 , which improves [14]. When the former algorithm for P mjoptjCmax with a competitive ratio of 13 8 the number of machines is small, another algorithm also given in [14] has a smaller 5m1 competitive ratio of 3mC1 . On the other hand, the lower bound of P mjsumjCmax is no longer valid for P mjoptjCmax . Though 43 is clearly a lower bound of P mjoptjCmax for any m  2 [14], no larger lower bound has been reported for m  3. There are fewer results on the max variant. He and Zhang [97] presented an optimal algorithm for P 2jmaxjCmax with competitive ratio 43 . Note that for the max variant, not only the processing times of all jobs are no more than pmax , but also at least one job with processing time pmax will arrive sooner or later. Hence, the first job with processing time pmax , which is denoted as Jmax , can be shifted to the beginning of the job sequence by assumption. Thus, the optimal algorithm given in [97] can be simplified by using above idea. Algorithm PLS 1. Let Jmax be scheduled on M1 from time 0 to pmax . 2. Assign the remaining jobs by LS rule. Theorem 6 The competitive ratio of PLS is Proof If maxfL1 ; L2 g  23 P , then

C PLS C



4 3

for P 2jmaxjCmax .

maxfL1 ;L2 g P 2



4 3

by (1). Otherwise,

maxfL1 ; L2 g C .L1 C L2 / C jL1  L2 j D 3 maxfL1 ; L2 g > 2P D .L1 C L2 / C maxfL1 ; L2 g C minfL1 ; L2 g: (6) Since the remaining jobs are assigned by LS rule, jL1  L2 j  pmax . Combining it with (6), minfL1 ; L2 g < jL1  L2 j  pmax . Recall that Jmax is assigned to M1 , L1  pmax . Hence, L2 < pmax and thus no job other than Jmax will be assigned to M1 , which implies that PLS produces an optimal schedule.  However, for general m, no algorithm performing better than the best known algorithm for pure online model has been obtained. It is also unknown that whether

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Table 4 Results of semi-online variants on arbitrary m identical machines Preemptive Makespan minimization

Non-preemptive Makespan minimization

Machine covering

Variant

Lower bound

Upper bound

Lower bound

Upper bound

Optimal bound

decr

1C

5 4

[43]

Open

4m2 3m1

max0km

8 5 8 5

D 1:6000 [41]

p 37 6

[154]

sum

1:585 [3]

opt

4 3

max

Open

 1:333 [14]

D 1:6000 [41]

Open

[47]

2m2 C2mk 2m2 Ck 2 Ck

m  1 [22, 166] 43 24

 1:791 [61]

11 6

 1:834 [61]

m  1 [22, 166]

[154] 1 [98]

1 [58] max0km 2m2 C2mk 2m2 Ck 2 Ck

[154]

the partial information is still valuable for sufficiently large m, since the effect of the information about the single job Jmax tend to reduce as m increases. As far as the problem of machine covering is concerned, the four semi-online variants differ hugely both in property and difficulty. Optimal algorithms for P mjsumjCmi n and P mjmaxjCmi n can be obtained, both having a competitive ratio m  1 for m  3 [22, 166]. However, the research on P mjoptjCmi n seems to be much more difficult than that of P mjsumjCmi n. Azar and Epstein [12] proposed an algorithm with competitive ratio 2  m1 , which is optimal for m D 2; 3; 4. A  1:834 was given in [61]. The more involved algorithm with competitive ratio 11 6 current best lower bound is 74 for m > 4 [12] and 43 24  1:791 for sufficiently large m [61]. These bounds are far smaller than those for P mjsumjCmi n. For the problem P mjdecrjCmi n, LS has a competitive ratio of 4m2 3m1 [47] and is optimal for m D 2; 3 [96]. Tables 4 and 5 summarize the current best results of different semi-online variants on identical machines for arbitrary m and small number of m, respectively. As is shown in the table, different semi-online variants are diverse in value. In addition, optimal algorithm for two machines has almost all been found, whereas for three or more machines, nearly all problems are waiting to be answered, which is quite thought-provoking.

3.2.2 Two Uniform Machines Similarly as pure online problems, the upper and lower bounds for most semi-online problems on two uniform machines are piece-wise rational or irrational functions of s. But the number of the pieces could be considerably large, often exceeding 10. There may be large difference among optimal algorithms for the same problem with different value of s, which make the analysis of these problems extremely difficult. Most results are obtained mainly by case by case analysis. Up till now, no universal or revolutionary method has been developed to handle these problems efficiently.

[164]

[167]

[111]

[97]

[14]

[111]

[164] p 2 [190]

4 3 4 3 4 3 4 3 4 3 6 5 6 5

[52]

[154]

Open

1 [65]

1 [98]

Open

6 5 4 3

1 [65]

1 [98]

[154]

6 5

7 6

[86, 154]

pmtn OB

15 [4] 11 4 [182] 3 4 [182] 3 3 [190] 2

c

Open

p 1C 37 [154] 6 p 1293 [9] 6 4 [14] 3

mD3 Makespan Non-pmtn LB

15 [4] 11 4 [182] 3 4 [182] 3 3 [190] 2

Open

27 [9] 19 7 [14] 5

p 1C 37 6

UB [43]

[154]

[52]

[154]

Open

1 [98]

1 [98]

Open

6 5 9 7

1 [58]

1 [98]

6 5

pmtn OB

2 [166]

[166] Open

decr & opt

decr & sum

[166]

3 2 3 2

sum & reassignFE sum & parallel(2)

Open

d

sum & buffer

reassignSQ

[47, 96]

reassignFE

parallel(2)

5 4

[12]

Modela

2 [166]

5 3

MC Non-pmtn OB

4 [55] 3 6 [53] 5 5 [132] 4 5 [53] 4 10 [164] 9 10 [65] 9

4 [111] 3 p 2 [167]

mD2 Makespan Non-pmtn OB

b

MC machine covering, OB optimal bound, LB lower bound, UB upper bound For most semi-online variants, the makespan minimization problem and the machine covering problem on two identical machines are equivalent. The competitive ratios r1 and r2 of the same algorithm for corresponding two problems satisfy r1 C r12 D 2. Therefore, results about machine covering problems on two identical machines are omitted. However, there does exist semi-online model that above two problems are not equivalent, such as UB&LB c The lower bound is 15 for any buffer size K [63], and there exists an algorithm with matched competitive ratio which uses a buffer of size 6 [112] 11 d The lower bound is 11 for any buffer size K, and the upper bound is 52 for K  2 [73] 6

a

eos

opt & max

sum & max

reassignFA

buffer

max

opt

sum

decr

Modela

m D 2b Makespan Non-pmtn OB

Table 5 Results of semi-online variants on a small number of identical machines

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Online and Semi-online Scheduling

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1.30

1.45 1.40

1.25

1.35 1.30

1.20

1.25 1.20

1.15 1.0

1.5

2.0

2.5

3.0

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Fig. 2 The optimal bounds (thick) and competitive ratios of LS (dash) for Q2jdec rjCmax (left) and Q2jdec rjCmi n (right)

Among the four basic semi-online variants of Type I, decr is the only one which optimal bounds for all values of s  1 are obtained for both minimization and maximization problems. Based on the worst-case ratio of LP T for the offline version [133], Epstein and Favrholdt proved that LSc is optimal for the majority value of s. For the remaining two intervals which is close to 1 and nearby 2:5 where LSc is not optimal, they designed three new algorithms and proved their optimality. Symmetrically, for Q2jdecrjCmi n, LS s is also optimal except for two similar intervals of s, and new optimal algorithms have been proposed for these intervals [39]. The figures depicting two optimal bounds (as functions of s) bear some similarities (Fig. 2). The other two problems which optimal algorithms for any s  1 have been obtained are Q2jsumjCmi n and Q2joptjCmi n. The optimal bound of Q2jsumjCmi n [163] is p 8 sC2 s 2 Œ1; 2, p ˆ < sC1 ; p s; s 2 Œ 2;p 1C2 5 ; p ˆ : s 2 CsC1C 5s 4 C6s 3 C3s 2 C2sC1 ; s 2 . 1C2 5 ; 1/; 2s.sC1/ while the optimal bound of Q2joptjCmi n [66] is 8 sC2 p 2, ˆ < sC1 ; s 2 Œ1; p 1Cp5 s; s 2 Œ 2;p 2 ; ˆ : 2sC1 1C 5 sC1 ; s 2 . 2 ; 1/: Note that the optimal bound of Q2jsumjCmi n is strict smaller than that of p 1C 5 Q2joptjCmi n for s > 2 . The overall optimal bound of Q2jsumjCmi n is also strict less than that of Q2joptjCmi n (See Fig. 3). These conclusions turn out to be opposite to both the results of Lemma 3 concerning problems with objective to minimize makespan and that of machine covering problems on m identical machines.

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1.65

1.40

1.60

1.38

1.55

1.36

1.50

1.34

1.45

1.32

1.40

1.30

1.35

1.28

1.30 1.0

1.26 1.5

2.0

2.5

3.0

1.0

1.2

1.4

1.6

1.8

2.0

Fig. 3 Left: the optimal bound of Q2jjCmax (thick) and the upper bounds (solid) and lower bounds (dash) of Q2jmaxjCmax (bottom) and Q2jeosjCmax (top). Right: the upper bounds (solid) and lower bounds (dash) of Q2jsumjCmax (top) and Q2jopt jCmax (bottom)

2.4 2.2 2.0

Fig. 4 The optimal bounds (thick) of Q2jsumjCmi n (bottom) and Q2jopt jCmi n (top) and the upper bounds (solid) and lower bounds (dash) of Q2jmaxjCmi n (bottom) and Q2jeosjCmi n (top)

1.8 1.6 1.4 1.2 1

2

3

4

5

6

7

8

For the remaining four problems, there are still gaps between upper and lower bounds for some values of s, though ceaseless efforts have been put into narrowing the gap. The gaps of some problems seem to be small, but it is usually the case that the gap might appear exactly be the place where the analysis becomes toughest. For the sake of conciseness, only best known bounds will be listed in the following. General information of these bounds is summarized in Table 6 (also see Figs. 4 and 3). It can be seen that Q2jsumjCmax and Q2joptjCmax are much more complicated than the corresponding machine covering problems. For Q2jmaxjCmax , the effect of the information about the single job Jmax declines when s is sufficiently large. There is a noteworthy fact that for majority makespan minimization problems, the overall bound is achieved at the point where the argument s values exactly the overall bound itself and lies in the interval Œ1:2; 1:5. For machine covering problems, the overall bounds all achieve at s D 1 except Q2jsumjCmax .

Cmi n

decr

Cmax

2

5 2

2

max eos

p 1C 5 2

2

p 2 1.4656

p 1C 17 4 p 1C 3 2 p 1C 3 2

2 2

1 1 1 5 2

p 1C 5 2

2

3 2

p 2

1 1

1

p 1C 5 2

1

p 2 2

p 1C 3 2

1:5062

1:3692 p 1C 3 2

p 1C 17 4

p 1C 17 4

Overall upper bound Upper bound s

p 1C 5 2

1

p 2 1.4656

p 1C 17 4 p 1C 3 2 p 1C 3 2

Overall lower bound Lower bound s

opt

sum

decr

max eos

opt

sum

Variant

Problem Objective

0:064 0.0624

p 1C 2 7.1519

.2:148; 3:836/ .5:4043; 8:2426/

;

1:688 2.8383

0

0

;

0

0

;

0

1.704 0:2663

0

.1:414; 1:558/ [ .2; 3:56/ .1:4657; 1:732/

0:3600

p 3

.1:071; 1:0946/ [ .1:3956; 1:732/

1C 2

0:0219 0.081 0:1000

0:3641

.1:071; 1:0946/ [ .1:3915; 1:732/

5 3

; 1:5744

Length 0

Intervals where gap exists Intervals

0:0176

0

Largest gap Gap s

Table 6 General information and current status of upper and lower bounds on two uniform machines

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The lower and upper bounds of Q2jsumjCmax [54, 65, 137] are 8 ˆ 3s C 1 ˆ ; ˆ ˆ 3s ˆ ˆ ˆ ˆ ˆ ˆ 8s C 5 ˆ ˆ ; ˆ 5s C 5 ˆ ˆ ˆ ˆ ˆ ˆ 2s C 2 ˆ ˆ ; ˆ ˆ 2s C 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ s; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2s C 1 ; ˆ ˆ 2s ˆ < 3s C 5

; ˆ ˆ 2s C 4 ˆ ˆ ˆ ˆ ˆ 3s C 3 ˆ ˆ ˆ 3s C 1 ; ˆ ˆ ˆ ˆ ˆ ˆ 4s C 2 ˆ ˆ ; ˆ ˆ 2s C 3 ˆ ˆ ˆ ˆ ˆ 5s C 2 ˆ ˆ ; ˆ ˆ 4s C 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ c.s/; ˆ ˆ ˆ ˆ ˆ ˆ sC2 : ; sC1

and 8 3s C 1 ˆ ; ˆ ˆ 3s ˆ ˆ ˆ ˆ ˆ 7s C 6 ˆ ˆ ; ˆ ˆ 4s C 6 ˆ ˆ ˆ ˆ ˆ 2s C 2 ˆ ˆ ˆ ˆ 2s C 1 ; ˆ ˆ ˆ ˆ ˆ ˆ ˆ < s;

" s 2 1;

5C

p

#

205

 1:0732 , 18 # p 5 C 205 1Cp31  1:0946 , 2 ; 6 18 # " p 1 C 31 1Cp17  1:280 2 ; 4 6 " # p p 1C 3 2 1C4 17 ;  1:36603 2 # " p 1C 3 p 2 ; 2  1:41421 2 " # p p 21 2; 2  1:52753 3 "p # p 21 5 C 193 2 ;  1:57437 3 12 " # p p 5 C 193 7 C 145 2 ;  1:5868 12 12 " # p p 7 C 145 9 C 193 2 ;  1:63509 12 14 " # p 9 C 193 p 2 ; 3 14 p 2 Π3; 1/ "

s s s s s s s s s s

s 2 Œ1; x1  1:071, " s 2 x1 ; " s2

p

1C

1C

p 12 p

12

145

#  1:0868 ,

145 1 C ;

;

4 p

17

#  1:280

# 17 1 C 3 ;  1:36603 4 2 # " p 1C 3 s2 ; x2  1:3915 2 "

s2

1C

ˆ 2s C 1 ˆ ˆ ; ˆ ˆ ˆ 2sp ˆ ˆ ˆ ˆ s C 29s 2 C 59s C 30 ; ˆ s 2 Œx2 ; x3  1:5062 ˆ ˆ 4s C 5 p ˆ ˆ 4 C 9s 3  32s 2  2s C 9 C 6s 2 C 9s C 3 ˆ 36s ˆ ˆ ; s 2 Œx3 ; x4  1:6932 ˆ 9s 2 C 7s ˆ p ˆ 4 C 8s 3 C s 2 C 4 C 2s 2 C 3s C 2 ˆ p 4s ˆ ˆ ; s 2 Œx4 ; 3 ˆ ˆ 2s 2 C 6s ˆ ˆ p : sC2 sC1

p

s 2 Π3; 1/;

;

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p s 2 x 2 C 4s.s C 1  x/ C sx , x is a root of the respectively, where c.s/ D 2s p s.2s C 2  x/ s 2 x 2 C 4s.s C 1  x/ C sx D , x1 ; x2 ; x3 ; x4 are the equation 2s .s C 1/.x C 2/ positive roots of 3s 2 .9s 2  s  5/ D .3s C 1/.5s C 5  6s 2 /; p s C 29s 2 C 59s C 30 2s C 1 D ; 2s 4s C 5 p p s C 29s 2 C 59s C 30 36s 4 C 9s 3  32s 2  2s C 9 C 6s 2 C 9s C 3 D ; 4s C 5 9s 2 C 7s p 36s 4 C 9s 3  32s 2  2s C 9 C 6s 2 C 9s C 3 9s 2 C 7s p 4s 4 C 8s 3 C s 2 C 4 C 2s 2 C 3s C 2 D ; 2s 2 C 6s respectively. The lower and upper bounds of Q2joptjCmax [54, 65, 137] are 8 3s C 1 i h p ˆ ; s 2 1; 5C18 205  1:0732 , ˆ ˆ 3s ˆ ˆ ˆ i h p p ˆ 8s C 5 ˆ 5C 205 1C 31 ˆ ;  1:0946 , ; s 2 ˆ 18 6 ˆ 5s C 5 ˆ ˆ ˆ i h p p ˆ 2s C 2 ˆ ˆ ; s 2 1C6 31 ; 1C4 17  1:280 ˆ ˆ 2s C1 ˆ " # p ˆ ˆ p ˆ 3 1C 17 1 C ˆ ˆ ; s; s2  1:36603 ˆ 4 ˆ 2 ˆ ˆ i h p p ˆ 2s C 1 ˆ ˆ ; s 2 1C2 3 ; 2  1:41421 ˆ ˆ 2s ˆ ˆ ˆ i hp p ˆ 3s ˆ C5 21 ˆ 2;  1:52753 ; s 2 ˆ 3 ˆ ˆ 2s C 4 < 3s C 3

hp

21

p 5C 193

i

 1:57437 ; s 2 3 ; 12 ˆ 3s C 1 ˆ ˆ ˆ i h p p ˆ 4s C 2 ˆ ˆ ; s 2 5C12 193 ; 7C12 145  1:5868 ˆ ˆ 2s C 3 ˆ ˆ i h p p ˆ ˆ 5s C 2 ˆ ˆ ; s 2 7C12 145 ; 9C14 193  1:63509 ˆ ˆ 4s C 1 ˆ ˆ i h p ˆ 7s C 4 ˆ 9C 193 5 ˆ ; ; s 2 ˆ 14 3 ˆ 7s ˆ ˆ i h ˆ p ˆ 7s C 4 ˆ ˆ ; s 2 53 ; 5C8 73  1:89300 ˆ ˆ 4s C 5 ˆ ˆ h p p i ˆ ˆ ˆ s C 1 ; s 2 5C 73 ; 3 ˆ 8 ˆ ˆ ˆ 2 ˆ p ˆ s C 2 : ; s 2 Œ 3; 1/ s C1

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Z. Tan and A. Zhang

8 3s C 1 ˆ ˆ ; s ˆ ˆ ˆ 3s ˆ ˆ ˆ ˆ ˆ 7s C 6 ˆ ˆ ˆ ; s ˆ ˆ 4s C6 ˆ ˆ ˆ ˆ ˆ ˆ 2s C 2 ˆ ˆ ; s ˆ ˆ 2s C1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ s; s ˆ ˆ ˆ ˆ ˆ < 2s C 1 ; s ˆ ˆ 2s ˆ ˆ ˆ ˆ ˆ ˆ 6s C 6 ˆ ˆ ; s ˆ ˆ 4s C 5 ˆ ˆ ˆ ˆ ˆ ˆ 12s C 10 ˆ ˆ ˆ ;s ˆ ˆ 9s C 7 ˆ ˆ ˆ ˆ 2s C 3 ˆ ˆ ˆ ; s ˆ ˆ sC3 ˆ ˆ ˆ ˆ sC2 ˆ ˆ : ; s sC1

2 Œ1; s8  1:071, " # p 1 C 145 2 s8 ;  1:0868 , 12 " # p p 1 C 145 1 C 17 2 ;  1:280 12 4 " # p p 1 C 17 1 C 3 2 ;  1:36603 4 2 " # p p 1 C 3 1 C 21 2 ;  1:39562 2 4 " # p p 1 C 21 1 C 13 2 ;  1:535187 4 3 " # p p 1 C 13 5 C 241 2 ;  1:71035 3 12 h p p i 2 5C12241 ; 3 p 2 Œ 3; 1/;

respectively. The lower and upper bounds of Q2jmaxjCmax [23, 24, 128] are 8 " # p ˆ 2s C 2 1 C 17 ˆ ˆ ; s 2 1;  1:280 , ˆ ˆ ˆ 2s C 1 4 ˆ ˆ ˆ h p p i ˆ ˆ ˆ s; s 2 1C4 17 ; 2 , ˆ ˆ ˆ ˆ ˆ p ˆ ˆ ˆ p 1 C 4s 2 C 1 ˆ ˆ ; s 2Œ 2; x5  1:558 ˆ ˆ ˆ 2s ˆ ˆ ˆ 2. However, if only one job can be reassigned, the competitive ratio of the best known algorithm is also the same as that of Q2jbufferjCmax with buffer size K D 1, but it is larger than that of Q2jreassignSQjCmax with G D 1 (Fig. 6). Another final one-off reassignment model is denoted reassignFE. After all jobs have been arrived, the last job assigned to each machine can be reassigned. For two p identical machines with objective to minimize makespan, the optimal bound is 2 [167]. In fact, in such situation at most one job is needed to be reassigned [132]. Cao and Liu [25] generalized the result to two uniform machines and obtain the optimal bound

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Z. Tan and A. Zhang

1.6 1.5 1.4 1.3

1.0

1.2

1.4

1.6

1.8

2.0

Fig. 6 The optimal bounds (thick) of Q2jjCmax (top), Q2jreassig nFEjCmax (middle), and Q2jbufferjCmax with K  2 (also Q2jreassig nSQjCmax with G  2 and Q2jreassig nFAjCmax with H  2) (bottom), the upper bounds (solid) and lower bounds (dash) of Q2jbufferjCmax with K D 1 (also Q2jreassig nFAjCmax with H D 1) (top) and Q2jreassig nSQjCmax with G D 1 (bottom)

8p < s C 1; 1  s  : sC1 s

;

s>

p 1C 5 ; 2

p 1C 5 : 2

Comparing it with the optimal bound of the pure online problem [72], reassignment p 1C 5 is useless when s > 2 (Fig. 6).

3.4

Results on Combined Semi-online Models

3.4.1 UB and LB Though UB and LB are both valueless, their combination can be valuable. Such situation is very common in practice, since schedulers tend to estimate the processing time of jobs. However, it would be of little significance if the estimation is too rough to neglect the error. Hence, it is necessary to take influence of a parameter ˇ, defined UB as ˇ D LB  1, into consideration when dealing with competitive analysis. Main results of this variant are summarized in Table 8. Note that LS remains optimal for any value of ˇ for P 2jUB&LBjCmax and P mjUB&LBjCmi n . It is not true for P 3jUB&LBjCmax , though LS is also the optimal algorithm for P 3jjCmax . He and D´osa [92] proved the competitive ratio of LS is

Online and Semi-online Scheduling

2227

8 3 2ˇC1 ˆ ˆ ; ˇ 2 Œ1; ; ˆ 3 ˆ 2 ˆ ˆ ˆ 3 p 2ˇC3 ˆ ˆ < ˇC3 ; ˇ 2 Œ ; 3; 2 p ˇC1 ; ˇ 2 Œ 3; 2; ˆ 2 ˆ ˆ ˆ ˆ 1 ˆ 2  ˇ ; ˇ 2 Œ2; 3; ˆ ˆ ˆ :5 ˇ 2 Œ3; C1/, 3; p and LS is optimal only when ˇ 2 Œ1; 32  [ Œ 3; 2 [ Œ6; C1/. When ˇ 2 Œ2; 6, there exists an improved 8 algorithm with competitive ratio 5 3 ˆ ˆ ; ˇ 2 2; 2 ; ˆ ˆ 2 ˆ ˆ ˆ < 4ˇ C 2

; ˇ 2 52 ; 3 ; ˆ 2ˇ C 3 ˆ ˆ   ˆ ˆ 5 1 6ˇ 3 ˆ ˆ :  ; ˇ 2 Œ3; 6: min ; 3 18 18 103 Nevertheless, the new upper bound matches the lower bound of the problem only when ˇ 2 Œ2; 52 . By the definition of the paradigm, the lower bound of the problem is obviously a nondecreasing function of ˇ. For the remaining intervals of ˇ where optimal algorithm has p not been obtained, nonconstant lower bound of the problem, equaling to

7ˇC4C

pˇ

2ˇC2C2

2 C8ˇC4

ˇ 2 C8ˇC4

, can be found only in the situation where ˇ 2 Œ 52 ; 3.

It is implied again from the results stated above that scheduling for two machines is of enormous difference in essence from that of three machines, while the latter is much more complicated. A similar model of LB and max is proposed by Cao et al. [26]. It is assumed that pmax ˇ  pj  pmax for any j , and there always exists a job of processing time pmax . They proved that the optimal bound of P 2jLB&maxjCmax is 8 ˇC1 4 ˆ ˆ 2 ; ˇ 2 Œ1; 3 ; ˆ ˆ ˆ p ˆ < 4ˇC4 ; ˇ 2 Œ 4 ; 2; 3ˇC4 3 p 2ˇ ˆ ˆ ˆ ˇC1 ; ˇ 2 Œ 2; 2; ˆ ˆ ˆ : 4; ˇ 2 Œ2; C1/. 3 It is smaller than that of P 2jLB&UBjCmax for some value of ˇ (Fig. 7).

3.4.2 sum=opt and max=UB Angelelli et al. [7, 8] studied the problem P 2jsum&UBjCmax . Lower bounds as functions of D 2UB P and several algorithms are proposed. Optimal bounds are achieved for majority value of (see Figs. 4–6 of [8]).

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Table 8 Main results for the UB&LB variant

Problem

Interval of ˇ where the partial information is valuable and the optimal bound

P 2jUB&LBjCmax [97]

ˇ 2 Œ1; 2,

PmjUB&LBjCmax , m  3 [92]b PmjUB&LBjCmi n [91] Pmjpmtn, UB&LBjCmax [94] Q2jpmtn, UB&LBjCmax [56, 93] Q2jpmtn, UB&LBjCmi n [105]

ˇ 2 Œ1; ˇ 2 Œ1; m, ˇ

1Cˇ 2 m , 1 m1

Interval of ˇ where the partial information is valuelessa ˇ 2 Œ2; C1

C

.m1/.r1/ m

c

ˇ 2 Œ1; 2s,d 2sC2Cˇ ˇ 2 Œ1; 2s, 2sC2

ˇ ˇ ˇ ˇ

2 Œm; C1/ m m1 2 Œ. m1 / ; C1/ 2 Œ2s; C1/ 2 Œ2s; C1/

a

Valueless implies that the optimal bound is the same as the optimal bound of the corresponding pure online problem b Only partial results about this problem have been obtained c Optimal algorithm can be obtained the   by using  general framework included in [59]. The m m1 analytical lower bound for ˇ 2 1; is at least m1  m k C .m  k  1/ˇ m1  ;  k .m  k/.m  k  1/ˇ m .2m  k  1/  1 C .m  k/ C m1 2m m

 m k  m kC1 ˇ ; k D 0; 1; : : : ; m  2: m1 m1 Whether it is the optimal bounds remains open except for m D 2; 3. d The optimal bound is 8 ˇs ˇ ˆ   ˆ 1CsC C ˆ ˆ 2 2 ; 1  s  2 and ˇ 2 Œ1; 2s, s > 2 and ˇ 2 1; 2 ˆ ˆ [ Œ2s  2; 2s ˆ ˆ ˇs s1 ˆ < 1CsC 2   .1 C ˇ/.1 C s/s 2 ˆ ˆ ; s  2 and ˇ 2 ; s ˆ ˆ ˆ 1 C s C s 2 C sˇ 2 s1 ˆ ˆ s2 C s ˆ ˆ : ; s  2 and ˇ 2 Œs; 2s  2: s2 C 1 If idle time is not allowed, then the (deterministic) lower bound will be larger, but no online algorithm with matched competitive ratio has been obtained [93]

There are other papers considering the combination of sum.opt/ and max. However, only overall bounds (the maximum bound among all combinations of P (or C  ) and pmax ) are obtained. For example, the optimal bound for P mjsum&maxjCmi n [166] is

Online and Semi-online Scheduling

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Fig. 7 The optimal bounds of 1.8 P 2jMj .GoS/; LB&UBjCmax (top), P 2jLB&UBjCmax (middle), and 1.6 P 2jLB&maxjCmax (bottom) 1.4

1.2

1.0

1.5

2.0

2.5

3.0

3.5

4.0

8 5 ˆ ˆ ; m D 2; ˆ pk , then reset k D h and reset ˛k and s accordingly. Let U.t 0 / D U.t/ [ fJh g. Reset t D t 0. 3. At time s, schedule all jobs in U.s/ as a single batch. If some new jobs arrives by s C pk , let t D s C pk ; otherwise, wait until a new job arrives and let t be the arrival time of such a job.pGo to Step 1. In fact, the lower bound 5C1 even holds when B D 2, and all jobs have only 2 two distinct arrival times [191]. It seems much more challenging to derive optimal algorithms for the bounded model. When all jobs have exactly two distinct arrival p time, Zhang et al. [191] gave an optimal one with competitive ratio 5C1 . When jobs 2 7 have arbitrary arrival times, Poon and Yu [145] obtained a 4 -competitive algorithm for B D 2. For the general B, we note there exist several 2-competitive algorithms. The first one is the greedy-like algorithm GRLPT, which was proposed by Lee and Uzsoy [113], and was shown to be 2-competitive by Liu and Yu [123]. Another two are due to Zhang et al. [191], which can be seen as a generalization of H 1 . Finally, Poon and Yu [145] claimed all the FBLPT-based (full-batch longest processing time) algorithms, including the above three, can achieve the same competitive ratio of 2. 5.2.2 Unbounded and Bounded Model on Identical Machines For batch scheduling on m parallel identical machines, researches are p mostly focused on the unbounded model. Zhang et al. [191] gave a lower bound mC2 2 and

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an online algorithm PH.ˇm / with a competitive ratio 1 C ˇm , where 0 < ˇm < 1 is a solution to the equation ˇm D .1  ˇm /m1 . In the same paper, the dense algorithms, which always immediately assign the currently available job to one idle machine as long as there are two or more idle machines, are proposed. Zhang p et al. proved that the competitive ratio of any dense algorithmpis no smaller than 2 and there exists one dense algorithm with competitive ratio 5C1 2 . In a later paper [192], Zhang et al. gave an optimal algorithm with competitive ratio 1 C m for the special case that all jobs have unit processing time, where m is the positive solution of the equation .1 C m /mC1 D m C 2. When jobs have arbitrary processingptimes, the optimal algorithm for twomachine case has a competitive ratio of 2 [139, 173]. The optimal algorithm for the general case is due to Liu et al. [126] and Tian et al. [172] independently, who p 2 both proved the competitive ratio to be 1 C m C4m . In addition, Tian et al. [172] 2 provided a dense algorithm with competitive ratio 32 and showed it is best possible. For the bounded model, Zhang et al. [192] observed that the optimal algorithm is p 5C1 -competitive if all jobs have equal processing times. Apart from this, neither 2 algorithm results nor lower bounds are found in the literature.

5.2.3 Other Variants There are some other discussions with respect to the online batch scheduling problems. One interesting variant is allowing to restart a batch, which means a running batch may be interrupted, losing all the work done on it, and the jobs in the interrupted batch are released and become independently unscheduled. Allowing restarts reduces the impact of a wrong decision. For the unbounded model on single machine, Fu et p al. [78] showed the competitive ratio of any online algorithm is 5 5 no less than 2 and gave a 32 -competitive algorithm. Shortly after that, Yuan p et al. [185] gave a 52 5 -competitive algorithm for the problem, and thus it is optimal. Fu et al. [79] studied the same problem with an additional assumption that any job cannot be restarted twice. Optimal algorithm with competitive ratio 32 is obtained. In another paper [81], Fu et al. extended the result to identical machines, p where a lower bound of 1:298, as well as a 1C2 3 -competitive online algorithm, was given. Nong et al. [140] introduced job families to the batch scheduling problem, which means jobs from different families cannot be processed in the same batch. Online algorithms with competitive ratio 2 are presented for both bounded and unbounded models on single machine. Besides, they showed that the algorithm is best possible for the unbounded model in the sense that jobs consist of an infinite number of families. For the bounded model, there is no online algorithm with competitive ratio p 1C 5 2B smaller than maxf BC1 ; 2 g. Fu et al. [80] revisited the unbounded model and p

gave a best possible algorithm with competitive ratio 17C3 for the special case of 4 two families of jobs. Yuan et al. [186] and Li et al. [119] studied a lookahead semi-online model. Yuan et al. [186] considered the unbounded problem on a single batch machine.

Online and Semi-online Scheduling

2239

With the information of the next longest job at anyptime t, they presented a best possible online algorithm with competitive ratio 52 5 . If jobs are of unit length and an online algorithm can foresee all the jobs that will arrive in the time segment .t; t C ˇ at any time t, Li et al. [119] presented a best possible online algorithm for the unbounded problem on m parallel-batch machines. Chen et al. [38] considered a batch scheduling problem on single machine to minimize the total weighted completion times of jobs. They developed a linear time online algorithm with competitive ratio of 10 for the unbounded model and an 3 efficient algorithm with competitive ratio 4C for any  > 0 for the bounded model. Also, they observed that there exists a 2:89-competitive and a .2:89C/-competitive randomized algorithm for unbounded and bounded models, respectively.

5.3

Online Scheduling with Machine Eligibility

Online scheduling with machine eligibility constraints has received much attention in recent years. Unlike the classical parallel machine scheduling, job Jj cannot be processed on any one among the machine set M in this model. Instead, it has a specific set Mj  M that can be assigned to. Set Mj is so-called the eligible processing set of job Jj . Depending on the structure of Mj , j D 1;    ; n, there are five classes of eligibility constraints that have been studied extensively [116]: 1. Arbitrary eligible processing sets 2. Tree-hierarchical processing sets 3. Grade of Service (GoS) processing sets 4. Interval processing sets 5. Nested processing sets. With tree-hierarchical processing sets, each machine is represented by a node, and the nodes are connected in the form of a rooted tree. Any job assignable to a node is also assignable to the node’s ancestors in the tree. The GoS processing set structure can be seen as a special case of the tree-hierarchical processing set structure where the rooted tree actually forms a chain. Problems with a GoS processing set are also called online hierarchical scheduling in the literature. With regard to interval processing sets, job Jj is associated with two integers aj and bj  aj such that Mj D fMaj ; Maj C1 ;    ; Mbj g. A special case of interval processing set structure is a nested processing set structure, where, for any pair of jobs Jj and Jk , we either have Mj  Mk or Mk  Mj or Mj \ Mk D ;. Clearly, the GoS processing set structure is also a special case of nested processing set structure. Problems with one of the above classes of eligibility constraints will be denoted as “Mj ,” “Mj .t ree/,” “Mj .GoS /,” “Mj .i nterval/” and “Mj .nested /” in the middle field of the three-field notation, respectively. It is clear that P jjCmax (QjjCmax ) is a special case of P jMj jCmax (QjMj jCmax ) with Mj D M for all Jj , while RjMj jCmax is a special case of RjjCmax . However, in the rest of this subsection, we will mostly consider the identical machine environment, unless stated otherwise.

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5.3.1 Online Over List For arbitrary eligibility, Azar et al. [13] contributed the first result. They consider a greedy algorithm, namely, AW , as follows: When a job arrives the algorithm assigns it to an eligible machine that has the smallest current load among all eligible machines. The competitive ratio of AW is shown to be at most dlog2 me C 1, and the lower bound is proved to be at least dlog2 .m C 1/e. Hwang et al. [102] improved the analysis of AW and obtained a slightly smaller competitive ratio of log2 m C 1. The best known upper and lower bounds for this problem are found by Lim et al. [120] recently. They showed that AW has a competitive ratio no greater than EUm D blog2 mcC 2blogm2 mc , and the lower bound of the problem is ELm , where EL1 D 1 and ELm D ELm C

1 m b m c m

( ; m D argmaxb m ci m1 ELi C 2

1 i b mi c

) :

The gap between the two bounds does not exceed 0:1967 and AW is optimal when the number of machines can be written as a sum of two powers of 2. Lee et al. [114] proved the optimal bound of Q2jMj jCmax is 1 C minfs; 1s g. Note that for Q2jMj jCmax , the index of M1 and M2 cannot be reordered, and thus the speed ratio s can be an arbitrary positive number. Bar-Noy et al. [16] analyzed online scheduling problem subject to treehierarchical eligibility. A 5-competitive online algorithm is proposed. Furthermore, if jobs are of unit size, then the competitive ratio can be improved to 4. Randomized algorithms are also proposed, where the competitive ratios are shown to be e C 1 and e, respectively. A lower bound of 1 C 12 blog2 mc for P mjMj .nested /jCmax was given in [120]. There is a large number of researches discussing problems with a GoS eligibility, since it is very natural and has application in the service industry [103]. In [16], BarNoy et al. gave an algorithm with competitive ratio e C 1 for the general problem P jMj .GoS /jCmax and showed that it is e-competitive if either jobs have unit size or can be fractionally processed. Moreover, they proved a lower bound of e for the fractional assignment case. However, the lower bound is valid only in the case when the number of machines tends to infinity. If the number of machines is fixed, Tan and Zhang [169] proposed an improved algorithm with competitive ratio GUm and lower bound GLm for the fractional assignment case; both are based on the solutions of mathematical programming. The algorithm can be modified to solve the general problem P mjMj .GoS /jCmax by using the same technique included in [16]. When the number of machines is small, algorithms for the general problem can be further improved. For m D 5 and 4, Tan and Zhang [169] proposed two algorithms with competitive ratios of 2:610 and 73 , respectively. Lim et al. [120] improved the two results to 2:501 and 2:294. For m D 3, Zhang et al. [193] showed that there exists an optimal algorithm with competitive ratio 2. For m D 2, Park et al. [141] and Jiang et al. [109] independently proposed an optimal algorithm with competitive ratio 53 . If machines have different speeds, the optimal

Online and Semi-online Scheduling

(

bound h1 .s/ D

2241

2

g; 0 < s  1, minf1 C s; 2C2sCs 1CsCs 2 1Cs 1C3sCs 2 minf s ; 1CsCs 2 g; 1  s < 1,

of Q2jMj .GoS /jCmax was given by Tan and Zhang [169] (Fig. 8). If preemption is allowed, Jiang et al. [109] designed a 32 -competitive algorithm for the problem P 2jMj .GoS /, pmtnjCmax and showed that it is optimal if idle 2m time is not allowed. For general m, D´osa and Epstein [50] proved that mC1 is a lower bound even if idle time is allowed. An algorithm for three machines with matched competitive ratio was also given. In the same paper, they also studied the problem Q2jMj .GoS /; pmtnjCmax . An online algorithm with competitive ratio  h2 .s/ D max

s.1 C s/2 .1 C s/2 ; 2 1 C s C s 1 C s2 C s3



is proposed and is shown to be optimal. Both algorithms need to introduce idle time. For Q2jMj .GoS /; f racjCmax , Chassid and Epstein [28] designed an optimal .1Cs/2 algorithm with competitive ratio 1CsCs 2 (Fig. 8). For semi-online scheduling, Park et al. [141] studied the problem on two machines with the knowledge of the total processing time of jobs, where an optimal algorithm with competitive ratio 32 is presented. Wu et al. [183] considered two semionline problems on two machines. The first one is known the optimal makespan, for which they showed an optimal algorithm with competitive ratio 32 . The second is known the maximum processing time, for which an algorithm with competitive p ratio 5C1 , as well as a matched lower bound, is given. Liu et al. [125] considered 2 the semi-online problem on two machines where upper and lower bounds on the processing time of jobs are known in advance. An online algorithm, as well as a lower bound of the problem, was presented. Optimal bound of the problem p 8 2ˇC1 51 ; 1  ˇ < , ˆ ˆp ˇC1 2 p ˆ < 5C1 p5C1 3 51 ; p2  ˇ < 2 , 2 h3 .ˇ/ D ˇC2 ˆ ˆ ; 3 251  ˇ < 3, ˆ : 53 ; r  3, 3 UB (Fig. 7). is given by Jiang and Zhang [107], where ˇ D LB There are several researches considering problems with two GoS levels (denoted as 2GoS ). In this problem, jobs that can be processed on first k machines are called high level and the other jobs which can run on all m machines are called low level. Jiang [104] first showed that the AW algorithm is at least .4  m1 /-competitive and p

2  2:522. then provided an online algorithm with a competitive ratio of 12C4 7 2 m m 7 Later, Zhang et al. [194] improved the result to 1 C m2 kmCk 2 < 3 . For machine covering problems, Chassid and Epstein [28] proved that no algorithm can have a constant competitive ratio even for the most special

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Fig. 8 The optimal bounds of Q2jMj jCmax , Q2jMj .GoS/jCmax , Q2jMj .GoS/, pmtnjCmax , and Q2jMj .GoS/, fracjCmax (from top to bottom)

2.0 1.8 1.6 1.4 1.2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

problem Q2jMj .GoS /jCmi n. For problems Q2jMj .GoS /; f racjCmi n and Q2jMj .GoS /; sumjCmi n, they gave optimal bounds of 2sC1 and maxf1; sg C 1s , sC1 respectively. Main results on online over list scheduling with machine eligibility are summarized in Tables 12–14.

5.3.2 Online Over Time In contrast with problems of online over list, there are very few results concerning problems of online over time. Lee et al. [115,116] considered the problems allowing restart or preemption of jobs. Various lower bounds, as well as a full study on two machines, are given. They also observed that the general method introduced by Shmoys et al. [157] that uses offline algorithms to obtain online algorithms for problems with job release times can work for scheduling with machine eligibility as well [116], although it produces algorithms with competitive ratio at least 2. All related results are summarized in Table 15. Wang et al. [178] introduced another interesting problem with respect to scheduling with GoS eligibility, which is so-called online service scheduling. The problem arises from the service industry where customers (jobs) are classified as either “ordinary” or “special.” Ordinary customers can be served on any service facility (machines), while special customers can be served only on a flexible service facility. The difference between their model and the problem with two GoS levels is that customers arrive over time and the order of the assignment should be consistent with the order of arrival time of jobs in the same class. For several service policies used in practice, Wang et al. [178] and Wang and Xing [177] analyzed and compared their performance in the sense of competitive ratios.

6

Conclusion

This chapter surveyed different paradigms of online scheduling, including online over list and online over time, and gave a relatively complete picture to the semionline scheduling problem. Most of detailed algorithms and proofs are not given in

Online and Semi-online Scheduling

2243

Table 12 Results of online over list scheduling with machine eligibility on two identical and uniform machines Problems

Optimal bounds

Problems

Optimal bounds

P 2jMj jCmax

2 [13]

Q2jMj jCmax

1 1 C minfs; g [114] s

P 2jMj .GoS /jCmax

5 [109, 141] 3

Q2jMj .GoS /jCmax

h1 .s/ [168]

P 2jMj .GoS /; fracjCmax

4 [169] 3

Q2jMj .GoS /, fracjCmax

.1 C s/2 [28] 1 C s C s2

P 2jMj .GoS /; pmtnjCmax a

4 [50] 3

Q2jMj .GoS /, pmtnjCmax

h2 .s/ [50]

P 2jMj .GoS /; sumjCmax

3 [141] 2

Q2jMj .GoS /jCmi n

1 [28]

3 2s C 1 [183] Q2jMj .GoS /; f racjCmi n [28] 2 sC1 p 5C1 1 maxf1; sg C [28] [183] Q2jMj .GoS /; sumjCmi n 2 s

P 2jMj .GoS /; optjCmax P 2jMj .GoS /; maxjCmax P 2jMj .GoS /; UB&LBjCmax a

h3 .ˇ/ [107]

If idle time is not allowed, the optimal bound increases to

3 2

[109]

Table 13 Results of online over list scheduling with machine eligibility on a small number of identical machines Problem P mjMj jCmax [13, 120] P mjMj (interval)jCmax [120] P mjMj (nested)jCmax [120] P mjMj .GoS/j Cmax [120, 169, 193] P mjMj .GoS/; fracjCmax [169] P mjMj .GoS/; pmtnjCmax [50]

mD3 LB 2:5 1C

mD4 UB LB 2:5 3

p 2

UB 3

p 3C 5 2

7 3

1C

mD5 LB UB 3:25 3:25

mD6 LB UB 3:5 3:5

mD7 LB UB 3:667 3:75

3

p 2

5 2

3

2

2

2

2:294 2

2:501 2

2:778 2

2:828

3 2

3 2

44 27

44 27

245 143

16 9

1;071 586

3 2

3 2

8 5

245 143 5 3

16 9 12 7

1;071 586 7 4

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Table 14 Main results on online over list scheduling with machine eligibility on general number of machines Problems P mjMj jCmax P jMj .t ree/jCmax P jMj .GoS/jCmax

P mjMj .GoS/jCmax

Jobs None None pj  1 None pj  1 frac None frac

Lower bound ELm e e e e e GLm GLm

[120] [16] [16] [16] [16] [16] [169] [169]

2 GoS

2

[104, 194] 1 C

2 GoS, pj  1 3=2

Upper bound EUm 5 4 eC1 e e GUm C 1 GUm m2 m m2 kmCk 2

3=2

[193]

[120] [16] [16] [16] [16] [16] [169] [169]

Gap 0:1967 5e 4e 1 0 0 1 0

[194] 1=3 [193] 0

Table 15 Main results on online over time scheduling with machine eligibility Problem

Jobs

Lower bound

Upper bound

P jrj ; Mj jCmax

None res

2 1:5687

4  2=m 3  1=m

pmtn

1C

None res pmtn

2 1:5687 1:125

P 2jrj ; Mj jCmax

P jrj ; Mj .GoS/jCmax

P 2jrj ; Mj .GoS/jCmax

P jrj ; Mj (nested)jCmax

P jrj ; Mj (tree)jCmax

pj D p None res pmtn None res pmtn pj D p None res pmtn None res

p

.m1/.m2/ 2m.2m3/

5C1 2

1:5550 4=3 1:0917 1:5550 4=3 1 p 2 1:5550 4=3 1:148 1:5550 4=3

2 2 2 2

p

5C1 2

2C 2C 2 2 2 1 p 2 2C 2C 2 8=3 7=3

the chapter; please refer to the reference for more details. Online and semi-online scheduling is relatively young when compared to offline scheduling. Yet they have generated tremendous interest and promise to have more results in the future.

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Online Frequency Allocation and Mechanism Design for Cognitive Radio Wireless Networks Xiang-Yang Li and Ping Xu

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 System Model, Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Network and System Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Requests Arrival Following a Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Spectrum Allocation and Mechanism Design with Known , ˇ and  Only. . . . . . . . . . . . 3.1 Upper Bounds on All Online Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Efficient Online Allocation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 More General Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dealing With Selfish Users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Simulation Studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Spectrum Allocation and Mechanism Design with Known Distribution on Requests. . . . 4.1 Semi-Arbitrary-Arrival Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Random-Arrival Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 General Conflict Graph Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Literature Reviews. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The research of authors is partially supported by NSF CNS-0832120, NSF CNS-1035894, National Natural Science Foundation of China under Grant No. 60828003, program for Zhejiang Provincial Key Innovative Research Team, program for Zhejiang Provincial Overseas High-Level Talents (One-hundred Talents Program). Any opinions, findings, conclusions, or recommendations expressed in this chapter are those of author(s) and do not necessarily reflect the views of the funding agencies (NSF, and NSFC). X.-Y. Li • P. Xu Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA e-mail: [email protected]; [email protected]; [email protected] 2253 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 29, © Springer Science+Business Media New York 2013

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Abstract

In wireless networks, we need to allocate spectrum efficiently. One challenge is that the spectrum usage requests often come in an online fashion. The second challenge is that the secondary users in a cognitive radio network are often selfish and prefer to maximize their own benefits. In this chapter, we review results in the literature that address these two challenges, namely, (1) TOFU, semi-truthful online frequency allocation method for wireless networks when primary users can sublease the spectrums to secondary users, and (2) SALSA, an online truthful mechanism when the distributions of user arrivals are known. In TOFU protocol [36], secondary users are required to submit the spectrum bid ˛ time slots before its usage. Upon receiving an online spectrum request, the protocol will decide whether to grant its exclusive usage or not, within at least  time slots of requests’ arrival. It is assumed that existing spectrum usage can be preempted with some compensation. For various possible known information, authors in [36] analytically proved that the competitive ratios of their methods are within small constant factors of the optimum online method. Furthermore, in their mechanisms, no selfish users will gain benefits by bidding lower than its willing payment. The extensive simulation results show that they perform almost optimum: their methods get a total profit that is more than 95 % of the offline optimum when  is about the duration of spectrum usage . In [39], Xu, Wang, and Li proposed SALSA, strategyproof online spectrum admission for wireless networks. They assume that the requests arrival follows the Poisson distribution. Preempting existing spectrum usage is not allowed. They proposed two protocols that have guaranteed performances for two different scenarios: (1) random-arrival case: the bid values and requested time durations follow some distributions that can be learned, or (2) semi-arbitrary-arrival case: the bid values could be arbitrary, but the request arrival sequence is random. They analytically proved that their protocols are strategyproof and are both approximately social efficient and revenue efficient. Their extensive simulation results show that they perform almost optimum. Their method for semi-arbitraryarrival model achieves social efficiency and revenue efficiency almost 20–30 % of the optimum, while it has been proven that no mechanism can achieve social efficiency ratio better than 1=e ' 37 %. Their protocol for the randomarrival case even achieves social efficiency and revenue efficiency 2–6 times the expected performances by the celebrated VCG mechanism.

1

Introduction

The current fixed spectrum allocation scheme leads to significant spectrum white spaces. Several new spectrum management models [28] have been proposed to improve the spectrum usage. One promising technology is the opportunistic spectrum usage. Subleasing is widely regarded as another potential way to share

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spectrum, if we ensure that primary users’ spectrum usage is not interfered with. Assume that there is a set of n secondary nodes V that will share a spectrum channel. Each secondary user vi may wish to pay (by leasing) for the usage of the available spectrum. Previous studies on spectrum assignment (e.g., [25, 38, 42, 43]) often assume that the information of all spectrum requests is known before allocation is made. However, the spectrum usage requests often arrive online and the central authority (typically a primary user) needs to quickly decide whether the requests are granted or not. Here, assume that upon receiving an online spectrum request, the protocol needs to decide whether to grant its exclusive usage or not, within at least  time slots. The rejection typically could not be revoked. Two different approaches of granting the request could be used: preemptive and non-preemptive. If the requests are non-preemptive, then the central authority could not stop the current running request(s) to satisfy this new request. In the first part of the chapter (Sect. 3), we review the results with preemptive requests where the central authority could potentially terminate the current running request(s) to satisfy this new request to potentially make more profits. Assume that requests terminated cannot be restarted or resumed later (so it is also called “abortion”). Terminating a running request ei , the central authority has to pay a penalty that is ˇ > 0 times of the amount paid for the remaining unserved time slots. In the second part of the chapter (Sect. 4), we assume that the central authority could not preempt existing requests to satisfy new requests. Without any known constraint on requests, it was shown in [35–37] that the competitive ratio of any online spectrum allocation method can be arbitrarily bad when preemption is not allowed. This is because any online method has to accept the first coming request (it is possible that no other requests come), therefore miss the following conflicting requests with an arbitrarily large value. In the first part of this chapter, we review the results in [36] on the case when the maximum time requirement is  time slots. Always assume that every request arrives at the beginning of a time slot and the number of time slots requested is an integer t 2 Œ1; . To the best of our knowledge, [35–37] are the first to study online spectrum allocation with cancelation and preemption penalty. Assume that a sequence of spectrum requests arrives online: the i th request ei arrives at time ai , requesting the usage of the spectrum channel for a duration ti (starting at time si ), with a bid bi . The central authority needs to design an online method which, upon receiving the i th bidding request, decides before time si ˛C whether the request will be granted or not, and if it is granted, how much will be charged. Here,  is called delay factor. If allowed, terminating a running request ei , the central authority has to pay some penalty, denoted as .bi ; `i ; ti / D ˇ `tii bi , to the user who initiated the request ei , where ˇ  0 is a constant. In other words, the penalty is proportional to the amount `tii bi paid by the requestor for the service of remaining `i unserved time slots. The main contributions of the protocol TOFU by Xu and Li [36] are as follows. They design several efficient online spectrum allocation methods, called TOFU , that

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perform almost optimum in all cases. When 0  ˇ < 1, their protocol has a constant competitive ratio depending on ˇ. When ˇ > 1, they show that the competitive ratio C1 of any online method is at most O.   /. They then show that their protocol TOFU  C1 achieves the bound .  /. When ˇ D 1, they show that the competitive ratio C1 1=2 of any online algorithm is at most O..   / /. They then show that their online method TOFU has competitive ratios that match the bounds asymptotically, i.e., C1 1=2 ‚..   / /. Their extensive simulations show that their methods have competitive ratios almost 95 % in most cases, even compared with the optimum offline spectrum allocation method that knows all future inputs. They also extend their protocol to a more general case in which the secondary users are distributed arbitrarily and the required service regions are heterogeneous disks. They show that the same asymptotic lower bounds and upper bounds apply to this general model. They also design efficient auction mechanism for spectrum allocation when secondary users are selfish. In such a mechanism, each user will be charged an amount, say pi , if its request ei is granted. They show that in their online mechanism, to maximize its profit, no secondary user will bid lower than its actual valuation. However, a user may have a chance to improve its profit by bidding higher than its actual valuation. They call this property as semi-truthful. Recall that a protocol is called truthful in the literature if no user can lie about its bid to improve its profit. Their methods still achieve similar results when the requested region of each user is a sectoral cell. The main contributions of the SALSA protocol [39] are as follows. For both requests arrival models, they designed an admission and charging protocol, called SALSA (for strategyproof online spectrum admission), such that the expected social efficiency and revenue are within small constant factors of that of the optimum offline method in which all information is known in advance. They also proved that their protocol is strategyproof, which implies that any secondary user cannot improve its profit by lying on his request (bid value and time requirement). They also extended their results to a more general model in which secondary users are distributed in a domain arbitrarily and the conflict graph formed by secondary users is growth-bounded. They showed that their results still hold for this general model. Their extensive simulations show that their methods perform almost optimum, even compared with the optimum offline methods that know all bids in advance. For example, for semi-arbitrary-arrival model, their protocol gets a total profit that is almost 20–30 % of the optimum, while it has been proven in [7] that no mechanism can achieve social efficiency ratio better than 1=e in the worst case. The spectrum utilization is also around 20–40 % for this model. For the random-arrival model, their protocol achieves social efficiency and revenue efficiency that is 2–6 times of that by the celebrated VCG mechanism. The spectrum efficiency is around 90 % even for slightly loaded systems. The rest of the chapter is organized as follows. In Sect. 2, we define in detail the problems to be studied. Section 3 reviews the TOFU protocol when the central authority only knows the time ratio . Specifically, in Sect. 3.1, we review upper bounds on the competitive

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ratios of any online allocation methods. Then, we review their solutions in Sect. 3.2 and their analytical proof on the performance bounds of their methods. Their methods for networks with general conflict graphs are discussed in Sect. 3.3. We review their mechanisms to deal with selfish users in Sect. 3.4. We summarize their simulation studies in Sect. 3.5. Section 4 reviews the protocol SALSA when the central authority knows some distribution information about the future requests. Specifically, in Sect. 4.1, the SALSA protocol with semi-arbitrary-arrival is presented, and Sect. 4.2 presents the SALSA protocol for the case with random-arrival. Section 4.3 reviews the SALSA protocols to networks with growth-bounded conflict graphs. The simulation results are reported in Sect. 4.4. The related work is reviewed in Sect. 5. We conclude the chapter in Sect. 6.

2

System Model, Problem Formulation

2.1

Network and System Models

Consider a wireless network consisting of some primary users U D fu1 ; u2 ;    ; up g who hold the right of some spectrum channels for subleasing. There is a central authority who decides the spectrum assignment on behalf of these primary users. In other words, here assume that each primary user will trust the central authority and is satisfied with what he/she will get from the auction based on the mechanism designed. If each primary user has an asking price for its spectrums, we need to design mechanisms (such as double auction in which buyers enter competitive bidders and sellers enter competitive offers simultaneously) that also take into account the selfish behavior of primary users. Note that Wang et al. [31] proposed some truthful double-auction mechanisms with known distributions on bids. The wireless network also consists of some secondary users V D fv1 ; v2 ;    ; vn g. Those secondary users may lease the spectrum usage in some region for a time period at any time. In this chapter, assume a simple scenario where there is only one channel available. It is interesting to design allocation and auction schemes when multiple channels are available and a secondary user may request for a number of channels at same time. Secondary users may reside at different geometry locations. Here, assume that each request is associated with a disk region. Whether a secondary user’s request of channels conflicts with the request of another secondary user depends on their locations, in addition to the required time period. This location-dependent conflict will be modeled by a conflict graph H D .V; E/, where two nodes vi and vj form an edge .vi ; vj / in H if and only if they cannot use same channel simultaneously. We will first study the networks with a complete conflict graph and then extend the methods to cases when requested disks are of arbitrary sizes. We will show that the methods have asymptotically optimum performance guarantees in both cases.

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Problem Formulation

Assume that secondary users V D fv1 ; v2 ;    ; vn g will ask for spectrum usage on the fly. A user could ask for spectrum usage at different time slots. Let e1 ; e2 ;    ; ei ;    be the sequence of all requests. Each request ei D .v; bi ; ai ; si ; ti / is claimed by a secondary user v at time ai , who bids bi for the usage of the channel from time si to time si C ti . Here, we omit the region requirement since this chapter will first address networks with complete conflict graph. For most discussions, we will omit the user v when it is clear from the context. In other words, ei D .bi ; ai ; si ; ti / denotes the i th request. Obviously, ai  si for all requests, which means only spectrum usage in future can be requested. Assume time is slotted and time requirement ti must be an integer. Assume that every user will ask for the spectrum usage for at most  time slots (ti   for all i ). They call  the time bound. Since the objective for spectrum auction/lease is to improve spectrum utilization and improve the revenue from the spectrum auction/lease, the central authority (i.e., spectrum auctioneer) can post a requirement that all requests must be submitted in advance of a certain time slots. In this chapter, assume that for every request ei , si ai  ˛ for a value ˛ given by the auctioneer. Hereafter, call ˛ the advance factor. Each request is claimed at least ˛ time slots before its required usage. Intuitively, a larger advance factor ˛ will give more advantage to the central authority. Later, we will show that the asymptotic performances of the methods do not depend on ˛ as long as ˛   , the delay factor. The advance factor puts some constraints on the secondary users on when they should submit their bids. To be fair, the system will also put some condition on the central authority about when the central authority should make a decision on each request. In this work, it is always required that the central authority should make a decision on a request within  time slots of its arrival time. Call  delay factor in the spectrum allocation problem. Obviously, they need   ˛ to make the system meaningful. When  D 0, central authority has to make decision immediately on request ei at time ai . When  D ˛, central authority could make decision as late as possible. If a request has been rejected, it will never be reconsidered and accepted later. Figure 1 illustrates the model of advance and delay factor. For each request, its arrival time ai must be before time t1 in the figure, and central authority must make a decision before time t2 in the figure. Since central authority is not able to estimate accurate price for spectrum usage, the model lets the central authority accept a reservation in advance and lets the secondary user get a reasonable guarantee. Crucially, to improve the spectrum usage and revenue, let the central authority terminate a reserved/running request at a later time for new request with higher priority at any time. Cancelation and preemption is necessary to take advantage of a spike in demand and rising prices for spectrum channels and not be forced to sell the spectrum below the market because of an a priori contract. When current spectrum usage is terminated, penalty should be paid to compensate the preemption. Here, assume that the penalty .b; `; t/ is a linear function of the unfinished time `  t of that request e.v; b; a; s; t/, i.e., .b; `; t/ D ˇ `t b for a constant ˇ  0. Notice that the cancelation is modeled as ` D t since the spectrum usage was not started at all. Figure 2 illustrates the unfinished time `

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t2 ei = (bi, ai, si, ti)

γ t1

α

ti

si

Fig. 1 Advance factor ˛ and delay factor 

e

e = (b, a, s, t) t l

Fig. 2 Unfinished time ` when e is preempted by e0 Table 1 Symbols used in this chapter

Symb. Meaning

Symb. Meaning

ei ai ti  

bi si `i ˛ ˇ

Request Arrival time Required service time Time bound Delay factor

Bid value Start time Unfinished service time Advance factor Penalty factor

when a request e is preempted by another request e0 . Here, the constant ˇ  0 is the penalty factor. Table 1 summarizes the symbols used in this chapter. As stated before, all requests arrive on the fly, thus any information about the future, e.g., the distribution of future bids, the arrival times, the start times, and the required time durations, are all unknown. The objective of the spectrum allocation is to find an allocation which maximizes the total net profit, i.e., the total payment collected minus the total penalties caused by preemption and cancelation. To evaluate the performance of an online algorithm, it is usually compared with an optimal offline algorithm. The following lemma indicates the hardness of the offline version of the problem. Lemma 1 Even knowing all requests, it is NP-hard to find an optimal spectrum allocation that maximizes the total bids of admitted requests. Proof Observe that the maximum weighted independent set (MWIS) problem in geometry graph is a special case of the spectrum allocation problem: assume that all spectrum requests are for the same time duration, the graph for MWIS problem is the conflict graph H , and the weight of a node is the bid value of the spectrum request by this node. Recall that MWIS in geometry graph itself is an NP-complete problem [24]. Thus, the lemma follows. 

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For all online spectrum allocation problems studied here, online algorithms with constant competitive ratios will be designed if advance factor is chosen carefully. The performance of an online algorithm A for spectrum allocation is measured A.I/ by its competitive ratio %.A/ D minI OPT.I/ , where I is any possible sequence of requests arrival, A.I/ is the profit produced by online algorithm A on input I, and OPT.I/ is the profit produced by optimum offline algorithm OPT on input I when the sequence I is known in advance by OPT. In this work, when we study the competitive ratio, we assume all users are truthful. Given a sequence of spectrum requests, at any time instant t, an online spectrum allocation method only knows all spectrum requests that arrived from time t   to time slot t. On the other hand, for the same sequence of spectrum requests, although the optimum offline method also has to make decision for a request within  time slots of its arrival, it does know all future requests it will get, thus makes a smarter decision. As we will see later that the performances of methods and the lower bounds on the performance of any online algorithm vary when the penalty factor ˇ, the advance factor ˛, and the delay factor  are different, for certain parameters ˇ, ˛, and  , call it .ˇ; ˛;  / problem in this chapter.

2.3

Requests Arrival Following a Distribution

In this chapter, two different scenarios will be reviewed. The first scenario assumes that the mechanism designer does not know any distribution information about requests, except the delay factor  , the penalty factor ˇ, the advance factor ˛, and the time ratio . Section 3 will focus on this scenario. The second scenario assumes that the mechanism designer does know some distribution information about the arrived requests, such as the expected bid values by all requests. Section 4 will focus on the second scenario. In the second scenario, it is assumed that the true bid values bi and time requirement ti are always independent variables. This assumption is reasonable in practice since bi is the amount a user is willing to pay for a unit of time. Also assume that arrival time ai is memoryless, i.e., the distribution of arrival time is Poissonian. The arrival rate  is not necessarily known in advance. In this chapter, two different cases, random-arrival case, and semi-arbitraryarrival case, will be studied. The simplest case is random-arrival case, in which the true bid values bi and required time slots ti are sampled i.i.d. from some stationary distributions that are not necessarily known to the central authority in advance. For example, true bid values bi of all requests could be randomly and independently drawn from a normal distribution with mean  and variance , and the values of  and  could be known or unknown to the central authority in advance. Or the central authority may also know that the arrival of requests follows the Poisson process with a rate , although the value of  may be unknown to the central authority in advance. Although motivated by this i.i.d. model, it may be more robust to work in semiarbitrary-arrival model. This model also implies some fundamental limits to this online spectrum allocation problem. Assume that bid values bi , and required time durations ti are generated by an adversary who wants to beat these protocols. Let n

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be the number of requests that will be posted. The adversary generates a (potentially random) set A D fa1 ; a2 ;    ; an g of arrival times that are produced from Poisson distribution with unknown rate and a set T D ft1 ; t2 ;    ; tn g of required time durations that are produced from an unknown distribution. And the adversary could generate an arbitrary set B D fb1 ; b2 ;    ; bn g which is not necessarily sampled from any stationary distributions. After these three sets are generated, the bid values, arrival times, and required time durations are matched together using a random permutation, which is not controlled by the adversary. Call this the random ordering hypothesis. The hypothesis is reasonable since we concern the expected performance of these mechanisms. When this matching is also controlled by the adversary, we can present examples of requests such that, for any online spectrum allocation and auction method, the expected social efficiency or revenue could be arbitrarily bad comparing with that of offline methods. The protocols need to compute how much each request needs to pay. When a request is accepted, the secondary user who issued the request will be granted the usage of channel, and will need to pay some monetary value for this. We will design mechanisms to compute the payment by secondary users. Let vi be the true valuation of the spectrum usage requested by ei and pi be the payment that request ei should pay. Then, the final profit (or the utility) of request ei is U.i / D xi  vi  pi . When ei is admitted, xi D 1; otherwise, xi D 0. Notice that bi ti D vi when secondary user issued request ei truthfully. In the online auction, the secondary users seek to place requests to maximize their individual utility in equilibrium. As a standard approach, in this chapter, we will only focus on direct revelation mechanisms in which secondary users directly submit their bid values and time requirement (although not necessarily truthfully). We assume that each secondary user cannot announce an earlier arrival time than its true arrival. At every time t, given the set of submitted requests that are not processed yet, the central authority decides an allocation xi . ; t/ 2 f0; 1g and a payment pi . ; t/ for request ei . We place the following natural conditions on the allocation and payments: allocations cannot be revoked, so xi . ; t/ is a nondecreasing function of t. The allocation and payments must be online computable, in that they can only depend on information available at time t. We are interested in mechanisms with dominant-strategy equilibrium, such that every secondary user has a single optimal strategy regardless of the strategies of others. This is a particularly robust solution concept. Notice that a secondary user may lie lower or lie higher its bid value bi (compared with its true bid value b 0 i ) and can only lie its time requirement ti to a larger value. Definition 1 A mechanism is value-strategyproof and time-strategyproof if a secondary user’s dominant strategy is to report its true bid value (called value-SP) and true time requirement (called time-SP). Competitive ratio is always used to evaluate the performance of an online algorithm. Most of the previous work focused on the worst-case competitive ratio of the online algorithm; however, the probability that the worst case will happen is very small in practice. Thus, we study the expected competitive ratio instead, which

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could better reflect the practical performance of an online algorithm. The average competitive ratio of an online algorithm is defined as the ratio between its average performance and the average performance of the optimal offline algorithm for all possible inputs. We focus on the average competitive ratio since it is easy to prove that without any known constraint, the worst-case competitive ratio of any online method can be arbitrarily bad. To evaluate the performance of an online mechanism, we usually compare it with offline Vickrey mechanism which will maximize the total social efficiency (but not necessarily strategyproof in the online model). Notice that offline mechanism knows all future inputs before it makes decisions, which surely improves the performance significantly. The performance of an online mechanism M is measured by its social efficiency Effs .M/ and revenue efficiency Effr .M/. The social efficiency of mechanism M is defined as the total true valuations of all winners, i.e., Effs .M/ D P i xi vi . For a mechanism M, let pi be the payment collected from secondary request ei . And the revenue efficiency of mechanism M is simply Effr .M/ D P p . The benchmark mechanisms that we adopt for the purpose of competitive i i analysis are the offline Vickrey mechanisms, denoted as V i ck. Notice that for offline Vickrey mechanism, it will always select a set of winners that maximize the social efficiency. Definition 2 The social efficiency competitiveness of an admission mechanism M is defined as the ratio of the total valuations of winners under mechanism M over the total valuations of winners under offline Vickrey mechanism. In other words, Effs .M; V i ck/ D Effs .M/=Effs .V i ck/. Similarly, the revenue efficiency competitiveness of mechanism M is defined as the ratio of the total payments of winners under mechanism M over the total payments of winners under offline Vickrey mechanism. In other words, Effr .M; V i ck/ D Effr .M/=Effr .V i ck/. 1 We say that mechanism M is Eff .M;V i ck/ -competitive for social efficiency and 1 Effr .M;V i ck/ -competitive

s

for revenue efficiency.

The objective is to design online strategyproof spectrum admission mechanisms in different cases which maximize the expected social efficiency and revenue efficiency. The expectations are computed over all possible permutations of bid values bi and all possible permutations of time requirements.

3

Spectrum Allocation and Mechanism Design with Known , ˇ and  Only

3.1

Upper Bounds on All Online Methods

In this section, we present upper bounds on the competitive ratios of online allocation methods in cases depending on whether ˇ is less than 1, equal to 1, or larger than 1. These bounds will be proved to be almost tight later.

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3.1.1 Upper Bound for .ˇ < 1; ; ˛/ Problem We first show upper bounds on the competitive ratios when ˇ < 1. Note that in this case, we assume that every preempted job must be executed for at least one time slot. The main approach of the proofs in this section is adversary argument: an adversary carefully constructs sequences of requests to make the performance of online algorithms as bad as possible. The trick is that the adversary can choose the future requests based on the current decision of online algorithms. Then, no matter what decision is made, certain performance cannot be achieved. Theorem 1 Let D

1 cˇ .

No online algorithm has a competitive ratio > for .ˇ
1=ˇ.

3.1.2 Upper Bound for .ˇ D 1; ; ˛/ Problem When the penalty factor ˇ D 1, there are two different cases:  D o./ or  D ./. (1) When  D o./, i.e., lim  D 0, we first show that no online algorithm p 1 can achieve competitive ratio more than 3 2. C 1/ 3 for .1; ˛;  / problem. p 1 Then, we improve this bound to 2. C 1/ 2 . (2) When  D ./, we show that .1; ˛;  / problem cannot have a competitive ratio 1  for an arbitrary > 0. p 1 Theorem 2 ([37]) No online algorithm has a competitive ratio > 3 2. C 1/ 3 for .ˇ D 1; ; ˛/ problem when  D o./. Theorem 3 ([37]) No online algorithm has competitive ratio > 1c for .ˇ D 1; ; ˛/ p p 1 1 problem, where 2. C 1/ 2 < 1c  3 2. C 1/ 3 for .ˇ D 1; ; ˛/ problem when  D o./. Theorem 4 ([37]) No online algorithm has a competitive ratio 1  for an arbitrary small > 0, for .ˇ D 1; ; ˛/ problem when  D ./.

3.1.3 Upper Bound for .ˇ > 1; ; ˛/ Problem For .ˇ > 1; ; ˛/ problem, we show that upper bound of competitive ratios is C1 O.   / when  D o./. Theorem 5 ([37]) No online algorithm has competitive ratio more than for .ˇ > 1; ; ˛/ problem, when  D o./.

2ˇ  C1 .ˇ1/2 

A surprising result from the analysis in this section is that the performance upper bounds do not depend on the advance factor ˛. In other words, no matter how many time slots the secondary users claim their requests in advance, the theoretical upper bounds will not be improved asymptotically if the delay factor  does not change.

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Efficient Online Allocation Methods

In this section, we present several methods for efficient online spectrum allocation. We then analytically study the performances of the methods and prove that they have asymptotically best competitive ratios. All results in this section assume that the conflict graph of the network is a complete graph.

3.2.1 Method K for .ˇ < 1; ; ˛/ Problem When ˇ < 1, whenever we admit a request ei at time t, even we preempt it at next time slot t C 1, we still receive a profit .1  ˇ/bi . Thus, we will adopt the following strategy K: at any time t, let ei be the request from the set of all currently available requests, denoted as Ra .t/, with the largest bid bi among all requests whose starting time is t   C ˛. We admit request ei at time t. It is easy to prove the following Lemma. Lemma 2 Strategy K is at least .1  ˇ/-competitive.

3.2.2 Method G for .ˇ D 1; ; ˛/ Problem We then consider the case ˇ D 1, i.e., the preemption penalty is exactly the remaining portion of the bid. Since we do not know anything about the future, to maximize the total profits, we shall accept request with large bid or large bid per time slot (called density) intuitively. The main difficulty is the trade-off between requests with large bid and requests with large density. The results on upper bounds of competitive ratio in Sect. 3.1 illustrate some intuitions. In this and following subsection, we design online algorithms whose competitive ratios match corresponding upper bounds. Consider current time t, and let Ra .t/ be all spectrum requests submitted before time t, which are not processed. Based on the delay requirement, we know that all requests in Ra .t/ must be submitted during the time slots Œt  ; t/, and their starting times must be in the time interval Œt   C ˛; t C ˛. Among all spectrum requests in Ra .t/, let R.t/  Ra .t/ be all spectrum requests whose starting times are in the time interval Œt; t C  . Recall that we always have   ˛. The online algorithms will only make decisions at time t using the information about requests R.t/, although it knows a superset of requests Ra .t/. See Fig. 3 for illustration, where Ra .t/ D fe1 ; e2 ; e3 ; e4 ; e5 g and R.t/ D fe1 ; e2 ; e3 g. We will show that this method can achieve a competitive ratio that is already asymptotically optimum. In other words, knowing the information ˛ will not enable us to get a method with a better competitive ratio asymptotically. Candidate Requests Set For requests R.t/, we will find some subsets, called candidate requests set, using dynamic programming to optimize some objective functions. This method will then make decisions on whether to admit these subsets of requests under some conditions involving the currently running spectrum usages.

Online Frequency Allocation and Mechanism Design Fig. 3 All requests Ra .t / and a subset of requests R.t / with starting time in Œt; t C  (denoted by green color)

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γ

e4

γ

e2

γ

R(t)

e1 e3

α

Ra(t)

γ t−γ

t

e5 t+γ

e4 b2 = 12 b1 = 10 C1(t)

b3 = 13 e5

Fig. 4 Strong candidate set C1 .t / at time t (color green)

t

t+α−γ

t+α+Δ

Definition 3 Candidate Requests Set: (1) A strong candidate requests set at time t, denoted as C1 .t/, is a subset of requests from R.t/ that has the largest total bids if C1 .t/ were allowed to run from time t   C ˛ to time at most t C ˛ C . See Fig. 4 for illustration where C1 .t/ D fe1 ; e3 g  R.t/ since the total profit during time interval Œt   C ˛; t C ˛ C  is maximized. We abuse the notation little bit here by also letting C1 .t/ denote the profit made by C1 .t/. Also let P.C1 .t/; t 0 / denote the profit made from C1 .t/ if C1 .t/ are admitted and then possibly being preempted at a time slot t 0 2 Œt   C ˛; t C ˛ C . (2) A weak candidate requests set at time t, denoted as C2 .t/, is a subset of requests from R.t/ that has the largest total bids if C2 .t/ were allowed to run during time interval Œt   C ˛; t C ˛ (assume they are preempted at time t C ˛ C 1). We abuse the notation little bit here by also letting C2 .t/ denote the profit made by C2 .t/.

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Algorithm 1 Find C1 .t/ by dynamic programming Input: e1 ; e2 ;    ; en.t/ such that s1  s2      sn.t/ . Output: Strong candidate set C1 .t /. 1: for i D 1 to n.t / do 2: for j D 1 to i do Q.j; si / D maxk2Œ1;n fQ.k; sj / C bj  .ek ; sj /g 3: 4: C1 .t / D maxk2Œ1;n fQ.k; sn.t/ /g

Briefly speaking, C1 .t/ maximizes the total bids at time t, and C2 .t/ maximizes the total bid per time slot in the predictable future (since the requests are claimed in advance). Observe that the starting times of candidate sets are in the interval Œt; t C ˛. Since   ˛, at time t, we should know all requests whose starting times are from t to t C  . So we could find the candidate requests set C1 .t/ and C2 .t/ by dynamic programming as following. Consider requests e1 ; e2 ;    ; en in the increasing order of their starting times, i.e., t   C ˛  s1  s2      sn  t C ˛. When we compute the strong candidate set C1 .t/, we will not consider any requests that are currently running or have been rejected before. Additionally, for the scheduling of requests in C1 .t/, we may use one request to preempt another request in C1 .t/. Let Q.j; si / denote the maximum possible profit made during time interval Œt; t C C from R.t/ when ej is the last accepted request before or at time si . If the starting time sj of ej satisfies sj > si , Q.i; sj / D 0 since it is not a feasible solution. Then, C1 .t/ D max Q.k; sn /: k2Œ1;n

For each i and j such that i  j , we have Q.j; si / D max fQ.k; sj / C bj  .ek ; sj /g: k2Œ1;n

Here, .ek ; sj / is the paid penalty when request ek is preempted at time sj . Algorithm 1 summarizes the method finding C1 .t/. For each i and j , the algorithm takes O.n.t// time to compute Q.j; si /. Therefore, it takes O.n.t/3 / time to find C1 .t/ where n.t/ is the total number of spectrum requests in R.t/. The method can be easily extended to find C2 .t/ with same running time. The Greedy Algorithm G We then propose an online algorithm G. At each time t, the input of our algorithm G is Ra .t/. Algorithm G should decide whether a request that arrived at time slot t   will be accepted or rejected. The basic idea of our method is as follows. Assume that time 0 is the reference point for time. First, before time  , we do not need to make decisions on any arrived requests. At time  , we need to decide whether to accept/reject requests that arrive at time 0. If channel is not occupied, we will simply accept the request from C1 . / with the earliest starting time being ˛, if there is any. We call C1 . / as virtual meta job requests currently being accepted.

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For any time t >  , our method will choose either a request from the strong candidate request C1 .t/, or a request from weak candidate request C2 .t/, or continue to run the request (which could be empty) chosen previously. The method deals with three complementary cases about the channel status at time t   C ˛, separately. Case 1: Non-occupied Channel If the channel will be empty at time t   C ˛, we find C1 .t/ with the maximum overall profit in R.t/. We accept the request in C1 .t/ with starting time t   C ˛, and we say that the channel is being used by candidate requests set C1 .t/. In other words, here we treat C1 .t/ as a large virtual meta spectrum request. Note that at current time slot t, we only admit the first spectrum request from C1 .t/ while leave the admission decisions on other requests in C1 .t/ pending. Whether these pending “admitted” requests will be actually admitted depending on future coming requests, if future coming requests are better, we will preempt this virtual meta request C1 .t/.1 Thus, those preempted pending admissions from C1 .t/ will not be processed at all. Case 2: Occupied by Weak Candidate Requests If the channel will be used by a weak candidate requests set C2 .t/ at time t   C ˛ and this candidate requests set C2 .t/ weakly preempted (exact definitions will be given later) some other candidate requests set before, all requests from Ra .t/ submitted at time t   will not be admitted. Case 3: Occupied by Strong Candidate Requests The last case is that the channel will be used by a strong candidate requests set C1 .t/ at time t   C ˛. Assume the request to be run at time t   C ˛ is ej from some virtual candidate requests set C1 .t1 /. There are three possible steps in this case. (1) We first find the strong candidate requests set C1 .t/. The first request ei 2 C1 .t/ such that si D t   C ˛ will be accepted only if C1 .t/  c1  C1 .t1 /;

(1)

where c1 > 1 is a constant parameter. In other words, we use a virtual meta request C1 .t/ to replace another virtual request C1 .t1 / if the potential profit from new virtual meta request is sufficiently larger. We call it a strong-preemption. (2) If strong-preemption cannot be applied, we find C2 .t/. The request ei 2 C2 .t/ such that si D t   C ˛ will be accepted only if C2 .t/ C P.C1 .t1 /; t   C ˛/  c2  C.t1 /;

(2)

where c2 > 0 is an adjustable control parameter. In other words, we use a virtual candidate requests set C2 .t/ to replace another virtual candidate requests set C1 .t1 / if the profit from this new meta request and the profit from the meta request C.t1 / being preempted at future time t   C ˛ are at least a constant

Preempting C1 .t / at a time, say t 0 , will be implemented by preempting the requests from C1 .t / that are supposed to be running at and after time t 0 .

1

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Algorithm 2 Online spectrum allocation G Input: A constant parameter c1 > 1, an adjustable control parameter c2 > 0, , ˛, , Ra .t /, R.t /, C1 .t /, and C2 .t /. Current candidate requests set C from time t 0 < t . Here, C D C1 .t 0 / if C1 .t 0 / strongly preempted others, or C D C2 .t 0 / if C2 .t 0 / strongly preempted others. Output: Whether requests submitted at time t   will be admitted and new current candidate requests set C . 1: if C D C2 .t 0 / then 2: if t  t 0   then C D ;; 3: 4: else 5: Accept request ei 2 C2 .t / such that si D t   C ˛. 6: if C D C1 .t 0 / or ; then 7: if C1 .t /  c1  C1 .t 0 / then C D C1 .t /; 8: 9: Accept request ei 2 C1 .t / such that si D t   C ˛. 10: else if C2 .t / C P .C1 .t 0 /; t /  c2  C1 .t / then C D C2 .t /; 11: 12: Accept request ei 2 C2 .t / such that si D t   C ˛. 13: else 14: Accept request ei 2 C1 .t 0 / such that si D t   C ˛.

c2 fraction of the profit by keeping running the meta request C.t1 /. We call it a weak-preemption. In this subcase, all requests in C2 .t/ will be accepted and the last request will be terminated at time t C ˛ C 1 automatically. (3) If the weak-preemption cannot be applied also, we accept the request in the previously used candidate requests set C.t1 /, whose starting time is t   C ˛ (if there is any) or continue the request ej that will run through the time slot t   C ˛. Algorithm 2 summarizes the online spectrum allocation method G. Theoretical Performance Study We then show that the algorithm G is asymptotically optimal if we choose constant c1 and control parameter c2 carefully. We use G.I/ to denote the profit made on requests sequence I by the algorithm G. Also use OPT.I/ to denote the profit made by the optimal offline algorithm. To analyze the performance of the method, we first give a definition candidate sequence. Definition 4 Candidate Sequence: A candidate sequence is a sequence of candidate requests sets C1 .ti /, C1 .ti C1 /,    , C1 .tj 1 /, C2 .tj / or C1 .ti /, C1 .ti C1 /,    , C1 .tj 1 /, C1 .tj / which satisfies all of following three conditions: (1) Strong candidate requests set C1 .ti / does not preempt another candidate requests set. (2) Strong candidate requests set C1 .tkC1 / strongly preempts strong candidate requests set C1 .tk /, for k D i;    ; j  2.

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(3) Weak candidate set C2 .tj / weakly preempts strong candidate set C1 .tj 1 /, or a strong candidate set C1 .tj / strongly preempts C1 .tj 1 / and C1 .tj / is not preempted by another requests set. Here, we use the parameters of the first and last candidate requests set to denote a candidate sequence, e.g., S.ti ; tj /. According to the definition, we can decompose the solution of algorithm G, i.e., fe1 ; e2 ;    ; em g, into multiple candidate sequences. Notice that each spectrum request e will appear in exactly one candidate sequence. We use G.S.ti ; tj // to denote the profit made on candidate sequence S.ti ; tj / by algorithm G. And we use OPTŒti ; tj  to denote the profit made by optimal offline algorithm on the requests whose starting times are in interval Œti ; tj . Lemma 3 For each candidate sequence S.si ; sj / in the solution given by algorithm G, we have G.S.si ; sj //  min.c1 ; c2 /  C1 .sj 1 /: Proof By definition of candidate sequences, there are two different cases: (1) C1 .sj / strongly preempted C1 .sj 1 / or (2) C2 .sj / weakly preempted C1 .sj 1 /. If request C1 .sj / strongly preempted C1 .sj 1 /, G makes at least C1 .sj /  c1  C1 .sj 1 /. If request C2 .sj / weakly preempted C1 .sj 1 /, G makes at least P.C1 .si 1 /; sj / C C2 .sj /  c2  C1 .sj 1 /. So the lemma holds for either case.  Lemma 4 ([37]) For each candidate sequence S.si ; sj / in the solution given by algorithm G, for each i  k < j , we have c2

!

C1 .sk /: OPTŒsk ; skC1   c1 C c2 C p .fl C 1/= Lemma 5 For each candidate sequence S.si ; sj / in the solution given by algorithm G, we have  OPTŒsi ; sj   c1 C 1 C

1 c1  1



c2

!

c1 C c2 C p C1 .sj1 /: .fl C 1/=

Pj1 Proof Obviously, OPTŒsi ; sj  D kDi OPTŒsk ; skC1 . From Lemma 4, we c2 C1 .sk /. Thus, OPTŒsi ; sj   have OPTŒsk ; skC1   c1 C c2 C p.flC1/= P  j1 c2 c1 C c2 C p.flC1/= kDi C1 .sk /. Based on the condition of strong-preemption,  we have C1 .sk /  c1  C1 .sk1 / for all i  k  j . The lemma then follows. p 1 Theorem 6 Algorithm G is ‚.  C 1 2 /-competitive. Proof Given a request sequence I, we decompose the solution by algorithm G into multiple candidate sequence S.ti ; tj /. Then, the total profit made by algorithm G is

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P

i ; tj /. On the other hand, the total profit made by the optimal offline 8S.ti ;tj / S.tP algorithm is 8S.ti ;tj / OPTŒti ; tj . From Lemmas 3 and 5, the competitive ratio of algorithm G is at least

P P

8S .ti ;tj /

S .ti ; tj /

8S .ti ;tj / OPTŒti ; tj 



.c1 C 1 C

min .c1 ; c2 / C c2 C

1 /.c1 c1 1

p

c2 /: . C1/=

It is easy to show that the right-hand side gets the maximum value c1 D 2 and c2 D     1.

p 2 . 1C =. C1/

p1 4.1C =. C1//

Thus, the competitive ratio is at least

1 8

q

 C1 

if

when 

Let n.t/ be the cardinality of Ra .t/. At time t, Algorithm 2 takes O.n.t/3 / time to find C1 .t/ and C2 .t/, then P takes constant time to make decision. The time complexity of Algorithm 2 is 8t n.t/3 . Notice that each request could appear in at most  C  different Ra .t/. Let n be the number of all online requests. We have P 8t n.t/  . C  /n and n.t/  n for all t, which implies following theorem. Theorem 7 Time complexity of Algorithm 2 is O.. C  /n3 /.

3.2.3 Method H for .ˇ > 1; ; ˛/ Problem In this subsection, we propose a greedy algorithm H for .ˇ > 1; ; ˛/ problem. At time t, the method H should decide whether a request whose start time is t C ˛   will be accepted. Then, we show that algorithm H is asymptotically optimal. The notations are used in previous subsection. Theorem 8 Algorithm H is

.cˇ1/  C1 -competitive. C C1 c2

Proof We divide the requests accepted by algorithm H into multiple candidate requests sets. Let us consider the profit made by H on each candidate requests set C1 .t/. For each admitted C1 .t/, we redistribute .1 C ˇ/C1 .t 0 / of the money from C1 .t/ to C1 .t 0 / if C1 .t/ preempts C1 .t 0 /. There are three complementary cases here. Case 1: C1 .t/ does not preempt others and is not preempted. H makes C1 .t/ profit on C1 .t/. Let ts D t C ˛   . For the optimal offline algorithm OPT, if time ts C i. C 1/ (for some integer i > 0) is before the end time of C1 .t/, we have OPTŒts C i.fl C 1/; ts C .i C 1/.fl C 1/  c  C1 .t//. Otherwise, the requests whose starting times are during time interval Œts C i. C 1/; ts C .i C 1/. C 1/ should preempt C1 .t/. We know that the total running time of C1 .t/ is no more C1 than  C  C 1. Thus, no more than cd C  C1 eC1 .t// profit will be made by OPT during the running time of C1 .t//. Case 2: C1 .t/ does not preempt others and is preempted in H. H makes C1 .t// profit on C1 .t/ since the candidate requests which preempted C1 .t/ will distribute .1 C ˇ/C1 .t/ profit to C1 .t/. And at most ˇC1 .t/, profit is paid as penalty.

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C1 No more than cd C  C1 eC1 .t// profit will be made by OPT during the running time of C1 .t//. Case 3: C1 .t/ preempts another candidate requests set in H. H makes at least cˇ1 C1 .t/ profit from C1 .t/ (after considering redistributing some to the candic date requests set being preempted by C1 .t/) since we distribute .1 C ˇ/C1 .t 0 /  1Cˇ C1 .t/ to the candidate set preempted by C1 .t/. If C1 .t/ is not preempted, we c make at least cˇ1 C1 .t/. If C1 .t/ is preempted, we will receive .1 C ˇ/C1 .t/ c from the future candidate requests set that preempts it. Thus, we always make at least C1 .t/. For OPT, it is easy to show that OPT will make no more than C1 cd C  C1 eC1 .t/ profit.

Thus, the competitive ratio for each candidate requests set is at least  C1 The competitive ratio of H is at least .cˇ1/ C C1 . We finish c2 the proof.  .cˇ1/  C1 d C C1 e. c2

When  D a  1, it is easy to show that Theorem 9 Method H is at least ˇ/), when  D a  1.

3.3

a -competitive 4.1Ca/.1Cˇ/

(by choosing c D 2.1 C

More General Networks

All studies in previous sections assumed that the conflict graph of networks, which models the location-dependent conflict, is a complete graph. In this section, we extend the algorithms for the networks where each request ei asks for spectrum usage in a disk-shaped region Di centered at .xi ; yi / with a radius ri . For the performance upper bounds, all Lemmas and Theorems in Sect. 3.1 still hold since complete conflict graph is a special case. If we can find the strong (weak) candidate sets C1 .t/ (C2 .t/) in polynomial time as we did by using Algorithm 1 in Sect. 3.2.2, then the method G (or H) still works since all previous analysis still holds. However, the main difficulty is that, in this case, the central authority may accept multiple requests at same time, which makes it hard to get the strong/weak candidate sets in polynomial time by using dynamic programming. Therefore, we propose to approximate strong/weak candidate sets at each time t in polynomial time by using shifting strategy [24] as follows. We divide the request disks into different levels according to their radii. At same level, the radii of all request disks are within a constant factor  of each other. Let d be the diameter of smallest request disk; level i includes all request disks whose diameters are within Œdi 1 ; di . First, we show that a PTAS can be achieved in each level i . In other words, here we approximate the strong/weak candidate sets when only request disks in level i are considered. Let k > 1 be an integer. At level i , a .p; q/ partition is defined as the region is partitioned by a set of horizontal lines,y D    ; 2i k C p; i k

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Cp; p; i k C p; 2i k C p;    , and a set of vertical lines, x D    ; 2i k C q; i k C q; q; i k C q; 2i k C q;    . In each partition, the region is partitioned into a number of i k  i k squares. We ignore the requests whose disks are hit by any lines of the partition and find the strong/weak candidate sets in each i k  i k square. Together with strong/weak candidate sets in all squares, we get strong/weak candidate sets for a partition. Lemma 6 Strong/weak candidate sets for each partition can be computed in polynomial time. Proof According to area argument, the number of independent requests that can be accepted at same time is no more than C D 42 k 2 = which is a constant. Then, we can find the strong/weak candidate sets in each i k  i k square in polynomial time by using dynamic programming as we did in Algorithm 1 . Assume na;b .t/ requests located in a i k  i k square whose top left corner is .a i k C p; b i k C q/ in a .p; q/ partition at time t. The main difference is that we need to define Q.e1 ; e2 ;    ; eC ; s/ as the maximum possible profit from R.t/ when e1 ; e2 ;    ; eC are the last independent requests accepted by central authority before or at time s. Then, C1 .t/ D max Q.e1 ; e2 ;    ; eC ; max s/;  .t / D where max s the latest start time among R.t/. Since there are at most nCa;bC1   O.na;b .t/C C1 / different Q, and each Q can be computed within na;bC.t / D O.na;b .t/C / time, the time complexity to compute strong/weak candidate sets in a kk square whose left top corner is .akCi; bkCj / at time t is O.na;b .t/2C C1 /. Let n.t/ denote the total number of requests in R.t/. The time complexity to compute strong/weak candidate sets in all i k  i k squares of a partition .p; q/ at time t is no more than O.n.t/2C C1 /.  In each partition, some requests may be ignored. To achieve a better approximation, we shift the horizontal and vertical lines and compute strong/weak candidate sets for k 2 partitions where p 2 Œ1; k and q 2 Œ1; k. We set the best strong/weak candidate sets among all partitions as the strong/weak candidate sets of time t. It takes O.k 2 n.t/2C C1 / time to find strong/weak candidate sets at each time. Then, the total running time is at most O.k 2 n2C C2 / where n is the number of total requests since we have to compute this at most n times. On the other hand, the strong/weak candidate sets we find are at least a 1  k2 approximation as following lemma. Lemma 7 At least one partition achieves at least 1 

2 k

approximation.

Proof We prove the lemma by using strong candidate set. The weak candidate set i;j case follows same proof. Let C1 .t/ denote the total profit made by all requests such that (1) appear in C1 .t/ and (2) not ignored in partition .i; j /. Then,

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i;j

C1 .t/  .k 2  2k/C1 .t/

8i;j

since each request is ignored in at most 2 partitions. i;j On the other hand, C1 .t/ is smaller than the profit of the strong candidate set for partition .i; j /, which is no larger than the profit of the approximated strong candidate set, called C10 .t/. In other words, k 2 C10 .t/ 

X

i;j

C1 .t/  .k 2  2k/C1 .t/

8i;j

which implies C10 .t/  .1  k2 /C1 .t/ and finishes the proof.



According to Lemmas 6 and 7, shifting strategy gives a PTAS for each level i . The result can be extended when request disks in all levels are considered. The main technology is dynamic programming which is discussed in Sect. 3.2 of [24]. We omit the details here. Since the strong/weak candidate sets can be approximated well, we have the following theorem. Theorem 10 The methods (K, G, H) will achieve asymptotically optimum competitive ratios in polynomial time for general networks modeled by disk graphs. The competitive ratios will have an additional factor 1  for any constant > 0 comparing with previous analysis, when we set D k2 .

3.4

Dealing With Selfish Users

In this section, we show how to design a mechanism based on the allocation algorithm described in Sect. 3.2 when secondary users could be selfish. For the spectrum allocation and auction problem, when each secondary user declares his request, he may lie on the bid and time requirement. We need to design rules such that each secondary user has incentives to declare his request truthfully. Each secondary user i has its own private information ti , including bi , ai , and ti . Let ai D .bi0 ; ai0 ; ti0 / be the value user i will report. For each vector of actions a D .a1 ; a2 ;    ; an /, a mechanism M D .A; P / (including allocation algorithm A and pricing scheme P ) computes a spectrum allocation A.a/ D .A1 .a/; A2 .a/;    ; An .a// and a payment vector p.a/ D .p1 .a/; p2 .a/;    ; pn .a//. Each user i will be allocated Ai .a/ and be charged pi .a/. A mechanism is said to be truthful if it satisfies both incentive compatible (IC) and individual rational (IR) properties. A mechanism is incentive compatible if every user i will maximize its utility by reporting its private type ti truthfully. A mechanism is individual rational if the utility of an agent winning the auction is not less than its utility of losing the auction. We assume that no user will delay his/her request and a user will not lie about ti and si . Unfortunately, in this work, we cannot

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Algorithm 3 Online spectrum allocation H Input: A constant parameter c > 1 C ˇ, , ˛, , Ra .t /, R.t /, C1 .t /. Previous current candidate requests set C D C1 .t 0 / where t 0 < t . Here, C1 .t 0 / may be empty. Output: Whether requests submitted at time t   will be admitted and new current candidate requests set C . 1: if C1 .t /  c  C1 .t 0 / then C D C1 .t /; 2: 3: Accept request ei 2 C1 .t / such that ai D t  . 4: else 5: Accept request ei 2 C1 .t 0 / such that ai D t  .

design an online spectrum allocation mechanism that is always truthful. Instead, we will show that under the mechanism TOFU , regardless of the actions of other users, no user i can improve its profit by reporting a bid bi0 that is smaller than its true bid value bi . We call this property as semi-truthful. For the spectrum auction problem, the final profit (or the utility) of a user i is 0 if its request is rejected. If its request is admitted, the final profit is utility.i / D bi  pi C .bi0 ; `0i ; ti /; where bi is the actual valuation, pi is the real payment, and .bi0 ; `i ; ti / is the potential preemption penalty, where bi0 is the actual bid of user i . It is a forklore result that the allocation method in a mechanism that can prevent lying must have the monotone property. Here, a spectrum allocation method A is monotone if a user i is granted the spectrum usage under A with a bid ei D .bi ; ai ; ti /, then the user i will still be granted the spectrum usage under A if the user increases bid bi and/or decreases the required time duration ti . First, the methods (Algorithms 2 and 3 ) presented in previous section do have the monotone property. In the algorithm, we need to find strong candidate requests set, and weak candidate requests set. Both sets will be found by dynamic programming. It is easy to show that these dynamic programming methods are monotone. Thus, it is possible to design a mechanism using the algorithms (G and H) as spectrum allocation methods. Consider a user i , and assume the bids of all other users remain the same. We first propose the following definition based on the monotone property of the methods. Definition 5 Let b i be the minimum bid that i has to bid to get admitted when its spectrum request is to be processed at  time slots later. Let b i be the minimum bid that i has to bid to get admitted and not get preempted later. Clearly, b i  b i . For a secondary user whose request is not granted, the value b i D 0. The values b i and b i clearly can be computed in polynomial time since the bid values of all other users are known. Here, b i can be computed at the time ai C  . To compute the value b i , we need to know whether request i will be preempted later. Since the latest job that can preempt ei must have starting time no later than si C ,

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Algorithm 4 TOFU semi-truthful online spectrum auction 1: Based on the setting, we use either Algorithms 2 or 3 to decide whether a request ei will be admitted or not. 2: The mechanism will charge user i a payment pi D b i

we must process the potential jobs that can preempt ei no later than si C   ˛ C  . In other words, the value b i can be computed within time  C  after job i arrived. Then, the protocol TOFU works as follows. Theorem 11 The mechanism TOFU is semi-truthful, i.e., to maximize its profit, every secondary user will not bid a price lower than its actual value. Proof Consider a user i arrived at time t. Let bi be its private willing bid and bi0 be its actual bid. When i bids some value bi0 2 .b i ; b i /, i will get preempted with some `0

unserved time `0i and receive a compensation .bi0 ; `0i ; ti / D ˇ tii bi0 .

Case 1: bi  b i . There are three strategies again for i . (1) If bi0  b i , then user i will be admitted and will not be preempted. It will be charged a payment pi < b i and thus will get a profit bi  pi > 0. (2) If bi0 < b i , it will not get admitted and thus get 0 utility. (3) If bi0 2 .b i ; b i /, then i will be admitted and then be preempted with unserved `0

time `0i . It will get a compensation .bi0 ; `0i ; ti / D ˇ tii bi0 . Its utility then is f  `0

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bi pi C.bi0 ; `0i ; ti / D .1ˇ tii /bi pi Cˇ tii bi0 D bi pi ˇ tii .bi bi0 /, which

is less than the profit bi pi that it will get when reported bi since bi0 < b i  bi . Thus, in all subcases, reporting untruthfully will not gain any benefit for i . Case 2: bi 2 .b i ; b i /. There are five strategies for i . (1) If bi0 D bi , it will be admitted and then preempted. Let `i be the unserved time slots when bidding bi . Then, its utility is .1ˇ `tii /bi pi Cˇ `tii bi D bi pi > 0.

(2) If bi0  b i , then user i will be admitted and will not be preempted. It will be charged a payment pi < b i and thus will get a profit bi  pi > 0. (3) If bi0 < b i , it will not get admitted and thus get 0 utility. (4) If bi < bi0 < b i , then i will be admitted and then be preempted with a shorter unserved time `0i  `i , due to the monotone property of the method. It will get a `0

compensation .bi0 ; `0i ; ti / D ˇ tii bi0 . Its utility then is f bi pi C.bi0 ; `0i ; ti / D `0

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.1  ˇ tii /bi  pi C ˇ tii bi0 D bi  pi  ˇ tii .bi  bi0 /, which is larger than the profit bi  pi that it will get when reported bi since bi0 > bi . (5) If b i < bi0 < bi , then i will be admitted and then be preempted with a longer unserved time `0i  `i . Its utility then is f  bi  pi C .bi0 ; `0i ; ti / D .1  `0

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ˇ tii /bi  pi C ˇ tii bi0 D bi  pi  ˇ tii .bi  bi0 /, which is smaller than the profit bi  pi that it will get when reported bi since bi0 < bi .

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Thus, in all subcases, reporting lower will not gain any benefit for i . Case 3: bi  b i . There are several strategies again for i . (1) If bi0  b i , it will not get admitted and thus has 0 profit. (2) If bi0  b i , it will be admitted and not be preempted. In this case, its profit is bi  pi < 0. (3) If b i < bi0 < b i , it will be admitted and be preempted with an unserved time `0i . `0

In this case, its profit is bi  pi  ˇ tii .bi  bi0 / < 0 since bi < pi and bi < bi0 . In all subcases, bidding untruthfully is not beneficial to user i . 

Observe that, in the scheme, the only scenario (case 2.4) that a secondary user can gain benefit is when bi 2 .b i ; b i / and it bids a value bi0 2 .bi ; b i /. The user i must bid larger than its true valuation bi , but not larger than b i . Observe that value b i is computed at time ai C  C  using the requests submitted during time interval Œai ; ai C , where user i does not know when it arrives at time ai . Thus, we have Theorem 12 If any user i does not know the requests within  time after its arrival, the mechanism TOFU is truthful, i.e., to maximize its profit, every secondary user will bid truthfully even it learned history. It is not difficult to construct examples in which a user can gain profit by bidding larger than its true valuation if it can know the bids of other users. It will be an interesting future work to study the total payment of a mechanism compared with the best possible payments collected by a truthful mechanism. Notice that, when we know the distributions of all bids, and there is only one spectrum, and no spatial and temporal reuse, Myerson [26] presented a mechanism that guarantees the payment is optimal. We note that it is very challenging to design a mechanism with a total payment close to optimum for the problem studied here.

3.5

Simulation Studies

We conducted extensive simulations to study the performance of the algorithm G, specifically, the competitive ratio of the algorithm in practice. We set c1 D 2 and q  C1 for algorithm G. Suppose the total service time is always 2,000 time c2 D  slots, which is large enough comparing with time requirements. In the experiment, we first generate n secondary nodes that are randomly placed. We randomly deployed 100 nodes in a 5  5 square area. Assume that all secondary users locate at these positions. The geometry location of a request is exactly the location of secondary user who claimed that request. We assume that any two requests within distance 1 will conflict with each other if they requested the same channel for some time slots. We then generate random requests with random bid values, and time requirement. The bid of each request is uniformly distributed in Œ0; 1; the time requirement of each request is uniformly distributed in Œ1; . Observe that the average load of a node in the simulation is JTD , where J is total n number of requests, D is the average duration of a request (thus, D D =2 in the

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setting), T is total duration of scheduling (thus, T D 2;000 time slots here), and n is number of secondary users (n D 100 here). Observe that the expected number of users in a unit area is n=52 D 4 in the setting. Thus, varying J and , we tested both lightly loaded system and also highly loaded system.

3.5.1 Competitive Ratio of Algorithm G A number of parameters may affect the performances of algorithm G, e.g., the number of total requests, the time bound , and the delay factor  . To study the effect of these parameters, we first fix the time bound  and study the competitive ratios of G while the delta factor  varies. Then, we fix the delta factor  and study the competitive ratios of G while the time bound  changes. We set three different total numbers of requests in these experiments to study how the degree of competition affects G’s performance. In Fig. 5, we plot the competitive ratios of G in various cases while the delay factor  is fixed (see Fig. 5a–c); we also plot the competitive ratios of algorithm G in various cases while time bound  is fixed (see Fig. 5d–f). In most cases, method G makes a total profit that is more than 95 % of the optimum offline method when  ' . Observe that although it is NP-hard to find the optimum offline solution, we implemented a method (omitted here due to space limit) that will find an almost optimum solution (within a factor 1  for an arbitrarily small > 0) when the conflict graph is a disk-intersection graph. The experimental results are much better than the theoretical bound we proved in previous sections. The competitive ratios decrease when  increases. This is exactly what the theoretical analysis implies. When  < , the competitive ratio of G increases significantly when df actor increases. This implies that increasing the delay factor is a good way to achieve better performance if the delay factor is relatively small. We also observe that G almost achieves the optimum offline solution when delay factor  is close to or larger than , which verifies the theoretical bound. Furthermore, the competitive ratio will not improve much when  > . Thus, we recommend to set  2 Œ=2; . The total number of requests also affects the performance of G. We observe that the competitive ratios decrease when the total number of requests increases. This is because G is conservative: the weak preemption in G just tries to satisfy the theoretical bound, which may lose some profit potentially. Thus, when there are more requests, the optimal offline algorithm will make more profit but the profit made by G will not increase so much. 3.5.2 Efficiency Ratio of Algorithm G Recall that to ensure that secondary users will not bid a price lower than their actual values, we do not charge what they bid. For secondary user whose request is accepted, we charge the minimum bid such that the request will still be accepted. Thus, the actual profit made is not the winners’ total bids but the winners’ total payments. Clearly, the total payments are no more than the total bids. Here, we define the efficiency ratio as the ratio between the profit made by the mechanism and the winners’ total bids computed by the method G. We also study the effect of

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some parameters such as the number of total requests, the time bound , and the delay factor  . In Fig. 6, we plot the efficiency ratios of the mechanism in various cases. In (a)–(c), we fix time bound  and plot the efficiency ratio while delay factor  changes; in (d)–(f), we fix delay factor  and plot the efficiency ratio while time bound  changes.

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We observe that the efficiency ratios increase when the  or the number of total requests increases. In this case, a request has more chance to conflict with other requests. Thus, it is easier to find a replacement which increases the minimum value a winner has to bid. We also find that the efficiency ratios first decrease then increase when the delay factor  increases. Comparing with Fig. 5d–f, we know that the

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efficiency ratios decrease because the competitive ratios increase dramatically when  is relatively small. The total payments do not increase as fast as the total bids do. When the delay factor  is larger than , the efficiency ratios increase because the competitive ratios increase slowly.

3.5.3 Spectrum Utilization Ratio of Algorithm G We also studied the spectrum utilization of the method. The spectrum utilization in a time interval P PT Œ1; T  under a spectrum allocation method A is defined as T t D1 `A .t/= t D1 n.t/, where `A .t/ is the number of users who are using the channel (at different locations) at time t without conflict in spectrum allocation produced by A and n.t/ is the maximum number of users who can use the channel concurrently without conflict. Figure 7 reports the results. We find that varying  does not affect the spectrum utilization that much. Observe that the measurement of spectrum utilization compared to the method with the offline method that can arbitrarily preempt the spectrum usage. When the system is highly loaded, the spectrum utilization of the method is about 80 %, while the spectrum utilization is about 50 % for lightly loaded system. 3.5.4 Compare Algorithm G with Simple Heuristics We then compare algorithm G with two simple greedy algorithms. One greedy algorithm B always satisfies the virtual spectrum candidate request with the largest C1 .t/ value. When request with larger bid is coming, B will terminate current assignments with smaller value if necessary. Another greedy algorithm D always satisfies the virtual spectrum candidate request with the largest C2 .t/ value. When request with larger bid is coming, D will terminate current assignments with smaller value if necessary. The competitive ratio of these two simple greedy algorithms could be arbitrarily bad theoretically. We conduct simulations to compare them with algorithm G which is asymptotically optimal theoretically.

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In Fig. 8, we plot the competitive ratios of algorithm G, simple greedy algorithm B and D to compare their performances. In Fig. 8a–c, we fix time bound  D 20 and vary the number of total requests for different delay factors  . In Fig. 8d–f, we fix time bound  D 100 and vary delay factor  for different number of total requests. We observe that the algorithm G outperforms these two simple greedy algorithms significantly in most cases.

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The effect of number of total requests is also investigated in Fig. 8a–c. We can see that the competitive ratio of algorithm G is not affected much when the number of total requests increases. On the other hand, the performance of algorithm B decreases significantly when the number of total requests increases and delay factor  D 0. The competitive ratio of algorithm D increases slightly when the number of total requests increases. However, the competitive ratio of algorithm D is always worse than that of the algorithm G. The effect of delay factor  is then investigated in Fig. 8a–c. We can see that the performances of all three algorithms increase significantly when the delay factor  increases. When delay factor  is increasing, the algorithm G is always the best one whose performance is close to the optimum.

4

Spectrum Allocation and Mechanism Design with Known Distribution on Requests

In this section, we review the results on online spectrum allocation and mechanism design when some additional distribution information about online spectrum requests are known. Two different models will be studied here. The first model assumed that semi-arbitrary-arrival case, while the second model assume a randomarrival case.

4.1

Semi-Arbitrary-Arrival Case

In this section, we consider the semi-arbitrary-arrival case. We first show the lower bounds of competitive ratios under the random ordering hypothesis. The lower bound of social efficiency ratio comes from the well-known secretary problem [7] which is a special case in the model. The lower bound of revenue efficiency ratio comes from the result in [16]. Consider the model used in [16] (Sect. 5), in which a single item is being sold and an adversary specifies a set of bids that are randomly matched with arrival and departure intervals. That model is actually a special case of the model when T D 1. So its lower bound on the efficiency is also a lower bound in the model. We have following theorems. Theorem 13 For the online spectrum admission problem, no online mechanism is e-competitive for social efficiency. Theorem 14 For the online spectrum admission problem, for any constant > 0, these is no strategyproof online mechanism which is .3=2  /-competitive for revenue efficiency. We are now ready to describe the online mechanism for spectrum allocation. The online mechanism M1 runs as following. It will divide the overall time times T into two phases. In the first phase (composed of the first ıT time slots), it studies some

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Algorithm 5 Learning algorithm M1 .L/ for mechanism M1 Input: Requests arriving during Œ0; ıT . Output: A parameter > 0. 1: for D 1 to b T2 c do 2: if Pr .  ti  2 /  12 then 3: Return parameter

properties of the distribution of the requested time durations. Then, in the second phase, the mechanism will start to allocate the channel to the coming requests. In the first phase, the mechanism M1 collects the information of coming requests and computes the information Pr .t1  ti  2t1 / for t1 2 Œ1; T =2. Here, we use Pr .t1  ti  t2 / to denote the estimated probability that the required time duration of a coming request is at least t1 and at most t2 . We then learn a parameter which will be introduced later by using the learning Algorithm 5 M1 .L/. Here, Œ ; 2  will be a time interval such that at least a constant proportion of the requests whose requested duration of time slots will be in this interval. During the first ıT time slots, we reject all coming requests. Later, we will show that it does not affect the performance significantly. Notice that the adversary can pick up a common distribution (such as uniform distribution, normal distribution, exponential distribution) as the distribution of required time durations. The learning method (Algorithm 5 ) can find a feasible parameter without knowing the exact distribution. Notice that here the constant 1=2 in Algorithm 5 could be replaced by any positive constant 2 .0; 1/. For example, if the required time durations are uniformly distributed in Œt1 ; t2 , then 2 D t1 Ct is a feasible solution. If the required time durations are normally 2 distributed with mean , then D  is a feasible solution. In the rest of chapter, we make the following assumption. Assumption 1 Given a distribution on the required time slots, we assume that a feasible value can always be found by Algorithm 5 . Since we only sampled the requests arrived in time Œ0; ıT , we first would like to bound the estimation error of these probabilities. Let Err be the maximum error Err D max kPr .t1  ti  2t1 /  Pr .t1  ti  2t1 /k; t1 T =2

where Pr .t1  ti  t2 / is the actual probability that the required time slots of a coming request are in Œt1 ; t2 . Lemma 8 The estimation error Err is neglectable. Proof According to the theory of VC-dimension [30], the bound on the test error of a classification model is

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r Pr Err 

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After the first ıT time slots, based on the results from learning algorithm, the decision algorithm M1 .D/ of mechanism M1 decides which requests will be admitted and how much each admitted request should be charged. The following concept is needed for presenting mechanism M1 . .s/

Definition 6 Let p denote the s-th highest bid value (for the usage of per unit time slot) among all requests received during time interval Œ0; . The main idea of algorithm M1 .D/ is to divide the time interval ŒıT; T  into multiple non-overlap decision phases. Figure 9 illustrates the protocol. In each decision phase, the decision algorithm waits for some time slots, then tries to admit a request with the highest bid value in this decision phase. The reason why we do so is that the adversary could generate any kind of bid values set. In some cases, to guarantee the performance, we have to admit the highest bid value, e.g., all bid values are 0 except the highest one maxi bi D 1. By waiting a certain number of time slots, we can ensure that we will see the highest bid with a constant probability. Later, we will prove that the decision algorithm M1 .D/ could admit the request with highest bid value in each decision phase with constant probability. Current decision

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Algorithm 6 Decision algorithm M1 .D/ for mechanism M1 Input: Parameter returned from Algorithm 5 , constant parameter ˛  1 and ˇ, requests arrived during ŒıT; T . Output: Allocation and charging for each coming request. 1: Start a new decision phase at beginning. 2: A request ei D .bi ; ai ; ti / will be admitted during a decision phase if the following conditions are met: 1) No conflict request is admitted and running, 2)  ti  2 , 3) Request ei arrives during a time interval Œsi ; si C ˇti  for a si such that ˛ti  si  ti where ˛  1 is a constant, and .1/ 4) The bid value bi by the user satisfies that bi  psi . 3: Decide if current decision phase will end or not. The clock of each new decision phase is reset as 0. .1/ 4: For each request ei admitted, it will be charged psi per unit time. The total charge will be .1/ pi D psi  ti . For each request ei which is not admitted, it will be charged 0.

phase will end if (1) either a request is admitted and finished (2) or no request is admitted after 2.1 C ˇ/ time slots for a given constant ˇ. Another new decision phase will start when a previous decision phase ends. For simplicity of presentation, we assume that the clock of each new decision phase is reset as 0. This means that, in Algorithm 6 , when we say a request arrived in time ti , it actually arrived ti time slots after this new decision phase started. Algorithm 6 summarizes the allocation method. Theorem 15 Mechanism M1 is strategyproof. Proof We should prove that no request could make more profit by bidding other than its true valuation or/and announcing larger time requirement. In other words, we need to prove value-SP, time-SP, and value- and time-SP. We will prove these properties separately. Value-SP: According to the pricing scheme of M1 , we know that each request ei .1/ .1/ will be charged psi ti and make vi  psi ti profit if it is admitted. Request ei will .1/ be charged 0 and make 0 profit if it is not admitted. Since psi does not depend on bi , for a request ei admitted by mechanism M1 , it cannot improve its profit by lying on its bid value bi . On the other hand, for a request ei which is not admitted by mechanism M1 , there are two possible reasons causing its rejection. The first reason is that ei does not appear in a time interval Œsi ; si C ˇti  that satisfies the conditions in mechanism M1 . For this reason, no matter how request ei lies on its bid bi , it will never be .1/ admitted. The second reason is that ei bids bi < psi while it bids truthfully, i.e., .1/ vi D bi ti < psi ti . For this reason, if request ei bids lower than its true valuation, i.e., bi ti < vi , it still cannot be admitted and its profit is still 0; if request ei bided .1/ higher than its true valuation, i.e., bi ti > vi , there are two cases. Case 1: bi < psi ,

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request ei is still not admitted and no more profit is made. Case 2: bi  psi , request ei is admitted and make vi  bi ti < 0 profit, which implies request ei loses profit. As stated above, no matter how request lies on its bid value, it cannot make more profit. .1/ Time-SP: Similarly, we know each request ei will make vi  psi ti profit when it is admitted; otherwise, ei will make 0 profit when it is not admitted. For request ei which is admitted by mechanism M1 , if it announces a larger required time duration ti0 > ti , ei can be still admitted or not admitted. If ei is not admitted due to lying on ti , it loses profit due to lying. If ei is still admitted, it needs .1/ to pay ps 0 per unit time where si0  si is the parameter described in mechanism i

M1 . According to the definition of ps 0 , we know that ps 0  psi when si0  si . .1/

.1/

i

i

.1/

Therefore, request ei makes no more profit by lying since vi  ps 0 ti0 < vi  psi ti .1/

.1/

i

when ps 0  psi and ti0 > ti . i For request ei which is not admitted by mechanism M1 , there are two possible reasons causing its rejection. The first reason is that ei does not appear in a time interval Œsi ; si C ˇti  that satisfies the conditions in mechanism M1 . For this reason, no matter how request ei announces a larger ti , it will never be admitted. The second .1/ .1/ reason is that ei bids bi < psi while it bids truthfully, i.e., vi D bi ti < psi ti . For 0 this reason, if request ei announces a larger ti > ti , ei may be admitted when bi  .1/ ps 0 where si0 is the new parameter when ti0 is announced. Since ti0 > ti , we have i si0  si according to the mechanism. Therefore, even if ei is admitted due to lying .1/ .1/ higher on its required time slots, it makes vi  bi ti0 < vi  ps 0 ti0  vi  psi ti  0 .1/

.1/

i

since ps 0  psi when si0  si . i Value- and time-SP: When request lies on its bid value and time requirement, we can consider it as a lie on bid value and another lie on time requirement. As we proved above, neither of these could make more profit, we finish this part.  .1/

.1/

Lemma 9 At any decision phase, mechanism M1 could admit the request with highest bid value and charge that request at second highest bid value during that ˛ˇ phase with a probability at least 2.ˇC2/ 2. Proof To admit a request ei in a decision phase, the required time duration ti should be within Œ ; 2 . We know that Pr .  ti  2 / D 12 according to the learning algorithm M1 .L/. To admit a request ei with highest bid value with a charge equal to the second highest bid during a decision phase, the request with second highest bid value should arrive during time interval Œ0; si  and the request with highest bid value should arrive during time interval Œsi ; si C ˇti . Since the arrival of requests is memoryless, the number of arrival during a time interval only depends on the length of that interval. The probability that the request si ˛ with second highest bid arrives during Œ0; si  is at least si Cˇt  2Cˇ . The i Cti probability that the request with highest bid value arrives during Œsi ; si C ˇti  is ˇti ˇ at least si Cˇt  2Cˇ . i Cti

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Thus, at each decision phase, the mechanism could admit the request with highest bid value and charge it at second highest bid value with constant probability ˛ˇ ˇ 1 ˛ D 2.2Cˇ/  2. 2 2Cˇ 2Cˇ 3

Lemma 10 At each decision phase, mechanism M1 is 2.2Cˇ/ -competitive during ˛ˇ that phase with respect to offline Vickrey auction for both social and revenue efficiency. ˛ˇ Proof According to Lemma 9, we know that M1 has a constant probability 2.2Cˇ/ 2 to admit the request with highest bid and charge it with second highest bid. We know that decision phase lasts at most si C ˇti C ti  .2 C ˇ/ti time when ei is admitted. The maximum total valuation is at most .2Cˇ/bi ti . The total valuation of winner of ˛ˇ M1 is at least bi ti with probability 2.2Cˇ/ 2 . Therefore, the social efficiency ratio is at

least

˛ˇ 2.2Cˇ/3

2.2Cˇ/3 -competitive for social efficiency. ˛ˇ 3 -competitive for M1 is also at least 2.2Cˇ/ ˛ˇ

which implies M1 is at least

Similarly, it is easy to show that revenue efficiency. We finish the proof.



This lemma shows that, in single decision phase, the protocol does manage to get a social efficiency and revenue within a constant factor of optimum. This does not directly imply that the protocol is overall a constant approximation for social and revenue efficiency because many decision phases exist in Œ0; T . Definition 7 We call a decision phase good decision phase if a request is admitted during this phase. Otherwise, we call it bad decision phase. 3

-competitive for both social We proved that for good decision phase, M1 is 2.2Cˇ/ ˛ˇ and revenue efficiency. According to the mechanism, the length of a bad decision phase is always 2.1 C ˇ/ . In following lemma, we prove that the expected length of a good decision phase is at least .1 C ˛/ . Lemma 11 The expected length of a good decision phase is at least .1 C ˛/ . Proof When a request ei is admitted in a decision phase, the length of that decision phase is at least si C ti  .1 C ˛/ti . So the expected length of such decision phase is at least .1 C ˛/ti  .1 C ˛/ since the decision algorithm guarantees that ti  for all admitted request ei .  Theorem 16 Mechanism M1 is ‚.1/-competitive with respect to offline Vickrey auction for both social and revenue efficiency. Proof According to Lemma 9, we know that during time interval ŒıT; T , for each ˛ˇ decision phase, with probability P D 2.2Cˇ/ 2 , we have a good decision phase with expected length at least .1C˛/ , and with probability 1P , we have a bad decision phase with length at most 2.1 C ˇ/ . Thus, after each decision phase, the expected total length of all good decision phases increases P .1 C ˛/ and the expected total

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length of all bad decision phases increases 2.1  P /.1 C ˇ/ . So the expected total P .1C˛/ length of all good decision phases is P .1C˛/C2.1P /.1Cˇ/ .1  ı/T D ‚.1/T . 3

We know that M1 is 2.2Cˇ/ -competitive for both social and revenue ˛ˇ efficiency during each of the good decision phases. Then, M1 is at least 2.2Cˇ/3 P .1C˛/C2.1P /.1Cˇ/ 1 -competitive for both social and revenue ˛ˇ P .1C˛/ 1ı efficiency. 

4.2

Random-Arrival Case

In this section, we consider the random-arrival case where the bid values and required time durations follow some known/unknown distributions. We will propose an online mechanism M2 with performance analysis. Note that in this section, the time is also assumed to be slotted. If the distributions or arrival rate is unknown, similar to previous protocol M1 , in the first ıT (ı < 1) time, we learn the distributions and arrival rate based on the information of arriving requests [21]. We already proved that the sample size is large enough to achieve relatively small estimate error even that ı is very small. So in this section, we mainly focus on the case where the central authority knows the distributions and the arrival rate in advance. Since we know all distributions, at each time t, we can compute the expected social efficiency of the offline Vickrey auction from the current time slot t to T of the spectrum channel by dynamic programming (DP). Let V .t/ denote this expected social efficiency from time slot t to T of offline Vickrey auction, and let event Xk denote that k requests arrive at same time, event Yt denote requested time duration of a request is t, and the event Zb denote bid value of a request is b. It is not difficult to derive that

V .t/ D

C1 X

Pr .Xk /

T t X

 Pr .Yti /.bm .k/  ti C V .t C ti // :

ti D1

kD0

Here, bm .k/ is the expected highest bid value among k requests’ bid values. It is R bm .k/ 1 Pr .Zx /dx D kC1 . And we have following lemma. easy to show that 1 Lemma 12 Function V .t/ is monotonic nonincreasing as t grows from 0 to T . Proof First, it is trivial to show that 8t 2 Œ0; T , V .t/  0. Let V k .t/ denote the expected social efficiency where k requests arrive at time t. For any k, we have (

V k .t C 1/ D k

V .t/

D

PT t 1

Pr .Yti /.bm .k/ti C V .t C 1 C ti //   tj D1 Pr Ytj .bm .k/tj C V .t C tj //: t D1

PTi t

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Algorithm 7 Decision algorithm M2 .D/ for mechanism M2 Input: Probability distribution of bid values, discrete probability distribution of requested durations, arrival rate  and requests arriving at time t . Output: Allocation and charging scheme 1: Compute V .t /, 8t 2 Œ0; T . 2: if the channel is being used then 3: Reject all coming requests. 4: else 5: Admit request ei s.t. i D arg max V 0 .t / and reject others. Here, V 0 .t / D bi ti C V .t C ti / if ei is admitted. 6: If a request ei is admitted, charge it bj tj C V .t C tj /  V .t C ti / where ej is the request that makes, the second highest V 0 .t /; if a request ei is not admitted, charge it 0.

  Then, V k .t C 1/  V k .t/ D Pr Y.T t / .bm .k/.T  t/ C V .T //  0. From above, we prove that for any k, V k .t/ is monotonic nonincreasing as t grows from 0 to T , which implies that V .t/ is monotonic nonincreasing.  Notice that if all secondary users announce requests truthfully, V .0/ is actually the expected social efficiency and revenue for Vickrey auction. Next, we are going to give a scheme that guarantees the truthfulness of both bid values and requested time durations. The admission method of the protocol is as following. We assume a virtual request e0 with bid value b0 D 0 and required time duration t0 D 1 arrive at each time. If the protocol admits e0 at some time, then it means no request will be admitted at that time. For example, if k requests arrive, together with the virtual request, we have totally k C 1 requests. Let V 0 .t/ denote the expected social efficiency from time t to T after a decision is made at time t. We always admit a request ei which maximizes the expected total social efficiency V 0 .t/ D bi ti C V .t C ti / and reject others. Assume that ej is the request which made the second highest expected social efficiency. If request ei (i ¤ 0) is admitted, then the secondary user who proposed ei will be charged: pi D bj tj C V .t C tj /  V .t C ti /; i.e., it will be charged as the second highest expected social efficiency excluding the expected social efficiency after he finishes. Algorithm 7 summarizes the method. Theorem 17 Mechanism M2 is strategyproof. Proof We will prove value-SP, time-SP, and value- and time-SP.

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Value-SP: If a request ei is admitted when the bid value is truthful, since the charging scheme does not rely on the bid value of the request itself, so lying on bid value will not bring more profit. If a request ei is not admitted when the bid value is truthful, there are two cases when it lies on the bid value. If ei is still not admitted, it is trivial that the profit will not increase (always be 0). If ei is admitted by bidding bi0 , then it will be charged bj tj C V .t C tj /  V .t C ti /. Since ei is not admitted if bidding is truthful, we know that bi ti C V .t C ti / < bj tj C V .t C tj / which implies the profit bi ti  pi by reporting untruthfully is no more than 0. Thus, no matter how a secondary user lies on its bid value, it cannot make more profit. Time-SP: Recall that we assume each request could only claim a longer required time duration than its actual requirement. If request ei is admitted when it claims its true time requirement, it will still be admitted when claiming a longer required time ti0 > ti . Then, it will have to pay: V .t C ti0 /  V .t C ti / more than before. Since V .t/ is monotonic nonincreasing by Lemma 12, it does not make more profit by lying. If a request ei is not admitted when claiming its time requirement truthfully, there are two cases when it lies on the time requirement. If ei is still not admitted, it is trivial that the profit will not increase (always be 0). If ei is admitted when it lies its time requirement as ti0 > ti , it will be charged bj tj CV .t Ctj /V .t Cti0 /. Since ei is not admitted if bidding is truthful, we know that bi ti C V .t C ti / < bj tj C V .t C tj /. Together with V .t C ti0 /  V .t C ti /, we have bi ti  bj tj C V .t C tj /  V .t C ti0 / which implies the profit is no more than 0. In brief, no matter how request lies on its time requirement, it cannot make more profit. Value- and time-SP: When a request lies on its bid value and time requirement, we consider it as a lie on bid value and another lie on time requirement. As we proved above, neither of these could make more profit, we finish this part.  Lemma 13 At each time, mechanism M2 will admit a request with constant probability if the channel is not being used. Proof When the channel is empty, the probability that request ei can be admitted is Pr .bi  ti C V .ti /  V .t C 1//. We can rewrite V .t/ as V .t/ D b

T t X

.Yti  ti / C

ti D1

T t X

.Yti V .t C ti //;

ti D1

P where b D C1 kD0 Pr .Xk /  bm .k/ is a constant. So the probability whether request ei can be admitted or not is only related to the current time t and the required time duration ti .

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When k requests arrive at the same time, the probability that the bid value of 1 one request is greater than the expected maximum bid value bm .k/ is kC1 . Given the distributions of bid values, required time durations, and the arrival rate, when k requests arrive at time t, we can find a c.ti / such that Pr .bi  ti C V .t C ti /  V .t C 1// 

c.ti / : kC1

Note that without the specific distributions, we cannot give the exact form of c.ti /; however, after the distributions given, c.ti / can be calculated. We use A1 to denote the event that a request could be admitted (the expected social revenue exceeded the reserved social revenue V .t C 1/) and A0 to denote the event that no request could be admitted. When k requests arrive at time t, the probability that no requests could be admitted is

Pr .A0 jXk / 

k  Y c.i /  cm k 1  .1  / ; k C 1 k C1 i D1

where cm D minTtD1 c.t/ is a constant. The probability that a request could be cm k admitted is Pr .A1 jXk /  1  .1  kC1 / . At time t, the expected probability that some request could be admitted is Pr .A1 / D

C1 X

Pr .Xk /  Pr .A1 jXk / 

kD1

C1 X

 Pr .Xk /  1  .1 

kD1

0 .cm C o.1//.k  1/ cm C o.1/ C1 B k  2.k C 1/ X B  e  2  D1e B @ kŠ kD1

which is at least a constant.

cm k  / k C1

1 C cm Co.1/ C C  1  e 2 A



Theorem 18 Mechanism M2 is ‚.1/-competitive with respect to offline Vickrey auction for both social and revenue efficiency. Proof According to Lemma 13, we know a request could be admitted with probability greater than a constant P , thus no request could be admitted with probability smaller than 1  P at each time. Then, it is easy to show that in Pt expectation, the channel is busy for P t C1P T time slots where t  1 is the expected time requirement of admitted request. When the channel is busy, the social efficiency and revenue efficiency of M2 is no less than that of Vickrey auction. Then, we know in expectation, M2 is P t C1P Pt competitive. 

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General Conflict Graph Model

In this section, we show that the protocols can be easily extended for networks where the conflict graph is growth-bounded by a polynomial function f , which implies the one-hop independent number of conflict graph H is f .1/.

4.3.1 Semi-arbitrary-arrival Case For the semi-arbitrary-arrival case, we do not need to change the learning algorithm of the protocol. Here is the new decision algorithm for the extended protocol M01 . .s/ We define p .R/ as s-th highest bid among all requests which is received during time interval Œ0;  and proposed by secondary users in set R. The theoretical analysis of performance is similar to that of previous section. We omit the details due to space limit. The main difference is that when H has a onehop independent number f .1/, with certain probability, the protocol may accept a request ei with highest bid value during certain time interval and miss at most f .1/bi bid values. Therefore, compared with what we did in Sect. 4.1, there is an additional factor f .1/ in both social and revenue efficiency ratio. 4.3.2 Random-Arrival Case For the random-arrival case, we have new decision algorithm M02 .D/ for the extended online mechanism M02 as following. We still use V .t/ to denote the expected revenue of offline Vickrey auction from time t to T . When the conflict graph H is not a completed graph, more than one requests could be running at same time. Assume R is a set of requests; we use VR0 .t/ to denote the expected revenue after a request is admitted at time t, and all requests in R are running before it is admitted. The expected revenue can be achieved by dynamic programming

Algorithm 8 Decision algorithm M01 .D/ for mechanism M01 Input: Parameter learned by learning algorithm and requests that arrive during ŒıT; T . Output: Allocation and charging scheme. 1: All secondary users start a new decision phase initially. 2: A request ei will be admitted during a decision phase if: 1) ei does not conflict with any running requests in geometry region and the secondary user proposed ei is in its decision phase. 2)  ti  2 . 3) Request ei arrives during a time interval Œsi ; si C ˇti  for a si such that ˛ti  si  ti where ˛ < 1 is a constant. .1/ 4) Bid value bi  psi .R/ where R is the set of secondary users in decision phase and whose requests will conflict with ei in geometry region. 3: When request ei is admitted, decision phases of the secondary user who proposed ei and those secondary users in R will end. Start service phases for these secondary users. These service phases end when ei is finished. 4: New decision phase will start when either service phase ends or no request is admitted after 2.1 C ˇ/ time slots. Time of each new decision phase is reset as 0. .1/ 5: For each request ei admitted, it will be charged psi .R/ per time slot; otherwise, it will be charged 0.

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Algorithm 9 Decision algorithm M02 .D/ for mechanism M02 Input: Probability distribution of bid values, required time durations and arrival rate , a set of requests R at time t , and a set of running requests C . Output: Allocation and charging scheme 1: while R is not empty do 2: Admit request ei such that i D arg max VC0 .t / where ei does not conflict with any request in C . 3: If ei is a dummy request, then R D ;. Otherwise, (1) let R D R  Ri where Ri is a set of requests which conflicts with ei in geometry region (including ei itself), and (2) C D C [ fei g. 4: For each request ei admitted, it will be charged bj tj C V .t C tj /  V .t C ti / where ej is the request that makes the second highest VC0 .t / before ei is admitted; otherwise, it will be charged 0.

as described in Sect. 4.2. Details are omitted due to space limit. In the process of dynamic programming, we may need to find maximum weighted independent set. Finding maximum weighted independent set problem is NP-hard [12] and hard to approximate within n1o.1/ factor [29]. However, using the growthbounded property, we could find a 1 C approximation in polynomial time [24]. Algorithms 9 summarizes the method. The theoretical analysis of performance is similar to that of previous Section. Again, compared with what we did in Sect. 4.2, there is an additional factor f .1/ in both social and revenue efficiency ratios.

4.4

Simulation Results

In this section, we conducted extensive simulations to study the performance of the online mechanisms M1 and M2 . In the simulations, we set the total time T D 1;000 time slots. We generate random requests with random bid values and time requirements. The arrival of these requests follows Poissonian distribution with arrival rate , which varies in the simulations to simulate lightly loaded or heavily loaded system. We omit the learning phase since we want to study the performance of the online mechanisms and the influence of learning is proven neglectable when the number of total time slots is large enough. The bid value and time requirement of each request are either uniformly distributed or normally distributed. Here, we generate four different sets of requests. In set 1, the bid value of each request is uniformly distributed in Œ0; 1, and time requirement of each request is uniformly drawn from all integers in Œ1; 50; in set 2, the bid value of each request is uniformly distributed in Œ0; 1, and time requirement of each request is uniformly drawn from all integers in Œ1; 500; in set 3, the bid value of each request is normally distributed with mean value bid D 0:5 and standard deviation bid D 2, and time requirement of each request is normally distributed with mean value t i me D 25 and standard deviation t i me D 3; in set 4, the bid value of each request is normally distributed with mean value bid D 0:5 and standard deviation bid D 2, and time requirement of each request is normally distributed

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with mean value t i me D 250 and standard deviation t i me D 3. The bid values are randomly matched with the time requirements.

4.4.1 Mechanism M1 for Semi-arbitrary-arrival Case We set ˛ D 1 and ˇ D 1 in the experiments. In Fig. 10a, b, we plot the competitive ratio for social and revenue efficiency when the arrival rate  varies for different requests sets. As we stated in Theorems 13 and 14, the social efficiency ratio is no more than 37 % and the revenue efficiency ratio is no more than 66 %. Therefore, the performance of the mechanism M1 in the simulations, both social efficiency ratio and revenue efficiency ratio are almost 20–30 %, is close to the theoretical upper bound. We also observe that the competitive ratios increase when the system load increases since it is easier for M1 to admit a request when the system load is high, i.e., the competition among requests is high. In Fig. 10c, we plot the spectrum utilization ratio of the method M1 . The spectrum utilization ratio is defined as the ratio between the busy time of the spectrum channel and the total time T . In all cases, the utilization ratio is no more than 50 %. According to mechanism M1 , a request with time requirement ti is admitted while at least ˛ti time slots are empty. Since we set ˛ D 1, then the utilization ratio is no more than 50 % which is corroborated by the simulation result.

a

0.4 0.3

b Uniform tmax = 50 Uniform tmax = 500 Normal ¹ = 25 Normal ¹ = 250

0.4 0.3

0.2

0.2

0.1

0.1

0

2

4

6

8

0

10

Uniform tmax = 50 Uniform tmax = 500 Normal ¹ = 25 Normal ¹ = 250

2

4

¸

6

8

10

¸

c

0.4

Uniform tmax = 50 Uniform tmax = 500 Normal ¹ = 25 Normal ¹ = 250

0.2

0

2

4

6

8

10

¸

Fig. 10 The performance of online mechanism M1 . (a) Social efficiency ratio. (b) Revenue efficiency ratio. (c) Spectrum utilization ratio

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4.4.2 Mechanism M2 for Random-Arrival Case In Fig. 11a, b, we plot the competitive ratio for social and revenue efficiency when the arrival rate  varies. Unsurprisingly, the performance of mechanism M2 is much better than that of M1 since M2 has more known information. We can see that the mechanism M2 even outperforms the average performance of Vickrey auction in all cases. The reason is that the protocol will choose the request maximizes the social efficiency and compare it with the average performance of Vickrey auction. In slightly loaded systems, the social efficiency and the revenue efficiency of the protocol are about 2–6 times of the performance by the Vickrey mechanism. The reason is that in slightly loaded system, the average performance of Vickrey auction is poorer, so the ratios are higher. Observe that the competitive ratios decrease when the arrival ratios increase. When the arrival ratio is big, the system load is high and the competition among requests is high. Then, it is more difficult to admit a request which outperforms the expected performance of Vickrey auction. In Fig. 11c, we plot the spectrum utilization ratio of the method M2 . When the system is highly loaded, the spectrum utilization ratio of M2 is more than 95 %, while the spectrum utilization ratio is more than 75 % for lightly loaded system. When the system load is low, the competition among requests is low and thus it is harder to admit feasible request than that in a heavily loaded system. In all cases, mechanism M2 improves the efficiency ratios without sacrificing the spectrum utilization.

a

b 6

Uniform tmax = 50 Uniform tmax = 500 Normal ¹ = 25 Normal ¹ = 250

4

6 Uniform tmax = 50 Uniform tmax = 500 Normal ¹ = 25 Normal ¹ = 250

4

2

2 1

2

3

4

5

1

2

3

¸

c

4

5

¸ 1

0.9

Uniform tmax = 50 Uniform tmax = 500 Normal ¹ = 25 Normal ¹ = 250

0.8 1

2

3

4

5

¸

Fig. 11 The performance of online mechanism M2 . (a) Social efficiency ratio. (b) Revenue efficiency ratio. (c) Spectrum utilization ratio

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Literature Reviews

The allocation of spectrums is essentially the combinatorial allocation problem, which has been well studied in the literature [2, 23]. Yuan et al. [40] introduced the concept of a time-spectrum block to model spectrum reservation in cognitive radio networks. Li et al. [25,38] designed efficient methods for various dynamic spectrum assignment problems. They also showed how to design truthful mechanism based on those methods. Xu et al. [34] first studied online spectrum allocation when secondary users could bid arbitrarily. They designed efficient mechanisms that could thwart the selfish behavior of secondary users. In [35], Xu and Li designed protocols for spectrum allocation when requests are submitted in advance and the central authority only needs to make decisions within a certain time period. Zhou et al. [42] propose a truthful and efficient dynamic spectrum auction system to serve many small players. In [43], Zhou and Zheng designed truthful double spectrum auctions where multiple parties can trade spectrum based on their individual needs. In [31], Wang et al. designed an online version of truthful double auctions for spectrum allocation where the requests arrive in an online fashion. Ben-Porat et al. [6] gave a scheme scheduling decisions on the Cumulative Distribution Function (CDF). In [33], Wu and Tsang studied the distributed multichannel power allocation problem for the spectrum-sharing cognitive ratio networks. All these results are based on offline models. In this work, we use the online model which is similar to the online job scheduling problems [15]. Various online scheduling problems focus on optimizing different objective functions. The most common objective function is makespan, which is the length of the schedule. The first proof of competitiveness of an online scheduling algorithm was given by Graham in 1966 [15]. This is one of the most basic problems in online scheduling. Suppose that we are given m identical machines, jobs arrive one by one, and no preemption is allowed. Graham [15] proved that greedy algorithm, which assigns a new job to the least loaded machine, is 2 m1 competitive. A number of results have been proved to improve the upper bounds [3, 10, 20] and lower bounds [19]. In 1992, Bartal et al. [3] gave an algorithm that is 1:986-competitive. Then, the bound was improved to 1:945 [20], to 1:923 [1], and finally to 1:9201 [10] which is the best upper bound known to date. And the best lower bound up to now for any deterministic algorithm is 1:88-competitive which was proposed by Rudin et al. [19]. Closing the gap between the best lower bound (1:88 [19]) and the upper bound (1:9201 [10]) is an open problem. Many authors [14, 27] also investigated the case where preemption is allowed without penalty. Phillips et al. [27] gave an .8 C /-approximation algorithm for preemptive singlemachine scheduling problem. The approximation ratio was improved to 2 [13], and to 1:47 [14]. However, all these results did not consider the penalty of preemption. Bartal et al. [4] considered a version of online scheduling problem in which we pay penalty for rejecting jobs. This was improved later in [18] by Hoogeveen et al: they gave .1 C /-competitive algorithm, where  is the golden ratio. Hoogeveen et al. [18] improved the ratio to 1:58. They assume that the penalty is job-dependent only and is not affected by the preemption time.

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Above results focus on minimizing makespan. When jobs have deadlines, however, it is usually impossible to finish all jobs. Thus, another model aims to maximize the profit or number of completed jobs. There are different variants: preemption-restart, preemption-resume, and preemption-discard. In preemptionrestart model, it is allowed to preempt a running job, and the preempted job has to be restarted again from the beginning. In preemption-resume model, it is allowed to preempt a running job, and the preempted job could resume from where it was preempted. And other models assume that a preempted job cannot be restarted or resumed. In 1991, Baruah et al. [5] proved that no online scheduling algorithm 1 p can make profit more than times the optimal. Koren et al. [22] gave an .1C D/2 algorithm matching the lower bound in [5]. Woeginger [32] studied an online model of maximizing the profit of finished jobs where there is some relationship between the weight and length of job. He provided a four-competitive algorithm for tight deadline case and gave a matching lower bound. Hoogeveen et al. [17] gave a 12 competitive algorithm which maximizes the number of early jobs. They assume that preemption is allowed while no penalties will be charged. Chrobak et al. [8] gave a 23 -competitive algorithm which maximizes the number of satisfied jobs that have uniform length in the preemption-restart model. p Fung et al. [11] addressed a general model of nonuniform length and gave a C2 C2-competitive algorithm. Zheng et al. [41] studied the preemption-restart model where a penalty is the weight of the preempted job. They gave a 3 C o./-competitive for  > 9. The work that is most similar to the work is a recent result by Constantin et al. [9]. They proposed and studied a simple model for auctioning ad slot reservations in advance. A seller will display a set of slots at some point T in the future. Until T , bidders arrive sequentially and place a bid on the slots they are interested in. The seller must decide immediately whether or not to grant a reservation. Their model allows the seller to cancel at any time any reservation made earlier with a penalty proportional to the bid value. The major differences between the model and their model are as follows. Their model considers online requests and offline services. The services (ad slots) start from a fixed time and last for one unit time. And the preemptions happen before services start. In the model, the services (spectrum usage) can start from any time, lasting for an arbitrary duration (subject to time bound ), and the preemptions happen after the spectrum usages are assigned. We also considered the spatial reuse of spectrum resources.

6

Conclusion

In this chapter, we studied online spectrum allocation for wireless networks where a set of secondary users will bid for leasing a spectrum channel for certain time duration in different locations. After the central authority received the bidding request, it has to make a decision within a certain time  . For a number of variants, we reviewed efficient online scheduling algorithms and analytical proof of the constant competitive ratios of these methods. Especially, when  is around

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the maximum requested time duration , the algorithm TOFU results in a profit that is almost optimum. For TOFU protocol, Xu and Li [36] showed that no user will bid lower than its willing payment under their mechanism TOFU . Under a simple assumption that the requests by secondary users arrive with the Poisson distribution and the willing payment per unit time slot is independent from the number of time slots required, Wang et al. [39] are able to prove that their SALSA protocols simultaneously approximately maximize the expected social efficiency and the expected revenue. They also analytically proved that every secondary user will maximize its expected profit if it proposed requests truthfully. There are a number of interesting questions left for future research. It remains open to design a truthful mechanism for online spectrum auction when we do not know the distributions of user arrivals. It is also interesting to extend the TOFU mechanism to deal with different models, such as OFDM networks. It is also interesting to study the performance of the protocols when first price auction is used. In this case, secondary users may have tendency to lower their bids. Another challenge is to relax the wireless interference model used in both results. They assumed that the conflicts of spectrum usage among secondary users can be characterized by interference graphs. This assumption although is widely adopted in the literature and gives us some leverage in designing efficient protocols, it cannot perfectly reflect the interference in practice. Thus, it is important to extend the design of these protocols to network models with more complicated interference models.

Recommended Reading 1. S. Albers, Better bounds for online scheduling, in STOC ’97: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing (ACM, New York, 1997), pp. 130–139 2. A. Archer, C. Papadimitriou, K. Talwar, E. Tardos, An approximate truthful mechanism for combinatorial auctions with single parameter agents. Internet Math 1(2), 129–150 (2004) 3. Y. Bartal, A. Fiat, H. Karloff, R. Vohra, New algorithms for an ancient scheduling problem. J. Comput. Syst. Sci. 51, 359–366 (1995) 4. Y. Bartal, S. Leonardi, A. Marchetti-Spaccamela, J. Sgall, L. Stougie, Multiprocessor scheduling with rejection, in SODA (Society for Industrial and Applied Mathematics, Philadelphia, 1996), pp. 95–103 5. S. Baruah, G. Koren, B. Mishra, A. Raghunathan, L. Rosier, D. Shasha, On-line scheduling in the presence of overload, in FOCS (IEEE Computer Society, Washington, DC, 1991), pp. 100–110 6. U. Ben-Porat, A. Bremler-Barr, H. Levy, On the exploitation of cdf based wireless scheduling, in IEEE, Rio de Janeiro (2009) 7. F.T. Bruss, A unified approach to a class of best choice problems with an unknown number of options. Ann. Probab. 12(3), 882–889 (1984) 8. M. Chrobak, W. Jawor, J. Sgall, T. Tichy´, Online scheduling of equal-length jobs: Randomization and restarts help. SIAM J. Comput. 36(6), 1709–1728 (2007) 9. F. Constantin, J. Feldman, S. Muthukrishnan, M. Pal, An online mechanism for ad slot reservations with cancellations, in SODA (Society for Industrial and Applied Mathematics, Philadelphia, 2009), pp. 1265–1274 10. R. Fleischer, M. Wahl, On-line scheduling revisited. J. Sched. 3, 343–353 (2000)

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11. S.P. Fung, F.Y. Chin1, C.K. Poon, Laxity helps in broadcast scheduling. Theor. Comput. Sci., 3701, 251–264 (2005) 12. M. Garey, D. Johnson, Computers and intractability: A guide to the theory of np-completeness (Freeman, San Francisco, 1979) 13. M.X. Goemans, A supermodular relaxation for scheduling with release dates, in Proceedings of the 5th International IPCO Conference on Integer Programming and Combinatorial Optimization (Springer, London, 1996), pp. 288–300 14. M.X. Goemans, J.M. Wein, D.P. Williamson, A 1.47-approximation algorithm for a preemptive single-machine scheduling problem. Oper. Res. Lett. 26(4), 149–154 (2004) 15. R. Graham, Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966) 16. M.T. Hajiaghayi, R. Kleinberg, D.C. Parkes, Adaptive limited-supply online auctions, in EC (ACM, New York, 2004), pp. 71–80 17. H. Hoogeveen, C.N. Potts, G.J. Woeginger, On-line scheduling on a single machine: maximizing the number of early jobs. Oper. Res. Lett. 27(5), 193–197 (2000) 18. H. Hoogeveen, M. Skutella, G.J. Woeginger, Preemptive scheduling with rejection. In Algorithms - ESA 2000 (2002), pp. 268–277 19. I.F.R. John, R. Chandrasekaran, Improved bounds for the online scheduling problem. SIAM J. Comput. 32(3), 717–735 (2003) 20. D.R. Karger, S.J. Phillips, E. Torng, A better algorithm for an ancient scheduling problem, Society for Industrial and Applied Mathematics, Philadelphia in SODA (1994), pp. 132–140 21. S. Kern, S. M¨uller, N. Hansen, D. B¨uche, J. Ocenasek, P. Koumoutsakos, Learning probability distributions in continuous evolutionary algorithms–a comparative review. Nat. Comput. 3(1), 77–112 (2004) 22. G. Koren, D. Shasha, Dover: An optimal on-line scheduling algorithm for overloaded uniprocessor real-time systems. SIAM J. Comput. 24(2), 318–339 (1995) 23. D.J. Lehmann, L.I. O’Callaghan, Y. Shoham, Truth revelation in approximately efficient combinatorial auctions, in ACM Conference on Electronic Commerce (ACM, New York, 1999), pp. 96–102 24. X.-Y. Li, Y. Wang, Simple approximation algorithms and ptass for various problems in wireless ad hoc networks. J. Parallel Distrib. Comput. 66(4), 515–530 (2006) 25. X.-Y. Li, P. Xu, S. Tang, X. Chu, Spectrum bidding in wireless networks and related, in COCOON (Springer, Berlin, 2008), pp. 558–567 26. R. Myerson, Optimal auction design. Mathematics of operations research 6(1), 58 (1981) 27. C.A. Phillips, C. Stein, J. Wein, Scheduling jobs that arrive over time (extended abstract), in WADS (Springer, London, 1995), pp. 86–97 28. J.A. Stine, Spectrum management: The killer application of ad hoc and mesh networking, in DySPAN, Baltimore, 8–11 Nov 2005 29. L. Trevisan, Non-approximability results for optimization problems on bounded degree instances, in STOC (ACM, New York, 2001), pp. 453–461 30. V. Vapnik, A. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–280 (1971) 31. S. Wang, P. Xu, X.-H. Xu, S. Tang, X.-Y. Li, X. Liu, Toda: Truthful online double auction for spectrum allocation, IEEE DySpan, IEEE, Singapore (2010) 32. G.J. Woeginger, On-line scheduling of jobs with fixed start and end times. Theor. Comput. Sci. 130(1), 5–16 (1994) 33. Y. Wu, D.H.K. Tsang, Distributed multichannel power allocation algorithm for spectrum sharing cognitive radio networks, IEEE INFOCOM, IEEE, Rio de Janeiro (2009) 34. P. Xu, X. Li, Online market driven spectrum scheduling and auction. In Proceedings of the 2009 ACM Workshop on Cognitive Radio Networks (ACM, New York, 2009), pp. 49–54 35. P. Xu, X. Li, SOFA: Strategyproof online frequency allocation for multihop wireless networks, in 20th International Symposium on Algorithms and Computation (ISAAC) (Springer, New York, 2009), p. 311

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36. P. Xu, X. Li, TOFU: Semi-truthful online frequency allocation mechanism for wireless networks. Netw. IEEE/ACM Trans. 99, 1 (2011) 37. P. Xu, X.-Y. Li, Oasis: Online algorithm for spectrum allocation in multihop wireless networks 38. P. Xu, X.-Y. Li, S. Tang, J. Zhao, Efficient and strategyproof spectrum allocations in multichannel wireless networks. IEEE Trans. Comput. 60(4), 580–593 (2011) 39. P. Xu, S. Wang, X. Li, SALSA: Strategyproof Online Spectrum Admissions for Wireless Networks. IEEE Trans. Comput. 59(12), 1691–1702 (2010) 40. Y. Yuan, P. Bahl, R. Chandra, T. Moscibroda, Y. Wu, Allocating dynamic time-spectrum blocks in cognitive radio networks, in MobiHoc (ACM, New York, 2007), pp. 130–139 41. F. Zheng, Y. Xu, E. Zhang, On-line production order scheduling with preemption penalties. J. Comb. Optim. 13, 189–204 (2007) 42. X. Zhou, S. Gandhi, S. Suri, H. Zheng, eBay in the Sky: strategy-proof wireless spectrum auctions, in MobiCom (ACM, New York, 2008), pp. 2–13 43. X. Zhou, H. Zheng, Trust: A general framework for truthful double spectrum auctions, in IEEE INFOCOM (IEEE, Rio de Janeiro, 2009)

Optimal Partitions Frank K. Hwang and Uriel G. Rothblum

Contents 1 Formulation and Classification of Partition Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Sets of Partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Number of Characteristics Associated with each Partitioned Elements. . . . . . . . 1.3 The Objective Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries: Optimality of Extreme Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Single-Parameter: Polyhedral Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Single-Parameter: Combinatorial Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Multiparameter: Polyhedral Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Multiparameter: Combinatorial Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2302 2302 2304 2304 2308 2313 2322 2333 2342 2354

Abstract

This chapter provides a survey on the problem of partitioning points in the d -dimensional space with the goal of maximizing a given objective function; the survey is based on the forthcoming book (Hwang FK, Rothblum UG (2011b) Partitions: optimality and clustering. World Scientific). The first two sections describe the framework, terminology, and some background material that are useful for the analysis of the partition problem. The middle two sections study the case of d D 1, and the last two sections study the case of general d . Results



Work of Uriel G. Rothblum was supported by The Israel Science Foundation (Grant 669/06).

F.K. Hwang National Chiao Tung University (Retired), Hsinchu, People’s Republic of China e-mail: [email protected] U.G. Rothblum Faculty of Industrial Engineering and Management Science, Technion - Israel Institute of Technology, Haifa, Israel e-mail: [email protected] 2301 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 40, © Springer Science+Business Media New York 2013

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are presented without proofs, but with motivations, comments, and references. Complete proofs can also be found in the forthcoming book.

1

Formulation and Classification of Partition Problems

Consider a finite set N of distinct positive integers (for most of the development N D f1; : : : ; jN jg). A partition of N is a finite collection of sets  D .1 ; : : : ; p / where 1 ; : : : ; p are pairwise disjoint nonempty sets whose union is N . In this case, p is called the size of , and the sets 1 ; : : : ; p are called the parts of . Further, if n1 ; : : : ; np are the sizes of 1 ; : : : ; p , respectively, the vector Pp .n1 ; : : : ; np / is called the shape of ; of course, in this case, j D1 nj D jN j. The prefix “p-” or “.n1 ; : : : ; np /-” is sometimes added to explicitly express the size or shape of a partition, referring to a p-partition of N or to an .n1 ; : : : ; np /partition of N . Further, for brevity, the reference to the set N as the partitioned set is frequently omitted, referring simply to partitions. In the forthcoming development, it is sometimes required that partitions’ parts are nonempty, while at other times, this requirement is relaxed. Partition problems are (combinatorial) optimization problems in which a partition  is to be selected out of a given set … so as to optimize (i.e., minimize or maximize) an objective function F that is defined over …. Such problems are classified by (i) the set of partitions … over which optimization takes place, (ii) the number of characteristics associated with each of the partitioned elements, and (iii) the objective function F that is defined over …. It is next elaborated on each of these classifiers.

1.1

Sets of Partitions

At times, attention is restricted to the set of all partitions or to the set of all partitions whose size or shape satisfies prescribed restrictions; these situations are, respectively, referred to as open, constrained-size, and constrained-shape sets of partitions. Situations where the restrictions on the size or shape are expressed by prescribing a single element are referred to as single-size or single-shape sets of partitions, and situations where restrictions are expressed through lower and upper bounds on the size or shape are referred to as bounded-size or bounded-shape sets of partitions, respectively. All of the above classes of sets of partitions can be treated as special cases of a constrained-shape class. Their respective names simply emphasize the kind of constraints on the shapes. For example, the class of p-partitions collects those partitions with p (nonempty) parts, and the class of open partitions collects all partitions without restriction on the number of (nonempty) parts. Thus, constrainedshape class is the most general class. Still, the general framework of constrainedshape sets of partitions does not appear in the forthcoming development, and whenever constrained-shape partition problems are mentioned, all shapes have the same size; consequently, the terminology “constrained shape” is used for sets of

Optimal Partitions Table 1 Classification of sets of partitions

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Open

Constrained-size Bounded-size Single-size

Constrained-shape (size given) Bounded-shape (size given) Single-shape

partitions with restricted shapes that have the same size (though formally, these are single-size constrained-shape sets of partitions). Sometimes, the above adjectives will be used casually to refer to partitions that belong to corresponding sets. Let the (partitioned) set N be given and let n  jN j. When a single-size set of partitions is considered, the prescribed single-size is given as a positive integer p. Given a positive Pp integer p and a set  of positive integervectors .n1 ; : : : ; np /, each satisfying j D1 nj D n, … will denote the corresponding constrained shape partitions, that is, all partitions with shape in . In particular, if  consists of a single vector .n1 ; : : : ; np /, the notation ….n1 ;:::;np / refers to (the single-shape set of partitions)P… . Also, if L and Pp U are nonnegative integer p-vectors satisfying p L  U and j D1 Lj  n  j D1 Uj ,  .L;U / will denote the set of nonnegative integervectors .n1 ; : : : ; np / satisfying Lj  nj  Uj for j D 1; : : : ; p; in this case, .L;U / . (The the notation ….L;U / is used for (the bounded-shape set of partitions) … restrictions on L and U assure that  .L;U / and ….L;U / are nonempty.) Of course, single-size and single-shape sets of partitions are instances of bounded-shape sets obtained, respectively, by setting Lj D 1 and Uj D jN jp C1 for all j or Lj D n for all j . Similarly, open and single-size sets partitions are instances of bounded-size sets. The hierarchy of the classification of partitions is summarized in Table 1. A multiset is a group of elements where each element is allowed to have multiple occurrence. The formal notation of a multiset has double brackets, for example, ff1; 1; 2; 2; 3gg, or is given as a bracketed list of distinct elements with superscripts designating their duplications, for example, f12 ; 22 ; 3g. However, at times, (the abused notation of) single brackets is used, for example, f1; 1; 2; 2; 3g. It is implicitly assumed in the above definitions that the parts of partitions are distinguishable. But, in some applications, the parts are indistinguishable and can be permuted without any restrictions. These situations are referred to as unlabeled partitions. Formally, an unlabeled partition of N is a finite collection of sets  D f1 ; : : : ; p g where the j ’s are as above. Again, p and the sets 1 ; : : : ; p are referred to as the size and the parts of , respectively. Further, if n1 ; : : : ; np are the sizes 1 ; : : : ; p , respectively, the shape of  is the multiset ffn1 ; : : : ; np gg; Pof p again, j D1 nj D jN j is required. In the literature, unlabeled partitions are commonly referred to as allocations. Herein, the term “partitions” is reserved to labeled ones; but, at times (when potential ambiguity may arise), there is reference to labeled partitions. The classification to sets of labeled partitions applies to unlabeled ones; see Table 1. Single-shape and bounded-shape sets of unlabeled partitions are defined, respectively, by a multiset ffn1 ; : : : ; np gg or a multiset of pairs ff.L1 ; U1 /; : : : ; .Lp ; Up /gg; the specification or the bounds on the sizes of the parts of partitions then hold for some labeling of the parts. Frequently, as parts of

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unlabeled partitions are indistinguishable, sizes and bounds of unlabeled partitions are uniform, namely, all parts have the same size and a multiset of bounds ff.L1 ; U1 /; : : : ; .Lp ; Up /gg consists of p identical pairs.

1.2

The Number of Characteristics Associated with each Partitioned Elements

It is assumed throughout that each element i of the partitioned set N is associated with a vector Ai 2 Rd where d is a fixed positive integer (independent of i ); the coordinates of Ai are referred to as parameters or characteristics associated with i . The n vectors Ai are part of the data for the problem and are given in the form of a real d  n matrix A. For a subset S of N D f1; : : : ; ng, AS is the submatrix of A consisting of the columns of A indexed by S , ordered as in A. Also, “bars” over matrices are used to denote the multiset consisting of their columns, for example, a subset S of N D f1; : : : ; ng, AS is the set of columns of AS , accounting for multiplicities.

1.3

The Objective Function

An objective function F ./ associates each (feasible) p-partition  with a value F ./, and F ./ is to be maximized (or minimized) over the given set of partitions. The value F ./ of a partition  is assumed to depend on the parameters of the elements that are assigned to each part. More specifically, for each positive integer v, there is a column-symmetric function hv W Rd v ! Rd , defined over multisets of v d -vectors; there are functions gj W Rd ! Rm , j D 1; : : : ; p, and a function fp W Rmp ! R. Then the value F ./ associated with partition  having shape .n1 ; : : : ; np / is given by   F ./ D fp g1 Œhn1 .A1 /; : : : ; gp Œhnp .Ap / :

(1)

The functions hnj can, in a more general context, depend on the location within the variables of fp , that is, on the index j ; also, the functions gj may depend on nj . In many common applications, each of the functions hnj is the summation function; in such cases, the problems are called sum-partition problems. When hv or gj is independent of the indexing parameter, the corresponding subscripts are dropped. Also, when referring to partitions of common size, the subscript “p” of fp is dropped. Functions fp are called additive when fp is the sum function. It is also possible to consider partition problems where the domain of the functions hnj consists of ordered lists (and the functions hnj are not symmetric). Some further notation is next introduced for a more concise form of sum-partition problems. For a d  n real matrix A and a p-partition  D .1 ; : : : ; p / of N , the -summation-matrix of A, denoted A , is given by

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2 A  4

X i 21

Ai ; : : : ;

X

3 Ai 5 2 Rd p ;

(2)

i 2p

where the empty sum is defined to be 0. When each of the gj ’s is the identity over Rd , the objective function F associates with partition  the value F ./ with the representation F ./ D fp .A /: (3) (As was already mentioned, when the optimization over partitions concerns only partitions of fixed size p, the dependence of the functions fp on p is suppressed). Of particular interest is the case where all Ai ’s have a common coordinate, say the first one, and it equals 1 for each Ai . In this case, row 1 of A is the shape of . It follows that ( 3) allows for the objective function F to depend on the shape of partitions. Of course, for single-shape problems, the part-sizes (i.e., the coefficients of the shape) are fixed and can be viewed as parameters of the objective function. In summary, the major classifiers of partition problems are: (1) The family of partitions … over which the function F (with representation as in (1) or (3)) is considered and optimized: Using the classification of families of partitions provided in Table 1, there are open, constrained-size, boundedsize, single-size, constrained-shape, bounded-shape, and single-shape partition problems. In addition, relaxed-size problems refer to single-size problems which allow for empty parts. The description of the set of feasible partitions has to specify whether or not empty parts are allowed. (2) The number of parameters associated with each of the partitioned elements: The forthcoming development refers to single-parameter problems, two-parameter problems, and multiparameter problems. (3) The objective (cost) function F as expressed by (1): Adjectives like “sum,” “max-,” or “mean-” of partition problems reflect properties of hv while properties of f , like “linear,” “convex,” “Schur convex,” and “separable,” are explicitly referred to fp , for example, sum-partition problems with f Schur convex. A partition that maximizes/minimizes F ./ over a prescribed family of partitions … is called optimal over …. For single-, bounded-, and constrained-shape families of partitions, the inclusion or exclusion of empty parts is implicit in the description of the set of feasible shapes – empty parts are prohibited when the set of feasible shapes consists only of positive vectors, whereas the inclusion of nonnegative shapes that are not (strictly) positive indicates that empty parts are allowed. In particular, results that apply to all single-, bounded-, or constrained-shape partition problems with empty parts allowed extend to corresponding problems with empty parts prohibited (by restricting attention to corresponding sets of shapes that consist only of positive vectors). Still, there are results in this chapter that apply to constrained-shape

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problems that require the exclusion of empty parts. For sets of partitions that are determined by restrictions on size, for example, the case of single-size, the inclusion or exclusion of empty parts, must be made explicit as neither variant captures the other; see the results of the forthcoming Sect. 3 (summarized Table 4) about the optimality of monopolistic partitions for size problems with empty parts allowed and the optimality of extremal partitions for size problems with empty parts prohibited. Default positions in different parts of this chapter differ – at times they allow for empty parts, at times they exclude them, and at times they allow either way (to be determined through the specification of the set of feasible shapes). The above classification refers to labeled partitions. Optimization problems over sets of unlabeled partitions can be embedded in the above framework, with … and F being invariant under part permutation. A major goal of this chapter is the identification of properties that are present in optimal partitions, thereby allowing one to restrict attention to corresponding subclasses of partitions. These properties are usually defined in terms of geometric/algebraic characteristics of the set of vectors associated with the indices assigned to the parts of the underlying partition. In particular, the focus is on properties that capture “clustering” of these vectors. The number of single-shape partitions is exponential in n (which implies the number of size partitions, open partitions, and other varieties of partitions are all exponential in n), so that it is not feasible to enumerate efficiently the partitions for selecting the optimal one. On the other hand, if a property has only polynomially many members, then it is possible to find an optimal partition by enumerating the subclasses of partitions having this property and comparing the values of the objective function. Formally, a property Q of a partition is a unitary relation over sets of partitions, and it can be identified with the sets of partitions that satisfy it. Properties that are considered when studying multiparameter problems depend on the Ai ’s; as such, it seems natural to consider the partitioning of the Ai ’s rather than the partitioning of the underlying index set. But, possible equality among the Ai ’s often obscures the partition property. This situation is easily handled when the partitions are of the set of distinct indices and not of the Ai ’s. A type 1 result for a partition problem is one that establishes a property for every optimal partition; a type 2 result is one that establishes a property for some optimal partition. The next lemma, observed by Golany et al. [29], records a useful implication for the presence of properties in optimal solutions for single-shape/size and constrainedshape/size partition problems. For simplicity, attention is restricted to partitions whose size p is fixed. Lemma 1 Consider an objective function F over partitions, a set  of integer p-vectors whose coordinate-sum is n and a property Q of partitions such that each single-shape partition problem corresponding to a shape in  and the objective function F , Q is satisfied by some (every) optimal partition. Then, for each constrained-shape partition problem corresponding to a subset of  and the

Optimal Partitions

2307

objective function F , Q is satisfied by some (every) optimal partition. Also, the above holds with “shape” replaced by “size.” For single-parameter problems, the n elements in the single row of A are usually denoted  1 ; : : : ;  n . In particular, for single-shape sum-partition problems, the -summation matrix associated with a partition  is a p-vector, which is denoted  and referred to as -summation vector, that is, 0   @

X i 21

i; : : : ;

X

1  i A 2 R1p :

(4)

i 2p

Evidently, when  i D 1 for each i ,  is the shape of . For single-parameter problems, the partitioned elements can be renumbered so that  i is monotone in i , that is,  1       n: (5) (Still, when analyzing the complexity of algorithm that solve partition problems, one should account for time required to sorting  i ’s, if needed). Much of the analysis focuses on partition problems with fixed size (single-shape, bounded-shape, constrained-shape, and single-size problems). When considering problems that allow for varying partition sizes, the objective function is, effectively, parameterized by an integer p representing the partition sizes; see (1). In particular, results about optimal partitions will then depend on some structure that expresses a connection between the objective function of p-partitions for different values of p. The most natural structure is the “reduction assumption” that asserts that F is invariant under the permutation/elimination of empty parts. For example, this is the case when fp in ( 1) is the sum- (max/min)-function and the value assigned to empty parts is 0 (1/1). A natural assumption that accompanies the “reduction assumption” is symmetry of F under part permutations. Open problems that do not satisfy the reduction assumption can be difficult to analyze: They may have infinitely many feasible partitions, and there need not be an optimal one; this situation is demonstrated in the next example. Example 1 Let N D f1; 2g and consider the open problem where Pempty sets Pp are allowed and objective function is given by F ./ D j i 2j i for j D1 partitions  of size p. Then every single-size problem will assign both elements of N to the part with the highest index. But, the open problem does not have an optimal partition as every p-partition is dominated by the best .p C 1/-partition.  When the “reduction assumption” is satisfied, each partition with p nonempty parts can, effectively, be viewed as (one of several) p 0 -partitions (with empty sets allowed) for any p 0  p. In particular, upper bounded-size problems with upper bound p and open problem can be embedded in single-size problems with,

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F.K. Hwang and U.G. Rothblum

respectively, p or n as the prescribed size (and with empty parts allowed). The amount of computation of solving n-size problems may be high, but the reduction may be useful in establishing properties of optimal solutions. Further, in cases where an optimal solution for a single-size problem with empty parts excluded is simple, optimizing over the number of nonempty parts may be computationally tractable. When the reduction assumption is in effect, allowing for empty parts is a technical notion that helps us address bounded-size and open problems. The following list provides a collection of applications of the optimal partition problem reported in the literature (number 11 has not yet appeared in the literature): 1. Assembly of systems (Derman et al. [17]; El-Newerbi et al. [22, 23]; Du [19]; Du and Hwang [20]; Malon [49]; Hwang et al. [41]; Hwang and Rothblum [36]) 2. Group testing (Dorfman [18], Sobel and Groll [68], Gilstein [28], Pfeifer and Enis [59], Hwang et al. [41]) 3. Circuit card library (Kodes [47]; Garey et al. [27]) 4. Clustering (Fisher [24]; Boros and Hwang [7]; Golany et al. [29]) 5. Abstraction of finite state machines (Oikonomou and Kain [58], Oikonomou [56, 57]). 6. Multischeduling (Mehta et al. [51] Rothkopf [62], Tanaev [69], Graham [31], Cody and Coffman [16], Chandra and Wang [13], Easton and Wong [21]) 7. Cashe assignment (Gal et al. [26]) 8. Blood analyzer problem (Nawijn [53]) 9. Joint replenishment of inventory (Chakravarty et al. [11, 12], Goya [30], Silver [67], Nocturne [55], Chakravarty [10], A.K. Chakravarty 1983, Coordinated multi-product purchasing inventory decisions with group discounts, unpublished manuscript; A.K. Chakravarty, 1982b, Consecutiveness rule for inventory grouping with integer multiple constrained group-review periods, unpublished manuscript) 10. Statistical hypothesis testing (Neyman and Pearson [54]) 11. Nearest neighbor assignment 12. Division of property (Granot and Rothblum [32]) 13. Consolidating farm land (Brieden and Gritzmann [8]; Borgwardt et al. [5])

2

Preliminaries: Optimality of Extreme Points

This section presents sufficient conditions for the optimality of vertices, conditions that unify classical quasi-convexity and of Schur convexity. Throughout, standard definitions and facts about polytopes and convex sets are used; in particular, the vertices of a polytope are its extreme points. Let C be a convex subset of Rp and let f W C ! R. The function f is (strictly) quasi-convex along a nonzero vector d 2 Rp , or briefly (strictly) d -quasi-convex, if the maximum of f over every line segment in C with direction d is attained (only) at one of the two endpoints of that line segment, possibly both. The function f is (strictly) quasi-convex along a set D  Rp , or briefly (strictly) D-quasi-convex, if f is (strictly) quasi-convex along every nonzero vector d 2 D. The function f

Optimal Partitions

2309

Fig. 1 Quasi-convexity

is (strictly) quasi-convex if it is (strictly) Rp -quasi-convex. Reference to (strict) directional quasi-convexity is an abbreviation of (strictly) quasi-convexity along sets. Recall that a function f is (strictly) convex on C if for every distinct x; y 2 C and 0 < ˛ < 1, f Œ˛x C .1  ˛/y  . 0 the restriction of f to the polytope x 2 R W x  0; i D1 xi D ˇ is edge-quasi-convex. Evidently, quasi-convex functions and Schur convex functions are asymmetric Schur convex. Thus, asymmetric Schur convexity unifies classical quasi-convexity and Schur convexity. The following example identifies functions that are asymmetric Schur convex but neither convex nor symmetric, let alone Schur convex. Example 2 Let f W R2 ! R be given by f .x1 ; x2 / D g.x1 C x2 / C h.x1  x2 / with g arbitrary and h convex. To see that f is asymmetric Schur convex consider a line with direction e 1  e 2 , say fx C .e 1  e 2 / W  2 Rg. It then follows that f Œx C .e 1  e 2 / D g.x1 C x2 / C h.x1  x2 C 2 /, and this expression is clearly convex in  . The function f need not be symmetric nor convex, for example, f .x/ D sin.x1 C x2 / C .x1  x2 /2 .  Theorem 4 Let P be a polytope all of whose edges have direction in fe ij W .i; j / 2 ƒg and let f W P ! R. If f is asymmetric Schur convex on P , then f attains a maximum over P at one of P ’s vertices. Further, if f is strictly asymmetric Schur convex, then every maximizer of f over P is a vertex of P ; in particular, such a maximizer exists. Pp Polytopes of the form fx 2 Rp W x  0, i D1 xi D ˇg with ˇ  0 clearly satisfy the assumption of Theorem 4 about the direction of their edges. Theorem 4 follows Hwang and Rothblum [37]. It streamlines the classical result about quasi-convexity being a sufficient condition for optimality of extreme points with a corresponding result about Schur convex functions. Interestingly, historically, these two conditions were considered distinct; in fact, Marshall and Olkin [50, p. 13] refer to the use of “convexity” in describing Schur convex functions as “inappropriate” and state that they adhere to the “historically accepted term ‘Schur convexity’.” Figure 2 below provides a graphical illustration of the relationships between quasi-convexity, directional quasi-convexity, classical Schur convexity (for symmetric functions), and asymmetric Schur convexity.

Optimal Partitions

2313

Quasi-convexity Schur convexity

Asymmetric Schur convexity Directional quasi-convexity

Fig. 2 Relationships among quasi-convexity, directional quasi-convexity, Schur convexity, and asymmetric Schur convexity

3

Single-Parameter: Polyhedral Approach

This section examines single-parameter partition problems. In particular, a class of polytopes, called partition polytopes, is introduced, and these polytopes are used to study partition problems. Throughout this section and the next one, let (the partitioned set) N D f1; : : : ; ng, (the size of partitions) p and (the real numbers associated with the partitioned elements)  1 ; : : : ;  n be fixed. Without loss of generality, assume that the  i ’s are ordered and satisfy 1  2      n I

(10)

also,  will stand for the vector . 1 ; : : : ;  n /. Recall from Sect. 1 that for a p-partition  D .1 ; : : : ; p / of N ,  is given by 



0

  1 ; : : : ; p D @

X

i 21

i; : : : ;

X

1  i A 2 Rp :

(11)

i 2p

Given a set … of ordered p-partitions, the …-partition polytope is denoted P …  conv f W  2 …g  Rp ; when the set … is either generic or clear from the context, the prefix “…” is omitted. In forthcoming sections, partition problems are examined where the partitioned elements are associated with vectors rather than scalars; in the broader context, the above polytopes are called single-parameter partition polytopes.  The abbreviated notation P  will be used P … when  is a set of nonnegative Pfor p integer p-vectors .n1 ; : : : ; np / satisfying j D1 nj D n, and such a polytope will be referred to as a constrained-shape partition polytope. Further, if  consists of a

2314

F.K. Hwang and U.G. Rothblum .n1 ;:::;np /

single vector .n1 ; : : : ; np /, P .n1 ;:::;np / will stand for P … and such a polytope will be referred to as a single-shape partitionPpolytope; if L and U nonnegative Pare p p .L;U / integer p-vectors satisfying L  U and j D1 Lj  n  j D1 Uj , P .L;U /

will stand for P … and such a polytope will be referred to as a bounded-shape partition polytope; finally, if … consists of all partitions of size p, P p will stand for p P … and such a polytope will be referred to as a single-size partition polytope. The vertices of a …-partition polytope P … have a representation  with  2 …. Thus, in seeking an optimal partition in … when the objective function F has the representation F ./ D f . / for each  2 …, it is useful to examine an extension of the functions f to P … (from f W  2 …g) and use results of Sect.2 to determine conditions that suffice for this extension to attain a maximum over P … at a vertex. Consider a single-shape problem corresponding to shape. If nj D 0 for some j D 1; : : : ; p, then for every partition  2 ….n1 ;:::;np / j is empty and . /j D 0; so, part j is redundant. Consequently, such indices j can be eliminated, and there is no loss of generality by imposing the restriction that all nj ’s are positive. But, such a reduction needs not be possible when considering constrained-shape problems. The analysis of partition polytopes usually allows for empty parts (diverting from the default position that empty parts are prohibited). The following example identifies an important class of partition polytopes. Example 3 Consider the single-shape partition Pp polytope defined by .n1 ; : : : ; np / D .1; : : : ; 1/. In particular, in this case, n D j D1 1 D p. A partition  with shape .1; : : : ; 1/ corresponds to a permutation of .1; : : : ; p/, and for such a partition,  is the p-vector obtained from the corresponding coordinate permutation of . 1 ; : : : ;  p /. The partition polytope P .1;:::;1/ is then the convex hull of these (permuted) vectors, and it is called the generalized permutahedron corresponding to . 1 ; : : : ;  p /. The standard permutahedron is the generalized permutahedron with  i D i for i D 1; : : : ; p.  Generalized permutahedra were first investigated by Schoute [63]; see also Ziegler [71, pp. 17–18 and 23]. A set of partitions … is next associated with two additional polytopes; it will be shown that the definition of these polytopes provides useful representations for partition polytopes under general conditions. First, let C … be the solution set of the following system of linear inequalities: X j 2I

8 9 1, a consecutive partition  is called polarized-extremal if it has two distinguished parts, say j˚ and j , such that: (i) (a) If i 2 j˚ and u > i , then u 2 j˚ . (b) If i 2 j and u < i , then u 2 j . (ii) (a) If j˚ is not a singleton, then j˚  fi 2 N W  i  0g. (b) If j is not a singleton, then j  fi 2 N W  i  0g.

Optimal Partitions

2321

(iii) If j … fj˚ ; j g, then j is a singleton; further, if either j˚ is not a singleton and  u D 0 for some u 2 j˚ , or j is not a singleton and  u D 0 for some u 2 j , then the single element in j , say i , has  i D 0. When   0 (  0), every extremal (reverse-extremal) partition is polarizedextremal and when  > 0 ( < 0), a partition is extremal (reverse-extremal) if and only if it is polarized-extremal. Example 4 Let n D 7 and  D .3; 2; 1; 0; 0; 1; 2/. Then .f1; 2; 3g; f4g; f5; 6; 7g/ and .f1; 2g; f3g; f4g; f5g; f6; 7g/ are polarized-extremal partitions, but .f1; 2g; f3g; f4g; f5; 6; 7g/ is not. Theorem 14 Suppose … is a single-size set of partitions with empty parts allowed, f W P … ! R is asymmetric Schur convex and   0 (  0). Then (i) there exists a partition  which is optimal, monopolistic, and has  as a vertex of P … ; (ii) if the  i ’s are distinct and f is strictly asymmetric Schur convex, then every optimal partition  is monopolistic and has  as a vertex of P … . Attention is next turned to the partition problems considered in Theorem 14 without the restriction that the  i ’s are one-sided. Theorem 15 Suppose … is a single-size set of partitions with empty parts prohibited and f W P … ! R is quasi-convex. Then (i) there exists a partition  which is optimal, polarized-extremal, and has  as a vertex of P … ; (ii) if the  i ’s are distinct and nonzero and f is strictly quasi-convex, then every optimal partition  is polarized extremal and has  as a vertex of P … . Table 4 Properties of optimal partitions (continued) …D…



bdd-shape bdd-shape Uniform bdd-shape Single sym-shape Single-size+ no ; parts Single-size+ no ; parts Single-size+ ; parts ok Single-size+ ; parts ok

f a-S-c S-c S-c a-S-c a-S-c a-S-c a-S-c a-S-c a-S-c q-c a-S-c a-S-c a-S-c

 0 0 0 0 0 0 0 0

0 0

Properties of  

  2 V .P … /

consec s-cons r-s-cons s-cons r-s-cons s-cons r-s-cons ext rev-ext polar-ext

+ + + + + + + + + +

+ + + + +

Theorem 12 Theorems 12, 6, and 7 Theorems 12, 6, and 7 Theorem 13 Theorem 13 Theorems 13, 6, and 7 Theorems 13, 6, and 7 Theorems 14, 6, and 7 Theorems 14, 6, and 7 Theorems 15, 6, and 7

monop monop Bipolar

+ + +

+ + +

Theorems 16, 6, and 7 Theorems 16, 6, and 7 Theorems 16, 6, and 7

P… Perm + +

Ref

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F.K. Hwang and U.G. Rothblum

When   0 or   0, the conclusions of Theorem 15 follow from Theorem 14. Theorem 16 Suppose … is a single-size set of partitions with empty parts allowed and f W P … ! R is asymmetric Schur convex. Then (i) there exists a partition  which is optimal, bipolar, and has  as a vertex of P … ; (ii) if the  i ’s are distinct and nonzero and f is strictly asymmetric Schur convex, then every optimal partition  is bipolar and has  as a vertex of P … . Further, if   0 or   0, then “bipolar” in the above conclusions can be replaced by “monopolistic.” Sections 2 and 3 are next augmented with a summary of the “some” results of Theorems 12–16 (Table 4).

4

Single-Parameter: Combinatorial Approach

The main goal in this section is to develop combinatorial tools for establishing that given a partition property Q and a set of partitions, some (every) partition in the set satisfies Q. Such results are useful for the analysis of partition problems when the set consists of all optimal partitions for a particular partition problem. The tool that is developed, referred to as local sorting, is to move from a partition in the given family that does not satisfy Q to a partition (in the family) that does by recursively rearranging (the elements in) a small number of parts at each step. Two conditions must be observed to guarantee that successive (local) sorting will produce a partition satisfying the given property Q have two parts: first, in each step, the sorting must produce a partition in the given set; and second, in order to avoid being trapped in a cycle, some statistic must decrease (or increase) monotonely as the sorting progresses. The first condition concerns the set of partitions, while the other concerns property Q. A useful approach of local sorting is the intuitive one which requires that the (sub)partition consisting of the sorted parts satisfies Q. These techniques will be applied to some specific optimization problems later, demonstrating the existence of optimal partitions with particular properties. Some properties of partitions were introduced in Sect. 3. These include (with abbreviations added in parenthesis) consecutiveness (C), size-consecutiveness (S ), reverse size-consecutiveness (S 1 ), extremalness (E), reverse extremalness (E 1 ), monopolisticness (Mp ), bipolar (BP ), and polarized-extremalness (PE). Let S and S 0 be two disjoint subsets of N . Then S is said to penetrate S 0 , written S ! S 0 , if there exists a; c in S 0 and b in S such that a < b < c. Note that if .S [ S 0 / ! S 00 then either S ! S 00 or S 0 ! S 00 , but if S ! .S 0 [ S 00 /, it is possible that neither S ! S 0 nor S ! S 00 . In addition, the convention that each part penetrates itself will be adopted. For a partition  D f1 ; : : : ; p g of a subset N 0 of N , define the digraph associated with , denoted G./, as the digraph with p vertices representing the parts of  and edges .i; j / for i; j 2 f1; : : : ; pg if i penetrates j . A partition  is called noncrossing (NC ) if i ! j implies j 6! i . Noncrossing partitions were first studied in Kreweras [48].

Optimal Partitions

2323

Table 5 Properties of partitions Noncrossing

NC

Consecutive

C

Nested Size-consecutive Extremal Polarized-extremal Monopolistic

N S E PE Mp

Nearly nested Reverse size-consecutive Reverse extremal Order-nonpenetrating Bi-polar

Ne N S 1 E 1 O BP

There are three special cases of noncrossingness that are of particular interest. The first concerns the consecutive partitions, introduced and studied in Sect. 3. A partition is called consecutive if and only if the penetration relation among its parts is an empty partial order, that is, no part penetrates any other part. The second instance of NC is nestedness (N ); a p-partition  is called nested if the penetration relation among its parts is linear. Finally, a partition  is called ordernonpenetrating (O) if its parts can be indexed (ordered) such that for i D 2; : : : ; p, i 6! [ij1 D1 j . Note that order-nonpenetrating was called order-consecutive by Chakravarty et al. [11]. Examples of partitions that satisfy the properties considered so far are next illustrated. Example 5 Let N D N 0 D f1; : : : ; 5g, n D 5 and p D 3. Then: (a) ff2g, f4g, f1; 3; 5gg is noncrossing but, is not order-nonpenetrating. (b) ff2g, f1; 3g, f4; 5gg is order-nonpenetrating, but neither nested nor consecutive. (c) ff1g, f2; 3g, f4; 5gg is consecutive. (d) ff3g, f2; 4g, f1; 5gg is nested.  A noncrossing partition is called nearly nested (Ne N ) if it is nested except for some parts of size 1. Example 6 Let N D f1; 2; 3; 4; 5; 6; 7g and p D 4. Then the partition .f1; 4; 7g; f2; 3g, f5g; f6g/ is nearly nested but not order-nonpenetrating.  Table 5 summarizes partition properties mentioned so far. The following result records implications among the above properties of partitions. The relationship between C , N , NC , and O was determined in Hwang et al. [42]. Theorem 17 The implications among the properties C , N , NC , O, Ne N , S , E, PE, Mp , and BP of partitions are represented precisely by the partial order illustrated in Fig. 3. Given a (single-parameter) partition problem, let nC , n , and n0 denote the number of ’s which are positive, negative, and equal 0, respectively. Following Hwang and Mallows [35] and [39], Table 6 provides the exact numbers of (single-)size partitions for the various properties listed in Table 5. Note that all the numbers are polynomial in n.

2324

F.K. Hwang and U.G. Rothblum

Mp

E

S

C

O

BP

PE

NC

NeN

N

Fig. 3 Implications among E, S, C , O, N , Ne N , NC , PE, Mp , and BP Table 6 The number of size partitions satisfying NC , C , N , Ne N , S, S 1 , E, E 1 , PE, O, Mp , and BP Q NC N S E PE Mp

Number  n  n  p1

p

 n1n p2 p1

n .p  1/pŠ 1 n0  p  1 if n0  p  2, nC > 0 and n > 0 (different numbers in other cases) 1

Q C Ne N

Number  n1  p1

Pp1

j D0

nj





nj 2 2p2j 2

S 1

np1 .p1/pŠ

E 1 O

1 Pp1 n1

BP

n0 C 1

j D0

j

n1j 2p2j 2



The focus will be on local sorting which changes only a prescribed number of parts. Local sorting requires that attention be given to subsets of the set of parts of a given partition which are then considered themselves as (sub)partitions. Let N D f1; : : : ; ng and E D f1; : : : ; pg. To facilitate reference to the indices of these (sub)partitions, it will be useful to consider a framework where a labeled partition of a set N 0  N over an index set E 0  E is a set  D f.j; j / W j 2 E 0 g where the j ’s are disjoint subsets of N 0 whose union is N 0 . In particular, E 0 , jE 0 j, and f.j; jj j/ W j 2 E 0 g will be referred to as, respectively, the support, size, and shape of . The notation  D .1 ; : : : ; p / for “(labeled) partitions,” used so far, captures the case where N 0 D N and E 0 D E; its use will be continued in those cases. Given a labeled partition of N 0 over E 0 , say  D f.j; j / W j 2 E 0 g, and a subset J of E 0 , J  f.j; j / W j 2 J g will be called the subpartition of  corresponding to J ; in particular, J is a partition of [j 2J j over J (having support J , size jJ j and shape f.j; jj j/ W j 2 J g). The reference to the partitioned set and/or the support will be sometimes suppressed, referring to partition, subpartitions, etc. Also, a k-partitions/subpartition is a partition/subpartiton of size k. Generically, “local” will refer to subpartitions of limited size.

Optimal Partitions

2325

Some definitions about properties of partitions that connect global satisfiability and local satisfiability are next introduced. Suppose Q is a property of partitions. Property Q is hereditary if every subpartition of a partition that satisfies Q must also satisfy Q. Also, for a positive integer k, property Q is k-consistent if for each p  k, a p-partition satisfies Q whenever all of its k-subpartitions satisfy Q. The minimum consistency index of a property of partitions is defined as the minimal positive integer k for which the property is k-consistent provided such an integer exists and as 1 if Q is not k-consistent for any positive integer k. The next lemma records an order for k-consistency. Lemma 6 Let k be a positive integer and let Q be a property of partitions such that a partition  satisfies Q if and only if every k-subpartition of  satisfies Q. Then Q is k 0 -consistent for all k 0 > k. Attention is next turned to transformations of partitions, where the image of a partition  consists of partitions obtained from  by sorting elements of subpartitions of  that do not satisfy Q to subpartitions that satisfy Q, while preserving the remaining parts of the partition. For a partition , let ./ denote its support and ./ denote its shape. Formally, three partition-transformations are defined for a given property Q of partitions and a given positive integer k. The transformations are T Q;k;open , T Q;k;support , and Q;k;t ./  ŒT Q;k;t J ./ is given by T Q;k;shape , where for J  E and partition , TJ 8 0 0 ˆ ˆ f W  ./nJ = ./nJ ; ˆ ˆ 0 ˆ ˆ [j 2 ./ j = [j 2 . 0 / j ; ˆ < 0 and  . Q;k;open 0 /nŒ ./nJ  2 Qg if J 62 Q and jJ j=k  jj TJ ./ D ˆ 0 ˆ ˆ f W  0 2 Qg if J 62 Q and jJ j=k > jj ˆ ˆ ˆ ˆ : ; otherwise; Q;k;support

TJ and

Q;k;shape

TJ

n o Q;k;open D  0 2 TJ ./ W . 0 / D ./

n o Q;k;open ./ D  0 2 TJ ./ W . 0 / D ./ :

For t 2 fopen, support, shapeg, partitions in T Q;k;t ./ are called .Q; k; t/-sortings of . Recall that N D f1; : : : ; ng and E D f1; : : : ; pg are assumed given. The natural universal domains for T Q;k;open , T Q;k;support , and T Q;k;shape are partitions without restriction, those with a prescribed support and those with prescribed shape, respectively (where no restriction is to be interpreted as p D n and the support of the partitions is restricted to subsets of E D f1; : : : ; ng). To index these potential domains, define X open  fopeng; X support  f W ; ¤ P  Eg, and X shape  f D f.j; nj / W j 2 g:  E; each nj is an integer and j 2 nj D jN jg Wg;

2326

F.K. Hwang and U.G. Rothblum

in particular, X support and X shape represent, respectively, all potential supports and b is the set of all partitions, then it will be denoted shapes of subpartitions over E. If … b open (the support of partitions in … b open is in the power set of f1; : : : ; ng). Further, … b , the set of the set of all partitions with a prescribed support will be denoted …  b , and finally, for positive all partitions with a prescribed shape  will be denoted … b p for … b f1;:::;pg . If … b is, respectively, … b open , … b for a particular integer p, write …  b for a particular , a corresponding transformation on … b is called open,

, or … support-preserving, or shape-preserving, respectively. The potential domains of the transformations T Q;k;t for given Q and t 2 fopen, support, shapeg are then the sets b x with x 2 X t (but degenerate instances are sometimes excluded). … For t 2 fopen, support, shapeg and x 2 X t , a property Q of partitions is t-regular for x, if for every N 0  N with …x .N 0 / ¤ ;, there is a partition in …x .N 0 / \ Q (with the intuitive interpretation of …x .N 0 /); Q is t-regular if it is t-regular for every x 2 X t . Lemma 7 Let Q be a t-regular property of partitions. Then for every x 2 X t , b x and J  E with jJ j D k,  2… Q;k;t

ŒJ 2 Q , ŒTJ

./ D ;I

(20)

further, if Q is also k-consistent and hereditary, then Œ 2 Q , ŒT Q;k;t ./ D ;:

(21)

Let k be a positive integer, let t 2 fopen, support, shapeg, let Q be a property of partitions, and let … be a set of partitions. Then: (i) … is Q-(strongly, k; t)-invariant if for each  2 … n Q: for every subset Q;k;t J of E having k elements and with J … Q, every  0 in .TJ /./ satisfies  0 2 … . (ii) … is Q-(sort-specific, k; t)-invariant if for each  2 … n Q: for every subset Q;k;t J of E having k elements and with J … Q, some  0 in .TJ /./ satisfies 0   2… . (iii) … is Q-(part-specific, k; t)-invariant if for each  2 … n Q: for some subset Q;k;t J of E having k elements and with J … Q, every  0 in .TJ /./ satisfies 0   2… . (iv) … is Q-(weakly, k; t)-invariant if for each  2 … n Q: for some subset J Q;k;t /./ satisfies of E having k elements and with J … Q, some  0 in .TJ  0 2 … . When referring to Q-(`; k; t)-invariance, the values of `, k, and t are called level, degree, and type of the corresponding invariance. The level ` can take four possible values, forming two pairs – fstrongly, weaklyg and fpart-specific, sort-specificg; if ` is a specific value, then and `1 is defined as the other value in the same pair. Also, type can take one of three possible values – “open,” “support,” and “shape” – and the possible values of degree are k D 1; 2; : : : . When a variable is not quantified

Optimal Partitions

2327

in forthcoming statements/results about .`; k; t/, it means that the statement is valid for all choices of that variable. A multivalued function on a set of partitions will be called a statistic. To compare values of a statistics, “ f .b/. As majorization and weak sub/super-majorization are invariant over permutations of vectors, these relations can be applied to multisets of real numbers (with the interpretation that the relation holds for any representing vectors). Lemma 10 Let f be Schur convex and nondecreasing (nonincreasing) and Q be .`,k,t/-sortable with ` 2 fstrongly, sort-specific, part-specific,weakg, k D 2; 3; : : : and t 2 fshape, sizeg. Suppose for any partition  not satisfying Q, there always exists an .`1 ; k; t/-Q-sorting from  to  0 such that, with K as the indices of the sorted parts, ffgj Œh.j0 / W j 2 Kgg w . w /ffgj Œh.j / W j 2 Kgg. Then every t-problem has an optimal partition satisfying Q. Further, under strict conditions, every optimal partition satisfies Q. Finally, the above conclusions hold if f is just Schur convex and the weak majorization requirement is replaced by majorization. Objective functions that are given by (22) with f Schur convex, gj independent of j and h as the sum-function, are next addressed. The result modifies Theorem 21 which considered problems with f convex and gj ’s allowed to depend on j .

Optimal Partitions

2331

Theorem 22 Suppose F ./ D f Œg.1 /; : : : ; g.p /;

(25)

where f is Schur convex and either of the following holds: (a) f is nondecreasing, and g is convex and nondecreasing (nonincreasing). (b) f is nonincreasing, and g is concave and nondecreasing (nonincreasing). (c) g is linear and nondecreasing (nonincreasing) (and no monotonicity assumption imposed on f ). Then every shape-problem has a consecutive optimal partition. Further, if in addition   0 (  0), then every shape-problem has a size-consecutive (reverse size-consecutive) optimal partition, every size-problem has an extremal (reverse extremal) optimal partition, and every relaxed-size problem has a monopolistic optimal partition. Finally, under strict conditions, the corresponding type 1 conclusions hold. The objective function of (25) with g as the identity and f Schur convex, but not necessarily monotone, was studied in Sect. 3. In fact, the conclusions of Theorem 22 for those cases were established in Theorem 11 for shape-problems and in Theorem 14 for size-problems (the latter with the Schur convexity of f relaxed to asymmetric Schur convexity). In a clustering problem, one wishes to partition the points into clusters such that under some distance measure, points in the same cluster are close to each other compared to points in different clusters. It is natural to formulate the clustering problem into a minimization problem. To preserve this spirit, consider through the end of this section partition problems where F ./ is to be minimized. The notion of “strict conditions,” will continue to refer to strict properties of functions appearing in the objective function, for parameters (the forthcoming wj ’s) to be distinct and nonzero and, in addition, for  i ’s being distinct. A general distance function (gdf) is a continuous symmetric function D W R2 ! R such that for x 0  x  y  y 0 , D.x 0 ; y/  D.x; y/ and D.x; y 0 /  D.x; y/ (the continuity requirement can be relaxed for much of the forthcoming results, but technical details are required). A gdf is strict if nondecreasing and nonincreasing are replaced by increasing and decreasing, respectively. A gdf is normalized if D.x; x/ is independent of x. Boros and Hammer [6] considered the strict and normalized gdf D.x; y/ D jx  yj and established the following theorem. Theorem 23 Suppose F ./ D

p X X

j i   u j :

(26)

j D1 i;u2j

Then every size problem has a noncrossing optimal partition. Further, under strict conditions, the corresponding type 1 conclusions hold.

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F.K. Hwang and U.G. Rothblum

It is not known whether or not the conclusions of Theorem 23 extend to shape-problems, except that the case of uniform shape was solved by Pfersky et al. [60]. Theorem 24 Suppose p divides n and consider the single shape-problem corresponding to .n=p; : : : ; n=p/ with F defined by (26). Then there exists a consecutive optimal solution. Further, under strict conditions, a corresponding type 1 conclusion holds. Assume the  i ’s are given. For a given gdf D and a set S  N , c is called a centroid P of S with respect to D, or briefly the D-centroid of S , if c 2 argminx i 2S D.i ; x/; continuity assures that every finite set has a centroid. When D.x; y/ D d.jx  yj/ where d./ is a continuous, nondecreasing function on the nonnegative reals with d.0/ D 0, a d -centroid will refer to the corresponding D-centroid. If d is the identity, then the d -centroid is the median. Chang and Hwang [14] proved the type 1 results of the following theorem. Theorem 25 Suppose F ./ D

p X

wj

j D1

X

d.j i  cj j/ ;

(27)

i 2j

where d./ is a nondecreasing function on the nonnegative reals with lg d concave and d.0/ D 0, and for each j , wj > 0 and cj is a d -centroid of j . Then every size problem has a noncrossing optimal partitions. Further, under strict conditions the corresponding type 1 conclusions hold. Boros and Hwang [7] considered shape problems with the objective function given by ( 27) where d is the identity function (and cj is a medium). They proved the following result. Theorem 26 Suppose F ./ D

p X j D1

wj

X

j i  mj j ;

(28)

i 2j

where for each j , wj > 0 and mj is the median of j . Then every shape problem has a noncrossing optimal partition. Further, under strict conditions, the corresponding type 1 conclusions hold. For uniform weights, the consecutive property can be preserved under more general distance functions. Theorem 27 Suppose F ./ D

p X X j D1 i 2j

Di . i ; cj / ;

(29)

Optimal Partitions

2333

where for each i , Di is a gdf and for each j , cj P is a centroid of j with respect to fDi W i 2 j g (in the sense that cj 2 min argx i 2j Di . i ; x/). Then every size problem has a consecutive optimal partition. Further, under strict conditions, the corresponding type 1 conclusions hold. Theorem 27 for Di . i ; cj / D wi d. i cj /2 where wi is a weight on element was proved by Fisher [24]. Hwang [34] extended the Fisher result to distance functions Di . i ; cj / D wi D. i ; cj /. To obtain a shape-version of Theorem 27, uniform Di ’s that satisfy some further structure have to be assumed. Specifically, a gdf D is submodular if for every x; x 0 ; y; y 0 2 R with x  x 0 and y  y 0 , D.x 0 ; y 0 / C D.x; y/  D.x 0 ; y/ C D.x; y 0 /. The following result was proved by Hwang et al. [41]. Theorem 28 Suppose 0 F ./ D f @

X

i 21

D. i ; c1 /; : : : ;

X

1 D. i ; cp /A ;

(30)

i 2p

where D is a submodular gdf, f is Schur concave nondecreasing, and for each j , cj is a D-centroid of j . Then every shape problem has a consecutive optimal partition. Further, under strict conditions, the corresponding type 1 conclusions hold.

5

Multiparameter: Polyhedral Approach

Sections Sect. 3 and Sect. 4 examined partition properties that facilitate a reduction of the size of the set of partitions through which one has to search for an optimal partition (for one-dimensional problems). This approach is continued in the current section and the next one but with d  1 instead of d D 1. The properties that were explored in those sections are mostly geometric in nature, though the geometric characteristics may be in disguise for some of them. For example, the property “consecutive” was defined as “each part consists of consecutive integers,” but it can also be defined as “the convex hulls of the parts are pairwise disjoint.” This section and the next one continue the study of geometric properties of partitions but when the partitioned points are vectors in a high-dimensional space. A unique feature of R1 vs. Rd for d > 1 is that it has a natural order. In particular, when ordering the partitioned elements  1 ;  2 ; : : : ;  n so that  1   2       n , it is possible to express all relevant geometric properties of the partitioned elements in terms of the corresponding partition of the index set N , except hiding the distinction whether two disjoint intervals of ’s touch at the boundary (see next paragraph on this issue). This conversion allows us to focus on “index-partitions.” However, when the dimension d exceeds one, there is no natural ordering of the points which allows the conversion of geometric properties that depend on the formation of the points to properties of partitioned indices. Consequently, it is more natural to define partitions on the set of vectors and not on the set of indices.

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F.K. Hwang and U.G. Rothblum

There is still the question why the focus so far was on partitioning index sets in the case of single-parameter partition problems. The reason is that “index-partitions” have an advantage over “vector-partitions” in situations where the columns of A have duplicate vectors. As the indices are always distinct, such situations do not require any attention when partitioning the index set. But, when considering “vector-partitions,” the presence of duplicate points requires one to account for the duplication of each vector in each part, that is, the parts are multisets, rather than sets, and more importantly, the same vector may appear in different parts. Partitioning the index set in the case where d D 1 voided the need of dealing with multisets and allowed us to focus on the main development of the theory. In Sect. 5, the focus is to prove that there exists an extreme point of the partition polytope which corresponds to an optimal partition with certain geometric property. When Ai ’s are distinct, then this geometric property does not allow two disjoint parts to touch at boundary; when Ai ’s are not distinct, then the geometric property is weakened to allow such touching. But this distinction does not affect the role the partition polytopes play. Therefore, partition of the index set can be continued so as to take advantage of the convenience it provides. However, in Sect. 6, the emphasis is to study these two and other geometric properties in detail, not only in their natures, but also in their differences. In order to understand these fine points, the framework will then revert to partitioning the set A. Throughout this section, it is assumed that d  1 and A1 ; : : : ; An 2 Rd are given. In the sum-partition problem, an objective function F .:/ is maximized over a family of p-partitions … where the objective value  D .1 ; : : : ; p / is P of a p-partition P i i 2 Rd p and given by F ./ D f .A /, with A D i 21 A ; : : : ; i 2p A

f .:/ a real-valued function on Rd p (or a relevant subset thereof). The approach that is followed in the current section is to consider the problem of optimizing (an extension of) f over the partition polytope PA… defined as the convex hull of fA W  2 …g. When the function f is guaranteed to obtain a maximum over PA… at a vertex, enumerating the vertices of PA… and selecting a partition corresponding to a vertex that maximizes f will produce a solution to the partition problem. This approach is enhanced when the vertices are associated with partitions having properties that are present only in a “small number” of partitions. The case that is considered has the function f .:/ linear. It will be shown how linear programming can be used to solve the corresponding partition problem. Here, the approach is to obtain a representation of the problem as a projection of an optimization problem over a (generalized transportation) polytope which has a simple linear inequality representation. The resulting algorithm is polynomial in the number of partitioned vectors, their dimension, and the number of parts of the partitions. The approach follows Hwang, Onn, and Rothblum [44]. Superscripts will be used to denote columns of matrices, subscripts for rows, j and double indices for elements, for example, Ut , U j , and Ut . For matrices U and V of common dimension, the inner product of U and V is defined as P say Pp d  p, j j the scalar hU; V i  dtD1 j D1 Ut Vt . For matrices U , V , and W of dimension d  p, d  q, and q  p, respectively,

Optimal Partitions

2335

hU; V W i D hV T U; W i D hU W T ; V i :

(31)

Let I be the n  n identity matrix. At times, the columns of I (the n standard unit vectors in Rn ) will be the vectors that are associated with the partitioned elements 1; : : : ; n. In such cases, for each p-partition , I 2 Rnp is the matrix associated with , namely, 1 if t 2 j and j (32) .I /t  0 otherwise . For a set of partitions …, PI… D convfI W  2 …g will be referred to as a generalized transportation polytope (associated with …) and to the corresponding partition problem as a generalized transportation problem. Classical transportation polytopes are obtained when … consists of a single shape. The special attention given to this class of partition problems is justified by the following properties that will be established: (i) The correspondence  ! I is a bijection; further, none of the matrices I are expressible as a convex combination of others. (ii) Explicit representing systems of linear inequalities are available for boundedshape generalized transportation polytopes. (iii) Constrained-shape partition problems are reducible to generalized transportation problems. These properties are next verified and then used to develop efficient computational methods for solving the (linear) partition problem. Lemma 11 Let … be a set of p-partitions. Then the correspondence  ! I is a bijection of … onto the vertices of PI… ; in particular, the vertices of PI… are precisely the matrices fI W  2 …g. Further, the inverse correspondence of vertices of PI… onto the partitions of … has vertex v corresponding to the partition  with j j D ft W vt D 1g for j D 1; : : : ; p. An important feature of the bijection  ! I of Lemma 11 is the fact that it is constructive in both directions, namely, it is easy to determine I from  and, vice versa,  from I . Linear-inequality representation for bounded-shape generalized transportation polytopes is next provided. Theorem 29 Let P L and U be positive integer p-vectors satisfying L  U and P p p .L;U / L  n  . Then PI… is the solution set of the j D1 j j D1 Uj , and let …  … linear system: j

.a/ Xt  0 for t D 1; : : : ; n and j D 1; : : : ; p: .b/

p X j D1

j

Xt D 1 for t D 1; : : : ; n:

(33)

2336

F.K. Hwang and U.G. Rothblum n X

.c/ Lj 

j

Xt  Uj for j D 1; : : : ; p :

t D1

When L D U , Theorem 29 specializes to the following inequality description of classical transportation polytopes. Pp Corollary 4 Let n1 ; : : : ; np be positive integers with j D1 nj D n and let …  ….n1 ;:::;np / . Then PI… is the solution set of the linear system: j

.a/ Xt  0 for t D 1; : : : ; n and j D 1; : : : ; p: X

(34)

p

.b/

j

Xt D 1 for t D 1; : : : ; n:

j D1

.c/

n X

j

Xt D nj for j D 1; : : : ; p :

t D1

 Theorem 30 Let A 2 Rd n and let … be a set of p-partitions. Then the partition polytope PA… is the image of the generalized transportation polytope PI… under the linear function mapping X 2 PI…  Rnp into AX 2 Rd p . In particular, for every p-partition , A D AI . The representation of partition polytopes as a projection of generalized transportation polytopes derived in Theorem 30 yields the following test for vertices of constrained-shape partition polytopes: Corollary 5 Let A 2 Rd n , let … be a set of p-partitions, and let V 2 PA… . Then V is a vertex of PA… if and only if every representation V D 12 A.X 0 C X 00 / with X 0 ; X 00 2 PI… has AX 0 D AX 00 .  Theorem 30 yields the following result that shows that partition problems over … may be lifted to optimization problems over generalized transportation polytopes. Corollary 6 Let A 2 Rd n , … be a set of p-partitions, f W Rd p ! R, and F W … ! R with F ./ D f .A / for each  2 …. Consider the function h W PI… ! R with h.X / D f .AX / for each X 2 PI… . If  is a p-partition such that I maximizes h over fI 0 W  0 2 …g, then  maximizes F ./ over …; further, if f is linear with f .X / D hC; X i for every X 2 Rkp , then h.Z/ D hC; AZi D hAT C; Zi

for all Z 2 PI… ;

and a partition  with I maximizing h over PI…  fI 0 W  0 2 …g exists.

(35)

Optimal Partitions

2337

Let A 2 Rd n , C P 2 Rd p and let L P and U be positive integer vectors p p .L;U / satisfying L  U and j D1 Lj  n  ! R j D1 Uj , and F W … .L;U / . The goal is to maximize F .:/ over with F ./ D hC; A i for each  2 … …  ….L;U / . Let h W PI… ! R with h.X / D hC; AX i D hAT C; X i for each X 2 PI… . In view of Corollary 6 and Lemma 11, the problem of maximizing F over ….L;U / reduces to finding a vertex of PI… that maximizes the linear objective j with coefficients f.AT C /t W t D 1; : : : ; n and j D 1; : : : ; pg over PI… ; with j j a  maxflg jAit j W Ait ¤ 0g and c  maxflg jCi j W Ci ¤ 0g, these coefficients are bounded by ke aCc and are computable in time OŒnpd.a C c/ (the availability of sophisticated fast algorithms for multiplying matrices is ignored); using the explicit representation of PI… provided in Theorem 29, the problem reduces to a structured linear program, and standard results show that an optimal vertex for PI… can be identified in strongly polynomial time OŒ.n C p/np C .n C p/2 u where u  maxi lg.Ui Li /  n (see Ahuja and Orlin [1] or Ahuja, Orlin, and Tarjan [2]). The above solution method applies (in fact, in a simplified form) to single-shape partition problems (with linear objective) and, consequently, to constrained-shape problems where the set of shapes is tractable. Thus, for a set of shapes … whose size is polynomial in the parameters k; n, and p, a polynomial solution method by solving j…j linear programs (each having L D U is available). It is next shown that, with linear objective and special structure, a solution to single-shape partition problems can be obtained explicitly without addressing the linear programming problem over PI… . Theorem 31 Let A 2 Rd n , C 2 Rd p , n1 ; : : : ; np be positive integers with Pp .n1 ;:::;np / ! R with F ./ D hC; A i for each j D1 nj D n, and F W … .n1 ;:::;np / . If A and C satisfy  2… .At  At C1 /T .C j  C j C1 /  0 for 1  t < n and 1  j < p ; then the p-partition  with j D

nP

j 1 uD1 nu

C 1; : : : ;

(36)

o n uD1 u for j D 1; : : : ; p

Pj

maximizes F .:/ over ….n1 ;:::;np / ; further, if the inequalities of (36) hold strictly, then  is a unique maximizer. Standard arguments show that condition ( 36) is equivalent to (the seemingly stronger assertion): 

At1  At2

T  j  C 1  C j2  0 for 1  t1 < t2  n and 1  j1 < j2  p : (37)

Theorem 31 facilitates the identification of optimal partitions when specified permutations of the columns of A and C satisfy (36). This is always possible when d D 1 by sorting the two sets of scalars A1 ; : : : ; An and C 1 ; : : : ; C p to increasing sequences, thereby yielding an explicit solution to the partition problem. Two properties of partitions – separability and almost separability – are next introduced. In particular, conditions are determined for these properties to be present

2338

F.K. Hwang and U.G. Rothblum

in partitions whose associated vector is a vertex of a constrained-shape partition polytope; results of Sect. 2 can then be used to deduce conditions which assure the existence of optimal partitions which are (almost) separable. Two subsets 1 and 2 of Rd are called separable if there exists a nonzero d -vector C such that C T u1 > C T u2 for all u1 2 1 and u2 2 2 :

(38)

Two subsets 1 and 2 of Rd are called almost separable if there exists a nonzero d -vector C such that C T u1 > C T u2 for all u1 2 1 and u2 2 2 with u1 ¤ u2 :

(39)

In either of these cases, the vector C is referred to as a separating vector. Of course, separable sets are necessarily almost separable and disjoint and for disjoint sets almost separability and separability coincide. When two sets are not disjoint, almost separability (as defined by (39)) is the strongest possible form of separation by a linear functional, as the condition asserts strict separation of the values of the functional for all pairs of points at which the sets do not overlap. In particular, (39) implies that 1 \ 2 contains at most a single point, for if u and w were distinct points u and w in the intersection, it would follow that C T u > C T w and C T w > C T u. The next lemma converts separability of finite sets to separability of their convex hulls, yielding as a corollary, a characterization of the former. Lemma 12 Let 1 and 2 be two finite sets in Rd and let C be a nonzero vector in Rd . Then: (a) C T u1 > C T u2 for all u1 2 1 and u2 2 2 if and only if C T w1 > C T w2 for all w1 2 conv 1 and w2 2 conv 2 . (b) C T u1 > C T u2 for all u1 2 1 and u2 2 2 with u1 ¤ u2 if and only if C T w1 > C T w2 for all w1 2 conv 1 and w2 2 conv 2 with w1 ¤ w2 . Call a partition  D .1 ; : : : ; p / separable (almost separable) if the sets fAi W i 2 j g for j D 1; : : : ; p are pairwise separable (almost separable). The next result and its corollary establish (almost) separability of a partition  whose associated matrix A is a vertex of a corresponding constrained-shape partition polytope. Under the restriction that the columns of A are distinct, the results are due to Barnes et al. [4]. Some of the results about separable partitions for the general case appear in Hwang et al. [43] and the results about almost separable partitions are from Hwang and Rothblum [38, 40]. Theorem 32 Let  be a nonempty set of positive integer p-vectors with coordinatesum n, and let  be a partition in … where A is a vertex of PA . Then for some matrix C 2 Rd p (with columns C 1 ; : : : ; C p )

Optimal Partitions

.C r  C s /T u > .C r  C s /T w for all distinct r; s 2 f1; : : : ; pg; u 2 conv fAi W i 2 r g and w 2 conv fAi W i 2 s g with u ¤ w:

2339

(40)

Corollary 7 Let  be a nonempty set of positive integer p-vectors with coordinatesum n, and let  be a partition in … where A is a vertex of PA . Then  is almost separable; further, if the columns of A are distinct, then  is separable. The next result shows that when d D 1, the necessary condition for the matrix (in fact, vector for d D 1) associated with a partition to be a vertex is also sufficient. Pp Theorem 33 Let A 2 R1n , let n1 ; : : : ; np be positive integers with j D1 nj D n and let  be a partition with shape .n1 ; : : : ; np /. Then  is almost separable if and .n ;:::;n / only if A is a vertex of PA 1 p .D PA for  D f.n1 ; : : : ; np /g/. The next result establishes the optimality of (almost) separable partitions. Theorem 34 Let  be a nonempty set of positive integer p-vectors with coordinatesum n and let f ./ be an (edge-)quasi-convex function on the constrained-shape partition polytope PA . Then there exists an optimal partition  which is almost separable and has A as a vertex of PA . Further, if f ./ is strictly (edge-)quasiconvex, then every optimal partition  is almost separable and has A as a vertex of PA . If A’s columns are distinct, the quantifier “almost” can be dropped in the above statements. Theorem 34 shows that when considering constrained-shape partition problems with f ./ (edge-)quasi-convex, it suffices to consider partitions that are (almost) separable (and whose associated matrix is a vertex of the corresponding partition polytope). Cone-separability, another property of partitions, it next introduced and studied. In particular, it will be shown that single-size partition problems have optimal partitions with this property and an enumeration algorithm for solving such problems will be developed. It is emphasized that the results require the assumption that empty parts are allowed. Of course, single-size problems with empty parts prohibited (allowed) are instances of constrained-shape problems corresponding to the set of all positive (nonnegative) shapes. In particular, problems of either type have optimal solutions that are (almost) separable, and enumerating (almost) separable partitions can be used to solve these problems. The following stronger conclusions do not apply to single-size problem when parts are required to nonempty. Two subsets 1 and 2 of Rd are called cone-separable if there exists a nonzero d -vector C such that C T u1 > 0 > C T u2 for all u1 2 1 n f0g and u2 2 2 n f0g

(41)

(if either 1 n f0g or 2 n f0g is empty and the other set is not, then (41) is to be interpreted as a requirement just on the elements of the nonempty set).

2340

F.K. Hwang and U.G. Rothblum

A relation between cone-separability and (almost) separability is determined in the next lemma. Lemma 13 Suppose 1 and 2 are subsets of Rd that are cone-separable. Then 1 and 2 are almost separable with the vector C satisfying (41) as a separating vector; in this case, 0 is the only possible vector in 1 \ 2 . Further, if 0 … 1 or 0 … 2 , then 1 and 2 are separable. The next example demonstrates that (almost)-separability does not imply coneseparability.     Example 7 For k D 1; 2, let k D f k0 ; k2 g  R2 . Then 1 and 2 are separable 1 (with C D 0 as a separating vector), but they are not cone-separable as a vector    C 2 R2 satisfies C T 10 > 0 if and only if C T 20 > 0. Recall that the conic hull a set  Rd , denoted cone , is defined as the Pof q set of linear combinations t D1 t x t with t  0 and x t 2 for t D 1; : : : q (with cone ; D f0g). A set is a cone if it equals its conic hull. A cone is pointed if for no nonzero vector x, both x and x are in the cone. A fundamental result about cones (see Rockafellar [61], Schrijver [64], or Ziegler [71]) shows that a set is the conic hull of a finite set in Rm if and only if it has a representation fx 2 Rm W Ax  0g with A as a real matrix having m columns; such a cone is called a polyhedral cone. Lemma 14 Let 1 and 2 be two finite sets in Rd and let C be a nonzero vector in Rd . Then C T u1 > 0 > C T u2 for all u1 2 1 n f0g and u2 2 2 n f0g if and only if C T w1 > 0 > C T w2 for all w1 2 .cone 1 / n f0g and w2 2 .cone 2 / n f0g. The next result provides characterizations of cone-separability for polyhedral cones. Lemma 15 Let 1 and 2 be two finite sets in Rd . Then the following are equivalent: (a) 1 and 2 are cone-separable. (b) cone 1 and cone 2 are cone-separable. (c) cone 1 and cone 2 are almost separable. (d) cone 1 and cone 2 are pointed cones and .cone 1 / \ .cone 2 / D f0g. Attention is next turned to partition problems. Here, fixed-size problems with a relaxation of the assumption that partitions’ parts are nonempty are considered. cp denote the set of p-partitions which allow empty parts. For  2 Formally, let … p c … , A has the natural definition with the empty sum taken as zero. The partition cp g. As in cp is defined by P …bp  conv fA W  2 … polytope corresponding to … A cp is all sum-partition problems, it is assumed that the objective function F over … p given by F ./ D f .A / with f as a real-valued function on PA…b .

Optimal Partitions

2341

Call a partition  D .1 ; : : : ; p / cone-separable if the sets fAi W i 2 j g for j D 1; : : : ; p are pairwise cone-separable. Lemma 13 shows that cone-separability implies almost separability, with 0 as the only potential overlapping vector for any pair of parts; moreover, a cone-separable partition with at most one part containing indices that correspond to the 0 vector is separable. The next two theorems extend results of Barnes, Hoffman, and Rothblum [1992] (which considered the case where the Ai ’s are distinct). cp and A is a vertex of P …bp , then for some matrix C 2 Rd p Theorem 35 If  2 … A .C r /T u > .C s /T u for all distinct r; s D 1; : : : ; p and u 2 .cone r / n f0g: (42) p Theorem 36 Let f ./ be an (edge-)quasi-convex on PA…b . Then there exists an optimal partition  which is both cone-separable and (almost) separable and has p A as a vertex of PA…b . Further, if f ./ is strictly (edge-)quasi-convex, then every p optimal partition  is cone-separable and has A as a vertex of P …b .



A

The next two results consider cone-separable partitions when d D 1 and d D 2. Theorem 37 Suppose A 2 R1n has no zero column. If p  2, then a p-partition  is cone-separable if and only if two of its parts are fi 2 N W Ai > 0g and fi 2 N W Ai < 0g (where either may be empty) and all other parts are empty. If p D 1, then the (only) 1-partition .N / is cone-separable if and only if A contains either just positive elements or just negative elements. To study cone-separable partitions with d D 2, associate each x 2 R2 n f0g with the angular coordinate of its polar-coordinate representation denoted .x/ (measured in degrees). It is observed that for ; ¤ C  R2 nf0g, C [f0g is a pointed polyhedral cone if and only if for some 0   : < 360ı , < C 180ı and ( C D

fx 2 R2 n f0g W  .x/  g if < 360ı fx 2 R2 n f0g W  .x/ < 360ı or 0  .x/   360ı g if  360ı :

Theorem 38 Suppose A 2 R2n has no zero column. A p-partition  is coneseparable if and only if for some q  p, there exist 0  1  1 < 2  2 <    < q  q < 1 C 360ı such that q < 360ı, t  t < 180ı for each t D 1; : : : ; q, and the nonempty parts of  are fi 2 N W t  .Ai /  t g for t D 1; : : : ; q  1 and (

fi 2 N W q  .Ai /  q g

if q < 360ı

fi 2 N W q  .Ai / < 360ı or 0  .Ai /  q  360ı g if q  360ı I

further, the t ’s and t ’s can be selected from f .Ai / W i 2 N g.

2342

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F.K. Hwang and U.G. Rothblum

Multiparameter: Combinatorial Approach

This section explores a combinatorial approach which can apply to partition problems that are more general than the sum-partition problems studied in Sect. 5. Here, vector partitions rather than index partitions are used – see the first three paragraphs of Sect. 5. A finite subset 1 of Rd penetrates another finite subset 2 of Rd , written 1 ! 2 , if 1 \ .conv 2 / ¤ ;; 1 strictly penetrates 2 if 1 \ .riŒconv 2 / ¤ ;. Some implications about penetration that are true in R1 do not hold in Rd with d > 1. For example, 1 ! 2 and 2 6! 1 no longer imply conv 1 conv 2 . Properties of partitions of vectors in Rd , with d > 1, are next introduced. The fact that some implications of penetration in R1 do not hold in Rd for d > 1 implies that some properties of partitions of points in R1 that were defined in terms of penetration have more than one counterpart for partitions of points in Rd . A sphere in Rd is defined as a set of the form S D fx 2 Rd W kx  ak  Rg for some a 2 Rd and R > 0, where k k stands for the Euclidean norm. The interior and boundary of such a sphere is given by int S D fx 2 Rd W kx  ak < Rg and bd S D fx 2 Rd W kx  ak D Rg. Consider a partition  D .1 ; : : : ; p / of a finite set of vectors in Rd . The following properties of such partitions will be considered; the first three were introduced in Sect. 5. • Separable .S / (also referred to as “disjoint” in the literature): For all distinct r; s D 1; : : : ; p, r and s are separable, that is, .conv r / \ .conv s / D ;. • Almost separable .AS /: For all distinct r; s D 1; : : : ; p, r and s are almost separable, that is, .conv r / \ .conv s / contains at most a single point, and if the intersection contains a point, it is a vertex of both conv r and conv s . • Cone-separable .Cn S /: For all distinct r; s D 1; : : : ; p, r and s are coneseparable, that is, .cone r / \ .cone s / D f0g. • Sphere-separable .S S /: For all distinct r; s D 1; : : : ; p, there exists a sphere S Rd such that either .U; W / D .r ; s / or .U; W / D .s ; r /, satisfies S  U and S \ W D ;. • Convex-separable .Cv S /: For all distinct r; s D 1; : : : ; p, there exists a convex set S Rd such that either .U; W / D .r ; s / or .U; W / D .s ; r /, satisfies S  U and S \ W D ;. (Evidently, without loss of generality, it is possible to assume that dim S D d .) • Almost sphere-separable .AS S /: For all distinct r; s D 1; : : : ; p, there exists a sphere S Rd such that either .U; W / D .r ; s / or .U; W / D .s ; r /, satisfies S  U and jS \ W j  1 and if x is a single point in S \ W , then U \ .bd S / D fxg. • Nonpenetrating .NP /: For all distinct r; s D 1; : : : ; p, r 6! conv s . • Noncrossing .NC /: For all distinct r; s D 1; : : : ; p, either .conv r / \ .conv s / D ; or one convex hull is contained in the other with no member of the larger part penetrating the smaller part.

Optimal Partitions

2343

• Acyclic .AC /: There do not exist q  2 parts of , say i1 ; : : : ; iq , such that i1 ! i2 ! : : : ! iq ! i1 . • Monopolistic .Mp /: One part has all elements. Note that if empty parts are prohibited, there is only one partition satisfying Mp and its size is 1, if empty parts are allowed it means that p  1 parts are empty. • Nearly monopolistic .Ne Mp /: At most one part of  has more than a single point. • Nearly cone-separable .Ne Cn S /: For all distinct r; s D 1; : : : ; p, r and s are either cone-separable or at least one of them is a singleton. A property defined by penetration is called “weak” if the penetration allows touching at boundary. Also note that two subsets 1 and 2 of Rd are separable if and only if there exists a (closed) half-space S such that S  1 and S \ 2 D ;, and 1 and 2 are cone-separable if and only if there exists a (closed) cone S of dimension d such that S  1 and S \ 2 D f0g. S and Cn S were first considered in Barnes et al. [4]; Cv S (called “noncrossing”) in Boros and Hwang [7]; AS in Hwang and Rothblum [38]; S S , NP (called “nested”), and AC (called “connected”) in Boros and Hammer [6]; NC (for d D 1) in Kreweras [48], and Ne Mp in Pfersky et al. [60]. When d D 1, both S and NP reduce to consecutiveness, while Ne Cn S , NC , S S , AC , and Cv S all reduce to noncrossingness. Further, when d D 1, partitions in Cn S can have at most one part containing positive elements, at most one part containing negative elements, and some parts each consisting of only the 0-element (but may have multiple copies). For a set of points with k 0-elements, this set is not size(p)-regular for p > k C 2 and not shape-regular for almost all shapes. Hence, Cn S is not studied for d D 1. For d D 1, when there are enough 0-elements, Cn S reduces to bi-extremal, but Ne Mp does not reduce to extremal since the singletons can be anywhere, not necessarily the smallest elements. Theorem 39 The relations among the properties of partitions that were introduced are represented in Fig. 4. Golany et al. [29] proved an unexpected relation between S and S S . Here, an “almost” version of the result is added. To state these results, the partitioned vectors are embedded in Rd C1 by defining b Ai 

Ai kAi k2

! 2 Rd C1 for i D 1; : : : ; n :

(43)

 be the p-partition for the problem Next, for each p-partition  D .1 ; : : : ; p /, let b 1 n b b with partitioned vectors A ; : : : ; A with b  j D fb Ai W Ai 2 j g for j D 1; : : : ; p; in particular, 3 2 X X b b b Ai ; : : : ; Ai 5 2 R.d C1/p : D4 (44) Ab  Ai 21

Ai 2p

Theorem 40 A partition  is (almost) sphere-separable if and only if b  is (almost) separable when the partitioned vectors are b A1 ; : : : ; b An .

2344

F.K. Hwang and U.G. Rothblum

Mp

NeMp

SS

S

NC

CvS AC

NP

NeCnS

WNP WAC

CnS

WCnS

WS

WNC

AS

ASS

WSS

WCvS

Fig. 4 Implications among properties of multiparameter partitions

The proofs of (Golany et al. [29]) of the equivalences of Theorem 40 are constructive and show how to convert separating vectors that verify (almost) separability into spheres that verify (almost) sphere separability and vice versa. The next result provides conditions that suffice for .A/S S ) .A/S . Lemma 16 Assume that all the Ai ’s lie on the boundary of a sphere. Then a partition is (almost) sphere-separable if and only if it is (almost) separable. Next, the number of Q-partitions for each property Q introduced earlier is bounded. Since vector-partitions are considered, the counts may depend on the partitioned vectors. For each A 2 Rd n , let #A Q .p/ be the number of p-partitions of the columns of A that satisfy Q, allowing empty parts. The emphasis is to examine whether #Q .n; p; d /  maxA2Rd n #A Q .p/ grows exponentially or polynomially in n, with p  2 and d fixed (in the latter case, enumeration algorithms exist for all relevant properties Q, see [39]. Counting of a polynomial class is usually done by providing an upper bound of the form O.nm / for some positive m. Every unlabeled p-partition corresponds to at most pŠ labeled partitions (the option for identical parts implies that the bound needs not always be tight). Thus, the number of labeled p-partitions is bounded by pŠ times the number of unlabeled p-partitions (for index-partitions, there are no identical parts, so the bound is

Optimal Partitions

2345

realized), a multiplier that is independent of n. Consequently, the O./ order of the classes of labeled and unlabeled partitions with a given property coincide. Here, it is convenient to count the labeled classes. Also, empty parts will be allowed. When columns of A are not distinct, a vector x 2 Rd is referred to as a multiple point (of A) if it appears more than once among the columns of A; the multiplicity of a multiple point x is the number of times x appears among the columns of A. As vector partitions are considered, two partitions are identified if their parts are assigned, respectively, the same number of copies of each multiple point. The next lemma records a standard combinatorial fact.  n1  ways of Lemma 17 Suppose a point in Rd has n copies. Then there are p1 nCp1 splitting the point into p nonempty (labeled) parts and p1 ways of splitting it into p parts when empty parts are allowed; in either case, the number is bounded by O.np1 /. A set of points in Rd is called generic, or in general position, if for k D 1; : : : ; d no k C 1 points of them lie in a k  1-dimensional hyperplane. A matrix is called generic, if the set of its columns is generic. Harding [33, Theorem 1] gave an exact count of the number of A-separable 2-partitions when A is generic. Harding’s result is next presented in a more general context which corrects [33, Theorem 2]; see Hwang and Rothblum [38]. For n; d  1, define the .n; d /-Harding number, denoted H.n; d /, by ! d X n1 ; (45) H.n; d /  j j D0    (where the standard convention that n0 D 1 for each n  0 and kn D 0 if k > n is used). For A 2 Rd n and E 2 Rd u , a p-partition  is .A; E/-separable if every pair of parts of  are separable by a hyperplane that contains the columns of E. Theorem 41 Suppose A 2 Rd n and E 2 Rd u , where 0  u  d and ŒA; E is generic. With empty parts the number of .AjE/-separable 2-partitions is  allowed, n  . With empty parts prohibited, the number of .AjE/2H.n; d  u/  2d uC1 d u separable 2-partitions is 2H.n; d  u/  2 if either u < d or if u D d and neither side of the unique hyperplane spanned by the columns of E contains all of A’s columns. Finally, if empty parts are prohibited, u D d and all columns of A are on one side of the unique hyperplane spanned by the columns of E, then the number of .AjE/-separable is 0. d Corollary 8 Suppose A 2 Rd n is generic. Then #A S .2/  2H.n; d / D O.n /. d Hwang et al. [43] proved the equality #A S .2/ D O.n / of Corollary 8 without using the Harding function. They also extended their result to arbitrary A in Rd n by perturbing A into A0 which is generic and proved that the set of A-separable

2346

F.K. Hwang and U.G. Rothblum

2-partitions is contained in the set of A0 -separable 2-partitions, thus bounding #A S .2/ A0 d by #S .2/ which equals O.n / by Corollary 8. Finally, they  gave a construction of each A-separable partition by merging a given set of p2 2-partitions ij of A, for all “1  i < j  p,” and showed that all A-separable p-partitions can be obtained from such constructions (by varying the given set of 2-partitions). A .p2/ D O.nd .p2/ /. Also, #S .n; p; d / D Theorem 42 For A 2 Rd n , #A S .p/  Œ#S .2/ p O.nd . 2 / /. p Alon and Onn [3] proved that the O.nd . 2 / / bound of Theorem 42 cannot be improved for either p  3 or d  3. Using Theorem 40, the above results can be used to derive bounds for sphereseparability.

Theorem 43 Suppose A 2 Rd n , p  2 and nN is the number of distinct columns  nN  .p/ p A .p2/  Œ2H.n;  2 . Also, N d C 1/. 2 /  Œ2d C1 d C1 of A. Then #A S S .p/  Œ#S S .2/ .d C1/.p2/ #S S .n; p; d / D O.n /. Given A 2 Rd n , Theorem 40 also shows that any enumeration method for separable partitions immediately yields a method for enumerating the A-sphereseparable partitions (with a complexity bound obtained by substituting d C 1 for d ). Hwang and Rothblum [39] obtained the following result. A .p2/  Theorem 44 For A 2 Rd n having no zero-column, #A Cn S .p/  Œ#Cn S .2/ p p p Œ2H.n; d  1/. 2 / D OŒn.d 1/. 2 / . Also, #Cn S .n; p; d /  Œ2H.n; d  1/. 2 / D p O.n.d 1/. 2 / /.

Almost separable partitions are next considered, starting with 2-partitions. The result follows Hwang and Rothblum [40]. n distinct columns, #A Lemma 18 For A 2 Rd n having e AS .2/  2H.n; d / C .n  e n/Œ2H.e n  1; d  1/  2H.n; d /. Also, #AS .n; 2; d / D #S .n; 2; d / D 2H.n; d /. Two difficulties prevent one from mimicking the perturbation argument and the merging argument used for separable partitions to obtain an extension of Lemma 18 to general p. First, using the same perturbation A to A0 , the set of A-almost separable 2-partitions is no longer contained in the set of A0 -separable partitions (though for generic A0 , almost separable and separable coincide). Thus, one cannot transform a bound of the latter into a bound of the former. Second, the straightforward merging construction for separable partitions is not delicate enough to cover the almost separable case. Hwang and Rothblum [40] showed how to overcome these two problems. The first problem is solved by using the second conclusion of Lemma 18, that is, #AS .n; 2; d / D 2H.n; d / for the need to bound #A AS .2/. They also gave a much more elaborated construction to deliver the merging successfully. The resulting bound of #AS .n; p; d / is given in the next theorem.

Optimal Partitions Table 8 Bounds of #Q .n; p; d /

2347

Q

#Q .n; p; d /  p  O nd . 2 /   p O n.d C1/. 2 /   p O n.d 1/. 2 /  p  O nd . 2 /   p O n.d C1/. 2 /

S SS Cn S AS ASS

Ref. Theorem 42 Theorem 43 Theorem 44 Theorem 45 Theorem 46

p p Theorem 45 #AS .n; p; d /  Œ#AS .n; 2; d /. 2 / D O.nd . 2 / /.

The following result is an immediate consequence of Theorems 40 and 45. p

Theorem 46 #AS S .n; p; d / D O.n.d C1/. 2 / /. The following Table 8 summarizes the bounds of #Q .n; p; d / for those properties Q discussed above. Several simple bounds on the number of partitions that satisfy  n other properties D O.np1 /, are next recorded. First, it is trivial that #MP .n; p; d / D p1  n  consequently, #NeMP .n; p; d / D pŠ p1 . Also, a careful combinatorial argument Pp1 n p qŠ#C nS .n  q; p  q; d / D O.n.d 1/. 2 /Cp1 /. yields #NeC nS .n; p; d / D qD0 q

Finally, for Q 2 fNP; AC; NC g, Hwang, Lee, Liu, and Rothblum [45] gave examples to show that even for d D p D 2, the number of Q-partitions is exponential. From the implication relation shown in Fig. 4, the number of C vS partitions must also be exponential. Finally, it is easily verified that all weak properties are exponential. Consistency and sortability, as introduced in Sect. 4, refer to properties of index-partitions, without reference to the representation of the partitioned elements. However, p-partition or k-subpartition is said to satisfy Q, and then satisfaction of a part or parts of indices always implies the same satisfaction of the part or parts of the corresponding points. Further, the extension of the notion of satisfaction from a 1-dimensional point to a multi-dimensional point is straightforward. Hence, consistency and sortability can still be discussed for vector-partitions, and the approach of using Q-sortability to prove the existence of a Q-optimal partition is valid as it was for the single-parameter case. The remainder of this section discusses geometric properties that are present in optimal partitions. Let fA1 ; : : : ; An g be a set of n vectors in Rd and let F ./ denote an objective function over partitions of these vectors. Throughout the remainder of this section, unless explicitly stated otherwise, F ./ is to be minimized in the partition problems that are considered (as is usually the case in “clustering problems”). Also, the results for size problems hold regardless of whether empty parts are allowed or not (for constrained-shape problems, prohibiting empty parts is captured by having the set of feasible shapes contain only positive integer vectors). However, if a result depends on allowing empty parts or on prohibiting such parts, then the condition will be stated explicitly.

2348

F.K. Hwang and U.G. Rothblum

Golany et al. [29] extended results of Barnes et al. [4] to the following (which extends Theorem 34): Theorem 47 Suppose F ./ D f .j1 j; : : : ; jp j; A /;

(46)

where for each fixed set of values .j1 j; : : : ; jp j/, f is (edge-)quasi-concave on the corresponding single-shape partition polytope (in the d  p variables of A ). Then every constrained-shape problem has an almost separable optimal partition. If furthermore, f is strictly (edge-)quasi-concave, then every optimal partition for a constrained-shape problem is almost separable. Recall the use of “b” introduced in (43) and (44). The following result is reported in Golany et al. [29]. Corollary 9 Suppose A / ; F ./ D f .j1 j; : : : ; jp j; b

(47)

where for each fixed set of values .j1 j; : : : ; jp j/, f is (edge-)quasi-concave on the corresponding single-shape partition polytope (in the .d C 1/  p variables of b A ). Then every constrained-shape problem has an almost sphere separable optimal partition. Further, if f is strictly (edge-)quasi-concave, then every optimal partition for a constrained-shape problem is almost sphere separable. If the function f in Corollary 9 is linear in the variables corresponding to b A and in the shape-variable, the results of Sect. 5 can be used; that is, bounded-shape partition problems with such objective functions can be solved by LP methods, applied in the enlarged .d C 2/-dimensional space where the partitioned vectors Ai  are b . 1

Following Golany et al. [29], Corollary 9 is next used to address some clustering problems. Theorem 48 Let w1 ; : : : ; wp be positive numbers. Suppose F ./ has either one of the following three expressions: p X X wj kAi  Ak k2 . (i) F ./ D j D1

X

Ai ;Ak 2j

p

(ii) F ./ D ANj D

j D1;j ¤; P Ak Ak 2 j

jj j

X p

(iii) F ./ D

wj

j D1

wj

. X

Ai 2j

X

kAi  ANj k2 , where for j D 1; : : : ; p,

Ai 2j

kAi  tj k2 , where t1 ; : : : ; tp are prescribed d-vectors.

Optimal Partitions

2349

Then every constrained-shape problem has an almost sphere separable optimal partition. c and The third objective function can be expressed as a linear function of A of the parts’ sizes. Consequently, the comment following Corollary 9 applies and corresponding bounded-shape partition problems can be solved by LP methods. Uniform versions of the objective functions that are listed in Theorem 48 are next considered, where uniform means that the wj ’s are constant. Theorem 49 Consider uniform versions of the objective functions that are listed in Theorem 48. Then: (i) For a constrained-shape problem with constant part-sizes and with the first function, every optimal partition is almost separable. (ii) For a constrained-shape problem with the second function, every optimal partition is almost separable. (iii) For a constrained-shape problem with the third function, there exists an almost separable optimal partition. Boros and Hammer [6] considered case (i) of Theorem 49 and proved that every single size problem has a weakly separable optimal partition. Theorem 49 strengthens their result in several ways: A stronger property is satisfied for every optimal partition, and the conclusion holds for a wider class of partition problems. Attention is next turned to single-size problems. For a p-partition  and i 2 N , let j .i / denote the index of the part of  that contains Ai (here, distinction must be made between multiple copies of the same vector). Lemma 19 Let t1 ; : : : ; tp be prescribed distinct vectors in Rd . Then, allowing for empty parts, there is a separable partition  such that jjAi  tj .i / jj  jjAi  tj 0 .i / jj for every partition  0 and i 2 N :

(48)

Further, a partition  satisfies ( 48) if and only if each vector Ai is assigned to a part j for which kAi  tj k D mi nuD1;:::;p kAi  tu k; each such partition is weakly separable. Theorem 50 Let t1 ; : : : ; tp be prescribed distinct vectors in Rd and consider the single-size problem, allowing empty parts, which has F ./ D f .kA1  tj .1/ k; : : : ; kAn  tj .n/ k/;

(49)

where f W Rn ! R is nondecreasing. Then the partition  which assigns each vector Ai to the part j with the lowest index j among those for which kAi  tj k is minimized is both optimal and separable. Further, if f is increasing, then a partition  is optimal if and only each vector Ai is assigned to a part j for which kAi  tj k D mi nuD1;:::;p kAi  tu k, and every optimal partition is weakly separable.

2350

F.K. Hwang and U.G. Rothblum

Corollary 10 The conclusions of Theorem 50 hold for the single-size problem, allowing empty parts, which has F ./ D

p X

X

h.kAi  tj k/;

(50)

j D1;j ¤; Ai 2j

where t1 ; : : : ; tp be prescribed distinct vectors in Rd where h W R ! R is nondecreasing/increasing. Theorem 50 and Corollary 10 specify a single separable partition that is uniformly optimal for all underlying partition problems, independently of the functions f and h that occur in (49) and (50). The characterization of optimal partition of Theorem 50 (and Corollary 10) allows one to construct all optimal solutions by assigning each vector Ai to any part j for which tj is the closest to Ai . More specifically, associate each vector Ai 2 Rd with the vector D i  .kAi  t1 k; : : : ; kAi  tp k/T 2 Rp . A partition is then optimal if and only if it assigns Ai to j with Dji being any minimal coordinate of D i . The amount of effort needed to compute each vector D i is O.dp/, so the total amount of effort to compute all of the D i ’s (and thereby solve the partition problem) is O.dpn/. Since the boundary of the convex hull of two parts can contain points of both parts, the optimal partitions identified in Theorem 50 and Corollary 10 need not be almost separable, let alone separable. Still, it is noted that breaking ties by using any part-ranking will produce separable partitions. The geometric figure of the partition of Rd into p convex subsets S1 ; : : : ; Sp with Sj consisting of all points in Rd closest to tj (a point on a boundary is closest to all tj whose Sj shares that boundary) is known in the literature as the Voronoi diagram. Efficient constructions of Voronoi diagrams are well studied in theoretical computer science (see Fortune [25] for a survey). In particular, hyperplanes representing a pC1 Voronoi diagram for p given points t1 ; : : : ; tp in Rd can be constructed in O.d 2 /time. For d D 2, Shamos and Hoey [65] showed that the Voronoi diagram can be constructed in O.p log p/-time. A tool is next developed for using results about properties of optimal partitions of problems with objective functions that depend on prescribed vectors to results where the prescribed vectors are replaced by some minimizing vectors. Two new definitions are needed to present a formal result. A function g W Rd ! R is inversebounded if for every K 2 R fx 2 Rd W jg.x/j  Kg is a bounded set of Rd . Evidently, an inverse-bounded function that is continuous is guaranteed to attain a minimum over Rd . Also, let P  .Rd / be the set of finite subsets of Rd , that is, P  .Rd / D f Rd W j j is finiteg. Theorem 51 Let Q be a partition property, f W Rp ! R be nondecreasing and g W P  .Rd /Rd ! R with the property that for every 2 P  .Rd /, g. ; / is inversebounded and continuous. Let  be a set of p-integer vectors with coordinate-sum n.

Optimal Partitions

2351

Assume that for any t1 ; : : : ; tp 2 Rd , some optimal partition of the constrainedshape problem corresponding to  and having objective function F t1 ;:::;tp ./, empty parts not allowed, given by F t1 ;:::;tp ./ D f Œg.1 ; t1 /; : : : ; g.p ; tp / for each partition 

(51)

satisfies Q. Then some optimal partition for the corresponding constrained-shape problem with objective function F ./ given by " F ./ D f

# min g.1 ; x1 /; : : : ; min g.p ; xp /

x1 2Rd

xp 2Rd

for each partition 

(52)

satisfies Q. Further, the above holds with “every” replacing “some.” Theorem 51 used to obtain Theorem 49 (ii) from Theorem 49 (iii) by Ppcan be P i 2 noting that is minimized over .t1 ; : : : ; tp / at Ai 2j kA  tj k j D1 wj P

k

Ak

A 2j for j D 1; : : : ; p). .AN1 ; : : : ; ANp / (with ANj D jj j The proof of Theorem 51 in [39] can, in fact, be applied to a partition that is not optimal. Specifically, if  is any partition and t1 ; : : : ; tp are minimizers of g.1 ; /; : : : ; g.p ; /, respectively, then any partition  # that satisfies F t1 ;:::;tp . # /  F t1 ;:::;tp ./ satisfies F . # /  F ./. In particular, any sorting method that is applicable to partition problems with objective function F t1 ;:::;tp ./ for fixed tj ’s which does not rely on a statistics depending on the tj ’s, applies to the partition problem with objective function F ./. This is the case for p-sortings for which no statistics is needed. ! R, an h-centroid of a finite set  Rd is a minimizer of P For h W R i d Ai 2 h.kA  xk/ over x 2 R . When h is continuous, bounded from below and inverse-bounded, each finite set has an h-centroid. (This definition extends the definition given in the paragraph preceding Theorem 25 from d D 1 to d > 1.)

Theorem 52 Consider the single-size problem which has F ./ D

p X X

h.jjAi  cj jj/;

(53)

j D1 Ai 2j

where h W R ! R is nondecreasing, continuous, and inverse-bounded, and for j D 1; : : : ; p, cj is the h-centroid of j and where the sum over an empty set of h./-values in (53) is defined to be 0. Then there exists a separable optimal partition. Further, if h is increasing, then every optimal partition is weakly separable. Corollary 11 Consider the partition problem of Theorem 52. Then there exists a separable optimal partition without empty parts. Further, if h is increasing, then every optimal partition has no empty parts.

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F.K. Hwang and U.G. Rothblum

Theorem 52 asserts the existence of a separable optimal partition that is specific to the function h which occurs in (53). This type of result is weaker than the uniform optimality of a specific separable partition asserted in Theorem 50. The second paragraph following Theorem 51 shows that the assignment of each vector to its closest tj can be used to replace a partition  that is not separable with one that is and has improved F -value where F ./ is given by (53). Partition problems that are next considered have objective functions that depend on the least radii of spheres that, respectively, include the parts. It is first shown that given a partition whose parts are included in prescribed spheres, there is a separable partition with the same property. The proof of this result follows an approach of Capoyleas, Rote, and Woeginger [9] who proved the case d D 2 (and claimed its extension to general d ). Lemma 20 Let t1 ; : : : ; tp be prescribed distinct vectors in Rd . Allowing partitions to have empty parts, for each partition , there exists a separable partition  0 such that max0 jjAi  tj jj  max jjAi  tj jj for j D 1; : : : ; p: (54) Ai 2j

Ai 2j

Further, if rj  maxAi 2j jjAi tj jj for j D 1; : : : ; p (i.e., rj is the smallest radius of a sphere that is centered at tj and contains j ), then  0 can be selected by the following rule: Ai 2 j0 if j is the lowest index of those for which jjAi tj jj2 rj2 D minu ŒjjAi  tu jj2  ru2  (of course, one may use any a priori permutation of the partindices). Capoyleas, Rote, and Woeginger [9] did not mention a tie-breaking rule; as such, the “power diagram” they used to repartition the vectors leads to weak separability rather than strict separability (earlier results in this section show that the number of weakly separable partitions is not necessarily polynomial in the number of partitioned vectors). The separating hyperplane between parts j0 and w0 of the partition  0 constructed in Lemma 20 is given by H  fx 2 Rd W jjx  tj jj2  rj2 D jjx  tw jj2  rw2 g D fx 2 Rd W 2.tw  tj /T x D jjtw jj2  jjtj jj2 C rj2  rw2 g:

(55)

This hyperplane is called the power-hyperplane of the underlying spheres fx 2 Rd W jjx  ts jj  rs g and fx 2 Rd W jjx  tw jj  rw g. Further, the construction of  0 from  in the proof of Lemma 20 is a sorting operation of all p-parts called power-hyperplane sorting. The use of p-sorting has the advantage that it achieves the goal in one step and therefore does not require a monotone statistics that shows iterative sortings will not cycle. A natural variant of the use of power-hyperplane p-sorting to prove the conclusions of Lemma 20 is to iteratively apply power-hyperplane 2-sorting (of pairs of parts). For this approach to work, that is to avoid the possibility of cycling, it

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is necessary to identify a statistics over partitions that is reduced at each iteration. Unfortunately, it is not known that such a statistics exists. Still, power-hyperplane 2-sortings can be used to prove a weaker result. Specifically, consider t1 ; : : : ; tp 2 Rd and r1 ; : : : ; rp 2 R. Lemma 20 implies that power-hyperplane p-sorting will convert a partition  whose parts are included, respectively, in the spheres fx 2 Rd W jjx  tj jj  rj g, j D 1; : : : ; p, into a separable partition having the same property. Here, one can use iterative power-hyperplane 2-sorting and assure that the process does not cycle because of strict lexicographic reduction of the statistics 0 s./  @

p X X

1 ŒjjAi  tj jj2  rj2 ; j1 j; : : : ; jp jA

j D1 Ai 2j

in each step. In particular, this proves that S is (sort-specific, 2, support)-sortable with “sort-specific” referring to the power-hyperplane method with fixed center and fixed radii. Theorem 53 Let t1 ; : : : ; tp be prescribed distinct vectors in Rd and consider the single-size problem, allowing empty parts, which has F ./ D f Œ max jjAi  t1 jj; : : : ; max jjAi  tp jj; Ai 21

Ai 2p

(56)

where f W Rp ! R is nondecreasing. Let  be a given partition. Then the partition  0 which satisfies the conclusions of Lemma 20 is separable and satisfies F . 0 /  F ./. In particular, the underlying single-size problem has a separable optimal partition. Theorem 53 provides a construction (appearing explicitly in the statement of Lemma 20) of a separable partition that improves on the objective function of a given partition . This result can be cast as p-sortability result, as were the results of Theorem 50 and Corollary 10. The construction of Theorem 53 is independent of the f appearing in ( 56), but unlike the construction of Theorem 50 (and Corollary 10), it is specific to the underlying partition . For a bounded set Rd , let r. / denote the minimum radius of a sphere that includes ; in particular, when is finite, r. / D infx2Rd Œsupy2 jjx  yjj, and the “inf” and “sup” are attained (and can therefore be replaced by “min” and “max”). For a p-partition , let r./ D .r.1 /; : : : ; r.p //. The next result considers partition problems with objective function that is determined by these partition characteristics; Capoyleas, Rote, and Woeginger [9] considered the case d D 2. Theorem 54 Consider the single-size problem which has F ./ D f Œr./ where f W Rp ! R nondecreasing. Then some optimal partition is separable. A result of Pfersky et al. [60] is next generalized; their result is then deduced as a corollary.

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Theorem 55 Consider the single-size problem, not allowing empty parts, which has

F ./ D

p X X X u;vD1 Ai 2u

u¤v

h.Ai ; Ak /;

(57)

Ak 2v

where h.; / is a metric. Then there exists a nearly monopolistic optimal partition. If h is strict, then every optimal partition is nearly monopolistic. Finally, when allowing empty parts, “nearly monopolistic” can be replaced by “monopolistic.” The reason that there is no strict version for Theorem 55 is that even if h is strict, when the Ai ’s are all the same point, then all p-partitions are optimal. The Pfersky, Rudolf, and Woeginger result is next derived. It is stated in the original framework of a maximization problem (of course, multiplying the objective function by 1 converts a maximization problem into an equivalent minimization problem). Corollary 12 Consider the single-size problem in which

F ./ D

p X

X

kAi  Ak k2 :

(58)

j D1 Ai ;Ak 2j

is to be maximized. Then there exists a nearly monopolistic optimal partition.

Recommended Reading 1. R.K. Ahuja, J.B. Orlin, A fast and simple algorithm for the maximum flow problem. Oper. Res. 37, 748–759 (1989) 2. R.K. Ahuja, J.B. Orlin, R.E. Tarjan, Improved time bounds for the maximum flow problem. SIAM J. Comput. 18, 939–954 (1989) 3. N. Alon, S. Onn, Separable partitions. Discret Appl. Math. 91, 39–51 (1999) 4. E.R. Barnes, A.J. Hoffman, U.G. Rothblum, Optimal partitions having disjoint convex and conic hulls. Math. Program. 54, 69–86 (1992) 5. S. Borgwardt, A. Brieden, P. Gritzmann, Constrained minimum-k-star clustering and its application to the consolidation of farmland. Oper. Res. (2009). Published Online First 6. E. Boros, P.L. Hammer, On clustering problems with connected optima in Euclidean spaces. Discret. Math. 75, 81–88 (1989) 7. E. Boros, F.K. Hwang, Optimality of nested partitions and its application to cluster analysis. SIAM J. Optim. 6, 1153–1162 (1996) 8. A. Brieden, P. Gritzmann, A quadratic optimization model for the consolidation of farmland by means of lend-lease agreements, in Operations Research Proceedings 2003: Selected Papers of the International Conferenceon Operations Research (OR2003), ed. by D. Ahr, R. Fahrion, M. Oswarld, G. Reinelt (Springer, Berlin, 2004), pp. 324–331 9. V. Capoyleas, G. Rote, G. Woeginger, Geometric clusterings. J. Algorithms 12, 341–356 (1991) 10. A.K. Chakravarty, Multi-item inventory into groups. J. Oper. Res. Soc. U. K. 32, 19–26 (1982)

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37. F.K. Hwang, U.G. Rothblum, Directional-quasi-convexity, asymmetric Schur-convexity and optimality of consecutive partitions. Math. Oper. Res. 21, 540–554 (1996) 38. F.K. Hwang, U.G. Rothblum, On the number of separable partitions. J. Combinatorial. Optimization. 21, 423–433 (2011) 39. F.K. Hwang, U.G. Rothblum, PARTITIONS Optimality and Cluster. Vol. 1: Single-Parameter, World Scientific, Singapore (2012) 40. F.K. Hwang, U.G. Rothblum, Almost separable partitions (2011, forthcoming) 41. F.K. Hwang, J. Sun, E.Y. Yao, Optimal set partitioning. SIAM J. Algorithms Discret. Methods 6, 163–170 (1985) 42. F.K. Hwang, U.G. Rothblum, Y.-C. Yao, Localizing combinatorial properties of partitions. Discret. Math. 160, 1–23 (1996) 43. F.K. Hwang, S. Onn, U.G. Rothblum, A polynomial time algorithm for shaped partition problems. SIAM J. Optim. 10, 70–81 (1999) 44. F.K. Hwang, S. Onn, U.G. Rothblum, Linear shaped partition problems. Oper. Res. Lett. 26, 159–163 (2000) 45. F.K. Hwang, J.S. Lee, Y.C. Liu, U.G. Rothblum, Sortability of vector partitions. Discret. Math. 263, 129–142 (2003) 46. F.K. Hwang, J.S. Lee, U.G. Rothblum, Permutation polytopes corresponding to strongly supermodular functions. Discret. Appl. Math. 142, 87–97 (2004) 47. U.R. Kodes, Partitioning and card selection, in Design Automation of Digital Systems, vol. 1, ed. by M.A. Breuer (Prentice Hall, Englewood Cliffs, 1972) 48. G. Kreweras, Sur le partitions non croisKees d’un cycle. Discret. Math. 1, 333–350 (1972) 49. D.M. Malon, When is greedy module assembly optimal. Nav. Res. Logist. Q. 37, 847–854 (1990) 50. A.W. Marshall, I. Olkin, Inequalities, Theory of Majorization and Its Applications (Academic, New York, 1979) 51. S. Mehta, R. Chandraseckaran, H. Emmons, Order-preserving allocation of jobs to two machines. Nav. Res. Logist. Q. 21, 361–374 (1974) 52. O. Morgenstern, J. Von Neumann, Theory of Games and Economic Behavior (Princetona University Press, Princeton, 1944) (A translation of a German 1928 text) 53. W.M. Nawijn, Optimizing the performance of a blood analyser: application of the set partitioning problem. Eur. J. Oper. Res. 36, 167–173 (1988) 54. J. Neyman, E. Pearson, On the problem of the most efficient tests of statistical hypotheses. Philos. Trans. R. Soc. Lond. A 231, 289–337 (1933) 55. D. Nocturne, Economic order frequency for several items jointly replenished. Manage. Sci. 19, 1093–1096 (1973) 56. K.N. Oikonomou, Abstractions of finite-state machines optimal with respect to single undetectable output faults. IEEE Trans. Comput. 36, 185–200 (1987) 57. K.N. Oikonomou, Abstractions of finite-state machines and immediately-detectable output faults. IEEE Trans. Comput. 41, 325–338 (1992) 58. K.N. Oikonomou, R.Y. Kain, Abstractions for node-level passive fault detection in distributed systems. IEEE Trans. Comput. C-32, 543–550 (1983) 59. C.G. Pfeifer, P. Enis, Dorfman-type group testing for a modified binomial model. J. Am. Stat. Assoc. 73, 588–592 (1978) 60. U. Pfersky, R. Rudolf, G. Woeginger, Some geometric clustering problems. Nord. J. Comput. 1, 246–263 (1994) 61. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970) 62. M.H. Rothkopf, A note on allocating jobs to two machines. Nav. Res. Logist. Q. 22, 829–830 (1975) 63. P.H. Schoute, Analytic treatment of the polytope regularly derived from the regular polytopes. Verhandelingen der Koningklijke Akademie van Wetenschappen te Amsterdam 11(3), 87p. (1911). Johannes M¨uller, Amsterdam 64. A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York, 1986)

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Optimization in Multi-Channel Wireless Networks Hongwei Du, Weili Wu, Lidong Wu, Kai Xing, Xuefei Zhang and Deying Li

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Multi-interface Assignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bandwidth Allocation and NIC Assignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Efficient Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Channel Assignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Minimum Interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Maximum Throughput. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Optimal Fairness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Network Routing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Multicasting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Data Collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Data Aggregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2360 2360 2361 2363 2365 2366 2368 2371 2375 2375 2380 2382 2386 2387

H. Du () Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, People’s Republic of China e-mail: [email protected] W. Wu • L. Wu • K. Xing • X. Zhang Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] D. Li Department of Computer Science, Renmin University, Beijing, People’s Republic of China e-mail: [email protected] 2359 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 55, © Springer Science+Business Media New York 2013

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Abstract

Although most of research efforts are made on single-channel wireless networks, multichannel environment already exists in real world systems. In this chapter, we survey current research works in multi-channel wireless networks, especially in multihop wireless networks, such as wireless ad hoc, sensor, and mesh networks. We will focus on those optimization issues, to explore the hardness of extensions from single-channel to multichannel and valuable research topics. The chapter is divided into four parts: (1) introduction, (2) multi-interface assignment, (3) channel assignment, and (4) network routing. For each part, we give a detail explanation of the latest research in multichannel wireless networks.

1

Introduction

Although most of research efforts are made on single-channel wireless networks, multichannel environment already exists in real world systems such as 2.4 GHz and 5 GHz spectrum. To utilize those resource and to take advantage of this environment, many research works have made contributions to extend techniques from singlechannel to multichannel environment and explore valuable new research problems. In this chapter, we survey those research works, especially on optimization issues.

2

Multi-interface Assignment

An interface is a network interface card equipped with a half-duplex radio transceiver, for example, a commodity IEEE 802.11 wireless card. Also, the definition of wireless network interface is a network card (NIC) connected to a radio-based computer network. There are several advantages in using multi-interface instead of a single interface. First, with multi-interface, a node can receive and transmit data in parallel so that the throughput on a multihop route can be increased, possibly doubled. Secondly, with multi-interface, the end-to-end latency can be reduced if different hops traversed are on different channels. Indeed, for each node with a single interface, if the interfaces of two nodes are on different channels, then they cannot communicate. Thus, when packets are traversing multihop paths, delaying may occur at each hop and hence the end-to-end latency can be increased. A multichannel wireless mesh network (WMN) consists of a number of stationary wireless routers, forming a wireless backbone. Each router is equipped with multiple network interface cards (NICs) [45, 48, 55]. Each interface operates on a distinct frequency channel in the IEEE 802.11a/b/g bands. A logical topology is comprised of the sets of routers and logical links. A problem for using a WMN as a backbone for large wireless access is the high bandwidth requirements, which cause capacity reduction. One solution to address this problem is to use multiple nonoverlapping channels on the interfering links to maximize the parallel transmission and throughput. Although the one-NIC

Optimization in Multi-Channel Wireless Networks

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architecture is able to support simultaneous transmissions as long as the interfering links operate in different channels, its limit on the parallel use of multiple channels cannot be ignored. Since different NICs can operate on different channels without interference, multi-NICs for mesh network have made it possible for one node to access multiple channels at the same time. Many researchers study on channel allocation issues. There are some exiting centralized and distributed algorithms to solve this problem. Marina et al. [41] and Tang et al. [51] proposed the joint channel allocation and topology control problem. Marina et al. [41] proposed a permanent channel allocation algorithm to solve the problem; this algorithm aims to minimize the interface among the links while maintaining the network connectivity. Tang et al. [51] proposed a permanently allocating frequency channels method to solve the problem; the method is to minimize the maximum number of interfering logical links within each neighborhood. The constraint is that the logical topology graph should be K-connected. In [45], the authors consider channel allocation and topology control as two separate but related problems and formulate the logical topology formation and interface assignment as a joint optimization problem. In [46] and [48], the authors assume that there is no system or hardware support to allow a radio interface to switch channels on a per-packet basis. They proposed a centralized joint channel assignment and multipath routing algorithm. This algorithm considers high load edges first. The routing algorithm uses both shortest path routing and randomized multipath routing algorithm. In [25, 27], they both assume that a radio interface is capable of switching channels rapidly and is supported by system software. In [25], the authors give a channel assignment and routing algorithm to characterize the capacity regions. In [27], the authors focus on how the capacity of multichannel wireless networks scale related to the number of radio interfaces and the number of channel as the number of nodes grow. The backbone of the WMN is modeled as a directed graph G D .V; E/, where V represents the set of nodes in the network and E represents the set of directed links. Each node u.u 2 E/ may be equipped with a number of NICs, where the number is denoted as I.u/.I.u/  1/. The total number of nonoverlapping channels is k, and the channel set is denoted as C.jC j D k/. Each NIC can be fixed on one channel during the entire transmission lifetime or can switch channels after several packets have been sent out. Two nodes can communicate with each other via wireless links when one pair of their NICs is tuned to be on the same channel.

2.1

Bandwidth Allocation and NIC Assignment

In this section, the bandwidth allocation and NIC assignment problem are addressed for the WMNs, which is shown in [55]. There are the following assumptions: (1) the topology of the WMN has been established; (2) each node u, due to physical constraints, can be equipped with at most ı.u/ numbers of NICs and all NICs are fixed; and (3) each node u has a bandwidth requirement of req.u/. The objective of the problem is to max–min bandwidth allocation ˛max , that is, maximize guaranteed bandwidth for each node. This problem is practically interesting because it answers

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the following question: If certain bandwidth allocation should be satisfied, what is the requirement to the NIC assignment? Before the problem formulation given by Wang et al. [55] is formally introduced, some terms and constraints which will be used in the formulation are introduced. Suppose every node u 2 V has a bandwidth request as req.u/ and it is allocated with bandwidth of g.u/, since the allocated bandwidth should not be greater than the required bandwidth, there is the bandwidth constraint as g.u/  req.u/;

8u 2 V

(1)

As in [55], for each node, the traffic gathered from local clients has to be relayed to at least one of its neighbors. The traffic flows can be transmitted over different channels. Let fout .u; i / denote node u’s traffic-out on channel i , and fin .u; i / the traffic-in on channel i . For a node u, every channel is assumed to have the same capacity, denoted as c.u/. In each channel, the traffic-in and traffic-out should not exceed the channel capacity, so there are the following channel capacity constraints: fin .u; i / C fout .u; i /  c.u/;

8u 2 V; i 2 C:

(2)

Since total traffic-in and traffic-out of node u over all channels should keep a balance, the flow balance constraints were also concluded in [55] as follows: X

fin .u; i / C g.u/ D

i 2C

X

fout .u; i /;

8u 2 V:

(3)

i 2C

Another constraint (NIC constraint) was raised in this work [55], that is, to enable the network connectivity, each node must be equipped with at least one NIC. Since the system has total k available channels, different NICs in one node must operate on different channels; hence, there are at most k NICs on a single node. That is the NIC constraint by channels 1  I.u/  k;

I.u/ 2 Integer;

8u 2 V:

(4)

In the case of Fix-NIC [55], R.u; i / is used to indicate whether there is a NIC operating on channel i , and R.u; i / D 1 indicates there is a NIC fixed on channel i ; otherwise, R.u; i / D 0. The relationship between I.u/ and R.u; i / is X

R.u; i /  I.u/;

R.u; i / 2 0; 1;

8u 2 V:

(5)

i 2C

For every node, a NIC can only transmit or receive from one neighbor at the same time [55]. It is fin .u; i / fout .u; i / C  R.u; i /; c.u/ c.u/

8u 2 V; i 2 C:

(6)

Optimization in Multi-Channel Wireless Networks

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The problem formally is stated as a linear programing problem. Given G.V; E/ and all parameters, maximize ˛ subject to: g.u/  req.u/;

8u 2 V

fin .u; i / C fout .u; i /  c.u/; 8u 2 V; i 2 C X X fin .u; i / C g.u/ D fout .u; i /; 8u 2 V i 2C

i 2C

1  I.u/  k; I.u/ 2 I nteger; X R.u; i /  I.u/; R.u; i / 2 0; 1;

8u 2 V 8u 2 V

i 2C

fin .u; i / fout .u; i / C  R.u; i /; c.u/ c.u/

8u 2 V; i 2 C

g.u/  ˛;

2.2

u 2 V:

(7)

Efficient Solution

As the number of nodes in the mesh network increases, the cost to compute the optimal solutions by LP will increase dramatically due to the problem complexity. Hence, the LP solution is only suitable for off-line decisions for building or analyzing the static WMN architecture. If the system would have to recalculate the allocation and assignment strategy online, according to the topology change or new traffic requirement in a large WMN, finding the optimal solution is not practical. In [55], the authors discussed the feasible solutions case by case.

2.2.1 Assign NICs to Satisfy a Bandwidth Allocation Given a feasible bandwidth allocation g.u/ for each node u, the first question is how to calculate the number of NICs .I.u// in order to satisfy the bandwidth requirement. First, the case of Switch-NIC is considered, in which one NIC can switch among different channels. If the channels are not fully utilized around, it can always switch to a less used channel to alleviate the impact of interference, and all NICs in the WMN can work cooperatively. So the NIC is able to fully consume the capacity of c.u/. The only constraint is the capacity and the number of NICs. Hence, the number of NICs for node u can be approximated as  Iswitch .u/ D

 f .u/ : c.u/

(8)

Here, f .u/ represents the total traffic of node u. When taking Fix-NICs into consideration, the proper number of NICs depends much more on the interference

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around. For instance, if nearby nodes have heavy traffic and every channel is equally utilized, then the transmission load by using just the same number of Fix-NICs cannot be accomplished in the case of Switch-NICs. Here, a compensation factor ˇ.u/ to reflect the interference impact is introduced. Let fint .u/ represent all the traffic flows that may conflict with the in and out traffic of node u; then ˇ.u/ is defined as fint .u/ ; k D jC j: (9) ˇ.u/ D k  c.u/ So the number of NICs required can be approximated as 

 f .u/ Ifix .u/ D .1 C ˇ.u//  : c.u/

(10)

These equations in [55] can be used to calculate I.u/ for each node.

2.2.2 Optimize Bandwidth Allocation with a Given NIC Assignment For this subproblem, [55] showed the solutions for two cases: (1) Case 1: Switch-NIC If each node is assigned a number of Switch-NICs, the algorithm to find a feasible bandwidth allocation for all nodes is presented in [55]. Here, consider two allocation strategies: equally allocating the bandwidth to each node or allocating the bandwidth proportionally to the request. In both scenarios, the algorithm calculates the maximum value. First, an algorithm is proposed to estimate the max–min bandwidth allocation ˛max under the assumption that the bandwidth is equally allocated to each node and the bandwidth is ˛. For a node u, the total available capacity is I.u/  c.u/, and the total traffic in and out should not exceed this capacity. So there must have f .u/  I.u/  c.u/;

u 2 V:

(11)

According to the aggregate traffic model, the total traffic-in can be deduced and traffic-out of a node u is fin .u/ D ˛  jEj and fout .u/ D ˛  jEj. So the total traffic of u is f .u/ D ˛  2  jV j: (12) According to Eqs. (11) and (12), there are as follows: ˛  2  jV j  I.u/  c.u/; or ˛

I.u/  c.u/ ; 2  jV j

u2V

u 2 V:

(13)

(14)

Equation (14) [55] gives an upper bound to the bandwidth allocation for each node. Now another scenario is considered: allocating bandwidth proportionally according to the request req.u/. So every node u will have different bandwidth

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allocation as   req.u/. Similar to the deduction above, there are f .u/ D   2 

X

req.u/

(15)

u2V

or 

I.u/  c.u/ P ; 2  u2V req.u/

u 2 V:

(16)

The optimal ˛ and  can be achieved by minimizing Eqs. (14) and (16) [55]. (2) Case 2: Fix-NIC When the NIC cannot switch channel freely, the total available capacity for each node u is less than I.u/  c.u/. As [55] mentioned before, ˇ.u/ is introduced to represent the impact of interference. Here, the total available capacity is I.u/c.u/ . So the bandwidth allocation g.u/ for Fix-NIC is 1=.1 C ˇ.u// of the 1Cˇ.u/ bandwidth allocated for Switch-NIC. According to Eqs. (14) and (16), the maximum bandwidth allocation for each node with equal allocation strategy and the maximum proportion value are presented in [55] as the following:

3

˛

I.u/  c.u/=.1 C ˇ.u// ; 2  jV j

u2V

(17)



I.u/  c.u/=.1 C ˇ.u// P ; 2  u2V req.u/

u 2 V:

(18)

Channel Assignment

Wireless interference significantly impacts the performance in multihop networks. This is because most of the current multihop networks use just a single channel to communicate so that neighboring nodes interfere with each other. The IEEE 802.11 standards provide several orthogonal channels that can be used for wireless networks. Usually, in the networks working in infrastructure mode, neighboring access points operate on orthogonal channels to eliminate interference. However, this kind of design does not apply to most of the multihop networks for some reasons. The most important reason is that most of the commodity mobile devices have only one network interface card, and they cannot operate simultaneously using multichannel. Recently, wireless mesh architecture is more and more used to provide highbandwidth wireless network services. But since this architecture is breaking long distances into a series of shorter hops, the interference problem still exists. Recently, many studies begin to focus on using multichannels in WMNs, and some studies have presented channel assignment algorithms to improve the network performance. Some of the channel assignment algorithms [1, 40, 46, 48, 52] change the channels whenever there are significant changes to traffic loads or network topology.

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These channel assignment algorithms are relatively static. But if the network topology or the traffic load changes frequently, the performance of the network would significantly drop because of the delay of the channel switching. Some other algorithms [5, 49, 59] are designed to change the channel on the interface very frequently. These algorithms use the current state of the medium to determine the channel for each packet. However, an important condition for these algorithms is that the channel switch delay of the devices in the network should be small enough (switching in a packet or a few packets). And this condition eliminates most of the off-the-shelf hardware.

3.1

Minimum Interference

3.1.1 Optimization Problem Description Recent studies of minimum interference in multichannel wireless networks provide channel assignment algorithms optimizing this problem. In these studies, the main issue is how to design the channel assignment algorithms. The main goal of the algorithms is to use the limited channels and the network interface in each node to minimize the interference. In [50], the authors present the problem of static assignment of channels to link in the networks with multi-radio nodes. The objective of the channel assignment is to minimize the overall network interference. This channel assignment algorithm is derived from a graph coloring problem. The authors assume a binary interference model, in which there are two links that either interfere or not. Though the interference level between two links depends on the traffic on the links, the paper assumes uniform traffic on all links first and then generalizes the algorithms to nonuniform traffic. 3.1.2 Mathematical Formulation The communication graph of the network is modeled as an undirected graph over the set of nodes N . Each node i 2 N denotes a router that has Ri radio interfaces. There is a total of K channels available in the network. Let K D f1; 2; : : : ; Kg denote the set of K available channels. Let the conflict graph Gc [22] represent the interference between nodes. A conflict graph Gc .Vc ; Ec / is an undirected graph in which Vc D flij j .i; j / is a communication linkg: And a conflict edge .lij ; lab / is used to represent that two communication links .i; j / and .a; b/ interfere with each other if they are on the same channel. Then the channel assignment problem converts to a coloring problem, which is to color the vertices Vc of the conflict graph Gc using K colors without violating the interface constraint and minimizing the number of monochromatic edges in Gc . To solve this problem, a function f : Vc ! K needs to be computed to minimize the overall network interference I.f /. The following constraint satisfies the interface constraint:

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8i 2 N; jfkjf .e/ D k for some e 2 E.i /gj  Ri where E.i / D flij 2 Vc g, that is E.i / is a set of vertices in Vc that denote the communication links incident on node i and the network interference I.f / is I.f / D jf.u; v/ 2 Ec jf .u/ D f .v/gj: The network interference is the number of monochromatic edges in the colored conflict graph.

3.1.3 An Existing Solution In this section, a channel assignment algorithm presented in [50] is introduced. This algorithm, which is based on tabu search [19] technique, consists of two phases. In the first phase, a tabu search-based technique is used to find a solution f without considering the interface constraint. Then the violations of interface constraint are removed to get a feasible channel assignment function f in the second phase. First Phase: First, each vertex in Vc is given a random color in K. And this assignment is taken as the initial solution f0 , then a sequence of solutions f0 , f1 , f2 , . . . , fj ; : : : is created to get a solution with minimum network interference. The next solution fj C1 in the j th iteration is created. The j th Iteration: Start by making a certain number (say, r) of random neighboring solutions of fj . A random neighboring solution of fj can be made by picking a random vertex u and assigning a different random color in .Kffj .u/g/ to it. Therefore, a neighboring solution of fj is different from fj in the color assignment of only one vertex. Such randomly generated neighboring solutions of fj is made a solution set. Then pick the neighboring solution which has the lowest network interference as the next solution fj C1 from the set. In order to avoid local minima, I.fj C1 / cannot be required to be less than I.fj /. Tabu List: A tabu list [19]  is maintained in order to avoid reassigning the same color to a vertex more than once. For example, if fj C1 is created by assigning a new color to a vertex u, then .u; fj .u// is put into the tabu list . According to the tabu list, the solutions, which assign the color k to vertex u if .u; k/ is already in , are discarded. Termination: When the iterations pass the allowed maximum number (say, imax without any improvement in I.fbest /, the first phase terminates. In this algorithm, let imax be jVc j. The network interference I.f / is an integer, and it is lower than .jVc j/2 . Let r denote the number of random neighboring solutions, and d denote the maximum degree of a vertex in Gc . At each iteration, the value of I.fbest / decreases by 1 or the first phase terminates. Therefore, the time complexity of the first phase is O.rd jVc j3 /. Second Phase: After the first phase, a solution of channel assignment is got. However, the number of channels on a node may be larger than the number of

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interface cards on a node. The goal of the second phase is to remove the violations by using the following “merge” procedure. Assume a solution f has been got in the first phase. Based on this solution on the conflict graph Gc , the communication graph G is used in the second phase. After the first phase, each edge in the communication graph is colored. To begin with, a node which the violation is maximum for the merge operation is chosen. Let i be the node picked, two colors k1 and k2 incident on i are chosen. Then the color of all k1 -colored links is changed to k2 . To avoid creating more interface constraint violations at other nodes, all the k1 -colored links which are neighboring (two of more links are incident on a common node) are iteratively changed to the link just changed. This procedure ensures that for any node j , either all or none of the k1 -colored links incident on j are changed to color k2 . The steps above can reduce the number of distinct colors incident on i by one and does not increase the number of violations on other nodes. Therefore, this procedure can be repeated to eliminate all the interface constraint violations on the communication graph. However, this procedure will increase the interference in the network. Therefore, those two colors k1 and k2 are picked for the merge operation that cause the least increase of network interference. Simulations in [50] show that this algorithm effectively reduces the interference of the network and therefore significantly improves the performance.

3.2

Maximum Throughput

3.2.1 Optimization Problem Description Recent studies present multichannel multihop network models to optimize the throughput. In these studies, how to design the channel assignment algorithms is the core problem because the number of channels available and the number of the interfaces in each node are limited. So it is needed to design channel assignment algorithms to get the maximum throughput in the proposed networks. In [3], the authors present a fixed channel assignment algorithm for multi-radio multichannel mesh networks. In [4], the authors formulate the joint channel assignment and routing problem mathematically into a Linear Programming (LP) problem. The LP takes into account the interference constraints, the number of channels in the network, and the number of radios available at each router. Then the authors use this formulation to develop a solution for our problem that optimizes the overall network throughput. The optimization algorithm in [4] will be introduced in the following sections. 3.2.2 Problem Description and System Modeling In [4], a multichannel multihop infrastructure WMN is used as a network model. The WMN consists of some static wireless mesh routers and some mobile terminal clients. These wireless routers form a multihop wireless backbone. The backbone network relays the traffic from and to the clients. Some routers are the gateway to the wired Internet.

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To simplify the problem, it is assumed that there is only outgoing traffic to the wired Internet and l.u/ represents the outgoing traffic. Then the mathematical formulation is given in [4].

3.2.3 Mathematical Formulation The given backbone of the wireless network is modeled as a graph .V; E/. V is the set of all routers. Each edge e 2 E denotes a link between two routers in V . For that at least l.u/ of throughput can be routed to the Internet, it requires to find the maximized . Let t denote the destination node which connects directly to the Internet. Some definitions are given as follows: 1. There are K channels available. 2. I.e/  E denotes the set of edges that e interferes with. 3. Each node u 2 V has I.u/ network interface cards. 4. Each node has a total demand l.u/ from all the connected users. To get l.u/ throughput for each node u, to compute the following is necessary: 1. A network flow that consists of edges e D .u; v/ and values f .e.i //, .1  i  K/ where f .e.i // denotes the rate of the traffic which is transmitted from node u to node v using channel i . 2. A feasible channel assignment F .u/, which is an ordered set where the i th interface of u operates on the i th channel in F .u//, such that, whenever f .e.i // > 0, i 2 F .u/ \ F .v/. This case is called that edge e is using channel i to communicate. 3. A feasible schedule S which decides the set of edge channel pair .e; i / (i.e., edge e is using channel i ) scheduled at time slot , for  D 1; 2; : : : ; T where T is the period of the schedule. If the edges of no two edge pairs .e1 ; i /; .e2 ; i / scheduled in the a time slot for the same channel i interfere with each other (.e1 … I.e2 / and e2 … I.e1 //), the schedule is called as the feasible schedule. Let c.e/ denote the maximum rate at which router u can communicate with router v in one hop on a same channel. And let .RT / denote the interference range, which is assumed to be q times a transmission range RT for some small fixed constant q. Then the following constraints are got: The Link Congestion Constraint: Any valid edge flows without interference must satisfy for channel 1(or other equivalent channels): X f .e 0 .1// f .e.1// C  c.q/: c.e/ c.e 0 / 0

(19)

e 2I.e/

Link Schedulability Constraint: If the edge flows satisfy for channel 1 (or other equivalent channels) the following link schedulability constraint, then an interference-free edge communication schedule can be found:

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X f .e 0 .1// f .e.1// C  1: c.e/ c.e 0 / 0

(20)

e 2I.e/

f .e/ is defined as the rate at which traffic is transmitted by node u for node v on the only available channel. These values of f .e/ form the edge flows for a max-flow on a flow graph H . And H is constructed by the following steps. First, a source node s and sink node t are introduced into the graph G. And every gateway node v 2 V has an unlimited capacity directed edge to the sink node t. Node s has a directed edge to every node v 2 V with a capacity of l.v/. Let the set Es denote these edges and H be the required flow graph with s and t. The flow on every edge e 2 E in H is f .e/, and every e 2 Es in H has a flow equal to its capacity. According to the definition, the flow is a valid network flow, and it is a max-flow on H for the given source and sink nodes. Flow Constraints: Let V H .E H / be the set of nodes (edges) in H and f .e/.c.e// denote the flow (capacity) on an edge e of H . At any node v 2 V H other than s or t, the total incoming flow is equal to the total outgoing flow: X

f ..u; v// D

.u;v/2E H

X

f ..v; u//; 8v 2 V H  fs; tg:

(21)

.v;u/2E H

A flow is required on each outgoing edge for the largest possible . This result in the constraint f ..s; sv // D l.v/; 8v 2 V: (22) Also, it is ensured that no capacities are violated: f .e/  c.e/; 8e 2 E H :

(23)

.e.i // Node Radio Constraints: Let ˛.e; i / D f c.e/ for any e 2 E and channel i . It comes to the following linear constraints for each node v 2 V .

X

X

1i K eD.u;v/2E

X f .e.i // C c.e/ 1i K

X eD.v;u/2E

f .e.i //  I.v/: c.e/

(24)

Link Congestion Constraints: The link congestion constraint using link flows is changed: X f .e 0 .i // f .e.i // C  c.q/; 8e 2 E; 1  i  k: c.e/ c.e 0 / 0

(25)

e 2I.e/

Objective Function: max 

(26)

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The LP above involves channel assignment, routing, and link scheduling constraints. However, all the constraints we listed are necessary conditions. This means if a solution satisfies these constraints, it may still not be a feasible solution. This LP also yields a lower bound on the  value, which can be used to establish the guaranteed worst performance case. In [4], the authors present a channel assignment algorithm used to adjust the flow on the flow graph (routing changes) to ensure a feasible channel assignment. This flow adjustment also keeps minimum interference for each channel. The authors also provide algorithms for routing and link scheduling to form a complete solution of the optimization issue.

3.3

Optimal Fairness

In above sections, channel assignments in multichannel multi-radio WMNs have been studied, with the objective of minimum interference and maximum throughput. Now the fairness channel assignment scheme to optimally achieve Nash equilibrium (NE) solution with focus on fairness consideration to equalize the throughput of flows will be discussed.

3.3.1 Optimization Problem Description Multichannel WMNs often consist of mesh clients, mesh routers, and gateways [2]. A mesh topology is made up of the sets of routers and logical links. To increase the capability of WMNs, each node should be equipped with multiple radios interfaces. Achieving fairness channel assignment of communications in WMNs is important. In some instances, fairness is even more crucial than system-wide throughout optimization [8]. A Nash equilibrium with perfect fairness among all players is provided. And it is proved by using this fair channel assignment; when system converges to a constant status by Nash equilibrium, all game players obtain the equal throughput. The fairness problem was pointed out by Bharghavan et al., which was caused mainly because of hidden terminal problem as well as the backoff scheme used in the DFWMAC protocol [6, 21]. In this subsection, a system model is firstly formulated and presented, with some concepts of game theory and a game model in channel assignment. Then a Nash equilibrium solution is provided with perfect fairness and abundant proofs is given. 3.3.2 System Model and Concepts In this section, the model which is going to be studied will formally be defined. Let K be the number of radio transmitters which is manufactured by each user in a single collision domain. Suppose that each user participates and only participates in one such communication. Let N be the set of communicating pairs and C be the channel set. We suppose that the same channels have the same conditions. And let Tc denote the total throughput of channel c and c denote the throughput of each

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radio on channel c. So c can be assumed as a decreasing function of the number of radios on channel c [8]. The following definitions are given in [60] about game theory: 1. n is the number of all players. 2. Si be a strategy set for player i . 3. S is the set of strategy profiles and S = S1 X S2 : : : X Sn . 4. f is the payoff function and f =.f1 .x/; : : : ; fn .x//.  5. xi is a strategy profile of all players except for player i . Then for each player i 2 1; : : : ; n, if strategy xi resulting in strategy profile x = .x1 ; : : : ; xn /, then player i obtains payoff fi .x/. Note that the payoff is decided by the strategy profile chosen. Definition 1 Nash equilibrium: A strategy profile x*2 S is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player; then   8i; xi 2 Si ; xi ¤ xi W fi .xi ; xi /  fi .xi ; xi /:

(27)

Definition 2 Perfect fairness [8]: In the channel assignment game, the strategy profile s is perfectly fair, if 8i; j 2 N , fi .x/  fj .x/;

and †c2C ri;c  †c2C rj;c :

(28)

3.3.3 Mathematical Formulation People who use network nowadays need to pay for their connection and communication with other people in the network. Therefore, it is fairly to assume that charging to players for the usage of the channels to transmit packets is necessary. More significantly, a Nash equilibrium with perfect fairness can be completed by evaluating and designing carefully the amount of payment what players should give on the basis of the usage method of the channels. Here there is an assumption that a certain kind of virtual currency exists in the system and it demonstrates the payment for using channels. Every player pays to the system administrator some virtual currency on the basis of the channels which has chosen to use. Therefore, let pi;c represent the payment of player i for using channel c; a definition for pi;c is given as ˇ ˇ  ˇ ˇ 1 1 ˇ (29)  pi;c D ri;c ˇˇki;c  a  cC1 bC1 ˇ j k in which a D jCKj ; b D K mod jC j. Therefore, the total utility of player i is defined as ui D †c2C ri;c  †c2C pi;c : A strategy profile x  is defined as for every player i 2 N ,

(30)

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   xi D fxi;1 ; xi;2 ; : : : ; xi;jC C 1; : : : ; a C 1; a; : : : ; ag: j g D fa „ ƒ‚ … „ ƒ‚ …

(31)

jC jb

b

Note that .a C 1/b C a.jC j  b/ D K, which is the total number of radios that each player has. Theorem 1 In the channel assignment game described before, x  achieves a Nash equilibrium. In order to clearly describe the theorem, the following definition for any player i [8] is given:  aC1 if 1cb  D (32) ki;c if bC1cjC j

a

Note that for each channel c 2 C , the identification number of that channel c is also used by us. Given that b > 0, c > 0, 

1 1  cC1 bC1



1

if 1cb

D 0

if bC1cjC j

:

(33)

From previous two equations, for 8c 2 C , there is ˇ ˇ  ˇ  ˇ 1 1 ˇk  a  ˇ D 0:  ˇ i;c cC1 bC1 ˇ so for xi ,

P c2C

(34)

pi;c = 0. Therefore,  /D fi .xi ; xi

X

ri;c :

(35)

c2C

Now the strategies other than xi will be taken into consideration. 8xi ¤ xi , let Ci0 denote the channel set on which the radio assignment of player i deviates from  xi . Since xi ¤ xi , 9c 2 C , ki;c ¤ ki;c . So jCi0 j > 0. Next, it will be proved that for player i , 8c 2 Ci0 ; pi;c  ri;c . Since ki;c is not a  negative integer and ki;c ¤ ki;c , 8c 2 Ci0 , ˇ ˇ  ˇ ˇ 1 1 ˇ  1: ˇki;c  a   ˇ cC1 bC1 ˇ

(36)

From the previous (1) and (8) equations, it can be obtained that 8c 2 Ci0 ; pi;c  ri;c .

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Eventually, there is   fi .xi ; xi / fi .xi ; xi / X X X  D ri;c  r  0 i;c c2C

 D D D

X X

c2C

Xc2C Xc2C

 ri;c   ri;c   ri;c 

r c2Ci0 i;c

X

r 0C 0 i;c

c62Ci

X

0

Xc62Ci

pi;c C

c62Ci0

c62Ci

X

X

r  0 i;c

c2Ci

c2Ci0

ri;c 

X



X 

c2Ci0

pi;c

r 0 i;c

c2Ci

ri;c

r c62Ci0 i;c

 0:

Hence, x  is a Nash equilibrium. Next, it will be demonstrated that S  is perfectly fair as well. Theorem 2 S  reaches perfect fairness. From ri;c D ki;c †.kc /, it can be observed that throughput of player i on channel c depends on ki;c and the sum of the radios number on the channel c; k. It is easy to see that in xi , the total number of radios on channel c.kc / is  kc D †i 2N ki;c D

n

.aC1/jN j ajN j

if 1cb if bC1cjC j

(37)

where jN j is the number of players. Given (7) and (9), 8i 2 N , there is fi D

X c2C

ri;c D

X c2C

 ki;c .kc /

D †1cb .a C 1/..a C 1/jN j/ C †b 0: jM j 

X

Xi j;k  Fi j ; 8i; j 2 V; i ¤ j:

k2Q

The equation in list 4 relates the Xi j;k variables to the flow variables Fi j . Note that the optimization procedure excludes the case that two edges (operating on different channels) between the same two nodes are both included in the multicast tree. To achieve the minimize cost of multicast tree, a set of variables is introduced: fYi;k W 8i 2 V; k 2 Qg : Yi;k equals to one if node i is a transmitting node on channel k in the resulting multicast tree, and zero otherwise. Thus, a set of constraints relate to Yi;k to the Xi j;k can be obtained: Yi;k  Xi j;k ; 8i; j 2 V; i ¤ j; 8k 2 Q Therefore, the objective function is minimize

XX

Yi;k

k2Q i 2V

where the multicast tree cost is the total number of transmitting nodes in the tree over all channels. The ILP formulation for the MCMT problem is summarized below. minimize

XX

Yi;k

k2Q i 2V

subject to

Xi j;k  Ci j;k I 8i; j 2 V; i ¤ j; 8k 2 Q †j 2V;j ¤s Fsj D jM jI X X

†j 2V;j ¤s Fj s D 0I X Fj i  Fi j D 1; 8i 2 M I

j 2V;j ¤i

j 2V;j ¤i

Fj i 

X

j 2V;j ¤i

j 2V;j ¤i

Fi j D 0; 8i 2 V; i … M; i ¤ sI

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jM j 

X

Xi j;k  Fi j ; 8i; j 2 V; i ¤ j I

k2Q

Yi;k  Xi j;k ; 8i; j 2 V; i ¤ j; 8k 2 QI Xi j;k 2 f0; 1g I 8i; j 2 V; i ¤ j; 8k 2 Q Fi j  0I 8i; j 2 V Yi;k 2 f0:1g I 8i 2 V; 8k 2 Q:

4.1.4 Solution Next, a polynomial-time approximation algorithm, called Wireless Closest Terminal Branching (WCTB), is proposed for the MCMT problem. The main idea of the WCTB is that: 1. For each step, one new branch is added to the current multicast tree to reach a new destination that is closest to the source node. 2. The cost of each link in the networks initials as one. The crux is that when a new branch is included into the multicast tree, the cost of links may become zero due to the broadcast advantage. 3. It is easy to implement based on the Dijkstra’s algorithm. Theorem 3 The WCTB approximation ratio is no more than .2  Dmax /, that is, cost.TWCTB /  2  Dmax cost.Topt / where Topt denote the optimal multicast tree and TWCTB denote the multicast tree generated by WCTB. And cost.TWCTB / and cost.Topt / denote the tree costs produced by the WCTB and the optimal tree, respectively. Given G D .V; EI s; M /, let t1 denote the source node and ti C1 denote the ith destination in M . Let dist.u; v/ denote the shortest distance in terms of hop count from node u to v in G. The proof of Theorem 3 needs two lemmas. Lemma 1 Suppose p1 ; p2 ; : : : ; pk is a permutation of 1, 2, . . . , k, then for every i; 2  i  k, there exists a unique critical pair .pj 1 ; pj / such that min.pj 1 ; pj / < i  max.pj 1 ; pj /; 2  j  k. Lemma 2 For any instance .G D .V; E/; s; M / of the MCMT problem, jM j

cost.Topt / 

†i D1 dist.tpi ; tpi C1 / 2  Dmax

 ˚  ˚ where tp1 ; tp2 ; : : : ; tpjM j C1 is a permutation of t1 ; t2 ; : : : ; tjM jC1 obtained by a preorder traversal of Topt starting from the source node s.

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Both lemmas proof can be found in [32]. From the above lemmas, for 2  i  jM j C 1, there is jM jC1 X

0

dist.ni ; ni / D

i D2

jM jC1 X j D2

dist.mpj 1 ;mpj /:

In the algorithm WCTB, to reach a new destination, say mi , a minimum cost path from s is integrated. The cost of the minimum cost path is cost.s; mi / D cost.m1 ; mi /; therefore,

cost.TWCTB / D

jM jC1 X

cost.m1 ; mi /:

i D2

Furthermore, since the link cost of edges in the current multicast tree T in WCTB is zero, cost.m1 ; mi / is no more than the distance between mi and any destination that is included in the current multicast tree, and thus no more than the distance between nearlier and ninoearlier , that is, for 2  i  jM j C 1, i ; ninoearlier /: cost.m1 ; mi /  dist.nearlier i Combining the above proof equations,

cost.TWCTB / 

jM jC1 X

dist.mpj 1 ; mpj /

j D2

is obtained. Accordingly, from Lemma 2 and above equation, there is jM j

cost.Topt / 

†i D1 dist.tpi ; tpi C1 / cost.TWCTB /  : 2  Dmax 2  Dmax

Thus, cost.TWCTB /  2  Dmax : cost.Topt /

4.1.5 Conclusion In this section, MCMT problem is considered in MR-MC WMNs. The problem is in concern of dynamic traffic model. An ILP model for solving this problem optimally has been provided. The polynomial-time near-optimal algorithm, WCTB, is proposed to solve the problem efficiently.

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Data Collection

Multichannel wireless sensor networks for supporting a wide range of applications such as environmental monitoring, military surveillance, and traffic control have been drawn attention to researchers in the current wireless research area. Such kind of networks can be used to collect sensory information from the real world. In order to collect data, there are two types of data gathering in multichannel wireless sensor networks. One is data collection [11, 38, 63] which gathering data from the source nodes and send them to the base station. Different from data collection, data aggregation [15, 24, 33, 57] rather than sending individual data from sensor nodes to base station is that multiple data are aggregated as a data to be forwarded by the sensor network. In data collection [23], the problem of collecting all the sensing information from the sensors at a particular time is defined as snapshot data collection. Continuous data collection is represented as the collection of multiple continuous snapshot. The network capacity [18] is one of the network measurement for the network performance evaluation, while wireless sensor networks suffer from the interference problem during data transmission. In [23], the data receiving rate at the sink node is used to measure the data collection capacity. Extensive works have been studied in distinct communication tasks such as broadcast capacity [36], multicast capacity [34, 35], unicast capacity [44], and snapshot data collection capacity [11, 38, 63]. Most of the previous works are based on single radio and single channel in wireless sensor networks which the sensor node is only working on the half-duplex mode. In such sensor network environment, data transmission suffers from the radio conflict [12, 27] and channel interference [7, 13] issues. In this section, a dual radio multichannel is applied which dramatically will reduce the interference. The problem considering in this section is based on the protocol interference model [54]. Two algorithms are presented which investigate the snapshot and continuous data collection problem in dual radio multichannel wireless sensor networks.

4.2.1 Network Model and Concepts The network is represented as a graph G D .V; E/, where V is the set of sensor nodes and E is the set of links in V . The network is composed of n sensor nodes and one sink node. Each sensor is equipped with two radios and each radio’s transmission radius is fixed to 1 and the interference range is defined as ,  > 1. In protocol interference model, a receiving node v successfully receives the data from the transmitting node u only if ku  vk  1 and furthermore any other node s which satisfies ks  vk   will not simultaneously transmit data packet over the same channel with u. Each radio has K nonoverlapping channels as !1 ; !2 ; !3 ; : : : ; !K . Each sensor can transmit at a rate of W bits/second over a channel [11]. It is assumed that all the transmissions are synchronized and the size of a time slot is t = b/W seconds, where b bits is the size of all the packet transmission in the network. The capacity  of data collection is the ratio between the size of the data successfully received by the sink and the time  used to collect these data. The capacity to collect

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N snapshots of data is  D Nnb , which is truly the data rate at the sink. A connected dominating set based on the mechanism used in [57] is built for constructing a routing tree. Vertex coloring terminologies have been derived in [43, 57].

4.2.2 Scheduling Algorithm for Snapshot Data Collection In this section, a novel scheduling algorithm is proposed with a tighter lower capacity bound and order-optimal. Since at any time slot, the sink can receive data from at most one sensor, the upper bound of the snapshot data collection capacity is W [11]. A new multipath scheduling algorithm based on the previous built routing tree in [57] is proposed [23]. First how to schedule a single path and extend it to the schedule of multipath in the routing tree is discussed. For simplicity, the concept of round, which indicates a period of time consisting of multiple continuous time slots, is introduced. The path is denoted by P, and P0 (resp., Pe ) denotes the set of links on P whose heads (resp., tails) are dominators and whose tails have at least one packet to be transmitted. Scheduling P has the following two steps. Step 1. In an odd round, schedule every link in P0 once, that is assign a dedicated channel and one dedicated time slot to each link in P0 . Step 2. In an even round, schedule every link in Pe once, that is assign a dedicated channel and one dedicated time slot to each link in Pe . Then the multipath scheduling on the routing tree is discussed. The basic idea of the multipath scheduling algorithm is as follows: Step 1. For the nonintersecting paths without wireless interference, schedule them by the single path scheduling algorithm concurrently. Step 2. For the nonintersecting paths with wireless interference, schedule them according to a certain order. The path with the smallest subscript has the highest priority. Step 3. For the intersecting paths, schedule them according to a certain order, for example, if Pi is the one of these intersecting paths with the smallest subscript, schedule Pi until there is no packet having to be transmitted on the sub-path and then continue to schedule the next path. Theorem 4 The capacity  at the sink of T of the multipath scheduling algorithm is at least W ; 3:63 2 K  C o./ which is order-optimal. The proof can be found in [23]. Compared with the recent result in [11], the W proposed algorithm has a lower bound of 8 2.

4.2.3 Scheduling Algorithm for Continuous Data Collection In this section, a novel pipeline scheduling algorithm base on the compressive data gathering (CDG) [38] in dual radio multichannel wireless sensor networks

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is presented. From the benefit brought by CDG, the continuous data collection problem with the pipeline technique [11] can be addressed, different from the multipath scheduling algorithm, in which a sensor sends one packet over a link in one time slot. A sensor sends M packets for a snapshot employing CDG. A concept of a super time slot (STS) is introduced which consists of M time slots. In a STS, a sensor can send M packets over a channel for a snapshot. For the links working simultaneously, channels and STSs are assigned using the similar way of the multipath scheduling algorithm. Theorem 5 For a long-run continuous data collection, the lower bound of the achievable asymptotic network capacity of the pipeline scheduling algorithm is nW 2 12M. 3:63 K  C o.// when e  12 or

nW Me . 3:63 2 C o.// K

when e > 12. Proof detail is explained in [23].

4.2.4 Conclusion Motivated from the no research work studying the continuous data collection before [23], the problem in the dual radio multichannel wireless sensor networks is investigated. Two algorithms in snapshot and continuous data collection are proposed. The snapshot data collection algorithm can achieve the network capacity of at least 3:63 W 2 Co./ , and the continuous data collection achieves asymptotic K  nW when e  12 or  nW when network capacity of 3:63 3:63 12M

K

2 Co./

Me

K

2 Co./

e > 12. The simulation results show that the proposed algorithms’ network capacity outperforms the existing works.

4.3

Data Aggregation

Recent advancement of radio technology has enabled the sensor devices which are capable of sensing, computation, and communication setting in a multichannel environment. A multichannel wireless sensor network (MWSN) communication does not rely on any infrastructure but hop-by-hop sensor communication. To collect the sensory information such as temperature, medical, and surveillance data in this type of network, an efficient and robust data collection mechanism is in need. Data aggregation is one of the data gathering paradigm to perform in-network fusion of data packets with small number and size of data transmission. The goal of data aggregation is to achieve the minimum interference and latency in the network.

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Lots of works focus on distinct communication tasks with the objective of minimum latency. Many existing works have been studied in the minimum latency interference-aware broadcast scheduling extensively [9, 39, 56]. In [9], Chen et al. studied the minimum latency interference-aware broadcast scheduling considering the transmission range and interference range. Mahjourian et al. [39] extended a much more consideration of carrier sensing range in the minimum latency conflict-aware broadcast scheduling. Wan et al. [56] investigated the minimum latency scheduling for group communications (broadcast, aggregation, gathering, and gossiping) in multichannel multihop wireless networks. Data aggregation shows out to be more effective in data gathering in MWSN. Previous research considers the conflict-free data aggregation issues [10, 20, 26, 57, 61] in single-channel wireless sensor networks. The works proposed in [10, 20, 61] are only considered the nodes’ transmission range, and all nodes in the existing networks have data packet sent to the base station. Wan et al. [57] studied the minimum data aggregation latency problem in a more general situation. An algorithm with latency bound ˇC1 .15R C   4/ was proposed in [57], when the interference range is  times of the transmission range. Krishnamachari et al. [26] viewed data aggregation from another aspect. They considered the case where there was a subset of nodes whose data need to be sent to the base station and regarded aggregating these data as a way to save energy. The work [62] is the only one existing for data aggregation considering the interference range and carrier sensing range of the wireless nodes. From the above existing works, this section addresses the more generic minimum latency conflict-free data aggregation scheduling problem in multi-channel multihop wireless sensor networks [37]. Given an instance in which the network consists of a set of sensor nodes and a base station, the discussed problem is formally defined as to find a schedule such that data from a set of sensors can be successfully transmitted to the base station without any wireless conflict and the latency of accomplishing data transmission is minimized.

4.3.1 Conflicts and Conflict-Free Communication In this section, the conflict-free communication model is formulated. Two parallel data transmissions can be successfully transmitted only if all the following types of conflicts are avoided. Those transmissions which avoid all the types of conflicts are called conflict-free transmissions: 1. Collision at receiver: If there are two or more nodes in node u’s transmission range to transmit using the same channel k, node u cannot correctly receive messages transmitted by its neighbors using channel k. 2. Interference at receiver: If there is a node in node u’s interference range to transmit using channel k, node u cannot correctly receive the messages from its neighbors transmitting using channel k. 3. Contention at sender: If there is a node in node u’s carrier sensing range transmitting using channel k, node u cannot correctly transmit messages to its neighbors using channel k.

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4.3.2 System Model and Problem Definition The instance network consists of n sensors and a base station. Each sensor is equipped with a RF transceiver with omnidirectional antennae to send or receive data. There are   1 available nonoverlapping channels for node transmitting and receiving data. The transmission range, interference range, and carrier sensing range which roughly represented as disks are considered during the data transmission. The mathematical formulation of the communication network is illustrated as a unit disk graph (UDG) G D .V; E/. Each node has the network factor of r, ˛r, ˛  1, ˇr, ˇ  1 which indicate the transmission range, interference range, and carrier sensing range, respectively. An edge exists between node u and node v if and only if d.u; v/  r, where d.u; v/ is the Euclidean distance between u and v. There exists an edge between node u and the base station b if and only if d.u; b/  r. The problem is then formulated as follows: Given instances of a UDG G D .V; E/ along with a base station b 2 V , a subset S  V  fbg, and parameters ; r; ˛; ˇ, the considering problem is how to find a conflict-free many-to-one data aggregation schedule with latency minimized. 4.3.3 Data Aggregation Scheduling Algorithms In this section, the data aggregation scheduling algorithm will be proposed and analyzed. The algorithm will mainly contain two phases. The first phase is to construct an initial many-to-one data aggregation tree. The second phase is to derive an aggregation schedule from the initial data aggregation tree. 4.3.4 Many-to-One Data Aggregation Tree Construction The main idea of the initial data aggregation tree is to firstly take a breath-first search of some nodes which are not taken action in the many-to-one schedule with minimum latency. After that, a many-to-one tree is constructed based on the subgraph induced by these nodes. The breadth-first search (BFS) tree is rooted at the base station denoted by TBFS for the network G D .V; E/ and eliminates the nodes from G and TBFS which are not on the paths from any node u 2 S to the base station in TBFS . Note that the nodes eliminated will not join in the schedule. The node elimination could be done as follows. (1) Let A D V  fbg and P .u/ be the set of all theS nodes on the path from node u to the base station in TBFS . Then the nodes in A  P .u/ need to be eliminated. (2) After eliminating the nodes, u2S

the obtained network and breadth-first search tree are denoted by Gp and TBFSp , respectively. The S fact isSthat the subgraph Gp of G and the subgraph TBFSp of TBFS are induced by P .u/ fbg. u2S

Based on the subgraphs Gp and TBFSp , we construct a many-to-one data aggregation tree. After pruning the network, we construct a maximal independent set (MIS) on the pruned network according to the breadth-first search ordering. Then

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we transform the MIS to minimum cover problem and compute the minimum cover to obtain the initial data aggregation tree. The data aggregation tree does not prevent any conflicts. More detailed explanation is in [37].

4.3.5 Many-to-One Data Aggregation Scheduling The assignment of many-to-one data aggregation scheduling is designed by node coloring. The fact in node coloring is that any graph G is able to produce a proper node coloring in G using at most 1 C ı  .G/ colors. Smallest-degree-last ordering of nodes is applied to assign proper color to each node in particular order [42]. The following useful lemmas are given to construct a conflict-free schedule. Lemma 3 Two parallel transmissions “t1 k1 r1 ” and “t2 k2 r2 ” are conflict-free !  !  (t1 ¤ t2 and r1 ¤ r2 ), if and only if at least one of the following two conditions are satisfied: 1. d.t1 ; r2 / > ˛r; d.t2 ; r1 / > ˛r; d.t1 ; t2 / > ˇr 2. k1 ¤ k2 The following lemma shows the sufficient conditions for avoiding the three types of conflict. Lemma 4 ([39]) In order for two parallel transmissions “t1 k r1 ” and “t2 k r2 ” !  !  using the same channel k to be conflict-free according to the network model, it is sufficient to have d.t1 ; t2 / > max.˛ C 1; ˇ/r _ d.r1 ; r2 / > .max.˛; ˇ/ C 2/r. Two conflict graphs which are denoted by GCt and GCr according to Lemma 4 are constructed. The  channels are denoted by integers 0; 1;    ;   1. There are two cases. (1) When  D 1, there is only one available channel. The only needed thing is to schedule the transmitting time slot of each node. After coloring the conflict graph, if two nodes have the same color, that is, their distance is bigger than max.˛ C 1; ˇ/r (or .max.˛; ˇ/ C 2/r), then the two nodes can be scheduled to transmit (receive) at the same time slot. The nodes which have distinct colors will not be scheduled to transmit (receive) at the same slot. (2) When  > 1, the nodes with distinct colors can schedule at the same time slot when assigning different channels to them. Thus, it is needed to schedule the transmitting time slot and the channel of each node. Fortunately, a uniform method is found to deal with the two situations. Two sub-schedules are derived to obtain conflict-free scheduling [37]. Basing on the two sub-schedules, the conflict-free data aggregation schedule is constructed; the details can be found in [37].

4.3.6 Performance Analysis In this section, the approximation ratio of the proposed algorithm, which is for the minimum latency conflict-free many-to-one data aggregation scheduling problem,

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is analyzed. The performance of proposed algorithm is compared against the trivial lower bound R, where R is the depth of the TBFS p . A known theorem [17] is used to prove Lemma 5. Theorem 6 ([17]) Suppose that C is a compact convex set and U is a set of points with mutual distances at least one, then

jU \ C j 

2area.C / peri.C / C 1; p C 2 3

where area.C / and peri.C / are the area and perimeter of C , respectively. Lemma 5 Let Gp D .Vp ; Ep /, S  Vp be independent set

 in Gp , the ˘ inductivities

of GCt ŒS  and GCr ŒS  have an upper bound of p h2 C 2 C 1 h and p k 2 C 3 3  ˘ C 1 k , respectively, where h D max.˛ C 1; ˇ/ and k D max.˛; ˇ/ C 2. 2 Lemma 6 ([57]) The following statements are true: 1. jC0 j  12. 2. For each 1  i  RU  1, each dominator in Ui is adjacent to at most 11 connectors in Ci . 3. For each 0  i  RU  1, each connector in Ci is adjacent to at most 4 dominators in Ui C1 . ˙  ˙ Theorem 7 The algorithm has a latency upper bound a e C 11 b R C .  ˘

2  ˘

2  ˙b  ˙a 23/    C12, where a D p h C 2 C1 h C1, b D p k C 2 C1 k C1, 3 3  and R are the maximum degree of pruned network Gp and the depth of the pruned BFS tree TBFSp , respectively. Corollary 1 For  D 1, the algorithm has a latency upper bound .a C 11b/R C .  23/b  a C 12 and thus it has a nearly constant .a C 11b/ ratio.

4.3.7 Conclusion In this section, the minimum latency conflict-free many-to-one data aggregation scheduling problem in multichannel multihop wireless networks is addressed. Nearly a constant .d a e C 11d b e/-ratio algorithm for the problem is introduced.

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52. J. Tand, G. Xue, W. Zhang, Interference-aware topology control and QoS routing in multichannel wireless mesh networks, in MOBIHOC, 2005 53. M. Thorup, U. Zwick, Compact routing schemes, in SPAA, 2001, pp. 1–10 54. P.-J. Wan, Z. Wang, H. Du, S.C.-H. Huang, Z. Wan, First-fit scheduling for beaconing in multihop wireless networks, in INFOCOM, 2010, pp. 2205–2212 55. J. Wang, H. Li, W. Jia, L. Huang, J. Li, Interface assignment and bandwidth allocation for multi-channel wireless mesh networks. Comput. Comm. 31(17), 3995–4004 (2008) 56. P.J. Wan, Z. Wang, Z.Y. Wan, Scott C.-H. Huang, H. Liu, Minimum-latency schedulings for group communications in multi-channel multihop wireless networks, in WASA, 2009 57. P.-J. Wan, Scott C.-H. Huang, L.X. Wang, Z.Y. Wan, X.H. Jia, Minimum-latency aggregation scheduling in multihop wireless networks, in ACM MOBIHOC, 2009 58. J.E. Wieselthier, G.D. Nguyen, A. Ephremides, Energy-efficient broadcast and multicast trees in wireless networks. Mobile Network Appl. 481–492 (2002) 59. S. Wu, C. Lin, Y. Tseng, J. Sheu, A new multichannel MAC protocol with on-demand channel assignment for multi-hop mobile ad hoc networks, in ISPAN, 2000 60. Wikipedia: http://en.wikipedia.org/wiki/Nashequilibrium. Wikipedia, 2011 61. X.H. Xu, S.G. Wang, X.F. Mao, S.J. Tang, X.Y. Li, An improved approximation algorithm for data aggregation in multi-hop wireless sensor networks, in ACM MOBIHOC, 2009 62. Q.H. Zhu, D.Y. Li, Approximation for a scheduling problem with application in wireless networks. Sci. China Math. 53(6), 1407–1662 (2010) 63. X. Zhu, B. Tang, H. Gupta, Delay efficient data gathering in sensor networks, in MSN, 2005

Optimization Problems in Data Broadcasting Yan Shi, Weili Wu, Jiaofei Zhong, Zaixin Lu and Xiaofeng Gao

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Scheduling Optimization for Pure-Pushed Data Broadcasting. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Scheduling Optimization for Single-Channel Data Broadcasting. . . . . . . . . . . . . . . . . . . 2.2 Scheduling Optimization for Multichannel Data Broadcasting. . . . . . . . . . . . . . . . . . . . . . 2.3 Scheduling Optimization for Dependent Data Broadcasting. . . . . . . . . . . . . . . . . . . . . . . . 3 Optimizations for On-Demand Data Broadcasting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Online Scheduling Algorithms for On-Demand Data Broadcast. . . . . . . . . . . . . 3.2 R  W Scheduling for Large-Scale On-Demand Data Broadcast. . . . . . . . . . . . . . . . . . 3.3 Scheduling Time-Critical On-Demand Data Broadcast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Scheduling Preempting On-Demand Broadcast for Heterogeneous Data. . . . . . . . . . . 4 Data Retrieval Optimization in Data Broadcast Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Indexing Techniques for Data Broadcast Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Indexing in Single-Channel Broadcasting Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Indexing in Multichannel Broadcasting Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Coverage Optimization for Data Broadcast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Mathematical Modeling and Problem Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Y. Shi () Department of Computer Science and Software Engineering, University of Wisconsin Platteville, Platteville, WI, USA e-mail: [email protected] W. Wu • Z. Lu Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA e-mail: [email protected]; [email protected] J. Zhong Department of Mathematics and Computer Science, University of Central Missouri, Warrensburg, MO, USA e-mail: [email protected] X. Gao Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, People’s Republic of China e-mail: [email protected] 2391 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 50, © Springer Science+Business Media New York 2013

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6.2 6.3 6.4 6.5

Classification and Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cell Broadcasting Scheme to Support Multilingual Service. . . . . . . . . . . . . . . . . . . . . . . . . Coverage Prediction and Optimization for Digital Broadcast. . . . . . . . . . . . . . . . . . . . . . . Data Delivery Optimization in Multi-system Heterogeneous Overlayed Wireless Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Broadcasting is an efficient way to disseminate information to a large number of users. With the fast development of wireless communication techniques, wireless data broadcasting has become a popular data dissemination method. The major performance concerns for any wireless data broadcast systems are response time and energy consumption. Various optimization methods for different aspects of data broadcasting systems have been developed in order to improve the broadcast system performance. This chapter gives a complete overview of different optimization issues in data broadcasting. In this chapter, optimization problems and corresponding algorithms and solutions for data scheduling, indexing, retrieval, and coverage are discussed in various types of data broadcast systems, including single-channel, multichannel, push-based, and on-demand data broadcasting.

1

Introduction

Data broadcasting is not a new concept. The term has been used for more than 30 years. Traditionally, data broadcasting refers to the ability to send data other than video/audio through broadcast television/radio transport [14, 54]. A bestknown example is captioning. Takenaga categorized this type of data broadcasting into independent data broadcasting and linked/supplementary data broadcasting. Independent data broadcasting includes only data, such as weather reports or stock information, while linked/supplementary data broadcasting is for the purpose of better facilitating regular television programs and can sometimes be interactive. Traditional data broadcasting reached its peak during the middle and late 1990s due to the digitalization of broadcasting and the massive use of home networks. A lot of data broadcasting services became available, for example, on-screen TV guide, SkyLife service in South Korea, and Teletext service in United Kingdom. Most of these services, however, still use traditional broadcast media (cable or television satellites) for data transportation. In recent years, the boosting of wireless technology and mobile communications (GPRS, 3G, 4G, etc.) has given data broadcasting a completely new meaning. In a wireless communication environment, data broadcasting means disseminating data to a large group of candidate clients in a broadcast manner. The beauty of wireless data broadcast is its scalability, flexibility, and bandwidth efficiency. In wireless data broadcast systems, data are broadcasted from a base station on some channels of particular frequencies within a transmission range. Clients in this

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range can access the channels and download the data. Several regulatively located base stations may work together to cover a certain area, each taking charge for its local cell named microcell (usually known as hexagon, quadrangle, or triangle, depending on the geometric layout of base stations). Clients can move freely among cells. In contrast to point-to-point data delivery, data broadcast allows a mass number of clients to simultaneously access the data and thus is more scalable. Broadcast technique also fits the asymmetric structures of most wireless systems because clients simply “listen” to the broadcast channels, filter, and pick the data they need, which alleviates the uplink channel load. Therefore, the system can make use of most bandwidth as downlink for data dissemination. Moreover, within the coverage range of a base station, no matter how many clients need to download the data, the network behavior remains the same. In general, there are three types of data broadcast: push-based, pull-based, and hybrid. Push-based [1] data broadcast uses all bandwidth for broadcast. It is usually used to disseminate public information such as stock price, news reports, or traffic information. One benefit of pure push-based broadcast is that it makes full use of the network bandwidth for data dissemination. It also has very little computational requirement for the server because the broadcast program can be organized offline. On the other hand, since there is no uplink in a push-based broadcast system, it is very hard to dynamically adjust the broadcast program to changes of clients’ requirement. Hence, push-based data broadcast is suitable for environments where the data and client access patterns are almost static. Pull-based data broadcast is also known as “on-demand” data broadcast. Compared to push-based data broadcast, it has an additional uplink through which clients can send their requests to the server. Requests are queued in the server. Instead of organizing a broadcast program off-line and then broadcast it periodically, the server will use some online algorithm to decide the broadcast content based on the service queue in real time. In such a way, the broadcast content will usually reflect the clients’ real requirements and can also be adjusted dynamically according to changes of requests. Therefore, on-demand data broadcast can be used in environments where the data and client access patterns change frequently. However, handling client requests and making broadcast decision in real time means a heavy work load for the server and often requires high computational capacity. Hybrid data broadcast takes advantage of both sides. It combines push- and-pull based mechanisms. Different hybrid systems have different integration methods. Some systems use push-based scheme to broadcast data with static access patterns and on-demand scheme for data with dynamic access patterns, while some other systems broadcast most popular data in a pure push way and handle not-so-popular data individually. Response time is a major concern for all kinds of data broadcast systems. The definition of response time is not exactly the same for different systems. In general, response time for a request is the time interval between the moment a request is generated (or the client first tunes in a channel) to the moment the requested information is completely downloaded. In push-based systems, it is also called access time. There are many factors that can affect response time. From the network point of view, an intuitive way is to increase the throughput of the broadcast. Using more

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channels for data broadcast can naturally increase the overall throughput. From the server side, different ways of scheduling the broadcast data may result in different response time, even with the same sets of requests. There have been intensive research efforts on scheduling algorithm design in order to optimize response time. Sections 2 and 3 will introduce different scheduling algorithms for different pushbased and on-demand data broadcast systems. From the client side, when requesting multiple items, the downloading sequence can also make a difference. Section 4 will discuss several algorithms trying to optimize the retrieval scheduling. In wireless data broadcast systems, since most clients are mobile devices operating on battery power, energy conservation becomes another important performance concern. A mobile device usually operates on two modes: the active mode and the doze mode. The energy consumed in the active mode can be 100 times of that in the doze mode [42]. Based on this observation, tuning time, which is the total time when a client is in the active mode, is defined as a performance criterium to evaluate energy efficiency of a system. In order to optimize energy consumption of a broadcast system, index is introduced. With the help of index information, clients do not need to stay active all the time to wait for requested data. Instead, they can change to the doze mode while waiting and be active again and tune in the broadcast channel right before the requested data arrive. This can significantly reduce the tuning time of clients. Section 5 will talk in detail about dominating index methods for data broadcast in existing literature. Another optimization issue related to data broadcast is the coverage problem. As discussed before, data are broadcasted from a base station. The range that a base station can cover is determined by the physical feature of both the base station and the terrain. Given a relative wide area, how to locate base stations in order to cover the whole area is an important optimization problem. Based on different objectives such as minimizing financial cost or enhancing reliability, the coverage problem can be formulated differently. In general, most of them are NP-hard and can be converted into combinatorial optimization problems. Section 6 will give the formal model of the coverage problem and introduce different variations of coverage optimization for data broadcast.

2

Scheduling Optimization for Pure-Pushed Data Broadcasting

In a pure-pushed broadcast network, there are two distinct sets of entities: servers and clients. The servers at the base station repeatedly disseminate a fixed set of data items via wireless channels, while the clients listen to the channel, wait for their desired data items, and download them when they are coming. An example is shown in Fig. 1. In general, the data items are scheduled over time, and the clients do not have to send requests to the server, but wait for the requested items to be broadcasted. Informally, the scheduling optimization problem for data broadcast is that given a set of data items, find a schedule for deciding the broadcast time for each data item of the set such that the expected average access time of the requests is minimized.

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D2...D4...D3

D1...D3...D5

Base station

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Mobile user

Radio coverage area

Fig. 1 Broadcast wireless network

In the past two decades, several variants for the scheduling problem of purepushed data broadcast have been discussed, and lots of works have been done. They can be roughly classified into three kinds: scheduling optimization for single-channel data broadcasting, scheduling optimization for multi-channel data broadcasting, and scheduling optimization for dependent data broadcasting. In Sect. 2.1 the works that focus on single channel are presented. Besides developing the single-channel data broadcast, the multichannel data broadcast is also an important area and recently more works focus on this issue. In Sect. 2.2 the research for scheduling multichannel data broadcast is introduced. In addition it is much more likely that a client query a set of data instead of only one data item. Therefore, some works recently discuss about broadcasting dependent data, whose access sequence is pre-ordered. How to schedule dependent data broadcast is discussed in Sect. 2.3.

2.1

Scheduling Optimization for Single-Channel Data Broadcasting

In the early stage of wireless data broadcast research, researchers first study on scheduling data on single broadcast channel. In 1995 Acharya et al. first proposed the scheduling problem for data broadcast [2]. A new technique called broadcast disks for data broadcast is introduced in that paper. In order to make an initial study of the scheduling problem for data broadcast, they defined many assumptions for the broadcasting environments. The assumptions include the following: (1) the data items of the broadcast program remains static; (2) each data item is read only;

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(3) clients download the data items of interest from the broadcast server on demand;and (4) there is no upstream communications channels, which means that the clients don’t have to send request to servers to get data. Based on these four assumptions, they presented two problems which need to be solved when designing data broadcast programs: the first one is that given a set of data items and the access probabilities for those data items by all the users, how to generate a broadcast program to satisfy as much as possible the needs of the users; the second one is that given the server with a fixed data broadcast schedule, how to manage the data items in cache for the individual users to maximize their own performances. Furthermore, three types of broadcast programs are demonstrated in that paper. They are flat data broadcast, skewed data broadcast, and regular data broadcast. Examples for the three types of data broadcast programs are shown in Fig. 2. The first program is a flat data broadcast. It is easy to see that the average expected access time for all the data items on a flat broadcast program, regardless of their access frequencies to the clients, is the same for all the items and it equals to half a broadcast period. If considering the access frequencies, the server can also overbalance some data items in a broadcast cycle. Such a program can enhance broadcasting of the popular items. For the second and third programs, both of them broadcast data1 twice as often as data2 and data3 . The second program is called a skewed data broadcast, in which the data item data1 is clustered together to be broadcasted, while, the third program is called regular data broadcast, where the same data items are separated by different data items with same intervals. Because the structure design of the third one is similar to the disk and memory design, they refer to regular data broadcast as multi-disk broadcasting. Based on the idea of broadcast disk, they developed an efficient algorithm for scheduling data broadcast given a set of uniform-length data items and their access pattern. The algorithm has the following steps: 1. Sort the data items in descending order of their popularity. 2. Classify all data items into small subsets, and each subset contains data items with same or similar access frequencies. They call each subset a broadcast disk. 3. Choose an appropriate integer as the broadcast frequency for each of the disks. 4. Calculate the least common multiple (LCM) of all the frequencies; then, split each diski into LCM small sub-disks. freqi 5. These sub-disks are called chunks and denoted by Cij . i; j refers to the j th chunk in diski . 6. Let Ni denote the number of chunks in diski . 7. Let M denote the number of disks. 8. Finally the broadcasting program is generated by Algorithm 1. The basic idea of the algorithm is to partition the data items into different kinds of disks, and the data items stored in smaller disks are broadcasted more frequently than the data items stored in larger disks. Thus, the data broadcast corresponds to the traditional notion of a memory hierarchy. They also pointed out that automatic determination of the parameters in the algorithm is an interesting optimization problem. Another topic they mentioned is the local cache management for clients.

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2397 One broadcast cycle

Flat Broadcast (1)

D1

D2

D3

Skewed Broadcast (2)

D1

D1

D2

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Regular Broadcast (3)

D1

D2

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D3

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Di

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Fig. 2 Three broadcast programs

Algorithm 1 [2] 1: h max Ni ; 2: for i D 0 to h  1 do 3: for j D 1 to M do 4: k i.mod Nj /; 5: Broadcast Cj;k . 6: end for 7: end for

Unlike traditional operating system, in the broadcast systems, a client should cache those data items for which the local probability of access (P) is greater than the frequency of broadcast (X). Based on this idea, they proposed two coast-based data replacement strategies. One is PIX (P Inverse X) which uses the ratio between the access frequency of a data item and its frequency of broadcast as the cost. It is an optimal policy but not easy to implement. The second one LIX (LRU-style) is an approximation for PIX. In terms of single-channel data broadcast, some researchers concentrate on uniform-length data broadcast; others focus on nonuniform-length data broadcast. Ammar and Wong studied periodic data broadcast with uniform length and proposed a greedy algorithm for it in [7]. Moreover they defined the average response time for nonperiodic schedules; they also analyzed the properties of the optimal schedules and proved that the optimal schedule for periodic data broadcasting is also optimal for arbitrary data broadcasting in [8]. In that paper they presented two algorithms for solving the arbitrary data broadcast scheduling problem by using the works in [36]. The first one is a finite-time algorithm which gives an optimal schedule, and the second one is a heuristic algorithm which gives a nonoptimal schedule. However, they did not provide any performance ratio analysis for the heuristic algorithm in that paper. The similar problem is also pursued by Anily et al. in [9], in which they obtained a 2:5-approximation algorithm for the problem. Later Barnoy et al. developed a 2-approximation algorithm and proved that the heuristic algorithm proposed in [8] has approximation ratio 9=8 in [11].

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In terms of nonuniform-length data broadcast, in [57, 58] Vaidya et al. investigated heuristic algorithms for it and showed some experimental results. Kenyon and Schabanel studied approximation algorithms for similar problems and gave theoretical analysis for the problem in both single-channel and multichannel environments, respectively, in [41]. The main results in [41] with respect to the case of single-channel and nonuniform-length data are that they proved the data scheduling problem with nonuniform length is NP-hard, and developed a randomized approximation algorithm for this problem. They first proved that there exists an optimal schedule for the problem which is periodic if broadcast costs are not considered; they also showed even in that case, the problem is NP-hard too. Then, they presented a randomized polynomial approximation algorithm for the problem. Its approximation performance ratio is 3. This is the first paper finding a constant-factor approximation algorithm for nonuniform-length data broadcasting scheduling.

2.2

Scheduling Optimization for Multichannel Data Broadcasting

Besides developing the scheduling mechanisms for single-channel data broadcast, multichannel data broadcast is also an important research area. Recently lots of works have been done for this problem. The multichannel broadcast can be achieved by applying techniques such as frequency-division multiplexing or timedivision multiplexing. Consequently multichannel broadcast becomes feasible and can significantly reduce the request response time. Ardizzoni et al. discussed the scheduling problem of data broadcast over multiple channels in [10]. Like other works, they also defined many constraints for the problem such as the data items are allowed to be skewed allocated to different channels but are flat scheduled per channel. The objective is to minimize the average access time for all the requests. The assumptions they defined in that paper for the multichannel data broadcasting environment are as follows: a database D D fd1 ; d2 ; : : : ; dN g, which has N data items, is broadcasted by a set of K channels. The bandwidth for each channel is the same. Each data item di has a probability pi and a length zi . The access frequency of item di being requested by clients is denoted by pi , and the time needed to download the data item di is represented by zi . They also assume that the access frequencies for the items never change P and the length zi for data item di is subject to an integer number. It is obvious that N i D1 pi D 1. When the lengths of all the data lengths are the same, it is called uniform-length multichannel data broadcasting, and the zi for 1  i  N is one time slot for the uniform-length data scheduling problem. Informally the target of this problem is to partition the N data items into K groups G1 ; : : : ; GK such that the average access latency is minimized. The group Gj denotes the set of data items assigned to the j th channel. So there is a integer Nj to represent the size of Gj ,

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P while the sum of its item lengths is denoted by Zj . It is clear that Zj D di 2Gj zi . Note that, since the data items in Gj are cyclically broadcasted according to a flat schedule, Zj is the schedule period on the j th channel. zi D 1 for 1  i  N for the problem with uniform-length data items, Zj D Nj for 1  j  K. Assume an item is assigned to the j th channel and clients start to listen at any time slot with the same probability, the client average waiting time for receiving the item equals to half of the period, which is Zj =2. Suppose with the help of indexes the clients know in advance the locations of data items on all the channels; then, the average expected delay (AED) for all the data items over all the channels can be computed by (1): 0 1 K X X 1 @Zj pi A : AED D (1) 2 j D1 di 2Gj

To study the uniform-length data scheduling problem for multichannel data broadcasting, Theorem 1 and some notations were proposed in [69]. Theorem 1 ([69]) Let d1 ; d2 ; : : : ; dN be N items with access probabilities pi  pN . Then, there exists an optimal solution for partitioning the data items into K groups G1 ; : : : ; GK , where K  N and each group consists of consecutive elements. Let d1 ; d2 ; : : : ; dN be N data items which are sorted in the descending order of the access probabilities. Partition them in K subsets. Then, an optimal partition OPTN;K D .B1 ; B2 ; : : : ; BK1 / is called leftmost optimal if for any other optimal 0 0 solution .B10 ; B20 ; : : : ; BK1 /, BK1 in OPTN;K is not bigger than BK1 in that solution, where Bi is the index of the last data item that belongs to group Gi and it is similar for Bi0 . In addition, an optimal solution .B1 ; B2 ; : : : ; BK1 / is called strict left-most optimal solution if .B1 ; B2 ; : : : ; Bi / for every i < K is a leftmost optimal solution. In the rest of this section, the leftmost optimal solution is denoted as LN;K , and the strict leftmost optimal solution is denoted as SLN;K . In the paper they proposed a dynamic programming (DP) algorithm for the multichannel data scheduling problem. The algorithm computes only the leftmost optimal solution for subproblems and then finds a strict leftmost optimal solution. The most important property of using the DP algorithm is that given the items which are sorted by nonincreasing probabilities, if an optimal solution for the first N  1 data items has been found, then to find an optimal solution for N data items, it only 0 has to check the area larger than BK1 to find the final border BK1 for SLN;K , 0 where BK1 is the border in the optimal solution for N  1 data items. Such a property can be applied to solve the problems of increasing sizes to reduce the time complexity. If Ll;k and Lr;k are obtained for some 1  l  r  N , then it is c l r clear that Bk1 is between Bk1 and Bk1 , for any l  c  r. If Ci;h denotes the

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cost of assigning P consecutive items di ; di C1 ; : : : ; dh to one of the channels, then Ci;h D hi2C1 hqDi pq . The recurrences of the DP algorithm can be represent as e in Eq. (2): Eqs. (2) and (3). Equation (3) is derived by choosing c D d lCr 2 optc;k D optd lCr e;k D 2

min

optl;k1 C ClC1;c :

min

optl;k1 C ClC1;d lCr e :

l2fBkl 1; :::; Bkr g l2fBkl 1; :::; Bkr g

(2)

2

(3)

Based on these two recurrences, it is easy to prove that the uniform-length multichannel data scheduling problem can be solved in O.KN log N / time by the DP algorithm, where K is the number of channels and N is the number of data items. Besides the DP algorithm, they also proposed a faster O.N log N / time algorithm for the uniform-length multi-channel data scheduling problem with the number of channels less than 4. If the data items are already sorted, it requires only O.N / time. The algorithm is based on the special case which has only two channels. When K D 2, adding a new item with lowest probability to an optimal solution for the subproblem can be done in constant time. This property can also be applied to find the optimal solution for the case K D 3. Actually, the solution for K D 3 can be obtained by combining the solutions for K D 2 and K D 1 in Eqs. (4) and (5): optN;2 D

min

l2fB1N l ;B1N 1 C1g

optN;3 D

min

l2f1; :::; N g

C1;l C ClC1;N :

opt1;2 C ClC1;N :

(4) (5)

For the case that data items have nonuniform lengths, they proved the problem is NP-hard even when K D 2. A pseudo-polynomial algorithm is presented with time complexity O.NZ/ where Z is the sum of the data lengths and N is the number of data items. They proved the problem is strong NP-hard for arbitrary K. An algorithm is proposed with time exponential to the maximum length of data items. Therefore, the time complexity of this algorithm is only acceptable for small data items. For larger data items, they proposed a heuristic algorithm which is a modified local search algorithm. Its time complexity is O.N.K C log N //, and if the items are already sorted, the time complexity becomes O.KN/. In the aspect of independent data scheduling, the queries issued by clients are assumed to only request one single item at one time. There are a lot of research that investigate how to minimize the average expected response time for multichannel data broadcasting. A brief summary is provided as follows. Yee et al. presented two algorithms to minimize the average expected access delay by partitioning data among multiple channels [69]. They continued their work in this filed to take into consideration the hopping cost in [70]. A near-optimal greedy algorithm for data scheduling of multichannel was proposed by Zheng et al. [73]. In addition, Prabhakara et al. suggested a new system model for multilevel and multichannel data broadcast with air-cache design to improve the server performance [48].

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The flat algorithm proposed in [56] adopts a simple allocation algorithm, equally allocating the items to each channel. However, the expected response time for the most popular item is the same as that for a data item not frequently asked by clients. Hsu et al. proposed the VFk algorithm, which skews the amount of data allocated to each channel [24]. The greedy algorithm in [69], which will be explained in detail later, can achieve a good performance when compared to the DP algorithm, and its time complexity is much better. Although all the algorithms have the same or similar objective function, they assign different weights to complexity and performance. The greedy algorithm is an efficient algorithm which is often applied to achieve an optimal or near-optimal performance in scheduling problems. In [69], the authors assumed that each data item has a unit length and each channel has the same bandwidth. They also assumed that Ni items were cyclically broadcasted via channel Ci , so the expected delay wj of receiving dj on Ci is N2i . Given these assumptions, the multichannel average expected delay (MCAED) can be derived by Eq. (6), where the access probability of data item di is denoted as pi , and K is the number of channels: K X X MCAED D wj pj : (6) i D1 dj 2Ci

The greedy algorithm is to allocate data items to K channels in a way that minimizes the MCAED. A simple allocation technique which is easy to implement, called flat design, allocates the same number of items to each channel. In the flat N design, Ni D N K , 1  i  K, and it is obvious that MCAED D 2K . In [48], the authors discussed skewed design, where items are placed on varying sized channels, depending on their access probabilities. Skewed design has been shown to be better than flat design at reducing the MCAED [2]. The principle of the greedy algorithm propped by in [69] is to find the split points among all the partitions that reduces the total cost of MCAED most significantly. This process is terminated when there are K partitions. In detail, the algorithm is shown in Algorithm 3 , where SCAED denotes the single-channel average expected delay for a channel containing data items from di to dj , and it is computed by Eq. (7): j i C1X pq ; .j  i; 1  i; j  N /: 2 qDi j

SCAED D Ci;j D

(7)

The greedy algorithm runs very quickly, in which k denotes the number of partitions. It has the time complexity of O..N C K/ log K/, which makes it fast and suitable for practical use. In [69], the authors also discussed the case that the broadcast environment is changed during broadcasting by changing the number of channels as well as adding new data items. They described how a broadcast server generated by the greedy algorithm can easily adjust to changes of the number of channels. They also investigated how to adjust the broadcast with new data items added. Actually adding a channel requires only one more iteration in the greedy algorithm, and removing a channel just requires combines the last two partitions

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Algorithm 2 Greedy [69] Input: a set of N items sorted by probabilities; Output: K partitions; k 1; while k < K do Let point k be the split point that most reduces SCAED for each partition Pk ; Find the point point j be the one in point1 ; : : : ; pointk that most reduces MCAED; Split the partition Pj at pointj ; k k C 1; end while

together. It is also possible that items may change their access frequencies since some new data items may be inserted into the broadcast. The new data item is added to the correct position Pin the broadcast, and the access probabilities are recalculated so that it still holds pi D 1. The good thing is that the scheduler only needs to update split points where necessary. In other words many old split results may still be reused. Although this technique has to recompute the entire solution in the worst case, its efficiency is much better on average.

2.3

Scheduling Optimization for Dependent Data Broadcasting

It is much more likely that a client query requests a set of data items at one time instead of only asking for one data item from the broadcast server. Therefore, besides the independent data broadcast, dependent data scheduling is more useful for developing broadcasting programs in practice. Some works focus on this area, and they usually have the following common assumptions: 1. Data is broadcasted over single or multiple channels. 2. A query requests more than one data item at a time. 3. The queries with multiple data items have different access frequencies. 4. Data items are not necessarily of same sizes. In [19], Chung et al. discussed the problem of scheduling dependent data to minimize the total access time of queries in wireless data broadcast systems. A query requests a subset of all data items with a query probability. They defined a measure called query distance (QD), which represents the coherence degree of the data set requested by a query, and then proposed the problem of wireless data placement. The authors also proved that the problem is NP-hard by transforming it from a optimal linear arrangement problem. A greedy scheduling algorithm was designed for this problem, in which the QD of queries with higher frequencies is guaranteed not to be changed when minimizing the QD for the queries with lower access probabilities. In this chapter, the data placement problem for wireless broadcasting is to find a broadcast schedule such that the total access time (TAT) is minimized. The TAT can be computed by Eq. (8), where Q is the set of all the queries q1 ; q2 ; : : : ; qN , S is

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a schedule of solution, and ATavg .qi ; S / is the average access time of the query qi based on S : X TAT.S / D ATavg .qi ; S /  freq.qi /: (8) qi 2Q

The access time of a query for dependent data broadcast not only depends on the starting time of the query but also depends on the data items in the query set. It may spend the whole broadcast cycle for a single query. Thus, in [19], for a given query qi , if P .x/ and F .x/ are the probability and the access time of the query qi issued at time x, respectively, then the estimated average access time of qi can be computed by the following equation, where k is the factor converting the sizes of data items into time lengths: B p.x/  F .x/dx D 0

B 1  F .x/dx B 0

n tj 1 X F .y/dy D B j D1 0 2 size.dj / n 1 X4  D F .y/dy C B j D1

2

0

size.dj / n 1 X4  D kB dy C B j D1

0

0

3

tj

F .y/dy 5

si ze.dj /

tj

3

k.B  y C size.dj //dy 5

size.dj /

1 n X 1 D k @B  k.tj  size.dj //2 A : 2B j D1

The measurement of average access time defined above is too difficult to apply when developing data scheduling methods for wireless broadcasting. Therefore, Chung et al. defined a new metric, called query distance (QD), which indicates the largest distance between any two consecutive data itemsP in a query. Let the total query distance denoted by TQD.S /, which is defined as qi 2Q QD.qi ; S /  freq.qi /. They showed that given a set of data items D and a set of queries Q, the problem of finding a broadcast schedule S such that TQD.S / is minimized is NP-complete. Thus, they proposed a data placement method based on a greedy algorithm for the dependent data scheduling problem. Definition 1 Suppose a query qi with data items d1 ; d2 ; : : : ; dn in a schedule S , and ıj is the interval between dj and dj C1 in the schedule S . Then, the QD of qi in S is defined as max.ık /, k D 1; 2; : : : ; n.

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The algorithm schedules the broadcast program by inserting the set of data items requested by each query based on a greedy strategy. That is, the query data set with higher access frequencies is guaranteed not to be changed when minimizing QD of the queries with lower access probabilities. For this purpose five basic operations are defined in [19] as follows: 1. “Move data to the Right of a Fragment (MRF).” 2. “Move data to the Left of a Fragment (MLF).” 3. “Move data to the Right in a Segment (MRS).” 4. “Move data to the Left in a Segment (MLS).” 5. “Move Data Closest (MDC).” Each of the five operations may be used for add new data items of a query into a given schedule. The operations MRF, MRS, MLF, and MLS move a set of data item to the most right end of the given schedule or most left end of the given schedule. They are used for minimizing the QD of the newly added data items of a query. The operation MDC is used for minimizing the QD when all data items for a given query are already included in the current schedule. The operation MDC moves the data items for a query as close as possible. The proposed algorithm constructs a broadcast schedule by adding data items of each query in a greedy manner after sorting the queries based on their frequencies. Thereafter, many approaches have been proposed for minimizing the average expected delay for dependent data broadcast, and some of them further investigated the dependent data scheduling problem over multiple broadcasting channels. To generate a dependent data broadcast schedule in a multichannel environment, the server should partition the data items on multiple groups based on the collection of the data items for each query and their access frequencies. In order to retrieve a set of data items, a client may switch from one channel to another to download the data items until all the items requested in the query list are downloaded. The requested data items in a query can be retrieved in an arbitrary order. Moreover, if more than one queried item appear in different channels simultaneously, only one of them can be downloaded at one time. The other items should be retrieved when they are coming again. In a dependent data broadcasting program, one data item may also be requested by multiple queries. A brief summary of relative works is provided as follows. Hurson et al., tried to minimize the overall access time, and proposed an allocation algorithm of dependent data based on some heuristics for multichannel broadcast systems in [32]. Lee et al. introduced an algorithm for queries with multiple data items in mobile environments in [44]. After that, Huang and Chen proposed a scheme based on a generic algorithm to handle the similar problem in [28]; and they explored the problem of broadcasting dependent data for ordered query over multiple broadcast channels in [27]. An ordered query means that the access sequence for the given query data set is preordered. Hung et al. studied the same problem and proposed a greedy algorithm called PBA, standing for placementbased allocation, in [31]. This algorithm will be explained in detail next. Huang et al. first proposed a genetic algorithms called GA in [28] for the multichannel dependent data scheduling problem. The GA algorithm has been proved by experimental results to be a good approach; however, there are still

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some weaknesses of the GA algorithm which may result in poor performance. For example, genetic algorithms will be affected by the initial setting. The initial setting is generated randomly, so it is not guaranteed to result in a good result constantly. To overcome such drawback, Hung et al. proposed a deterministic algorithm called PBA to schedule the dependent data items onto multiple channels. In PBA, the broadcast cycle length is assumed to be the same for all channels. Since each query contains multiple items in a dependent data broadcasting environment, the main policy of the algorithm PBA is to avoid the occurrence of the conflicting items in a broadcast cycle, which are the situations that multiple data items in a query are located in the same time slot on different channels. In order to explain the algorithm formally, two definitions in [31] need to be introduced. Definition 2 Given a query qi , t cyclei is the number of complete broadcast cycles needed for a mobile user to download the items in qi . The refined query qi0 denotes the set of remaining items after the cyclei cycles. Definition 3 Given a refined query qi0 , let TStartup .qi0 / be the time duration since the query qi0 starts until its first data item appears. Its retrieval time, denoted by TRetr .qi0 /, is the time duration since the first data item appears until all the data items in qi0 are downloaded. The access time of a refined query qi0 , TAccess .qi0 / is the sum of the start-up time and the retrieval time. Assume all data items are of the same size. Let s and B represent the size of data item and the bandwidth of each channel, respectively. The access time of a query qi , denoted by TAccess .qi /, equals to TAccess.qi0 / C cyclei  lengthc , where lengthc is the time length of a broadcast cycle. The average access time of the query set Q is denoted as TAccess.Q/. It is obvious that a query will wait for multiple broadcast cycles to download all the data items required if there are many conflicting data items. The conflicting data items are in the same time placement on different broadcasting channels, so clients cannot download them in one cycle. Hence, the problem of generating a broadcast program equals to the problem of allocating the items of each query into L groups such that the number of conflicting items is minimized and each group contains items no more than K data items, where K is the number of available channels. The algorithm PBA in detail is outlined in Algorithm 3 . To allocate the data items, the algorithm PBA first sorts the queries in descending order Q according to their access probabilities. A query with higher access probability has higher priority to be scheduled. After that, a broadcast program P , which contains L sets of data items, is constructed. Each data item set Di contains the data items allocated in the i th time slot, but not the i th channel. Obviously if a query qj is processed, the data items requested by qj cannot be processed again. To prevent one data item be allocated more than once, they partition the data items in qi into two disjoint subsets, qiC and qi , which store the already allocated and remaining items, respectively. The algorithm PBA terminates when all the items in D are allocated in P . A query with high access probability is processed earlier than the lower one. Therefore, it is a greedy approximation algorithm.

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Algorithm 3 PBA [31] Input: the query profile Q, the broadcast database D, and the number of channels K; Output: the broadcast program P ; 1: Sort elements in Q according to the access probability in descending order; size.D/ 2: Let L d L e; 3: for each qi 2 Q do 4: Unmark all item sets; 5: Find qiC and qi which store the scheduled and unscheduled items in qi , respectively; 6: Mark the item sets which contain K items and the item sets which contain the items in q C ; 7: if all item sets are marked then 8: curr argi min size.Di / for all Di ; size.Di / < K; 9: else 10: curr argi min size.Di / for all unmarked Di ; 11: end if 12: for each item dj 2 qi do 13: Insert dj into Dcurr ; 14: mark Dcurr ; 15: if all item sets are marked then 16: Select next from Di , size.Di / < K such that DIST.curr; next/ is minimized; 17: else 18: Select next from the unmarked item sets such that DIST.curr; next/ is minimized 19: end if 20: Set curr D next; 21: end for 22: end for

3

Optimizations for On-Demand Data Broadcasting

In pure push-based data broadcast systems, data access patterns are usually assumed to be static in a certain period. Data are organized into broadcast programs according to their access patterns. The programs will then be broadcasted in cycles and updated periodically. In such systems, individual clients are passive listeners, which means they have no direct influence over the content of a broadcast program. This network structure brings up several issues. Since there is no feedback of clients in the form of requests, it is very difficult for pure push-based systems to get accurate data access patterns. Moreover, in circumstances where the client request pattern changes frequently, the static access pattern assumption will not be true anymore. This may result in longer system response time. In the worst case, the program may not be able to satisfy client requests anymore. “Pull-based” data broadcast is a different data broadcast method which does not have the aforementioned drawbacks of push-based data broadcast method. It is also referred to as “on-demand” data broadcast [13, 63]. A typical on-demand data broadcast system structure is shown in Fig. 3. As in Fig. 3, the network connection in an on-demand data broadcast system consists of two types of links: uplink and downlink. Clients send requests through uplink to the server. These requests are queued up at the server upon arrival.

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down link

Optimization Problems in Data Broadcasting

Base Station

...

server

clients

uplink

Fig. 3 A typical on-demand data broadcast system structure

The broadcast channel(s) serves as the downlink to broadcast data to clients. The server continuously puts data items onto the broadcast channel(s) according to the queued requests and removes corresponding requests from the queue. Clients will listen to the channel(s) and download their requested data when they arrive. This data dissemination process shows the influence of clients’ requests over the choice of broadcasted data. On-demand data broadcast can dynamically adjust broadcast programs according to the client requests and thus is suitable for situations when data access patterns change frequently. A good online scheduling algorithm to choose data items for broadcast is critical for the good performance of an on-demand data broadcast system. Similar as pushbased broadcasting, the main objective of scheduling on-demand data broadcast is also to reduce the response time. Different from push-based broadcasting, the response time for a request in an on-demand broadcast system is the time interval between the time the client sends the request through uplink and the time the client downloads the data from broadcast channels. There have been a lot of literature on scheduling on-demand data broadcast . This section will focus on different on-demand data scheduling algorithms for different communication environments. Section 3.1 introduces several basic scheduling algorithms. Section 3.2 presents the R  W algorithm which can handle largescale databases. Section 3.3 gives a scheduling algorithm for time-critical requests with deadlines. Section 3.4 discusses how to schedule heterogeneous data with preempting technique. All on-demand scheduling algorithms discussed in Sect. 3 are based on single broadcast channel.

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Basic Online Scheduling Algorithms for On-Demand Data Broadcast

Earliest studies on online scheduling for on-demand broadcast can be traced back to the late 1980s. In [21, 62], the following algorithms were proposed: • First come, first served (FCFS): always broadcast the data item of the request that arrives at the earliest time. Ignore a new request if the same request already exists in the queue. • Most requests first (MRF): always broadcast the data item of the maximum number of requests in the queue. • Longest wait first (LWF): always broadcast the data item that the sum of waiting times of all its requests in the queue is the longest. Intuitively, FCFS emphasizes fairness and gives preference to “cold” items, while MRF considers data popularity and prefers “hot” items. LWF tries to balance the waiting time of all requests and also prevents possible “starving” requests. Simulation results in the original papers show that FCFS results in poor average response time when the data access distribution is nonuniform, and LWF performs better than FCFS and MRF in general. A more theoretical analysis of these algorithms were presented in [47], which gives performance ratios of FCFS and MRF. Assume there are m data items in the database for broadcasting and n requests. Each request contains only one data item. All items are of same Plength. An instance is a data item on the broadcast channel. For any instance, let niD1 wi ./ represent the total response time of the solution given by  scheduling algorithm. OPT represents the optimal solution. [47] proved the following theorems: Theorem 2

Pn

i D1 wi .FCFS/

m

Pn

i D1 wi .OPT/.

Theorem 3 There exists at least one instance that satisfies P m niD1 wi .OPT/.

Theorem 4

Pn

i D1 wi .MRF/

m

Pn

i D1

i D1

wi .FCFS/ D

wi .OPT/.

Theorem 5 There exists at least one instance that satisfies P m niD1 wi .OPT/.

3.2

Pn

Pn

i D1

wi .MRF/ D

R  W Scheduling for Large-Scale On-Demand Data Broadcast

When the database size is getting large, simply considering average or total response time as the performance criteria for an on-demand data broadcast system is obviously not enough. In [4, 5], Aksoy and Franklin categorized the performance criteria for on-demand scheduling algorithms as follows:

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1. Responsiveness (a) Average waiting time: the average amount of time from the moment that a client request arrives at the server to the moment the requested item is broadcast. (b) Worst case waiting time: the maximum amount of time that any client request will have to wait in the service queue to be satisfied. 2. Scheduling overhead (a) Request processing: the overhead of the server deciding whether to add an entry to the service queue for the requested item and/or updating the queue (b) Scheduling decisions: the overhead to choose an item to broadcast at the beginning of each broadcast time slot. 3. Robustness: whether the scheduling algorithm can adjust to the changes of the workload or environment. Based on the observations of FCFS, MRF, and LWF’s performances with respect to the above criteria, Aksoy and Franklin developed a scheduling algorithm named R  W in [4, 5]. The intuition is to combine the benefits of FCFS and MRF in a way that ensures scalability while preserving low overhead. They first proposed an exhaustive algorithm and then improved it with a pruning technique to reduce the search space. To further reduce the scheduling overhead for large-scale databases, they developed an approximate, parameterized version of R  W algorithm at the expense of possible suboptimal broadcast decisions.

3.2.1 The Exhaustive R  W Algorithm The idea of R  W algorithm is to broadcast an item either because it is popular or because it has at least one request that has been waiting for a long time. Instead of a straightforward service queue, it maintains a service table that has an entry for each data item with outstanding requests. The structure of the service table is shown in Fig. 4. Each entry in the service table consists of the following elements: • PID: identifier of the data item in the database • 1stArv: the arrival time of the earliest outstanding request for this item • Prev in W-List: a pointer to the previous entry in the W-List • Next in W-List: a pointer to the next entry in the W-List • R: the total number of outstanding requests for this item • Prev in R-List: a pointer to the previous entry in the R-List • Next in R-List: a pointer to the next entry in the R-List The service table is hashed on PID. W represents the longest waiting time of all requests for a particular data item. W can be computed as W D clock  1stArv, where clock is the current time. All time used in the rest of Sect. 3 is represented in broadcast ticks, that is, the number of items broadcasted so far. The algorithm uses two sorted lists threaded through the service table: W-List in descending order of W and R-List in descending order of R. An R-Index is introduced to reduce the maintenance overhead of the R-List. For each distinct R value, there is a pointer to the first entry in the R-List of the same R value. The algorithm can be divided into two steps: request process and broadcast decision. The procedures of these two steps are described in Algorithms 4 and 5.

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first

Next in W-List

Prev in R-List

Prev in W-List

Next in R-List

PID

1stArv

R

b

10

1

n

20

1

y

50

2

a

90

40

k

95

25

f

98

80

z

99

20

R value

80 …...

1 R-Index first

Service Table

Fig. 4 The service table structure of R  W algorithm

Algorithm 4 Request process of exhaustive R  W Input: a request of item with identifier P ID; Output: updated service table; 1: perform a hash lookup on P ID in the service table. 2: if an entry exists then 3: increment the R value of the entry. 4: else 5: create a new entry with R D 1 and 1stArv D current time. 6: end if

Algorithm 5 Broadcast decision of exhaustive R  W Input: service table; Output: updated service table; 1: find the entry with the maximum R  W value in the service table. e 2: broadcast the corresponding data item. 3: remove this entry from the service table.

head

tail of W-List

head

tail of R-List

Optimization Problems in Data Broadcasting

2411

A theoretical analysis of the R  W algorithm in [5] concluded that for an arbitrary data item b, the expected waiting time is W0 C Wb D

PN 1

q

P C ib1 D0 i Wi : q i Pb1 h 1  i D0 i 1  ppbi i DbC1 i

pi pb

(9)

In Eq. (9), W0 is the mandatory wait for the completion of an already initiated page broadcast; i is the utilization of the bandwidth to broadcast item i ; and pi is the access probability of item i .

3.2.2 R  W Algorithm with Pruning The scheduling overhead of the exhaustive R  W algorithm is O.N /, where N is the total number of data items to broadcast. As N increases, the search space of Algorithm 5 is getting larger, which consequently increases the overhead. Aksoy and Franklin proposed a pruning technique for the R  W algorithm in [5]. The pruning technique uses the “sorted” feature of the R-List and W-List to reduce the search space when looking for the maximum R  W value. Suppose the examination starts at the top entry of the R-List. MAX is used to record the current maximum R  W value. Since the R-List is in descending order, the next entry in the list R0 is smaller than R. Naturally, for any unexamined entries with R  W value greater than MAX, their W value must satisfy W >

MAX : R0

Therefore, the search range of the W-List can be truncated by limit.1stArv/ D clock 

MAX : R0

That is, no entry with 1stArv greater than limit.1stArv/ in the W-List will have R  W larger than MAX. Similarly, if the examination starts at the top entry of the W-List, the search range of the R-List can also be restricted by limit.R/ D

MAX ; W0

where W 0 is the next entry in the W-List. Based on this observation, the scheduling algorithm starts examining the top entry of the R-List and keeps on alternating between the two lists. It can be described in Algorithm 6 . This pruning technique is effective in reducing the search space. Experiment results in [5] showed that there is a 73 % reduction in entries examined to find the maximum R  W for their Zipf-based workload.

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Algorithm 6 Broadcast decision of R  W with pruning Input: service table, R-List and W-List; Output: updated service table; 1: examine the top entry of the R-List; set MAX and limit.1stArv/. 2: while neither limit.1stArv/ nor limit.R/ is reached do 3: examine the next entry of the previously examined entry in the other list. 4: Set MAX D R  W if R  W > MAX. 5: update limit.1stArv/ and limit.R/. 6: end while 7: broadcast the corresponding data item. 8: remove this entry from the service table.

3.2.3 The Approximate R  W Algorithm Although the pruning technique discussed in Sect. 3.2.2 can help reducing the search space, the reduction may not be enough to confine the scheduling overhead within a reasonable range when the database scale is really large. Therefore, an approximating, parameterized variation of the R  W algorithm was proposed in [5] to further reduce the search space at the expense of possible suboptimal scheduling decisions. In the approximating algorithm, instead of finding the maximum R  W value, it chooses the first item encountered with RW  ˛threshold. ˛ is the parameter of the algorithm, which is greater than 0. threshold is the self-adaptive average, whose value can be computed as threshold.t C 1/ D

threshold.t/ C last R  W .t/ 2

threshold(0) is set to 1. last R  W .t/ is the R  W value of the item broadcasted at time t. ˛ is parameter of the algorithm that can be adjusted. It determines the trade-off between responsiveness and scheduling overhead. The smaller ˛ is the less entries need to be examined, but the more likely the scheduling decision is not optimal. In such cases, the scheduling overhead is reduced but the responsiveness may be jeopardized.

3.3

Scheduling Time-Critical On-Demand Data Broadcast

The work [4,5] by Aksoy and Franklin and many other literature on on-demand data broadcasting use the responsiveness performance criteria as discussed in Sect. 3.2, which aim at minimizing the average or worst waiting time. This implies an assumption that once a client generates and sends a request, it will wait until the request is satisfied. However, this assumption is not always true. In some circumstances, clients may lose patience and drop the request if they have waited

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for too long. In order to take the time constraint into consideration for time-critical requests, the concept of deadline is introduced [39, 40, 66]. In a time-critical on-demand data broadcast system, each request has an additional parameter deadline. An outstanding request will be dropped after its deadline is reached. Motivated by the EDF (earliest deadline first) scheduling algorithm in unicast and the MRF algorithm in broadcast, Xu et al. proposed a scheduling algorithm for time-critical on-demand data broadcast in [66]. The objective is to satisfy as many requests as possible. A new performance criterion is defined to measure this objective: request drop rate D

number of requests missing their deadlines : total number of requests

The idea of the algorithm can be summarized into two rules: 1. If two data items have the same number of outstanding requests, broadcast the one with a closer deadline. 2. If two data items have the same deadline, broadcast the one with more outstanding requests. The first rule gives priority to urgent requests, while the second rule gives preference to requests that are more productive. Both rules are for the sake of reducing the request drop rates. The algorithm proposed by Xu et al. is named as SIN-˛ (slack time inverse number of outstanding requests). For each data item with at least one outstanding request, there is a sin:˛ value attached to it: sin:˛ D

slack 1stDeadline  clock D ; ˛ num num˛

where slack represents the remaining time before the earliest deadline of all requests for this item, and num is the total number of requests for this item. ˛  0 is a parameter determining the relative importance of productivity over urgency in the broadcast decision. The larger ˛ is the more influential the number of outstanding requests in computing sin:˛. An exhaustive version of SIN-˛ is to compute the sin:˛ value of all items in the service queue at each broadcast tick and broadcast the item with minimum sin:˛ value. The complexity of this implementation is O.N /, where N is the total number of data items with outstanding requests. Inspired by the pruning technique discussed in Sect. 3.2.2, Xu et al. applied a similar pruning method during the searching for the minimum sin:˛. Two doublelinked sorted lists are maintained: S-List and N-List. Each data item has one entry in both lists. S-List maintains each item’s associated earliest request deadline in ascending order, while N-List contains the number of outstanding requests for each item in descending order. Similar as Algorithm 6 , the scheduling algorithm starts examining the top entry of the N-List and keeps on alternating between the two lists.

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Two limits are also maintained to reduce the search space for minimum sin:˛: limit.N / for the N-List and limit.1stDeadline/ for the S-List. During the searching, let MIN denote the current minimum sin:˛ value. If an entry in the S-List has just been checked, then any unexamined item with sin:˛ < MIN should have its num value exceeding  limit.N / D

Next S MIN

 ˛1 ;

where Next S is the slack of the next entry in the S-List. On the other hand, if an entry in the N-List has just been examined, then the 1stDeadline of any unexamined item should be less than limit.1stDeadline/ D .NextN/˛  MIN C clock; where NextN is the num of the next entry in the N-List. In [66], Xu et al. also gave a thorough analysis of the theoretical bound of request drop rate when the request arrival rate is approaching infinity. The request drop rate for item i arriving in interval Œ0;  is ./ D

 1 F .  t/dt; 

(10)

0

where F .t/ is the probability that a relative deadline is shorter than t. Relative deadline refers to the time interval between current time and the deadline. Xu et al. proved that the request drop rate of an item is minimized when instances of the item are broadcasted with equal intervals. Therefore, assume the spacing of any item i is uniform, denoted as si . The total request drop rate can be computed as .s1 ; : : : ; sN / D

N X i i D1



.si / D

 N X pi 1  F .  t/dt; s  i D1 i

(11)

0

where i is the request rate of item i and  is the total request rate. Their ratio pi is exactly the access probability of item i . If a broadcast tick is l, it means sli of the bandwidth is used to broadcast item i , which means N X l D 1: (12) s i D1 i Xu et al. proposed the following theorems about the lower bound of request drop rate as computed in Eq. (11) under three different relative deadline distributions. Theorem 6 Under exponential distribution of relative deadlines with a mean value of M , the total request drop rate is minimized when .s1 ; : : : ; sN / satisfies

Optimization Problems in Data Broadcasting

2415

  si si  si e M  e M D K; .i D 1; : : : ; N /: pi 1  M Theorem 7 Under uniform distribution of relative deadlines between 0 and L, the minimum total request drop rate is given by 8
1 Ll g : 2Lf l 

I m g L

  N X L 1f CpI C1 1   C pi 2 l i DI C2 (

where m D max

Px

)

!2 pi

C

i D1

I X pi 2 i DmC1

; )

p

pj j D1 xj f p . l  I x L / px

L :

Theorem 8 Under fixed relative deadlines of C , the total request drop rate is minimized when .s1 ; : : : ; sN / satisfies 8 ˆ byc C 1;

. The minimum request drop rate is .s1 ; : : : ; sN / D pbycC1 .1  y C byc/ C

N X

pi :

i DbycC2

3.4

Scheduling Preempting On-Demand Broadcast for Heterogeneous Data

All scheduling algorithms discussed in Sect. 3 so far have one common assumption: the size of all data items are equal. In many real applications of on-demand broadcast, data of requests are heterogeneous, which means that data items can be of different sizes. In [3], Acharya and Muthukrishnan first brought up the problem of scheduling on-demand broadcast for heterogeneous data and proposed the usage of preemption during scheduling. Preemption is usually used on heterogeneous workload in order to prevent backlog of pending requests when servicing a long job. In a broadcast environment, preemption means the server can abort the current broadcasting item and broadcast items for more valuable requests. Two models allowing preemption have been

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proposed: the restart model in [40] and the resume model in [39]. However, the treatments of the aborted broadcast in these two models are different: • Restart: the server can restart a preempted broadcast from the beginning • Resume: the server can resume a preempted broadcast from the point of preemption Ting [55] designed a scheduling algorithm named balance for on-demand broadcast utilizing preemption, under the heterogeneous data assumption. The design is based on the restart model. Given a set of data items, assume the length of the shortest item is 1 and the length of the longest item is 4. For any item P , l.P / is the time needed to completely broadcast P . A broadcast, schedule, and request are defined respectively as following: Definition 4 A broadcast B D .P; t/ is to broadcast item P at time t. Definition 5 A schedule is a sequence of broadcasts S D h.P1 ; t1 /; : : : ; .Pm ; tm /i, where t1 <    < tm . For any broadcast .Pi ; ti /, it is a complete broadcast if ti C l.Pi / < ti C1 ; otherwise, it is preempted by .Pi C1 ; ti C1 /, and .Pi C1 ; ti C1 / is a preempting broadcast. Definition 6 A request r is a four-tuple .P; a; d; v/, where P is the requested item, a is its arrival time, d is its deadline, and v is its value. Given a sequence of requests , the scheduling problem for time-critic ondemand data broadcast is to find a schedule S with maximum profit. Let RS; .P; t/ denote the set of outstanding requests of P at time t. The profit .S; / of a schedule S can be computed as .S; / D

X

vS; .P; t/;

(13)

.P;t /2S;.P;t / is complete

where vS; .P; t/ is the value of P at time t with respect to .S; t/, which equals to P r2RS; .P;t / . The motivation of balance algorithm is to reach a good balance between goodprofit broadcasts and short-length broadcasts. Good-profit broadcasts are those whose completion will increase the profit of the schedule, while short-length broadcasts are those whose completion will reduce the total time needed for the schedule. All complete broadcasts are considered to be good-profit broadcasts. Except for the preemption, balance has a “greedy”-like strategy. The scheduling process can be illustrated in Algorithm 7 . The preemption in balance follows a relatively complicated set of rules in order to achieve better performance ratio. The procedure is described in Algorithm 8 . In his paper [55], Ting also derived an upper bound and lower bound on the performance ratio of balance. Suppose S is the schedule generated by balance:

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Algorithm 7 Balance 1: while there are still unsatisfied requests do 2: find the item of highest total value and broadcast it; 3: check for possible preempting; 4: remove the requests of the completely broadcasted item from the service queue. 5: end while

Algorithm 8 Preemption 34 Input: S D hB1 ; : : : ; Bgp ; : : : ; Blast i, t 2 .tgp ; tgp C l.Pgp //,  D log . 4 p 1: if 9P; s.t. vS; .P; t / > 4vS; .Pgp ; tgp / then 2: preempt Blast and broadcast P . 3: else  !

4:

 D fQjt C l.Q/ < tlast C l.Plast / ^ vS; .Q; t / >

2

tc tgp 

41=3

vS; .Pgp ; tgp /g.

5: if  ¤ then 6: Q0 D argminQ2 fl.Q/g. 7: preempt Blast and broadcast Q0 . 8: end if 9: end if

Theorem 9 .OPT; / 



Theorem 10 .OPT; / 

64 log 4



C O.45=6 / .S; /.

4 2 ln 4

 1 .S; /.

Theorem 9 shows that balance breaks the 4-barrier for all previous schedulers without preemption. This demonstrates the effectiveness of introducing preemption into scheduling heterogeneous data. The combination of Theorems 9 and 10 also proves that balance is optimal within a constant factor.

4

Data Retrieval Optimization in Data Broadcast Systems

Scheduling data onto broadcast channels is a key optimization research issue on the server side in data broadcast. Similarly, scheduling the data retrieval from broadcast channels for a mobile client is also an important optimization problem. When there is only one broadcast channel, or the client request contains single data item, the data retrieval scheduling problem is usually trivial. With the fast development of mobile communication technology, multichannel data broadcast is gaining more popularity because the utilization of multiple channels can dramatically enlarge the bandwidth for the broadcast downlink. In a multichannel broadcast environment, how to schedule the downloading of multiple data items is the focus of this section. In an indexed multichannel data broadcast system, the data retrieval for a client is usually composed of three steps:

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1. Initial probing: the client tunes in a broadcast channel 2. Searching: the client follows the guide of embedded index information on broadcast channels and gets the locations of requested data. 3. Downloading: the client tunes in the channel and downloads all requested data. All literature on data retrieval scheduling so far consider only the downloading phase, that is, assume locations and arrival time of all requested data are known. Same as data broadcast scheduling, response time is also a primary concern when scheduling the data retrieval process for a client. Energy conservation is another performance criterion for data retrieval scheduling because of the limit battery lives of mobile clients. Therefore, the data retrieval scheduling problem can be summarized as follows: Definition 7 (Data retrieval scheduling sroblem) suppose there is a request for multiple data items. Given the channel locations and arrival time of all requested data items, how to scheduling the downloading process so that • The overall response time and/or • The energy consumption of the client can be minimized. The reason why a good retrieval schedule can reduce the response time is straightforward, while why it can reduce energy consumption is not so obvious. Hurson et al. [33] pointed out that switching among broadcast channels during retrieving data also consumes energy: in general, on switching costs about 10 % of one unit time active mode power consumption of a mobile client. Therefore, the energy conservation objective can be converted to minimizing the total number of switchings during the data downloading. One important issue that needs to be considered during data retrieval scheduling is conflicts. When a client just finished downloading one item at time t on channel A, it is impossible to start downloading another item at time t on channel B ¤ A. This is because switching from one channel to another also takes time. Although it is usually faster than one broadcast tick, it is not neglectable. Hence, when a client just downloaded one item at time t on channel A, it can only start downloading at time t C 1 from any channel other than A. An illustration of conflicts can be found in Fig. 5. Hurson et al. [33] first proposed the problem of downloading multiple data items from multiple broadcast channels with single retrieving process. They also explained the concepts of conflict during retrieval. In their work, they treated the two objectives, minimizing access time and minimizing number of switchings, separately, and proposed a heuristic algorithm for each of them. With the development of antenna technique, adding MIMO antenna in mobile devices is already feasible. As a consequence, it is possible for clients to use multiple processes to retrieve data in parallel. Shi et al. discussed the retrieval scheduling problem for such a communication environment in [50], which will be introduced in the rest of this section.

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Channel A

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t+1

di

dj

dk

: Shaded bucket means the item is downloaded

Fig. 5 An illustration of conflicts

Instead of treating the two objectives separately, Shi et al. defined a new cost function that combines the two objectives together: Definition 8 The cost of a parallel data retrieving schedule is evaluated as c D ˛  AT C ˇ  S , where AT is the access time of the data retrieving process from the starting time (initial probing and searching processes are not included); S is the number of switches among channels during the retrieving process. ˛ and ˇ are adjustable parameters which can reflect the user’s preference. If the priority for the user is to get the requested data faster, ˛ should be increased; if energy conservation is more important, ˇ should be increased. The retrieval scheduling only considers the downloading phase, so only channels with requested data on them are of interest, defined as requested channels. Suppose the number of requested channels are K, and the number of available retrieving processes for a client is M . When K  M , it is easy to prove that the optimal schedule is to assign one process to download all requested data on each requested channel. When K > M , the problem becomes nontrivial. Shi et al. designed two greedy heuristic scheduling algorithm in the case of K > M . The first algorithm is least switch. The idea is to guarantee minimum number of switching during downloading. Algorithm 9 describes the scheduling process. Note that in least switch, every time a process finishes downloading one channel, the scheduler will assign a most costly channel to it to continue downloading. This is because the overall access time is determined by the process which takes longest time to finish its task, so balancing the workload among processes is important in order to reduce the overall access time. The other algorithm is called best first. It is a greedy algorithm in that the scheduler always assignes to a process the item which costs least for a process to download. Algorithm 10 demonstrates the scheduling process. Best first can be divided into three phases: initial assignment, best-first assignment, and channel assignment. As the request list is shrinking during the downloading process, the number of requested channels is also getting smaller. When K D M , it is optimal to assign one channel to one process, so the best-first assignment will stop.

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Algorithm 9 Least switch 1: Assign M requested channels to M retrieving processes; K D K  M ; 2: while K > 0 do 3: whenever a process finishes downloading, assign to it the most costly channel to download; 4: K  ; 5: end while

Algorithm 10 Best-first 1: 2: 3: 4: 5: 6: 7:

assign M requested items with least cost to download to the M retrieving processes; remove these item from the request list and update K; while K > M do whenever a process finishes downloading, assign to it an item which costs least to download; remove this item from the request list and update K; end while assign remaining M requested channels to the M processes.

5

Indexing Techniques for Data Broadcast Optimization

Utilizing indices into the broadcasting program is another way of optimizing data broadcast problem by means of reducing the tuning time so as to improve the energy efficiency of a wireless data [16] broadcast system. An index is a specific data structure with pointers storing the location information that directs to the data items. Due to the nature of data broadcasting, such location information of an index usually indicates the time offset of the target data. Once a client gets this time offset, it could be aware of the total waiting time before the target data item is broadcasted on the broadcasting channel. Then, the client could turn into doze mode (or sleep) to save some energy and then tune back to the broadcasting channel before the data item appears. Indexing techniques could dramatically reduce the average tuning time for clients within a wireless data broadcasting system. However, if indices are inserted between data items, then the total length of a broadcasting program will increase, which may cause a longer access time. Therefore, when discussing about an indexing scheme, researchers will always consider the balance or trade-off between tuning time and access latency [37]. Since different indexing schemes have different searching efficiencies, how to optimize the tuning time while maintaining acceptable access latency is a critical issue in wireless data broadcasting systems.

5.1

Indexing in Single-Channel Broadcasting Environment

There are a number of works trying to apply traditional disk-based indexing approaches onto air indexing by converting location information or physical address into time offset. A lot of existing air indexing schemes, such as distributed index,

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hash-based scheme, and exponential index, were designed for flat broadcast on single-channel broadcasting environment, in which all data items are broadcasted under the same frequency [34, 35, 65, 67]. Imielinski et al. [34] were the first to discuss air indexing schemes, and they proposed flexible index. Furthermore, they customized BC -tree index in [35] and proposed .1; m/ index as well as distributed index. Distributed index was extended by many other researchers to fit different system requirements. For example, Hsu et al. [24] modified distributed index to deal with data under nonuniform access frequencies. Moreover, Lee et al. [45] discussed a signature-based approach for information filtering in wireless data broadcasting. Later, Hu et al. [25] designed a hybrid indexing scheme combining both distributed index and signature-based index. Xu et al. [67] gave an idea of exponential index that shares links in different search trees and allows clients to start searching at any index node, which is error resilient due to its distributed structure. More indexing schemes are further developed to deal with skewed access patterns on single-channel broadcasting environment. For instance, Huffman tree is a skewed index tree which takes into consideration the access probability of each data item, where the more popular data is stored at a higher level of the tree which has a shorter path from the root node. Thus, it minimizes the average tuning time in the search process. However, almost all existing works discussed Huffman tree on multichannel environment [17, 52]. It is hard to compare the performance of Huffman-tree index with other single-channel indexing methods, because when it comes to multichannel data broadcast, how to allocate index and data will produce heavy impact on the performance of the indexing techniques. A certain allocation method could be helpful to specific indexing structure, but at the same time it might reduce the efficiency of another indexing scheme. Recently, Zhong et al. [74] proposed an alphabetic Huffman-tree distributed index in the single-channel data broadcast environment, which minimizes both average access time and average tuning time. The experiments show that it outperforms the BC -tree distributed indexing approach. Besides Huffman-tree index, Imielinski et al. [34] presented hash-based scheme. Another work by Yao et al. [68] proposed MHash scheme to facilitate skewed data access probabilities and reduce access latency.

5.1.1 Flexible Index As Imielinski et al. [34] proposed, flexible index simply divides the data file into p segments and provides indices to help the clients “navigate,” so as to reduce the tuning time. The parameter p makes this scheme flexible since it can be adjusted to allow users choosing between either good tuning time or good access time. In flexible index, data records are sorted in ascending order. All the data buckets are divided into p groups called data segments, numbered from 1 to p, the first bucket of which stores the control index and possibly the data records if space allows. Control index is a binary index used for locating the data bucket which contains the given key. Figure 6 shows an example of flexible indexing with 66 data buckets. Let p = 9, with the length of eight buckets for all data segments except the last one. The first bucket of data segment contains the index. Three examples of such buckets are

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1 2 3 4 5 6 7 8 9 DS1 Data bucket # 1 non , 67 33 , 33 binary control 17 , 17 index control 09 , 09 index 07 , 07 local 05 , 05 index 03 , 03

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 DS2

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DS4 Data bucket # 25 25 , 43 binary 49 , 25 control index control 33 , 09 index 31 , 07 local 29 , 05 index 27 , 03

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Data bucket # 49 49 , 19 binary 57 , 09 control index control 55 , 07 index 53 , 05 local index 51 , 03

Data records

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Fig. 6 An example of flexible index

shown in Fig. 6. Each index entry is a pair of the key value and the offset (offset means the relative distance from this current bucket to that pointed bucket), which tells the clients when to tune in again to find the requested data. For instance, as shown in Fig. 6, the index of the data bucket #25 indicates that for all key values smaller than 25, the client should tune to the bucket #1 of the next bcast that is 43 buckets away (in this case the client missed the key and has to wait for the next broadcast). If the key is greater than 49, the client has to tune in again to bucket #49 (which the offset # = 25 points to). That data bucket will provide another index to help the client continue searching. Not all data buckets contain the index, and that the index information is distributed among different data buckets. The control index is divided into two parts: the binary control index and the local index. The binary control index has dlog2 i e tuples, where i is the number of data segments in front of (including) this one. The kth tuple contains the key of the first data item of the blog2 i=2kl c C 1th data segment and an offset to the first data bucket of that data segment. The local index has m tuples. Parameter m depends on how many tuples a bucket can hold, the access time desired, etc. The local index further partitions a data segment into m C 1 data subsegments D1 ; D2 ; : : : ; DmC1 , where the kth tuple contains the key of the first data item of DmC1k and an offset to the first data bucket of DmC1k . Following is the access protocol of flexible indexing scheme [34] for a record with key K. 1. Initial probe, get the offset to the next data segment, and go to doze mode. 2. Tune in again to the designated data segment. (1) If the search key K is less than the first tuple’s first field in the binary control index, then go to doze mode for the offset time (as the second field indicates) and repeat step 2; else (2) search the rest entries of the binary control index to check if K is greater than or equal to each tuple’s first field. If so, follow the pointer of the first such tuple and repeat step 2; else (3) search the local index and check if K is greater than or equal to each tuple’s first field. If so, follow the pointer of the first one of such tuples, tune in at the designated bucket and go to step 3.

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3. Search the following L data segme nt=.m C1/ buckets and sequentially locate K, where L data segme nt indicates the length of a data segment. In general, as shown in [34], the average tuning time using the flexible indexing technique is .†dlog2 i e C Data=.2  .m C 1///=p, 81  i  p, and Data denotes the size of the entire broadcast. The average access time using flexible indexing is 0:5  .F i le C p  .dlog2 pe C m/=n/  .1 C p/=p. Later on, Yu et al. [71] extended the flexible indexing method with another parameter, the index base r, and proposed a parameterized flexible indexing scheme (PFInd), which can be used to tune the trade-off between access efficiency and energy conservation based on the specific requirements of clients.

5.1.2 (1, m) Index and Distributed Index Imielinski et al. [35] also developed a method for efficient multiplexing of data file with clustering index, called distributed index, which becomes one popular air indexing approach and was extended by many other researchers to meet different system requirements. Before introducing distributed index, this section starts with a simple index allocation method called (1, m) index. (1, m) Index. (1, m) indexing scheme is an index allocation approach that broadcasts each index m times during one broadcasting cycle, in front of every fraction (1=m) of the file, as illustrated in Fig. 7. The first bucket of each index segment contains a tuple. The first field of this tuple is the attribute value of the last record in one bcast, and the second field is the offset to the next bcast. This tuple is used for guiding clients to the next bcast, if they missed the required record in the current bcast. Here is the access protocol of (1, m) indexing scheme [35] for the request with attribute value K: 1. Initial probe: tune into the broadcast channel, and get the pointer to the next index segment. 2. Go to doze mode and tune back to the broadcast channel to get the index segment. 3. Follow the index to determine when the data bucket of the first record with attribute value K will be broadcasted, and then follow the pointers to direct successive probes (client may doze between two successive probes). 4. Tune in again when the first record with attribute K is broadcasted and download all the records with attribute K. Data denotes the average file size, and Index indicates the total number of buckets in the index tree. Let C denote the coarseness of the attribute. As shown in [35], the tuning time of (1, m) indexing method is 1 C k C C . The access latency of (1, m) indexing method is     1 1  .m C 1/  Index C C 1  Data C C: 2 m

(14)

Optimum m. In order to minimize the latency for the (1, m) indexing, there is a formula to compute the optimum m (denoted as m ), which is obtained by

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Y. Shi et al. Data_1

Data_2

Previous Bcast

Data_m Next Bcast

…… 1

2

index segment

m

data segment

CAT(Data_1, Data_2, Data_3,……, Data_m) = FILE

Fig. 7 An example of (1, m) indexing

differentiating the formula of access latency with respect to m, equating it to zero and solving for m. r Data  : (15) m D I ndex Therefore, just divide the file into m data segments and broadcast each segment after each index segment on that channel. Distributed Index. (1, m) indexing scheme could be improved by cutting down the replication times of an index. Distributed index is such approach that index is partially replicated, since the portion of index that indexes the following data segment is not necessary to be replicated. Figure 8 shows the difference between full index replication as in (1, m) indexing scheme and the relevant index replication as in distributed index [35]. Distributed index takes an index tree T and subdivides it into two parts: the replicated part and the non-replicated part. Replicated part consists of the top r levels of T , and non-replicated part contains the bottom .k  r/ levels. Index nodes on the .r C 1/th level are non-replicated roots, denoted by NRR. Any index subtree rooted at non-replicated root in NRR appears only once in one bcast, in front of the data segments it indexes. On the other hand, the replication times of each index node in the replicated part are equal to the number of its children. As in [35], I indicates the root of T . B represents an index bucket in NRR, and Bi means the i th index bucket in NRR. Path.C; B/ is the path sequence from index bucket C to (excluding) index bucket B. Data.B/ shows the set of data buckets indexed by B. Ind.B/ indicates the part of T below (including) B. LCA.Bi ; Bk / represents the least common ancestor of Bi and Bk . NRR D B1 ; B2 ; : : : ; Bt , and Rep.B1 / D Path.I; B1 /, where B1 is the first bucket in NRR. Rep.Bi / D Path.LCA.Bi 1 ; Bi /; Bi / for i D 2; : : : ; t. Thus, Rep.B/ indicates the replicated part of the path from the root to index segment B, while I nd .B/ refers to the non-replicated part. Figure 9 shows the values of Rep, Ind, and Data for each index buckets in NRR. The bcast contains a sequence of triples: < Rep.B/; Ind.B/; Data.B/ > 8B 2 NRR.

Optimization Problems in Data Broadcasting Data segment

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Full Index Replication Data segment

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Fig. 8 Index distribution

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NRR = { b1, b2, b3, b4, b5, b6, b7, b8, b9 } Rep(b1) = { I, a1 } Ind(b1) = {b1, c1, c2, c3} Ind(b2) = {b2, c4, c5, c6} Rep(b2) = { a1 } Ind(b3) = {b3, c7, c8, c9} Rep(b3) = { a1 } Ind(b4) = {b4, c10, c11, c12} Rep(b4) = { I, a2 } Rep(b5) = { a2 } Ind(b5) = {b5, c13, c14, c15} Ind(b6) = {b6, c16, c17, c18} Rep(b6) = { a2 } Ind(b7) = {b7, c19, c20, c21} Rep(b7) = { I, a3 } Ind(b8) = {b8, c22, c23, c24} Rep(b8) = { a3 } Ind(b9) = {b9, c25, c26, c27} Rep(b9) = { a3 }

48

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Data(b 1 ) Data(b 1 ) Data(b 1 ) Data(b 1 ) Data(b 1 ) Data(b 1 ) Data(b 1 ) Data(b 1 ) Data(b 1 )

60

= = = = = = = = =

{ { { { { { { { {

63

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0 ,… …, 8 } 9 ,… …, 1 7 } 1 8 ,… …, 2 6 2 7 ,… …, 3 5 3 6 ,… …, 4 4 4 5 ,… …, 5 3 5 4 ,… …, 6 2 6 3 ,… …, 7 1 7 2 ,… …, 8 0

72

75

78

} } } } } } }

Fig. 9 An example of the index tree for distributed index

Control index is stored in those indices of replicated part. Suppose P1 ; P2 ; : : : ; Pr is the sequence of buckets in Path.I; B/. Last.Pi / indicates the attribute value in the last record indexed by Pi . NEXT B .i / represents the offset to the next occurrence of Pi (Fig. 10). Let l be the attribute value in the last record before B, and let begin be the offset to the next bcast. Control index in Pi have the following i tuples [35]:

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I a1 b1 c1 c2 c3 0 i_2

second_a1

first_a1

first_a2

……

9 a1 b2 c4 c5 c6 9 second_a2

third_a1 …… … 26 … … 17 a1 b3 c7 c8 c9 18 … third_a2

C C C … … 35 a2 b C C C 36 … … 44 a2 b C C C 45 … … 53 I a2 b 4 10 11 12 27 5 13 14 15 6 16 17 18 i_3

first_a3

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third_a3

C C C 54 … … 62 a3 b8 C C C 63 … … 71 a3 b9 C C C 72 … …… … 80 I a3 b7 19 20 21 22 23 24 25 26 27 begin/i_4 I

…………………………………… ( Next Bcast )

Fig. 10 An example of distributed index

ŒI; begi n ŒLast.P2 /; NEXTB .1/ ŒLast.P3 /; NEXTB .2/  ŒLast.Pi /; NEXTB .i  1/ An example of the control index is illustrated in Fig. 11. Control index of bucket Pi is used in this way: for the required data with key value of K, if K < 1, then the client is directed to the next bcast. If K  1, then (K > Last.Pj /) is checked to find the smallest j . If j  i , then follow NEXT B .j  1/ to the next occurrence of Pi ; otherwise search the rest of the index in bucket Pi . Here is the access protocol [35] for a data record with key value K: 1. Initial probe: tune into the bcast, and get the pointer to the next control index. 2. Go to the designated bucket with control index. Based on K and the control index, determine whether to do the following: (a) Wait for the next bcast and go to step 3. (b) Look for the appropriate higher level index, following one of the “NEXT” pointers, and go to step 3. 3. Follow the designated index and pointers to check when the first data bucket with key value K will be broadcasted, and doze between two successive probes. 4. Tune in again to download all the data with key value K. The tuning time of distributed indexing is bounded by 2CkCC as shown in [35]. The access time of distributed index is   1 Index  IndexŒr Data  C C Data C Index C 4Indexr C C: (16) 2 LevelŒr C 1 Level Œr C 1

Optimization Problems in Data Broadcasting

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44, begin 53, i_3

Fig. 11 An example of control index

Optimizing the Number of Replicated Levels. The authors proposed that optimizing the number of r is related to minimizing the access time. Choose r such that the following expression is minimal:   Data Index  IndexŒr : (17) C 4Indexr C LevelŒr C 1 LevelŒr C 1 Evaluate the above expression (Eq. 17) for a given index tree with r ranging from 1 to the number of levels in the tree. The value of r which results in the minimal value for (Eq. 17) is the optimal number of replicated levels.

5.1.3 Huffman-Tree Index Among traditional indexing structures, alphabetic Huffman tree is another important tree-based indexing technique, which has been applied to the wireless broadcast environment ever since the last decades. It is an efficient indexing technique because it takes into consideration the access probabilities of data items when constructing the Huffman tree. The popular data with higher access probability resides closer to the root in Huffman tree, which helps clients reduce search time when traversing from the root. However, almost all existing Huffman-tree indexing methods were discussed under multichannel broadcasting environment, which makes it hard to compare the performance between Huffman tree and other single-channel air indexing methods. Recently, Zhong et al. [74] proposed a novel Huffman-tree-based distributed indexing scheme (HTD) on single-channel broadcasting environment. They applied the feature of distributed indexing onto Huffman-tree indexing, which is an innovative idea that has not been considered before. The experiment results show that under a uniform communication environment, HTD outperforms BC -tree distributed indexing (BTD) and can replace BTD in most existing wireless data broadcast systems.

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Fig. 12 An example data set of Huffman-tree distributed indexing

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Fig. 13 The first step of constructing T0

Here is the construction of Huffman-tree-based distributed indexing scheme (HTD). The structure of index bucket and data bucket of HTD is almost the same as that of BTD. The first stage is to construct a k-ary alphabetic Huffman tree as in [52]. Figure 12 gives an example of the data set and the corresponding access frequencies. The construction process of alphabetic Huffman tree is illustrated in Figs. 13 and 14. Stage 1: choose data nodes di , dj to merge when: 1. There are no data nodes between di and dj . 2. The sum of their frequencies is the minimum. 3. They are the leftmost pair of nodes among all candidates. Create a new index node di0 with frequency equal to the sum of di ’s and dj ’s. Replace di and dj with di0 in the construction sequence. This first stage produces tree T0 without alphabetic ordering of the data nodes. Figure 13 gives an example, where the frequencies of each index node are shown inside the circle as index key values. Stage 2: record the level of each data node of T0 as Li for data di , while the root node is level 1. After this, rearrange pointers from bottom to the root, such that for each level the leftmost two nodes have the same parent, and then the next two, so on, and so forth. Generate an alphabetic Huffman tree T like this, while keeping the level of each node in T0 , as shown in Fig. 14.

Optimization Problems in Data Broadcasting

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Fig. 14 The final Huffman-tree T

To construct k-ary Huffman tree, just extend this algorithm by merging at most k nodes in stage 1 and combining up to k nodes in stage 2. After generating the alphabetic Huffman tree T , cut it at level l, and perform a distributed traversal as in distributed indexing method. The index nodes above cutting level l arecontrol index, and index nodes below l are search index. Next, append control tables onto control index in the same way as distributed indexing. The major differences between Huffman-tree and BC -tree indexing are that (1) the positions of leaf nodes are different and that (2) the subtree sizes below l are different. There might be data items above l in a Huffman tree, depending on which level is chosen for l, because data items are not restricted to reside at bottom level of Huffman tree. Also the sizes of subtrees below l may vary a lot in Huffman tree, but for BC -tree the subtrees have similar sizes. The final broadcast sequence B of this example is shown in Fig. 15. Control table contains a number of entries, each of which is a tuple of key value and offset. Here is the access protocol of Huffman-tree-based distributed indexing: when searching in a control table, client compares the key value of the first entry in control table with the request key value K: • If request key K is less than or equal to the first key in control table, wait until the root of the next bcast. • If request key is greater, go on to compare the next entry in control table. – Turn to doze mode for offset time if request key K is not greater than the next key in control table. – Otherwise, continue comparing with the rest entries of control table until finding such an entry, or go to default bucket (the next index) if not found.

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As proven in [74], the expected average access time of alphabetic Huffman-treebased distributed indexing scheme is 0 0 1 R R2 i Cw1 X 1 X @ X @ vi C ı i ıi Cw A C E.AL/ D .vj C ıj / C vi Cw C kBk i D1 wD1 2 2 j Di C1  P.i Cw/%R .vi C ıi / C

vi C ı i 2



! R X Pi vi C .vi C ıi /Pi ıi : (18) i D1

The expected average tuning time for alphabetic Huffman-tree-based distributed indexing scheme is E.T T / D

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5.1.4 Exponential Index Xu et al. [65] proposed the exponential index, which facilitates replication by sharing links in different search trees to minimize storage overhead and allows clients to start searching at any index node. Exponential index can be adjusted to optimize the access time with the tuning time bounded by a given limit, and vice versa. The nature of distributed structure makes it easy to handle link errors. When dealing with data updates, exponential index works well, especially for applications where the non-key values may change but the search key values do not change often. Exponential index was created based on clustered broadcasting, which means that data items are broadcasted in ascending order of their attribute values. Figure 16 shows an example of exponential index, that a server broadcasts stock information periodically. Each bucket stores one stock item in its data part and some index information in its index table. The index table contains four entries, each of them is a tuple fdisInt; maxKeyg, where distInt indicates the distance range, whose size grows exponentially, and maxKey is the maximum key value of these buckets. The first entry of index table shows the next bucket. For each i > 1, the i th entry indicates the bucket segments 2i 1 to 2i  1 away of totally 2i 1 buckets. Thus, the .i  1/th and i th entries’ maxKey values give the range of buckets indexed by the i th entry. Here in this example the sizes of the indexed segments exponentially increase by a base of 2. Exponential index has a linear and distributed structure, which enables searching from any index bucket. An index link is shared by different index buckets. In addition, by adjusting the exponential base, the indexing overhead can be controlled. Exponential index could be generalized by changing the index base to any value r  1. The i th entry of an index table describes the maximum key value of buckets P Pi 1 j r i 1 1 r i 1 j that are b ij2 D1 r C 1c D b r1 C 1c to b j D1 r c D b r1 c away. To reduce the indexing overhead, each group of I buckets are merged into a data chunk, and the exponential index is built for each chunk. Therefore, the size and the number of index tables are decreased. Furthermore, index links are constructed

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Fig. 16 An example of exponential index A Bcast Data Chunk 1

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Fig. 17 An example of generalized exponential index

for all buckets within each chunk, called local index. Those index entries directing to other data chunks are named global index. An example of such generalized exponential index is shown in Fig. 17, where the index base r and the chunk size I are both set to 2. Algorithm 11 presents the searching algorithm of exponential index [65]. Two major parameters for exponential index are index base r and chunk size I , which offer the flexibility between access time and tuning time. Generally, decreasing index base r leads to the increasing number of index entries and indexing overhead, while tuning time decreasing with r. If the chunk size I is larger, the tuning time becomes less, but the initial probing time might be longer. As shown in [65], the initial probing time plus half of the length of bcast makes the average access latency: I IC C : 2 2

(20)

C 1 I 1 B.I  1/ 1 X C C t.l/: I B.I  1/ C B 0 C

(21)

E.d / D The average tuning time is

E.t/ D

lD0

Here t.l/ is the tuning time for a chunk that is l chunks away [65]:

1; if l D 0I t.l/ D t.l  x/ C 1; if l > 0;

(22)

1 where x is the maximum value in the set of f1; 2; br C 2c; : : : ; b r r1 c C 1g that is less than or equal to l. nc

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Algorithm 11 Index searching algorithm for exponential index 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

wait for the first bucket of next chunk to be broadcasted; for each data item in the bucket do if it is the requested data then stop searching and finish the query; end if end for check local index in the index table; if the requested data is within the key range of the i th entry then go to doze mode until the i th bucket appears, retrieve data and finish the query; end if check global index in the index table; if the requested data is within the key range of the i th entry then i 1 i 1 go to doze mode for .b r r11 C 1c  I  1/ buckets until the first bucket of the .b r r11 C 1c/th chunk appears, and repeat this search process; 14: end if

5.1.5 Hashing Schemes Hashing-based schemes do not require additional index to be broadcasted together with the data. Hashing parameters are included in the control part of data buckets, which helps in reducing access latency and tuning time. Control part for the first N buckets contains the following: 1. Hash function h.K/ 2. Shift: the pointer to a bucket with key K that h.K/ = address(Bucket) In most cases the hashing function is not perfect, and there might be collisions. The colliding records will be stored immediately after the bucket assigned to them, pushing the rest of the file “down,” and create offsets for those buckets. Imielinski et al. [34] proposed two hashing protocols: hashing A and hashing B. The access protocol of hashing A [34] for record with key K is like this: 1. Initial probe, read the control data from the current bucket, and calculate h.K/. 2. If Current bucket # < h.K/, then go to doze mode and tune in again to slot h.K/; Else wait till the next bcast, and repeat protocol again. 3. At h.K/, get the Shift value, go to doze mode and tune in again after Shift number of buckets. 4. Listen until either the record with key K is found (success) or with key L (h.L/ ¤ h.K/) is found (failure). Figure 18a is an example of searching key K = 15 in the file, hashed Modulo 4. Initial probe is at the second bucket. Follow the hashing function, client first goes to bucket four which may contain key K = 15 if there is no overflow. However, the keys are shifted. Client reads the shift value, goes to doze mode, and tunes in again later. It must look at all the overflow buckets of that bucket to find the key K = 15. Figure 18b shows the cases when client needs to wait for the next bcast, which may occur due to either the data miss or the directory miss. Hashing A was further extended to hashing B, which significantly reduced the directory miss by a minor modification of the hashing function. Let d be the size of the minimum overflow chain. The hashing function h was modified as follows [34]:

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Fig. 18 An example of Hashing A (the numbers in the upper right corner of some buckets indicate the shift value in the control part)



h.K/; if h.K/ D 1: .h.K/  1/.1 C min overf low/ C 1; if h.K/ > 1: The new hashing function h0 takes into consideration the shift caused by overflow. Figure 19 shows the improvement of hashing B. The reduction is substantial if the sizes of the overflow chains do not differ much. When the sizes of overflow chain are same for all buckets, h0 becomes the perfect hashing function. However, perfect hashing function does not guarantee the minimal access latency. Let Displacement.h; K/ denote the distance between the physical bucket where K resides (Physical bucket.k/) and the designated bucket for k (h0 .k/): Displacement.h; K/ D Physical bucket.K/  h0 .k/. As presented in [34], the expected access time of hash B is obtained by calculating the access time for each key K and then averaging it out. The expected access time “per key” is combined with two factors: 1. Data miss occurs if the initial probe is between the designated bucket and the physical bucket. The client has to wait an extra bcast. Thus, 0

h .K/ D

.Displacement.h; K/=Data.h//  .Data.h/ C 1=2  Displacement.h; K//: 2. If the initial probe occurs outside the displacement area, it has to wait on average between the Di splace me nt .h; K/ and the file size. Then, .1  .Displacement.h; K/=Data.h///  .Data.h/ C Displacement.h; K//=2: The sum of expected access latency per key divided by total number of keys becomes the expected access latency. For tuning time, the more logical buckets, the lower the tuning time. When the number of logical buckets grows, the tuning time could go down to 3 [34].

Optimization Problems in Data Broadcasting

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Fig. 19 Displacement comparison

Later, Yao et al. [68] proposed MHash, which optimizes both tuning time and access latency. MHash uses a two-argument hash function H.k; l/, which allows each data to be mapped to a number of slots. Popular items are broadcast more frequently, which enables non-flat broadcast to reduce access time. The authors refined MHash by investigating the design of hash function, and analyzed the optimal bandwidth allocation for MHash indexing. They generated hash functions that produce broadcast schedules without unoccupied slots. Such a hole-free hash function was constructed by injecting an offset. MHash reduces access latency and tuning time, since it enables non-flat broadcast, properly spacing the instances of each data, and optimally allocates the bandwidth among data items. Under skewed access distribution, MHash achieves the access latency that is close to the optimal broadcast scheduling.

5.2

Indexing in Multichannel Broadcasting Environment

Multichannel broadcasting techniques enable further reduction of access time by partitioning data onto multiple channels. Several works [10, 15, 48, 49, 69] deal with efficient data allocation for multichannel data broadcast systems. Furthermore, various indexing techniques and index allocation methods have been developed for multichannel data broadcast. Due to the feature of multichannel broadcasting, how to efficiently allocate index has attracted more attentions. There exist two popular categories of index allocation techniques. One approach is to interleave index with data on multiple channels. Examples of such approach include [6, 46, 70]. In [6], Amarmend et al. derived external indices from the scheduled data to determine which channel broadcasts the required data, and allocate them over upper channels. They also extended the exponential index as the local index for each channel. Lo et al. [46] used a k-ary search tree as the

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index tree and discussed the optimal data and index allocation problem on multiple broadcast channels. In [70], indices are striped over multiple channels in such a way that index levels are sequentially broadcasted in sub-strips at certain frequencies. Hsu et al. [24] extended the distributed indexing scheme on multiple channels. Vijayalakshmi et al. [59] presented a hashing scheme for information access through multiple channels in which hash functions are used to index broadcast information across multiple channels. Another approach is to assign certain channels as designated index channels and others as data channels. Jung et al. [38] gave a tree-structured index allocation method with replication over multiple broadcast channels. Their allocation method designed index blocks based on the number of index channels, which contain null elements. Waluyo et al. [60] proposed a global indexing scheme in which each index channel has a different part of the entire index tree while preserving its overall structure, with replicated tree nodes among index channels. Wang and Chen [61] proposed an index allocation method named TMBT for multichannel data broadcast, which creates a virtual distributed index for each data channel and multiplexes them on the index channel. So far, almost all literature on multichannel data broadcast have the assumption that all channels are synchronized, that is, all channels have the same throughput and they begin and finish a broadcast cycle at the same time. Shivakumar et al. [52] discussed the construction of Huffman tree that is similar to Huffman code construction. Chen et al. [17] proposed algorithms for constructing the skewed Huffman tree. However, there exists a problem that the clients may fail to find the desired data item by traversing the Huffman tree. Another kind of Huffman tree proposed in [26], called alphabetic Huffman tree, serves as a binary search tree, which is further extended to k-ary search tree in [52], so that a tree node will fit in a wireless packet of any size by adjusting the fanout of the tree. In [51], Shivakumar and Venkatasubramanian proposed an unbalanced index tree allocation method based on alphabetic Huffman tree, which partitioned the multiple channels into data channels and index channels. They assigned one index channel to each level of the index tree, which might be inefficient when the number of index channels is less than the depth of the index tree. Jung et al. [38] studied an efficient alphabetic Huffman-tree index allocation method for broadcasting data with skewed access frequencies over multiple physical channels. The rest of Sect. 5.2 will introduce their method in detail. To cope with the two major problems of previously proposed indexing allocation methods that (1) require equal size for both index and data and (2) the performance degrades when the number of given physical channels is not enough, the proposed tree-structured index allocation method minimizes the average access time by broadcasting the hot data and their indices more frequently than the less hot data and their indexes over the dedicated index and data channels. Here are the assumptions in [38]. Let DCi denote the i th data channel. Let L.v/ denote node v’s level number in an alphabetic Huffman tree. IL1 is the first ordered list that consists of the index sequence along the path from the root to the data in DC1 . Then, ILi (2  i  the number of data channels) denotes the i th ordered

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Hottest DB Server

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list, excluding the index nodes in ILj for j D 1; : : : ; i  1. Each ordered list ILi D hI1 ; I2 ; I3 ; : : : ; Iq i satisfies two requirements: 1. L.I1 /  L.I2 /  L.I3 /      L.Iq / 2. When L.Ij / D L.Ij C1 /, index node Ij is on the left of Ij C1 in the index tree. Indexes in ILi point to the hotter (more popular) data than indices in ILi C1 , since data at DCi are hotter than data at DCi C1 . Thus, to minimize the access latency, indices in ILi are broadcasted as many times as the corresponding data in DCi . Let DXi denote the number of data items in channel DCi . LCM means the least common multiple of DX1 ; DX2 ; : : : ; DXND , where ND is the number of data channels. Then, the repetition frequency that ILi has to be broadcasted is RFi D LCM=DXi [38]. An example of an alphabetic Huffman tree with data nodes fA; B; C; D; E; F; G; H; I; J; K; L; M; Ng of Fig. 20 is shown in Fig. 21. The set of index nodes is f1; 2; 3; : : : ; 11; 12; 13g. The hottest data are E and N, and the oldest data are F and G, since they reside at the highest and lowest level of the tree, respectively. Figure 22 shows the examples of ILi and RFi from that Huffman tree of Fig. 21. Here, for data nodes (A, D, K) in channel DC2 , the corresponding ordered list of index is IL2 D h4; 5; 6; 7i. RF2 D 20 means the indices in IL2 will be repeated for 20 times in one index broadcast cycle. Given RF1 ; RF2 ; RF3 ; : : : ; RFND , the normalized repetition frequencies are NRF1 D RF1 =RFND ; NRF2 D RF2 =RFND ; : : : ; NRFND D RFND =RFND [38]. And the value of NRFi is rounded off. Figure 23 illustrates the index blocks for the index ordered lists h1; 2; 3i; h4; 5; 6; 7i; h8; 9i, and h10; 11; 12; 13i of the Huffman tree shown in Fig. 21. Since the number of index channels is 2, the size of each member of index block should be 2, and each member’s element is not related to each other with an ancestor-descendent relationship in the tree. Figure 24 shows the index allocation method on 2 index channels for the index blocks of Fig. 23. This index allocation method could also be extended to k channels.

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Figure 25 gives an example of broadcasting with two index channels and four data channels in the system. The gray boxes represent the index nodes pointing to indices, while the white boxes with letters represent the index nodes pointing to data. Assume that data node size is ten times larger than index node size and that a client requests data E which is broadcasted on data channel 1. The client first tunes into index channel 1 to get the root of the tree, and then traverses the index tree to reach the index node pointing to data E. Assume that when t D 14, the index node 2 is being broadcasted over index channel 1, which has the information of data E. With the location of E, the client goes to doze mode until E arrives, and then downloads data E. In [38], the average access time of alphabetic Huffman-tree indexing allocation method for multichannel environment is as follows: cd cd ISi ze X ISi ze X AveAT D NRFj  jILj j C 2  ci  NRF1 j D1 ci j D1

)

"( j 1 X kD1

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jILk j C

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Just compute a set of AveAT s for all possible values of ci and cd , and then choose the pair with minimal AveAT. Since the time complexity of computing AveAT is O..cd /2 / and cd < c, then the time complexity of this method is O.c 3 /. Although the optimal split might need to be recomputed with the changes of data set, it is not a big issue since the time complexity O.c 3 / is low.

6

Coverage Optimization for Data Broadcast

Coverage optimization is a special topic for data broadcasting problem in wireless communications. Usually, this topic deals with data broadcast and data transmission problems in wireless ZigBee networks [20, 53], wireless/mobile Ad-Hoc networks [64], cellular networks [75], heterogeneous overlayed wireless networks [30], and other types of data broadcast services. Most of the coverage problems are discussing whether the data broadcast service provider guarantees reliable coverage for a certain area or region and how to make the broadcast system more energy efficient, time efficient, or fault tolerant (more reliable). As described in previous sections, the researches for data broadcast are cross-disciplinary works among multimedia technology [43], wireless networking [22], data engineering [18], and other related fields [29]. Accordingly, majority of the studies on coverage optimization belong to wireless networking. This type of problems can be converted into combinatorial optimization problem and most of them are NP-hard problems reducible to known NP problems, which cannot be solved in polynomial time unless P D NP [23]. Therefore, to solve this type of problems, people usually design approximation algorithms and heuristic approaches to provide the near-optimal solutions. This section will briefly introduce several existing problems and optimization methods of coverage optimization for data broadcast in wireless communications. The organization of this section is shown as follows: Sect. 6.1 will describe the mathematical modeling and problem definition of coverage optimization, Sect. 6.2 will introduce the classification and reduction of this problem, while Sects. 6.3–6.5 will illustrate different variants of coverage optimization for data

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broadcast and its corresponding solutions. Finally, Sect. 6.6 will give a summary for coverage optimization and some discussions for the future directions.

6.1

Mathematical Modeling and Problem Definition

To formally define coverage optimization problems for data broadcast, the problem formulation and reduction have to be introduced. More specifically, as shown in Fig. 26, in a typical data broadcast system, one or more base stations (BSs) (also called “nodes” in combinatorial geometry) will continually broadcast information among a certain area/region. Within this area/region, there are hundreds/thousands of fixed-line or mobile clients that have the requirements for these information. As a consequence, how to economically, efficiently, and reliably cover the whole area such that each client can successfully receive the required messages/data streams becomes an important issue for all service providers. In mathematical description, given an area A in Euclidean space, a group of nodes N D fn1 ; : : : ; nk g, each node ni covers a certain area described as V .ni /. Each node can also communicate with other nodes in this area. In A, there are T D ft1 ; : : : ; tm g clients. In different mathematical models, V .ni / has different representations. If considering signal interference like interrupt, collision, and contention V .ni / usually denotes a circle whose center is ni . If each ni has the same capability, then the whole graph becomes a unit disk graph (UDG) otherwise the graph is a disk graph (DG). Figures 27 and 28 are examples of UDG and DG correspondingly. If not considering signal interference, then V .ni / usually represents a Voronoi division with the following definition: V .ni / D

\

fp W jp  ni j  jp  nj jg:

1j k; j ¤i

Fig. 26 An example of coverage for data broadcast

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Fig. 27 Unit disk graph model for wireless network

Fig. 28 Disk graph model for wireless network

It means that each BS ni only charges for its domain V .ni /. Such graph is named as Voronoi graph (VG). An easy example can be seen from Fig. 26, in which the dashed lines represent the boundary of covered area V .ni / for each ni in this case. The problems of coverage optimization are usually defined according to the above given factors. They can be reduced as a minimization-optimization problem with a coverage constraint and some cost constraint together. The next subsection will discuss the classification and reduction of coverage problem.

6.2

Classification and Reduction

There are two types of coverage problems. If not considering the data transmission among nodes, the object of coverage optimization is as follows: 1. Cover the whole area A using the minimum cost (defined as region coverage). 2. Cover the target set T using minimum cost (defined as target coverage).

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s5 s4

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Fig. 29 An example to reduce region coverage to target coverage

Here the cost can be defined as a customized function, according to different requirements in different scenarios. The region coverage problem can be simply converted to the target coverage problem in UDG or DG [72]. Figure 29 is an example to illustrate this reduction. The square in Fig. 29 is the potential region for broadcasting. For each point in this area, let A denote the set of nodes that can cover this point. Partition the area into different parts, each with different node coverage sets. As a consequence, area coverage problem can be reduced to target coverage problem when considering each part as a target. This problem can further reduced to a set cover problem. Consider each small division as a target mark it with a number from 1 to 14, and then insert covered targets into each set of sensors. Say, S1 D f7; 9; 11; 12; 13; 14g, S2 D f6; 9; 13g, S3 D f3; 6; 7; 8; 10; 13; 14g, S4 D f3; 4; 5g, S5 D f2; 1; 3; 4; 8; 12; 14g, and S6 D f2g. The coverage problem can be reduced to the problem of finding a minimum set cover from Si to cover T D f1; : : : ; 14g. In VG, target coverage is widely studied, since due to the definition of Voronoi division, area A has already been divided into different parts, and each part is dominated by some node. If further considering the data transmission among nodes, then the coverage problems are more complicated. It will either involve the traditional broadcast tree optimization, or deal with selecting a subset of broadcasting sources to reduce the

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energy consumption and latency cost. Moreover, based on different requirements from real world applications, there are additional constraints or benefits for coverage optimizations, for example the multilingual message broadcast service, the TV data streaming requirements, the multichannel data broadcast, and the multi-input multioutput (MIMO) antenna technologies. The following subsections will introduce specific coverage optimization problems according to various system models and various client requirements.

6.3

Cell Broadcasting Scheme to Support Multilingual Service

This subsection will describe a coverage problem for cell broadcast to support multi-lingual service [75]. It solves a coverage problem named EEML-CB, which is described as follows: Definition 9 The energy-efficient multilingual cell broadcasting (EEML-CB) problem can be described as Given: C D fc1 ; : : : ; cn g as client set, BS D bs1 ; : : : ; bsm as set of broadcast stations, H D fh1 ; : : : ; hn g as the index of the BS communicating q with ci , L D fl1 ; : : : ; lk g as language set, CL D fcli g as client acceptable language set, W as language weight set, Rmax as the maximum broadcasting radius, and A as target area. For each bsj , assign an efficient j strategy Sj , containing several couples as hlq ; rq i, which means that bsj j should broadcast language lq with distance rq . Find: A minimum energy consumption strategy S D fsj g; .0  j  m/ for every BS, such that each client ci can receive at least one copy of the required messages with its acceptable languages. In short, this topic deals with a target coverage problem for each BS specifically. Within V .ni / for each node, there locates a group of customers with different language requirements. Sending messages in different language will result different weights. And the object of this topic is trying to assign a minimum energy-cost schedule for the BS, such that all the clients within its domain can receive as least one copy of message in its acceptable language set. The authors first proved that EEML-CB is a NP-hard problem, by reducing it to an instance of set cover problem, and then provide the inter linear programming (ILP) as follows: min s:t:

m P k P

q

Ej

j D1 qD1 k P q xi  1 qD1 q q xi  cli A  .dihi /2  wq  q q dihi  xi  rhi ;

8ci I q

q

xi  Ehi ;

8ci ; lq I 8ci ; lq I 8ci ; lq :

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Algorithm 12 Smart multi-lingual cell broadcasting (SMCB) Input: C, BS, H, L, CL, W rmax and A. Output: S and E. 1: For each bsj , select its client set Cj . q j 2: Find the farthest client ct 2 Cj with it least weight language lq . Assign rj D dt . Mark this client. 3: Mark clients whose acceptable language set includes lq . 4: Repeat steps 2–3 until no client left unmarked in Cj .

The objective of this ILP model is to minimize the total energy consumption for a BS. The first two constraints guarantee that each client receives at least one language from its accepted language set. The third constraint calculates the energy consumption for each language, while the last one guarantees that for each given language, the energy is large enough for the BS to cover all target clients. The ILP is acceptable in small-scale networks. However, when network size grows, the time consumption of ILP grows exponentially. Therefore, the authors designed another greedy algorithm shown as follows: The main idea of this algorithm is to find the farthest client for a BS, broadcast with its least weight acceptable language lq , and then remove all clients accepting lq . Repeat this greedy step until no client is left without getting a copy of the content. Finally, they evaluated the performance of their designs by numerical experiments. The ILP was solved by CPLEX 7.0 for small-scale network, while the greedy algorithm was tested by simulation for large-scale network, with 5 base stations, 100,000 clients, and 50 languages. The results proved the efficient of their design. Their heuristic saves more than 50 % of the energy consumed by traditional broadcasting scheme without any optimizations.

6.4

Coverage Prediction and Optimization for Digital Broadcast

Digital terrestrial broadcasting services (e.g., DVB-T and DAB) are proving a big success in Europe, as improved service quality, low setup costs, and increased content offered by the broadcasters attract more viewers and listeners to these new platforms. In order to reach their target subscriber levels, digital operators not only face the challenge of making their content attractive but also need to know more accurately the quality of terrestrial coverage they can provide to their potential customers. DVB/DAB requires higher prediction accuracy than traditional analogue networks because digital services are planned with tighter margins on the signal strength and interference. This subsection will introduce new prediction models that offer higher accuracy and which can be used to optimize digital broadcasting networks [12]. Accordingly, broadcasters for DVB/DAB services need optimize their new digital networks to provide the following: • Accurate modeling of the impact of new digital services on analogue availability. • Prediction of the coverage that can be provided by a new digital network.

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• Selection of channels to best accommodate digital services. • Creation of efficient and effective networks to achieve migration from analogue. • Optimization of digital networks to maximize the number of customers. To meet these requirements, the authors provided a high-accuracy coverage prediction models to estimate the maximum coverage of their service, and optimize the following parameters for their network to deliver the best possible coverage. • Selection of site locations. • Antenna heights. • Antenna types and orientations. • Channel selection. This model can provide higher accuracy than traditional semiempirical models, yet can be computed in a reasonable timescale with a lower requirement on data accuracy than full 3D deterministic solvers. It relies on three important elements: 1. High-resolution data, representing the locations and heights of terrain and buildings in built-up areas, and medium-resolution data for more open, rural areas. They are usually collected by stereo aerial photography, laser interferometry, or satellite collection. The authors for this model require only a raster of terrain heights, avoiding the need to process the data into a vector form that represents the full geometry of the buildings. 2. Propagation prediction algorithms, to account for the improved data. The main mechanism for predicting the attenuation of signals across obstructions such as buildings and hills is multiple diffraction. The authors use slope diffraction to produce solutions that are as accurate as much more complex techniques in short run times. If radio wave propagation occurs over a clearly defined path (the greatcircle path) between the transmitter and receiver, then it is necessary to account for the finite width of buildings and the possibility of off-path propagation around building edges. 3. Optimized model, to be implemented efficiently in order to make the most of available computing resources. It also includes (a) the extraction of suitable path profiles to suit the requirements of the model (b) post processing of the prediction results to include the effects of antenna radiation patterns, and (c) computation of the availability, based on the relevant planning margins. The authors also provided example predictions on behalf of a digital broadcaster. The measurements covered a large area consisting of 6 km2 , served by a transmitter approximately 30 km away. Field-strength measurements on four channels were made at a height of 10 m using a vehicle equipped with a pump-up mast. An average of 20 measurements per square kilometer were made, resulting in a total set of 468 measurements. In order to assist with the prediction process, the authors have also developed a unique automated network planning and optimization service, known as SmartPlan, and have used it successfully in real networks. Their results prove that this model has consistently performed well and has proved to be very effective in (a) reducing the standard deviation of the prediction error, and (b) significantly increasing the hit rate. It has already been used in real situations, either to address optimization issues in existing networks or to choose the optimum parameters for a new network.

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Data Delivery Optimization in Multi-system Heterogeneous Overlayed Wireless Network

This subsection will employ data broadcast on multisystem heterogeneous overlayed wireless networks, due to the development of 4G networks [30]. The authors provided detailed two-step construction for data broadcast system and proved the efficiency of their algorithms by simulations. The system model can be described as follows: Consider a multisystem heterogeneous overlayed network N D fN1 ; N2 ;    ; NjN j g consisting of jN j subnetworks, and suppose these subnetworks are ordered by the sizes of their coverage areas in ascending order. Each Ni allocates Ci channels to provide the data broadcast service, and the bandwidth of each channel is Bi . The data program contains D D fD1 ; D2 ; : : : ; DjDj g data items. Each Di has data size si and access probability pi . For instance, Fig. 30a is a general system model of multisystem heterogeneous overlayed networks, while Fig. 30b is a practical example with regional-area subnetwork, metropolitan-are a subnetwork, and campus-area subnetwork. According to the phenomenon that wireless networks with larger coverage areas are usually of higher connection fee and of lower bandwidth than those with smaller coverage areas, the authors want to build a communication system with lowest cost such that all the subscribers can receive the required messages with minimum access time within the corresponding coverage area. The main idea for data broadcast system in this type of wireless network is as follows: they divided broadcast data program into several subsets Dcuti for subnetwork Ni , such that the largest service area will broadcast the whole data set, while other subnetworks will be viewed as supplements to reduce the minimum overall access time. Each subnetwork will only broadcast a subset of D correspondingly. As a consequence, the problem of broadcast program generation on heterogeneous overlayed wireless networks can be formulated as follows: Definition 10 (Data broadcast in multi-coverage network) Given a multisystem heterogeneous wireless network N D fN1 ; N2 ;    ; NjN j g, the number of allocated channels Ci in each subnetwork Ni , the number of data items, and the access probabilities and sizes of all data items, for each subnetwork Ni , determine: 1. Which data items will be broadcast in subnetwork Ni (i.e., a proper cutting configuration). 2. How these data items are broadcast in subnetwork Ni (i.e., one proper broadcast program for each subnetwork), such that the overall average access time for subscribers within the service area will be minimized. To solve this problem, the authors firstly designed an algorithm named bruteforce to enumerate and select the best cutting configuration and the corresponding broadcast programs for all subnetworks. When evaluating one cutting configuration, they generate one broadcast program for each subnetwork by executing algorithm

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b

a

Regional-Area Subnetwork Metropolitan-Area Subnetwork N1 N2 .......

Campus-Area Subnetwork

N|N| General System Model

Example

Fig. 30 Multisystem heterogeneous wireless networks

BPG-Single (a single-system network algorithm for data allocation) once so that the average access time of the subnetwork is minimized. Due to the high time complexity of this exhausted search, they next provided a heuristic algorithm named layeredcutting to determine a suboptimal cutting configuration and the corresponding broadcast programs for all subnetworks. Layeredcutting is a two-phase algorithm consisting of inter-network data allocation phase and intra-network data allocation phase. The objective of inter-network data allocation phase is to determine a proper cutting configuration so that the overall average access time of the whole multisystem network is minimized. Then, in intra-network data allocation phase, layeredcutting will generate one broadcast program for each subnetwork according to the resultant cutting configuration. It is an iterative algorithm which merges several subnetworks and determines the value of one cutting point in each iteration. The threshold is the probability estimation of each subnetwork for each data item. In each iteration, inter-network-data-allocation uses a divide-and-conquer procedure named test-and-prune to determine cuti , and then returns all cuti group, while intra-network-data-allocation applies execute algorithm BPG-Single on subnetwork Nj and returns broadcast programs of all subnetworks, which become the final broadcast program. The authors proved that the time complexity of layered-cutting is O.jN j  .log jDj C O.BPGSingle ///. They also illustrated their heuristics by simulation with 1,000 data items, following a Zipf distribution with parameter . There are jN j subnetworks, and the rate between the sizes of the service areas of subnetwork Ni and subnetwork Ni C1 is equal to 0.8. Finally, according to their simulation results, algorithm layered-cutting is able to efficiently generate broadcast programs of high quality for a multisystem heterogeneous overlayed wireless network and cover the whole service area.

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Summary

To summarize, this section describes the coverage optimization for data broadcast in wireless communications. Coverage problems are introduced in different types of networks like wireless ZigBee networks, wireless/mobile ad hoc networks, cellular networks, and heterogeneous overlayed wireless networks. Firstly, the formal definition of coverage optimization is presented, including the prerequisite factors, the classification, relation among various types of coverage problems, and many possible variants. The researches of coverage optimization belong to wireless networking field. These problems can be reduced to some known NP problem and cannot be easily solved in polynomial time unless P D NP. Next, several coverage problems are discussed in details, including the problem formulation, the construction of approximation or heuristic algorithms, and the performance of the solutions. Future study of coverage optimization could get benefit from the latest technologies, especially the new ideas and technologies in the 4G wireless networks. Moreover, researchers can combine knowledge from different disciplines and think about the requirements from the server side and client side wisely. Even for the same number of BS’s with the same capacity, different clients/service providers may have different requirements about the coverage issues. Clients always want to receive their requests quickly and correctly, while servers always hope to provide a reliable and efficient service. Therefore, the coverage problem is always a critical issue for data broadcast system, which makes coverage optimization an active and interesting research topic in both theoretical and practical fields.

7

Conclusion

This chapter discusses various optimization problems in all aspects of data broadcasting. Scheduling is a big topic in data broadcast. In order to reduce response time, push-based broadcast systems need optimized off-line data scheduling (or allocation) algorithms to organize broadcast programs; on-demand broadcast systems need optimized and efficient online scheduling algorithms to decide broadcast contents in real time, while clients may also need scheduling algorithms to decide the downloading sequence of multiple data items. Energy optimization in wireless data broadcast systems relies on good index schemes. Index design should be customized based on different features of broadcast data. Coverage optimization is also an important research area of data broadcast. Within this big picture, it also introduces different techniques during algorithm design to handle different features of data, communication environment, and user type. For example, data can be homogeneous or heterogeneous; broadcast can be on single channel or on multiple channels; request can be for single data item or multiple data items; and clients may have deadlines on responses to their requests. Data broadcasting is a mature topic with a new life. Existing research works on data broadcast establish a solid foundation for further research in this area.

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On the other hand, with the fast development of mobile communication techniques, new problems are keeping emerging waiting to be solved by future researchers. Hope this chapter can be a good guide for them to better understand the area and an inspiration for new questions and research topics.

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Optimization Problems in Online Social Networks Jie Wang, You Li, Jun-Hong Cui, Benyuan Liu and Guanling Chen

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Influence Spread. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 NP-completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 #P-Hardness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Other Influence Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Finding Communities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Graph-Partitioning-Based Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hierarchical Clustering-Based Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Random Walker Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Data Collection from OSNs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Measuring and Analyzing OSNs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Sampling Methods of OSNs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 MySpace Data Collection and Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Member Profile Patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Temporal Usage Patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Blog Content Patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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J. Wang () • Y. Li • B. Liu • G. Chen Department of Computer Science, University of Massachusetts, Lowell, MA, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] J.-H. Cui Department of Computer Science and Engineering, University of Connecticut, Storrs, CT, USA e-mail: [email protected] 2455 P.M. Pardalos et al. (eds.), Handbook of Combinatorial Optimization, DOI 10.1007/978-1-4419-7997-1 83, © Springer Science+Business Media New York 2013

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5 Multilayer Friendship Modeling in Location-Based OSNs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A Multilayer Friendship Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Friendship Ranking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter introduces and elaborates four major issues in online social networks. First, it describes the problem of maximizing the spread of influence in social networks and the problem of computing the spread. It shows that the problem of maximizing influence is NP-hard and the problem of computing the spread is #P-hard. It also presents a number of algorithms for finding approximations to the problem of influence maximization. Second, it describes the problem of detecting network communities and presents a few heuristic algorithms. In particular, it introduces and elaborates a new algorithm that uses random walkers to find communities effectively and efficiently. Third, it discusses how to collect data from online social networks and it presents the recent work on data collection. Forth, it discusses how to measure influence (friendship) in online social networks and elaborates the recent results on the location-based three-layer friendship modeling.

1

Introduction

Influence maximization (IM) is a combinatorial optimization problem in social sciences with important applications. Kempe et al. [20] formulated and studied IM as a discrete optimization problem and subsequently published additional results in [21]. Since then, a number of researchers have investigated the problem from different angles, including Kumura and Saito [23], Leskovec et al. [26], Narayanm and Narahari [32], and Chen et al. [6–8]. Domingos and Richardson [10] and Richardson and Domingos [38] were among the first to study the social influence problem from the algorithmic standpoint in connection to viral marketing. Suppose a company wants to promote a new product to a large group of people. To do so, the company plans to offer free samples to a small group of selected individuals who are more influential than the others, with the hope that these key individuals will help spread the information of the product to the largest number of individuals through their influence. How to find such key individuals motivated the study of the problem of influence maximization, which finds a small set of most influential individuals. Social networking in the cyber space has become increasingly popular in recent years. Online social networks (OSNs), such as Facebook and MySpace, provide

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a convenient platform for users to interact with each other through the Internet. Studying IM in the setting of OSNs may help reveal how social activities would take place in the electronic world. On the other hand, conducting research on OSNs has an added advantage: it is easier to collect data from OSNs than physical social networks. For example, one could either develop a crawler to collect information by crawling through the targeted OSN [11, 12] or simply use data dumps or the Application Programming Interface provided by the service provider. This chapter is focused on OSNs, but some of the results may also apply to general social networks. To measure influence of one individual to another, an immediate solution is to determine how close the relationship would be between them. In OSNs, a user can declare other users as friends, even though they may not know each other personally. This chapter follows the notion of online friendships. Thus, one way to measure how influential an individual would be is to determine how many friends they have. The individual with the largest number of friends may be considered as the most influential individual. On the other hand, this individual may not have the most influence on a specific friend. For example, Alice is considered as the most influential individual because she has the largest number of friends, including Bob. But Bob is only a casual friend of Alice. Bob, on the other hand, has a close longtime friend Christie whom he trusts. Thus, Christie would be more influential to Bob than Alice. Simply going through the data collected from OSNs, however, it is difficult to differentiate such level of friendships, and so most studies still rely on the loose concept of online friendships adopted by OSNs. Related to the problem of influence maximization is the problem of finding network communities. A network community is a collection of nodes within which connections are dense, while connections between communities are sparse. Finding communities in online social networks, biological networks, metabolic networks, and other large-scale networks has important applications, and it has been studied intensively and extensively in computer science, mathematics, biology, and other disciplines. Intuitively, network communities are formed as a subset of nodes with similar characteristics. In an OSN, a community may form from users with similar hobbies or interests. For example, users who like classical music may form a community; users who are interested in iPhone, iPad, and other Apple products may form another community. If a company wants to advertise a new product to as many people as possible effectively and efficiently, it would naturally want to choose communities with a keen interest in similar products. This chapter describes a number of heuristic algorithms for finding network communities, including a new algorithm that uses random walkers to search for communities. This algorithm was recently devised by Li et al. [28], which has only been reported in a technical report. Studying how users are connected through OSNs would help to measure influence of OSN users. Intuitively, people sharing mutual interests, being geographically close to each other, or belonging to the same social communities are more likely to become friends. This chapter elaborates multilayer metrics recently devised by Li and Chen [27] to quantitatively model these intuitive friendships.

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Social networks are often modeled as a weighted directed graph G.V; E/, where nodes represent individuals and edges represent relationships and interactions between individuals. To study social influence, each edge u ! v is associated with a weight p.u; v/ 2 Œ0; 1 that quantifies the level of influence on v by u. Such graphs are referred to as influence graphs. Node v is said to be node u’s neighbor if u ! v 2 E. Influence maximization seeks a small set of nodes that will influence the maximum number of nodes in the given influence graph. Kempe et al. [20] introduced two stochastic cascade models, also known as diffusion models, for studying the problem of influence maximization. They are independent cascade (IC) and linear threshold (LT), where each node in the graph is either active or inactive. Initially, both models specify an initial set of active nodes, referred to as seed set, while the rest of the nodes are inactive. Once a node is made active, meaning that it has been successfully influenced by its active neighbors, it will remain active throughout the entire process. The process of both models proceeds at discrete steps. Discrete steps are also referred to as rounds or iterations. In the IC model, each active node u is only given one chance to activate each of its currently inactive neighbor v with the success probability p.u; v/, where u ! v 2 E. This formulation is motivated by Goldenbert et al.’s work on viral marketing [16, 17]. If v has multiple active neighbors, their attempts to activate v are sequenced in an arbitrary order. In particular, if u is newly activated in round i , u ! v 2 E, v is inactive, and u is given a chance to activate v in round i C 1, then v is activated by u with probability p.u; v/. If u succeeds, then v becomes active at time i C 1. If u fails to activate v, then u cannot activate v in subsequent rounds. The process continues until no more activations are possible. In the LT model, each node v 2 V satisfies the following inequality: X

p.u; v/  1:

u!v2E

At the beginning, each node v chooses a threshold value v 2 Œ0; 1 uniformly and independently at random. An inactive node v becomes active in round i if X u!v

p.u; v/  v :

and u is active

The process continues until no more activations are possible. As noted by Kempe et al. [20], selecting thresholds at random is the only reasonable choice when the exact influence to v from its neighbors is unknown from the given data set. This chapter is organized as follows. Section 2 defines the IM problem, shows that it is NP-hard for the independent cascade and linear threshold models, shows that the influence spread of an initial seed set is #P-hard for both models, and provides algorithms to approximate the IM problem. It also introduces other influence models. Section 3 introduces a number of heuristic algorithms for finding network communities. Section 4 describes how to collect data from online social networks. Section 5 discusses how to determine friendships. Section 6 concludes the chapter.

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Influence Spread

Let S be an initial set of active nodes in G and IG .S / the expected number of active nodes at the end of the process in an influence model (IC or LT). Function IG is referred to as the influence function of S and the value of IG .S / the influence spread of S , or simply spread.

2.1

Formulation

The influence maximization problem can be formulated as follows: Influence maximization (IM) Input: An influence graph G.V; E/ and a positive integer k. Output: An initial seed set S of k nodes such that IG .S / is maximized for all k-node initial seed set. The decision version of IM is defined below: Decisional influence maximization (DIM) Input: An influence graph G.V; E/ and positive integers k and N . Question: Does there exist an initial seed set S of k nodes such that IG .S /  N ? It is straightforward to see that if IM is polynomial-time computable, then so is DIM. But DIM is NP-complete for both the IC and LT models. Thus, IM is NP-hard on both models. Counting how many k-node seed sets S that will maximize IG .S / is at least as hard as solving IM. Moreover, even computing the value of IG .S / for a fixed seed set S is #P-hard for both the IC and LT models. Given a seed set S , computing the exact value of IG .S / is referred to as the influence spread computation (ISC) problem, formulated as follows: Influence spread computation (ISC) Input: An influence graph G.V; E/ and a seed set S  V. Output: IG .S /.

2.2

NP-completeness

The following results were obtained by Kempe et al. [20]. Theorem 1 (([20])) DIM is NP-complete for both of the IC model and the LT model. Proof First consider the IC model. The goal is to construct a polynomial-time reduction from a known NP-complete problem to DIM for the IC model. The set cover (SC) problem is a suitable NP-complete problem for this purpose. Given a collection of subsets S1 ; : : : ; Sn of set U D fu1 ; : : : ; um g, SC decides whether there

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exist k subsets such that the union of these sets is equal to U . Without loss of generality, assume that k < m < n. Given an instance of SC, construct a directed bipartite graph .V1 ; V2 I E/ of m C n nodes such that each subset Si corresponds to a unique node i 2 V1 and each element vj corresponds to a unique node j 2 V2 , where i and j are connected by a directed edge i ! j 2 E with weight pij D 1 if vj 2 Si . Because pij D 1, the activation of a node is deterministic. It is straightforward to see that the instance of SC is positive if and only if there is a k-node seed set S  U1 such that IG .S /  m C k. Thus, SC is polynomial-time reducible to DIM. Now consider the LT model. The goal is to construct a polynomial-time reduction from a known NP-complete problem to DIM for the LT mode. The vertex cover (VC) problem is a suitable NP-complete problem for this purpose. Given an n-node undirected graph G and an integer k, VC decides whether there is a k-node set C such that every edge has at least one endpoint in C . Converting G to a directed graph by directing every edge in G to both directions and setting p.u; v/ D 1 for every edge u ! v, it is straightforward to see that there exists a k-node set C to cover G if and only if IG .A/ D n. Thus, VC is polynomial-time reducible to DIM. This completes the proof. 

2.3

#P-Hardness

The following result was obtained by Chen et al. [7]. Theorem 2 ([7]) ISC for the IC model is #P-hard. Proof It suffices to show that if ISC is solvable in polynomial time for the IC model, then so is a known #P-complete problem. The two-terminal-network-reliability problem is an appropriate problem, which was shown to be #P-complete by Valiant [40]. This will establish that ISC is #P-hard. The two-terminal-network-reliability problem is to count, on a given directed graph G D .V; E/ and two nodes fs; tg  V , the number of subgraphs in which there is a path from s to t. This is equivalent to computing the probability that there is a path from s to t under the condition that each edge in E has an independent probability of 1/2 to be selected and an independent probability of 1/2 to be discarded. Let S D fsg and p.e/ D 1=2 for all e 2 E. Compute I1 D IG .S /. Next, add a new node t 0 62 G and a directed edge t ! t 0 to obtain a new graph G 0 D .V [ ft 0 g; E [ ft ! t 0 g/. Let p.t; t 0 / D 1. Compute I2 D IG 0 .S /. By assumption, both I1 and I2 are polynomial-time computable and so is I2  I1 . Let p.S; v; G/ denote the probability that node v is influenced by the seed set S in G. Since t influences t 0 deterministically, it is straightforward to see that I2 D IG .S / C p.S; t; G/. Therefore, p.S; t; G/ D I2  I1 is the probability that s is connected to t, and it is polynomial-time computable. This completes the proof. 

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To show that ISC for the LT model is also #P-hard, it helps to consider the distribution over sets reachable from S via certain types of edges. Recall that in the LT model, each node v receives an influence weight p.u; v/  0 from node u for u ! v 2 E. Suppose that v selects at most one such incomingP edge at random with probability p.u; v/ and selects no edges with probability 1  u p.u; v/. The selected edge u ! v in this way is referred to as a live edge and the other edges as blocked edges. It can be shown that the distribution over active sets (i.e., sets of active nodes) by running the LT process starting from S is equivalent to the distribution over sets of nodes reachable from S via live-edge paths. A live-edge path is a path where all edges on the path are live edges. To motivate the proof idea, consider a simple case that G is an acyclic graph. In an acyclic graph, nodes can be listed in a fixed topological ordering: v1 ; v2 ; : : : ; vn , such that all edges are from nodes of smaller indexes to nodes of larger indexes. It suffices to look at the distribution over active sets following this order. For each node vi , suppose that the distribution over active subsets of its neighbors is already determined. Following the LT process, the probability that vi will become active, given that a subset Si of its neighbors is active, is equal to X

p.u; vi /:

u!vi 2E

This is the probability that the live incoming edge selected by vi lies in Si , and so it is straightforward to show that the two processes produce the same distribution over active sets by a simple induction on nodes in this order. For general graphs G, however, such an ordering of the nodes may not exist. Instead, an induction will be carried out over the number of rounds of the LT process. The following lemma was shown by Kemple et al. [20]. Lemma 1 ([20]) For a given initial seed set S in an influence graph G, the distribution over active sets from S under the LT process is the same as that over sets reachable from S via live-edge paths. Proof Let Si be the set of active nodes at the end of round i under LT process (i D 0; 1; 2; : : :), where S0 D S . If node v has not become active by the end of round i , then the probability that it becomes active at round i C 1 equals the probability that the total influence from Si  Si 1 exceeds the threshold, given that its threshold was not exceeded already. This probability is P Pu;v D

u2Si Si 1 & u!v2E

1

P

u2Si 1 & u!v2E

p.u; v/ p.u; v/

:

The following procedure can be used to identify reachable sets from the initial seed set S : Starting from the set S , for each node v with at least one edge u ! v 2 E and u 2 S , if one of these edges becomes live, then v is reachable from S , resulting

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in a new set S10 of reachable nodes from S . Repeat the same procedure recursively on edges from S10 to obtain a new reachable set S20 , from S20 obtain S30 . If node v is not yet reachable by the end of round i , then the probability that it will become reachable in round i C 1 is equal to the probability that its live edge comes from Si  Si 1 , P given that its live edge has not  come P from any of the earlier sets, which is equal to u2Si Si 1 & u!v2E p.u; v/= 1  u2Si 1 & u!v2E p.u; v/ . This is the same as Pu;v . Thus, a straightforward induction over LT rounds can establish that the live-edge process produces the same distribution over active sets by the LT process. This completes the proof.  The following result was obtained by Chen et al. [8]. Theorem 3 ([8]) ISC for the LT model is #P-hard. Proof It suffices to show that if IG .S / is polynomial-time computable, then so is the problem of counting simple paths (CSP) in a directed graph. CSP is to count, given a directed graph G D .V; E/, the total number of simple paths in G, which was shown to be #P-complete by Valiant [40]. Given an influence graph G, construct a new influence graph G 0 D .V 0 ; E 0 / by adding a new node s 62 G and connecting s to all the nodes in V . That is, the node s is a source node. Let d be the maximum in-degree of any node in G 0 . Assign a weight of p.e/ D w  1=d to each edge e 2 E 0 , where w is a constant, and it will be determined later. Clearly, for any node v 2 V 0 , X

p.u; v/  1;

u!v2E 0

and so the weight requirement of the LT model is satisfied. Let the initial seed set be S D fsg. By assumption, IG 0 .S / is polynomial-time computable. Let P denote the set of all simple paths starting from s 2 G 0 . It follows from Lemma 1 that XY IG 0 .S / D p.e/: 2P e2

Let Bi be the set of simple paths of length i in P, for i D 0; 1; : : : ; n, and let bi D jBi j. Then

I .S / D

n X Y X

G0

i D0 2Bi e2

p.e/ D

n X X i D0 2Bi

i

w D

n X

wi bi :

i D0

With the above equation, set w to n C 1 different values w0 ; w1 ; : : : ; wn , where n D jV j. For each value wi , by assumption, IG 0 .S / is polynomial-time computable. Note that this process produces a set of n C 1 linear equations with b0 ; : : : ; bn variables. The coefficient matrix M of these equations is a Vandermonde matrix

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j

with Mij D wi for i; j D 0; : : : ; n. The system of these equations therefore has a unique solution and is easy to compute. Finally, notice that for each i D 1; : : : ; n, there is a one-to-one correspondence P between paths in Bi and simple paths of length i 1 in graph G. Therefore, niD1 bi gives the answer to the total number of simple paths in graph G. This completes the proof. 

2.4

Approximations

Approximation algorithms for the IM problem are based on the observation that the influence function IG for both the IC and LT models is a submodular function.

2.4.1 Submodularity A submodular function f is a function that maps a subset of a finite ground set U to a nonnegative real number satisfying the following inequality: .8A  B  U /.8x 2 U  B/ W f .A [ fxg/  f .A/  f .B [ fxg/  f .B/: A function f is monotone if for all A  B W f .B/  f .A/. Submodular greedy algorithm (SGA) [9]: 1. Given a function f W 2U ! RC [ f0g. 2. Initially set S ;. 3. Repeat the following for k times: Choose u 2 U  S such that f .S [ fug/ is the largest among all possible u’s. Set S S [ fug. 4. Return S . The following result was shown by Nemhauser et al. [33] and is well known. Its proof is omitted. Lemma 2 ([33]) Let f be a monotone submodular function. Let S be a k-element set obtained from SGA. Let S  be a set that maximizes the value of f over all k-element sets. Then f .S /  .1  1=e/f .S  /. That is, S is a .1  1=e/approximation to S  . It is straightforward to see that IG is monotone. It can be shown that IG is submodular for the IC model and the LT model, which implies that the algorithm stated in Lemma 2 provides a .1  1=e/-approximation ratio to IM. Lemma 3 ([20]) The influence function IG is submodular for the IC model. Proof Consider a point in the IC process when node u, just becoming active, attempts to activate its neighbor v with a success probability p.u; v/.

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The outcome of this random event may be thought of as the outcome of flipping an independent coin of bias probability p.u; v/. Because coin flips are independent, it does not matter whether the coin was flipped at the moment when v becomes active or it was flipped at the very beginning of the whole process and is being revealed now. Thus, assume that for each edge u ! v in G, a coin of bias probability p.u; v/ is flipped at the very beginning of the process, independently of the coins for all other edges, and the result is stored for later use in the event that u is activated and v is inactive. Flipping coins in advance, edges in G with coin flips indicate that an activation will be successful and declared live. The remaining edges are declared blocked. Fix the outcomes of the coin flips and then initially activate a seed set S . It is clear how to determine the full set of active nodes at the end of the cascade process [20]: A node x ends up active if and only if there is a live-edge path from some node in S to x. Consider the probability space in which each sample point specifies one possible set of outcomes for all the coin flips on the edges. Let X denote one sample point in this space. Let IG;X .S / denote the total number of nodes activated by the process when S is the initial seed set and X the set of outcomes of all coin flips on edges. Because X is fixed, it is straightforward to see that IG;X .S / can be deterministically expressed as follows. Let R.u; X / denote the set of all nodes reachable from u on a live-edge path. Thus, IG;X .S / is the number of nodes that can be reached on live-edge paths from any node in S , and so it is equal to the cardinality of the union [u2S R.u; X /. For each fixed outcome X , the function IG;X .S / is submodular. To see this, let S and T be two sets of nodes such that S  T , and consider the quantity IG;X .S [ fug/  IG;X .S /. This is the number of elements in R.u; X / that are not already in the union [u2S R.u; X /. It is at least as large as the number of elements in R.u; X / that are not in the (bigger) union [u2T R.u; X /. It follows that IG;X .S [ fug/  IG;X .S /  IG;X .T [ fug/  IG;X .T /: Since the expected number of nodes activated is just the weighted average over all outcomes, it implies that IG .S / D

X

ProbŒX   IG;X .S /:

X

But a nonnegative linear combination of submodular functions is also submodular, and hence IG is submodular, which completes the proof.  To show that IG is submodular for the LT model, one may try to use a similar technique of the proof of Lemma 3. For example, it may be reasonable to consider the behavior of the process after all node thresholds have been chosen. Unfortunately, for a fixed choice of thresholds, the number of activated nodes may not be a submodular function of the initial seed set. Thus, a more subtle technique is needed.

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Lemma 4 ([20]) The function IG is submodular for the LT model. Proof Recall P that each node v has an influence weight p.u; v/ from its neighbor u, where u!v2E p.u; v/  1. Without loss of generality, extend this notation by letting p.u; v/ D 0 if u is not v’s neighbor. Suppose that v picks at most one live edge u ! v 2 E with P probability p.u; v/ and declares the rest of the edges blocked with probability 1  u!v2E p.u; v/. It follows from Lemma 1 that the LT model is equivalent to the reachability via live-edge paths. Similar to the proof of Lemma 3, it can be shown that IG is submodular for the LT model. This completes the proof.  It follows from Lemmas 2–4 that if the influence function IG is polynomial-time computable for the IC and LT models, then SGA is polynomial-time computable and provides a .1  1=e/-approximation ratio to IM. However, computing IG .S / is #P-hard for both the IC and LT models, and so in general, IG is unlikely to be polynomial-time computable. (There are some exceptions. For example, Chen et al. [8] showed that for the LT model, IG can be computed in linear time for acyclic influence graph G.) Thus, one would naturally want to obtain a good estimate of IG , which is justified by the following extension of Lemma 2, with a straightforward modification of the same proof as in [33]. Lemma 5 ([20]) Let f be a monotone submodular function. Let S  be a set that maximizes the value of f over all k-element sets. Suppose that for any " > 0, there is a  > 0 such that S is a k-element set obtained from SGA based on a g that is a .1 C  /-approximation to f . Then g.S /  .1  1=e  "/f .S  /. That is, S is a .1  1=e  "/-approximation to S  . Kemple et al. [20] used Monte-Carlo simulations to obtain a close estimate of, on a given seed set S , the influence spread IG .S / with a large number of runs, resulting in high time complexity. How to obtain a good estimate of IG .S / efficiently remains a challenge. Instead of finding close approximations to IG .S /, researchers have devised new algorithms without needing to evaluate IG .S / (e.g., see [6–8, 26]). Research in this line typically follows two directions: (1) improving the time complexity of submodular greedy algorithm and (2) devising heuristics. Presented below is an example of improving the time complexity of SGA. Note that each edge u ! v in an influence graph G D .V; E/ is determined once whether it is activated with probability p.u; v/. Let G 0 be an induced subgraph of G as follows: Determine, for each u ! v in the graph, whether it is selected and remove all edges that are not selected from G. Let RG 0 .S / denote the set of nodes reachable from S in G 0 . Performing a linear traverse of G 0 , depth first or breadth first, one can obtain RG 0 .S / and jRG 0 .fug/j for all u 2 V .

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The following greedy algorithm was devised by Chen et al. [6]. NewGreedyIC.G; k/ 1. Set S ; and r a positive integer (e.g., r D 20;000) 2. Repeat the following steps for k times 3. Set su 0 for all u 2 V  S 4. Repeat the following steps for r times 5. Construct G 0 by removing each edge u ! v from G with probability 1  p.u; v/ 6. Compute RG 0 .S / 7. For each u 2 V  S : 8. If u 62 RG 0 .S / then set su su C jRG 0 .S /j 9. Set su su =r for all u 2 V  S 10. Set s arg maxu2V S fsu g 11. Set S S [ fsg 12. Output S It is straightforward to see that NewGreedyIC.G; k/ runs in O.krjEj/ time. Simulation results [6] confirm that NewGreedyIC.G; k/ produces an initial seed set S with an influence spread that is very close to what is produced by SGA, but is much more efficient. For an example of heuristics, consider a simple degree discount for the IC model. According to the experimental results obtained by Kempe et al. [20], selecting nodes with maximum outgoing degrees as seeds could lead to larger influence spread than other previously known heuristics but still smaller than that of SGA. Chen et al. [6] presented a degree discount heuristics for the IC model based on this observation, but they discounted an edge from the degree. In particular, suppose that u ! v 2 E and u has been activated as a seed. To consider whether v should be included as a seed by looking at its degree, the edge v ! u will not be counted in the degree. Simulations show the degree discount heuristics, when properly tuned, can achieve almost the same influence spread as SGA in significantly shorter time.

2.5

Other Influence Models

In addition to the IC and LT influence models, other influence models have also been proposed, including weighted cascade (WC) [20], maximum influence arborescence (MIA) [7], and local directed acyclic graph influence (LDAGI) [8]. The WC and MIA models are related to the IC model, while the LDAGI model is related to the LT model. In the WC model, if u is activated in round i and u ! v 2 E, then v is activated by u in round i C1 with probability 1=dv, where dv is the degree of node v. Similar to the IC model, each neighbor of v may activate v independently. Thus, if an inactive node v has m neighbors activated in or before the round i , the probability that v is activated in round i C 1 is 1  .1  1=dv /m .

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In the MIA model, the calculation of the influence from one node to another node depends on the maximum influence path (MIP). In particular, given an influence graph G D .V; E/ and two nodes u and v, let PG .u; v/ denote the set of all paths from u to v in G. Let P D hu D u1 ; u2 ; : : : ; um D vi be a path from u to v. Define the propagation probability of P , denoted by pp.P /, as follows: pp.P / D

m1 Y

p.ui ; ui C1 /:

i D1

Let MIPG .u; v/ denote the path from u to v with the largest propagation probability. That is, MIPG .u; v/ D arg max fpp.P /g: P 2PG .u;v/

Given an influence threshold , define the maximum influence in-arborescence of node u, denoted by MIIA.u; /, as follows: MIIA.u; / D [u2V;pp.MIPG .u;v// MIPG .u; v/: The maximum influence out-arborescence of u, denoted by MIOA.u; /, is as follows: MIOA.u; / D [u2V;pp.MIPG .v;u// MIPG .v; u/: Given an initial seed set S , let the activation probability of any node v 2 MIIA.u; /, denoted by apS .v; MIIA.u; /, be the probability that v is activated when S is the seed set and influence is propagated in MIIA.u; /. In the MIA model, let IG denote the influence spread of S , defined as X IG .S / D apS .v; MIIA.u; //: u2V

As in the IC model, it is straightforward to show the following result. Theorem 4 It is NP-hard to compute a k-node seed set S such that IG .S / is maximized for the MIA model. On the other hand, IG is submodular and monotone, and so MIP can be approximated by SGA within a .1  1=e/-approximation ratio. For more information on efficient greedy algorithms, the reader is referred to [7]. In the LDAGI model, each node v in G is associated with a local DAG LADG.v/, a subgraph of G containing v. Given a seed set S , it is assumed that the influence from S to v is only propagated within LADG.v/ according to the LT model. Let apS .v/ be the activation probability of v when influence is propagated within LADG.v/. Then the influence spread of S in the LDAG influence model, denoted by IG .S /, is given by X apS .v/: IG .S / D v2V

As in the LT model, it is straightforward to show the following result.

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Theorem 5 It is NP-hard to compute a k-node seed set S such that the influence spread IG .S / is maximized for the LDAGI model. For more information of efficient greedy algorithms, the reader is referred to [8].

3

Finding Communities

This section introduces and elaborates a number of heuristic algorithms for finding communities, including a new algorithm of low complexity, which uses random walkers to search for communities. This algorithm has only been reported in a technical report [28]. Simulation results show that the random walker method is both effective and efficient in finding network communities. The existing algorithms for finding network communities can be roughly categorized into graph-partitioning-based algorithms and hierarchical clustering-based algorithms [35]. The Kernighan–Lin algorithm (K–L, in short) is one of the most popular graph-partitioning algorithms, and the agglomerative method and the divisive method are the common hierarchical clustering-based algorithms.

3.1

Graph-Partitioning-Based Algorithms

On a given (weighted) graph G D .V; E/ and a positive integer k, the graph partitioning problem tries to partition V into k subsets such that the size difference of any two subsets is at most 1 and the number of edges (or the summation of edge weights) between different components is minimized. This is an NPhard problem. Graph partitioning has important applications in task scheduling under multiprocessor systems, sparse matrix reordering, parallelization of neural net simulations, particle calculation, and VLSI design, to name just a few. The K–L algorithm is a popular heuristic algorithm in VLSI design, which could be used to obtain rough network communities.

3.1.1 The Kernighan–Lin Algorithm Let G D .V; E/ be a weighted graph with a weight function w W E ! R that maps each edge to a real value. Initially, the algorithm partitions V at random into two subsets V0 and V1 such that their size difference is at most 1. Let w.V0 ; V1 / D

X

w.u; v/

u2V0 ;v2V1

denote the cost of the partition. For each node u 2 Vi .i D 0; 1/, let Eu denote the external weight of u on edges from u to nodes in the other set, that is,

Optimization Problems in Online Social Networks

Eu D

X

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w.u; v/;

v2Vi ˚1

where 0 ˚ 1 D 1 and 1 ˚ 1 D 0. Let Iu denote the internal weight of u within Vi , that is, X Iu D w.u; v/: v2Vi

Let Du D Eu  Iu be the difference between the external and internal weights of u. For each edge .u; v/ 2 E with u 2 A and v 2 B, let Iu;v D Du C Dv . Swap u with v 0 and obtain a new Iu;v . Then 0 Iu;v  Iu;v D Du C Dv  2w.u; v/:

The heuristic of the K–L algorithm swaps nodes between u 2 V0 and v 2 V1 0 iteratively to maximize Iu;v  Iu;v . The time complexity of the K–L algorithm is 3 O.jV j /.

3.2

Hierarchical Clustering-Based Algorithms

The basic idea of the hierarchical clustering-based method is to find relations between nodes in the network and then extract communities from the network by adding or deleting edges. The hierarchical clustering method can be further divided into two subcategories: the agglomerative method and the divisive method. The agglomerative method follows a bottom-up approach. It starts with a completely disconnected network with no edges and proceeds in iterations. In each iteration, an edge is added based on a chosen measure. Pairs of clusters are merged as the algorithm moves up the hierarchy. The divisive method follows a top-down approach. It starts from the original network and proceeds in iterations. In each iteration, it deletes an edge and splits a current subgraph recursively as the algorithm moves down the hierarchy. The process of a hierarchical clustering method results in a tree structure known as “dendrogram” shown in Fig. 1.

3.2.1 The Agglomerative Method This method uses a chosen measure of “structural similarity” of two nodes in the graph. The use of a different similarity measure may result in different communities. Such a similarity measure could either represent metrics of similarities or the strength of connection between nodes. For instance, one may use the distance between two points in a two-dimensional space as a similarity measure. In this case, using different notions of distance, such as the Euclidean distance, Manhattan distance, and maximum distance, gives rise to different variants of the similarity measure. A commonly used similarity measure is based on the comparison of

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Fig. 1 The dendrogram

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

the neighboring nodes. More specifically, if two nodes have the same set of neighbors (or similar sets of neighbors), then they are considered “structurally equivalent” (or “structurally similar”) and should belong to the same community. For example, in an online social network, two persons in a friendship network are considered structurally equivalent if they have exactly the same set of friends [34]. But structural equivalence is rare in reality. Thus, structural similarities are often used instead. In particular, the following “Euclidean distance” [4] is often used to measure the similarity of two nodes u and v in a network. This Euclidean distance is not the same “Euclidean” commonly known:

xu;v D

sX

.au;z  av;z /2 ;

(1)

z¤u;v

where au;z D 1 if z is connected to u, and 0 otherwise. Clearly, if xu;v D 0, then u and v have the exact same set of neighbors. The agglomerative method adds edges from the network in iterations to increase the values of xu;v . As more edges are added, connected subsets begin to form, which represent the forming of network communities. The components produced at each iteration are always subsets of other structures. The tree diagram, that is, the dendrogram, can be used to represent these subsets. Horizontal slices of the tree at a given level indicate the communities that exist above and below the value of the dissimilarity.

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Running the agglomerative method requires a predetermination of how large each community should be. This is a drawback for finding natural communities.

3.2.2 The Divisive Method Different from the agglomerative method that starts from a completely disconnected network and keeps adding edges to it in iteration, the divisive method starts from the original network and keeps deleting edges iteratively. The algorithm introduced by Girvan and Newman [13] is one of the most popular ones. The Girvan and Newman algorithm defines “edge betweenness” of an edge as the number of shortest paths between the two end nodes of the edge. If the network contains communities within which nodes are highly connected and between which nodes are sparsely connected, then all the shortest paths between different communities must go through one of these few edges between communities. Thus, at least one of the edges connecting communities will have a high edge betweenness. Removing such an edge, the groups of nodes are separated from one another, which may help reveal the underlying community structure of the network. Thus, the dendrogram is created from root to leaves. The Girvan and Newman algorithm involves intensive computation. For each iteration, the time complexity of calculating the betweenness of all existing edges in the network is O.jV jjEj/. In the worst case, O.jEj/ edges may finally be removed, and so the worst-case running time is O.jV jjEj2 /.

3.3

The Random Walker Algorithm

This section describes the random walker (RW) algorithm devised by Li et al. [28]. Different from the early algorithms, this algorithm uses random walkers to search for communities. Given a graph G D .V; E/, the RW algorithm first generate, for each node, a number of walkers equal to its degree. The node for which a walker is generated is referred to as the owner node of the walker. Each walker is labeled with its owner node. The node where a walker is currently located is referred to as the current node of the walker. The RW algorithm runs in iterations. In each iteration, each walker chooses from two possible actions to proceed with certain probability. That is, with probability pmove , the walker chooses a node from the neighboring nodes of its current node and moves to it. With probability pjump D 1  pmove , it jumps back to its owner node. Run the algorithm for a few iterations, where the number of iterations t is a parameter chosen by the user who runs the algorithm. When the algorithm stops, each node will have a certain number of walkers. If two nodes have the same set of walkers (or similar sets of walkers), then they are considered structurally equivalent (or structurally similar), and so they should belong to the same community. The algorithm uses the Pearson correlation to measure similarity of two nodes [34]. For any node u and v, if when the algorithm stops node u has a walker from node v, let au;v D 1; otherwise, let au;v D 0. The Pearson correlation of node u and v, denoted

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by x.u; v/, is given below: P xu;v D

k .au;k

 u /.av;k  v /=n ; u v

(2)

where n D jV j; 1X au;k ; u D n k

1X .au;k  u /2 u D n k

are the mean and variance of node u. A larger value of xu;v would indicate that node u and v are more likely to be in the same community. The RW algorithm forms communities one at a time according to the similarities of nodes. In particular, it starts from an empty set S . It selects a pair of nodes with the highest similarity among all possible pairs that has not been selected and adds them to S . Next, expand S by repeatedly selecting new nodes in the remaining set of nodes, until a certain halting criterion is met. The halting criterion can be that either the size of the community constructed reaches a predefined threshold or the similarity between arbitrary node in the community and any node outside the community is smaller than a predefined threshold. Save S as a community and repeat the same process for the remaining nodes to find the next community, assuming that the nodes added into S will not belong to any other community. The RW algorithm stops after all nodes are considered. It is evident that the time complexity of the algorithm is O.t.jV j C jEj//. Extensive simulation using both halting criteria was executed, which show that the RW algorithm is highly effective. The accuracy of the RW algorithm is computed as follows: Nr X Nc 1 X fi Avg. accuracy D ; Nr i D1 i D1 s i where Nr is the total number of simulation runs, Nc the number of communities, f i the number of members that were correctly found for community i , and s i the size of the community i . For example, one set of simulations runs on two types of graphs of 1,000 nodes, with five communities of equal size. The edges in the first type are constructed with plow D 0:05 and phigh D 0:9, while in the second type with plow D 0:1 and phigh D 0:75. Each simulation runs for t D 5 iterations with ptravel D pjump D 0:5. For the first type of graphs, the RW algorithm achieves on average near 100 % accuracy using both halting criteria, where the size threshold is set to 200 and the similarity threshold is set to 2:0. In the second graph, the algorithm achieves on

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average near 100 % accuracy if the first halting criterion is used, where the threshold of the community size is set to 200. If the second halting criterion is used, where the similarity threshold is set to 1:5, the RW algorithm achieves on average 93 % accuracy.

4

Data Collection from OSNs

Online social networks are popular platforms for people to connect, share information, and interact with each other. There have been intensive efforts to measure and analyze various characteristics of online social networks in recent years, including topological characteristics, user behaviors, and interactions between users. Popular OSNs, such as Facebook and Twitter, are continuously growing to garner hundreds of millions of users. The sheer size of these OSNs and the dynamics of user activities make it impractical to measure the entire network. As a result, researchers often collect a small but representative sample of the network to study the properties of the network and test their algorithms. This section reviews some of the recent studies on the measurement and analysis of OSNs, describes the graph sampling methods that have been used to collect data over OSNs, and presents an example of collecting data from MySpace with interesting statistical results.

4.1

Measuring and Analyzing OSNs

Mislove et al. [31] carried out a large-scale study on measuring and analyzing the structure of multiple online social networks, including Flickr, YouTube, LiveJournal, and Orkut. Their results confirmed that these OSNs are small world and scale free, and the degree distributions satisfy the power law. They also offered the following observations: (1) The in-degree of user nodes tends to match the out-degree. (2) Each OSN contains a densely connected core of high-degree nodes. (3) This core links small groups of strongly clustered, low-degree nodes at the fringes of the network. Ahn et al. [1] measured and compared the structures of the following three online social networking services: Cyworld, MySpace, and Orkut, with respect to the degree distributions, clustering coefficients, degree correlations, and average path length. Krishnamurthy et al. [22] identified distinct classes of Twitter users and their behaviors, geographic growth patterns, and size of the network. Gauvin et al. [11] measured and analyzed the gender-specific user publishing characteristics on MySpace using the technique of uniform sampling. They showed that there are recognizable patterns with respect to profile attributes, user interactions, temporal usages, and blog contents for users of different genders. In particular, gender neutral profiles (most of them are music bands, TV shows, and other commercial sites) differ significantly from normal male and female profiles for several publishing patterns. Caverlee and Webb [5] analyzed the frequencies of

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words occurred in MySpace wall posts according to genders and ages. They looked at various statistics of male and female profiles in MySpace, including the degree distribution of friends, user demographics, language, and privacy preference. Chun et al. examined the question of whether interactions between OSN users would follow the declaration of friends using the guestbook logs of Cyworld, the largest OSN in Korea. They constructed an activity network that reflected the online interactions between users and found that its structure is similar to the friend relationship network. However, when only reciprocated links are considered, the degree correlation distribution exhibited much more pronounced assortativity than the friends network. Wilson et al. [41] studied the relationship between friends and publishers and whether social links (social network friendships) were valid indicators of real user interaction in Facebook. They built a graph based on user interactions (i.e., messages exchanged between users). They observed that the interaction graph has much lower average node degree than the Facebook friends graph. Benevenuto et al. [3] studied user workloads on four popular OSNs: Orkut, MySpace, Hi5, and LinkedIn. They analyzed key features of the social network workloads such as how frequently people were connected to social networks and for how long, as well as the types and sequences of activities that users conducted on these sites. They showed that browsing, which cannot be inferred from crawling publicly available data, accounted for 92 % of all user activities. Consequently, compared to using only crawled data, considering silent interactions like browsing friends’ pages increases the measured level of interactions among users.

4.2

Sampling Methods of OSNs

A number of graph sampling methods have been adopted and developed for measuring large-scale OSNs. These methods can be classified into the categories of breadth-first search, random walk, and uniform sampling, among others.

4.2.1 Breadth-First Search (BFS) The breadth-first-search traversal method was commonly used in recent measurement studies of OSNs (see, e.g., [1, 31, 41]). It is a classic graph search algorithm that begins from an initial seed node and explores all the neighboring nodes. The process is repeated, and at each iteration the earliest explored but not-yetvisited node is selected and its neighbors are explored. It is easy to see that the BFS method will discover all nodes within some distance from the seed node, but the sampled nodes are constrained within the connected component of the seed node. An incomplete BFS will cover and yield a full view (all nodes and edges) of some particular region in the graph. While the resulting sample graph readily enables the study of topological properties of the network (e.g., clustering coefficient, community structure), it is known that BFS introduces a bias toward nodes with high degrees [2, 25].

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Kurant et al. [24] quantified the degree distribution bias of BFS sampling. For a random graph RG.pk / with a given (and arbitrary) degree distribution pk , the node degree distribution expected to be observed by BFS as a function of the fraction of covered nodes (qk ) is given by pk .1  .1  t.f //k / qk D P ; l l pl .1  .1  t/.f //

(3)

where t.f / is the time that a fraction f of the nodes is covered, whose numerical value is straightforward to find. The bias of BFS monotonically decreases with increasing fraction of sampled nodes f , and there is no bias when BFS is complete (f D 1). Kurant et al. also showed that, for RG.pk /, all commonly used graph traversal techniques (breadthfirst search, depth first search, forest fire, and snowball sampling) lead to the same bias. Given a biased sample; they also showed how to correct the bias by deriving an unbiased estimator of the original node degree distribution.

4.2.2 Random Walk The random walk technique, successfully used for the World Wide Web (WWW) [19] and peer-to-peer network (P2P) [15, 36, 39] sampling, has been recently applied to OSN sampling [14]. In the classic random walk approach [29], a random walker starts from a node (u) and visits its neighboring nodes uniformly at random. The process continues and the given graph is sampled by the random walker. Denote the degree of node u by du , the probability that a node v is chosen as the next-hop node is given as follows: ( Pu;v D

1 du

if v is a neighbor of u, 0 otherwise.

(4)

For a network with m edges, the steady-state distribution of the random walker being at a node v is

.v/ D

dv : 2m

(5)

A linear bias exists on high-degree nodes. Nodes with higher degrees will be visited more often than those with lower degrees in proportion to the node degree. To correct this bias, one can apply the Metropolis–Hastings algorithm [18, 30] to make the random walk converge to a desired probability distribution. In particular, it can be shown that the following transition probability achieves the uniform distribution among nodes [24]:

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Pu;w

  8 du 1 ˆ if v is a neighbor of u, min 1; < du dv D 1P y¤u Pu;y if v D u, ˆ : 0 otherwise.

(6)

It can be seen from the above equation that, to correct the degree bias in the classic random walk, the probability of visiting a node has been adjusted to reduce the transition to high-degree nodes. In a random walk, the accuracy of estimating an underlying characteristic depends on the graph structure and the characteristic being estimated. The estimates may be inaccurate when the random walker gets “trapped” in a certain region of the graph. To mitigate this problem, one can start multiple independent random walks (MultipleRW) in different parts of the network [14]. This will reduce the chance of the sampling process being “trapped” in a region and has the advantage of being amenable to parallel implementation from different machines or different threads in the same machine. However, it was shown [37] that the MultipleRW approach may fail to reduce estimation errors. Starting vertices of MultipleRW in a component proportional to the number of edges of the component is a key factor to determine the accuracy of MultipleRW. While this approach can successfully mitigate estimation errors caused by disconnected components, it is not clear how it can be implemented in realistic OSNs such as MySpace and Facebook. Ribeiro and Towsley [37] devised a new m-dimensional random walk sampling method called Frontier sampling. It performs m-dependent random walks in a graph. Starting from a collection of m randomly sampled vertices, the Frontier sampling method preserves all of the important statistical properties of a regular random walk (e.g., nodes are visited with a probability proportional to their degrees). For a given sampling budget of B steps, let c be the cost of randomly sampling a node. The sampling algorithm is given as follows: Frontier Sampling 1 n 0 (n is the number of steps) 2. initialize L D .v1 ; : : : ; vm / with m randomly chosen nodes (uniformly) 3. repeat P 4. select u 2 L with probability du = 8v2L dv 5. select an outgoing edge of u, .u; v/, uniformly at random 6. replace u by v in L and add .u; v/ to the sequence of sampled edges 7. n nC1 8. until n  B  mc In Frontier sampling, while nodes are visited with a probability proportional to their degree, it can be shown that the joint steady-state distribution (the joint distribution of all m nodes) is closer to being uniform (the starting distribution) than k-independent random walkers, for any k > 0. This property has the potential to dramatically reduce the transient period of random walks. In simulations using realworld graphs, Frontier sampling can indeed effectively mitigate the large estimation

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errors caused by disconnected or loosely connected components that can “trap” a random walker and distort the estimated graph characteristic in a variety of scenarios.

4.2.3 Uniform Node Sampling As the name suggests, the uniform node sampling technique samples all the nodes in a graph uniformly and independently. There is no inherent measurement bias regardless of the distribution of the user IDs. When applicable, uniform node sampling can be used to establish the “ground truth” of the network properties and has the advantage of sampling disconnected graphs. Several recent studies have used the uniform node sampling technique to measure the network properties of MySpace and Facebook [11, 14]. For the uniform node sampling to work, however, an OSN must allow access to user profiles using unique numeric IDs. As such, it does not serve as a general solution for OSN sampling. Also, this approach may become highly inefficient when the space of user IDs is large and sparsely occupied, which was observed in both MySpace and Facebook [11, 14]. Since access to OSNs is often rate limited, the sparse distribution of the ID space will significantly increase the time to collect the data required for sampling.

4.3

MySpace Data Collection and Analysis

When sampling an online social network, the walker (a.k.a. crawler) collects the user profile information such as age, gender, education, work, location, friends/followers, and blogs, among others. The exact information that can be obtained from a user profile depends on the specific OSN design and the user’s privacy setting. The study carried out by Gauvin et al. on MySpace [11] provides an example of collecting and analyzing data of OSNs. MySpace is a major online social networking site. Since launched in 2004, it has attracted a large number of users. Typical MySpace users are teenagers, music bands, artists, and commercial sites. MySpace is particularly popular among young people and music communities. Each MySpace user owns a profile containing basic personal information including name, age, gender, and location. A user profile whose owner’s gender is neither female nor male is treated as a gender “neutral” profile. A neutral profile in the data sets occurs when the gender is not specified. Many of the gender neutral profiles were commercial sites for music bands and TV shows. Despite the importance of these profiles, Gauvin et al. were the first to look at the publishing tendencies of gender neutral, commercial profiles. In particular, Gauvin et al. [11] developed a crawling tool to sample MySpace, uniformly at random, to collect user profile attributes (name, gender, age, location, friends, among other things) and their corresponding blog information. Six million possible user profiles were randomly sampled between September 2008 and May 2009. When blogs for these random profiles were found, each blog was scanned to obtain the publisher profile attributes, the time stamp, and the content information.

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A profile may be public (the default option) or private depending on its privacy setting. Private profiles only share the owner’s name, gender, age, and location. No blog or friend information could be retrieved from these sites. As a result, the analysis was limited to the blogs posted on public profiles. The results of this data collection process produced 546,829 public and 68,141 custom profiles of which 92,653 sets of blogs were obtained and 1,867,299 blogs were collected. The number of blogs contained within a randomly scanned profile ranged from 0 to 85,643. To determine the publisher characteristics, 1,768,884 profiles were scanned.

4.4

Member Profile Patterns

4.4.1 Age, Gender, and Privacy In the collected data set, female, male, and neutral profiles constitute, respectively, 52.8 %, 37.7 %, and 9.5 % of the total valid IDs. A larger percentage of female users (57.1 %) were privacy aware than male users (39.2 %). Almost all (99.999 %) gender neutral profiles were public as they were typical commercial sites trying to attract people to visit their pages. Figure 2 shows the age distribution of private account profiles. The majority of profile owners declared their ages in the range of 15–30. Beyond this range the distribution falls off rapidly, only to increase around age 69 and age 100 due to a

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Fig. 2 MySpace private account and age distribution

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common practice among underage users to elude the age restriction by selecting an exaggerated age. Two age groups deserve a special mention. The first is the absence of ages from 1 to 12. MySpace protects users in this age group and does not create their profiles. The second is the age group under 16. MySpace automatically sets user profiles in this group to be private. This restricts the information that can be obtained for these sites. It can be noted that there was a large number of zero-age profiles. This is probably because accounts of underage owners were kept private. Also note that the data collected do contain some user profiles with ages under 16. This is contradictory to the current MySpace policy. This inconsistency may be the result of MySpace’s policy change, where an old policy may actually allow users under age of 16 to publish their ages.

4.4.2 Publishers Versus Friends In an online social network, friend relationships do not necessarily reflect real online interactions between users. The data sets depict an interesting relationship between the total number of friends of a user and the number of friends that actually publish on the user’s blog wall. Figure 3 plots two sets of complementary cumulative distribution functions (CCDF), one set corresponding to the number of friends and the other corresponding to the number of distinct publishers among those friends. In each set, a CCDF is obtained for each profile gender: female, male, and neutral. It can be seen that the friendship and publisher distributions for male and female profiles exhibit similar behaviors except at the tail, where the female distribution extends farther and the male distribution falls off sharply. The distributions for gender neutral profiles lie above the male and female distributions. This indicates that neutral profiles tend to have more friends and publishers than normal male and female profiles. This trend is consistent with the observation that neutral profiles were generally music bands or other commercial sites with a large number of friends publishing comments on the music, movie, or product being promoted. Figure 4 plots the number of distinct publishers versus the number of friends of MySpace users. It is observed that for most users, as the number of friends increases, only a small fraction of the users actually interact with each other through blogs. This is consistent with the observations made in [41] for Facebook users. Interestingly, the publisher-versus-friend plots for male and female profiles nearly coincide with each other, while the plot for gender neutral profiles falls far below. This indicates that although gender neutral profiles tend to have more friends than normal male and female profiles, an even smaller fraction of these friends actually write on their blog walls. A possible explanation for this difference is that many users befriend commercial sites just to browse or receive information from the sites. The relationships between friends and gender neutral commercial sites are less “active” than those between normal users. It is worth noting that some profiles contain more distinct publishers than friends. This is a result of profile owners removing friends from their profiles and occurs rather rarely.

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Fig. 3 Distribution of number of friends and publishers

4.4.3 Profile Owner and Publisher Age Difference Figure 5 shows a density graph of the ages of publishers versus those of the profile owners. It can be observed that the density is greatest along the diagonal line (zero age difference between publishers and profile owners)and drops quickly as the age difference increases. This indicates that users tend to interact with friends of similar ages. A large fraction (75 %) of publishing activities occurs between people within a 5-year age difference. The shape of the age difference distribution can be described by an exponential function f .x/ D a exp.bkxk/, where a D 0:334 and b D 0:470, for age differences between 0 and 8 years. The shape of known distributions is unlikely to match the distribution of age differences larger than 8, likely because a fraction of the young users tend to lie about their age. Note that the other small islands in the figure are artifacts caused by the following two groups of users: underage users below 14 whose age information is not available and set to 0 and users with exaggerated ages around 69 and 100.

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100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Profile Owner Age

Fig. 5 Publisher age versus owners

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Fig. 4 Distribution of friends versus publishers

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Temporal Usage Patterns

Three specific elements of time are analyzed. They are time of day, day of week, and monthly usage patterns, as shown in Fig. 6. All blog posting times are normalized to the local wall-clock time of the individual profiles. It can be observed from Fig. 6a that there is a significant increase in publishing activity from morning to afternoon, with the peak usage time at 10pm and a significant drop at midnight. The low activity between midnight and 12pm may reflect the times of rest and busiest time for labor in the day as far as school and work are concerned. As for the daily publishing pattern characterized in Fig. 6b, weekdays have a higher degree of activity than weekend days. Gender analysis for diurnal publishing patterns does not indicate any significate pattern difference between genders. The total number of blogs and content per day for each month was also collected. The reason for gathering this information was to ascertain whether trends in publishing patterns can be used to predict the load of a social network server and determine peak usage times. This information may be used to determine false positives during spam detection and other security filtering policies. Many months displayed evenly distributed activities on each day where the daily blog counts did not deviate by more than 200 blogs for the collection set analyzed. There were some months (February, November, December), however, that stood out with respect to predicting publishing patterns, mainly due to the specific calendar events within the months. An example is given in Fig. 6c for February, which is a low publishing month in general, but there is a sharp contrast between the publishing activities on the 14th and those on other days. Similar high publishing activities are observed for other major calendar events including New Years Day, Mother’s Day, 4th of July, Halloween, Thanksgiving, and Christmas.

4.6

Blog Content Patterns

A blog’s entry contains four types of content, namely, words, hyper-references (HREFs), images, and objects. In most cases the objects were ColdFusion scripts or other animated objects. ColdFusion objects allow users to create animated objects used to make their profiles more attractive. Each blog is a composition of these four content types. The inclusion and makeup of these different content types is a reflection of each individual’s writing style. The goal is to determine if specific content patterns can be identified. Figure 7 depicts the complementary cumulative distribution functions (CCDF) of the counts of words, HREFs, images, and objects in a blog entry. It can be seen that the counts of these content types decay roughly according to power laws in certain regimes. In a blog entry, word count dominates the counts of other types. Object count is the smallest among all types. HREF and image counts are similar to each other and lie somewhere in-between. Of course, the counts of the different content types do not reflect their actual sizes (bytes). The size of an image is typically

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significantly larger than that of a word. In general, users tend to post short messages. For example, only 10 % of the blog entries contain more than 50 words; 4 % of the blog entries contain 3 or more images. The use of objects is even less frequent. There are a number of recognizable patterns with respect to the count of these content types across different genders. The word count distribution of gender neutral profiles drops off drastically around word count 410. This may be due to the need of a commercial site to have very specific and focused contents (e.g., advertisements) that fit people’s attention spans. Another recognizable pattern is that females tend to publish more HREFs (hyperlink references) and images than male and neutral profiles.

5

Multilayer Friendship Modeling in Location-Based OSNs

How users are connected through an online social network is an interesting question. Intuitively, the formation of online friendship may reflect to certain degree the real-world counterpart. People who have mutual interests, are geographically close, or belong to the same social communities (e.g., churches, volunteer groups, and special interest groups) are more likely to become friends. The challenge is to devise adequate metrics to represent this intuitive understanding with a quantitative model. While there are a number of studies on structural properties of social networks, such as the distribution of node degrees, friendships (or connections) between users may be modeled with other user attributes.

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In the context of influence maximization, a quantitative friendship model can provide an interesting perspective for studying influence propagation in OSNs, for the strength of the friendship often reflects the likelihood of how people influence each other. For example, for any two people who are OSN friends, it would be reasonable to combine their behaviors, such as whether they would buy the similar products or like the same brands, with the friendship model to come up with an empirical influence propagation model. For any two people who are not OSN friends, it would be useful to predict the possibility and the strength of their connection if they will ever become friends. Such information would be useful to find the most influential individuals in the network. This section presents a friendship model for Brightkite, a location-based OSN. The primary service mode of Brightkite is to let users post and share updates with their friends or with the public. A user update could be a text note with a 140character cap, a photo, or a check-in to announce a location. The user performs a check-in when they come to a new location. This is the typical usage scenario of Brightkite. All note and photo updates will be associated with the location of the user’s latest check-in. Brightkite assumes that the user stays at the same location until they perform another check-in somewhere else. To study how Brightkite users share location and how they become connected, Li and Chen [27] collected Brightkite user updates posted from March 21, 2008, to December 15, 2009. There were 4,460,161 updates posted by 70,337 users. They also collected each user’s profile and friend list. They then built a threelayer friendship model to correlate the relationship between friend connections with attributes of their profiles, social graphs, and mobility patterns. For convenience, this model is referred to as the three-layer friendship (TLF) model. To validate the TLF model, they used new friend connections during the 168 days after the training data were collected. They showed that the TLF model can better predict user friendships than the classical naive Bayes and J48 tree algorithms.

5.1

A Multilayer Friendship Model

In social communities, friend connections have strong correlations to the social graph structures, profile attributes of individuals, and geographic locations. For social graph structures, the social distance between friends can be used to derive the empirical model of connections. For the profiles of individuals, the common interests between friends can be used to infer their friendships, which are reflected to some degree by the Brightkite tags in user profiles. For the geographic locations, it was observed that friends tend to send updates within a small geographic range with high probability. The TLF model exploits the correlations of friendship distance, tag attributes, and mobility patterns. In a social network Gs .Vs ; Es / where each node vs denotes a user and each edge es denotes the friendship between two nodes. For any node (user) pair .a; b/, its attribute(s) can be calculated. Given the attribute values of two users, the empirical probability of them being friends can be computed as follows: Assuming that mv is the attribute value, let Pf .mv / denote the probability of value mv for friend pairs and

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Pnf .mv / denotes the probability of value mv for non-friend pairs. Using the Bayes’ rule of conditional probability, the probability P .abjmv / of existence of an edge between the two users (meaning they are friends) while observing attribute value mv is

P .abjmv / D

P .mv jab/  P .ab/ ; P .mv /

(7)

where P .mv jab/ denotes the conditional probability that the attribute value mv occurs given that .a; b/ is a pair of friends, P .ab/ denotes the probability that node a and node b are friends, and P .mv / denotes the probability that mv occurs in all possible pairs. Thus, P .mv jab/ D Pf .mv /; P .ab/ D

jEs j ; CjV2 s j

(8) (9)

P .mv / Š Pnf .mv /;

(10)

where CjV2 s j denotes all possible user pairs. Equation 10 is satisfied because CjV2 s j is much larger than jEs j. In the social graph retrieved on April 14, 2009, jEs j D 175; 309 and CjV2 s j D 50; 842  50; 842=2 Š 1:29  109 : The probability of mv in non-friend pairs is approximately equal to that in all possible pairs. It follows from Eqs. 8–10 that Eq. 7 becomes

P .abjmv / D

Pf .mv /  jEs j Pnf .mv /  CjV2 s j

D F .mv / 

jEs j : CjV2 s j

(11)

Let F .mv / denote the ranking factor defined as follows: F .mv / D

Pf .mv / : Pnf .mv /

(12)

Since jEs j=CjV2 s j is a constant for all user pairs, to determine which pair has a higher probability to become friends, it suffices to calculate F .mv / and compare its value over different user pairs.

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Identifying the existing 175,309 friend pairs from the data set, it is straightforward to compute Pf .mv / using this empirical data. Since the number of non-friend pairs is huge, it would be hard to explore the entire sample space. Instead, Li and Chen [27] randomly selected 340,000 non-friend pairs and calculated Pnf .mv / from these samples. In the real world, a single attribute is insufficient to recommend user’s friendship. For example, potential friends could share the same tag(s) because of mutual interest(s), but may never send updates in nearby locations. Thus, if one only uses the location-based attribute, it would be difficult to accurately model the pair relationship. Instead, Li and Chen selected three attributes (details later) to calculate the probability of friends. Let P1 , P2 , and P3 denote the probabilities that were calculated by three different attributes, respectively, and if they are independent, then P .abjmv1;v2;v3 / D 1  .1  P1 /  .1  P2 /  .1  P3 / D 1  .1  P1  P2 C P1  P2 /  .1  P3 / D P1 C P2 C P3 C O.P 2 /  O.P 3 / Š P1 C P2 C P3 :

(13)

The values of O.P 2 / and O.P 3 / are extremely small, and so they can be ignored. For Eq. 11, jEs j=CjV2 s j D 1:36  104 in the current social graph, and each rank factor F .mv / is usually no more than 300. Ignoring the constant value, user friendship can be ranked by comparing the summation of the ranking factors of the three attributes: F D F1 C F2 C F3 :

(14)

The three attributes used in this calculation will be discussed below. Note that in Eq. 14, if one or two attributes are missed, the model is still valid.

5.1.1 The Social Graph Layer In the constructed social graph Gs .Vs ; Es / obtained from Brightkite, where jVs j D 50;842 and jEs j D 175;309, each node vs is a Brightkite user and each edge es represents a pair of friends. Since the “friendship” in Brightkite is mutual,1 Gs is an undirected graph. Define the friends distance DSi;j of vi and vj in Eq. 15, where ei;j is the link between vi and vj :

1

At the time of retrieving the data trace, a friendship in Brightkite was mutual. Starting from December 2009, friendships were changed to directional. In other words, like Twitter, each user has a “friends list” and “fans list” corresponding to the “following” and “follower.”

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Fig. 8 Histogram of distance DS of friends/non-friends in the social graph

DSi;j D geodesic distance.vi ; vj / in Gs0 .Vs ; Es  ei;j /:

(15)

In Eq. 15, geodesic distance is the shortest path of two nodes in the social graph. Equation 15 means if vi and vj are friends, then remove the existing edge between i and j and calculate their geodesic distance in the resulting social graph. Figure 8 depicts the histogram of normalized DS including non-friend pairs (the bar at the left-hand side) and friend pairs (the bar at the right-hand side). It turns out that the distance of more than 49.9 % friend pairs was 2 and more than 34.5 % was 3. On the other hand, the distance of less than 7.8 % non-friend pairs was less than 4, and 28.2 % of non-friend pairs were disconnected nodes in the social graph, which suggested that the social graph in Brightkite was quite sparse. The ranking factor of the social graph layer was calculated by dividing the friend pairs’ proportion by non-friend pairs’ proportion (Eq. 12). Consider the situations that the ranking factor is larger than 1 and ignore the other situations. Then 8 < 94:81 Fs .ms / D 4:75 : 0

ms D 2 ms D 3 others:

(16)

5.1.2 Tag Graph Layer To model the relationship between friendships and users’ tags, Li and Chen [27] selected 1,000 popular tags to build a tag graph, denoted by Gt .Vt ; Et /. Since Brightkite allows users to use any words as their tags, it makes sense to only focus on the popular tags to obtain statistical results. The graph Gt is an undirected complete graph, and each node corresponds to a tag word. Connect each pair of tags with a nonnegative value wt as weight to indicate the closeness of them, which is defined as follows:

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wt1 ;t2 D

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Pf .t1 ; t2 / Pa .t1 ; t2 /

D

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D

Pf .t1 ; t2 / ; 2  Pa .t1 /  Pa .t2 /

(17)

where Pf .t1 ; t2 / denotes the probability that tag t1 and tag t2 are shared by friends in all friend pairs. This probability can be calculated by first counting the number of pairs, where one user had tag t1 and the other had tag t2 , to get CNTf .t1 ; t2 /, then dividing it by the total number of friend pairs jEs j. That is, Pf .t1 ; t2 / D

CNTf .t1 ; t2 /: jEs j

(18)

Note that t1 and t2 could be the same. Let Pa .t1 ; t2 / denote the probability that for one user pair, one user has tag t1 and the other has tag t2 , in all possible user pairs, and Pa .t1 / denote the probability that tag t1 occurs in all users. The probability Pa .t1 / can be calculated by first counting the number of users who used tag t1 in all users to get CNTa .t1 / and then dividing it by the total number of users jVs j. That is, Pa .t1 / D

CNTa .t1 /: jVs j

(19)

To rank how close tags are to each other, let wt1 ;t2 D

CNTf .t1 ; t2 / ; CNTa .t1 /  CNTa .t2 /

(20)

which represents the weight of the tag pair of t1 and t2 . If neither t1 nor t2 is on the list of top 1,000 tags, then let wt1 ;t2 D 0. It turns out that the tag pairs with the highest weight are music versus photography and photography versus traval. The tag pairs with the lowest weight are film versus php and linux versus snowboarding. Let user i ’s tag list be .ti1 ; ti 2 ; : : : ; tiM / and user j ’s tag list be .tj1 ; tj 2 ; : : : ; tjN /. Define the tag metric mt .i;j / as follows: mt .i;j / D

M N X X

wti m ;tj n :

(21)

nD1 mD1

Figure 9 shows the histogram of tag metrics including both friend pairs (the bar on the right-hand side) and non-friend pairs (the bar on the left-hand side). It turns out that for most metric values, the tag metric distribution of friend pairs is

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Fig. 9 Histogram of tag metric Mt of friends and non-friends

Fig. 10 Rank factor of tag metric

larger than that of non-friend pairs, especially in large value ranges, which means that friend pairs tend to share closer tags. The ranking factor of tag metric can be calculated as follows: Pf .mt / : (22) Ft D Pnf .mt / Figure 10 shows samples of Fm by cross markers. To guarantee the statistical significance of Pnf .mt /, the tag count was required to be at least 50 for each sample in the figure. That is, values of Pnf .mt / for mt > 0:12 were ignored. An exponential function was applied to fit the samples for producing a fitting function, shown as the

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line of points. The two dotted lines depict the 95 % prediction bounds by the fitting function. When the tag metric is larger than 0.1, the values of the samples tend to be infinitely large, and so an arbitrary large number, 100, was assigned. The function Ft can be denoted by 8 0 0 otherwise:

(25)

Figure 12 shows the probability density function of the location metric for both friend pairs and non-friend pairs. The friend pairs tend to have smaller values of the location metric. To keep the statistical significance, samples with location metric larger than 500 were ignored, for there were few non-friend pairs in this range. Similar to the tag metric, define the ranking factor of location metric as follows: Fl D

2

http://www.census.gov/.

Pf .ml / : Pnf .ml /

(26)

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Fig. 11 An example of map splitting

Fig. 12 Probability distribution of location metric 1

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Fig. 13 Rank factor of location metric 1

Figure 13 depicts the samples of Fl by cross markers. The power law function was used to fit the samples for producing a fitting function, shown as the line of points. The two dotted lines show the 95 % prediction bounds by the fitting function. The situations where ml > 500 were ignored for statistical significance, and an arbitrary large number, 300, was assigned for ml > 500. 8 500:

(27)

Lacking population data outside the USA, another location metric was used to calculate the user pairs who posted updates outside of the United States. Assuming that user i sent updates in the locations .li1 ; li 2 ; : : : ; liN / and each location contained a number of updates .ci1 ; ci 2 ; : : : ; ciN /, respectively, let Dist.l1 ; l2 / denote the geographical distance of locations l1 and l2 . Define the location metric m0l.i;j / between user i and j as follows: m0l .i; j / D

8m2.1;M /;8n2.1;N /

min when;



Dist.li m ;lj n / ci m Ccj n

Dist.li m ;lj n / 14 were ignored because in these situations Fl0 < 1. An arbitrary large number, 100, was assigned for m0l < 0:01: 8 < 100 0 0 Fl .ml / D 0:5568  m0 0:8867 C 1:612 l : 0

m0l < 0:01 0:01  m0l  14 m0l > 14:

(29)

It can be seen that the second location metric is not as good as the first one, since it considers the two closest update locations and ignores the others. On the other hand, the first location metric reflects all update closeness between any two users. Finally, if the majority of updates of both users were in the conterminous 48 states of the USA, the first location metric was used; otherwise, the second location metric was used.

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Fig. 15 Rank factor of location metric 2

5.2

Friendship Ranking

Given a user pair, its friendship ranking can be calculated as follows: F .ms ; mt ; ml / D Fs .ms / C Ft .mt / C Fl .ml /:

(30)

A larger value of F .ms ; mt ; ml / indicates that the two users would have a higher probability to be friends. For any given user, the system can calculate the friendship ranking value with other users and returns a list of high rank users as recommended friends to that user.

5.3

Evaluation

The proposed friendship model was validated using the changes of friends lists from April 14, 2009, to October 4, 2009. A total of 19,092 new friend pairs were found. Then randomly select another 200,000 non-friend pairs that have no overlaps with the 340,000 non-friend model training pairs. The two groups of user pairs were test sets for model evaluation. Figure 16 shows the distributions of the ranking values of the two test sets. It is obvious that the distribution of ranking values of friend pairs and non-friend pairs is different. Given a threshold of 3, there were 93.01 % friend pairs whose ranking value is larger than 3. On the other hand, there were 86.32 % non-friend pairs whose ranking value is smaller than the threshold.

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5.3.1 Model Prediction Results One of the applications of this friendship model is friend recommendation. Given a large user set (there were over 70,000 users in Brightkite, for example), one can use this model to rank all users against a specified user and present a recommended user list to the user as suggested friends based on the ranking values. On the other hand, this model can also be used to predict friendship in user pairs. To evaluate the model, if most of user pairs with high ranking value indeed became friends after a period of time, then the model is valid and effective. The receiver operating characteristic (ROC) curve was used to compare the ranking performance of the proposed model with other classical data mining algorithms, as shown in Fig. 17. The ranking performance can also be considered as accuracy of prediction. To plot the ROC curve, first calculate the ranking values of all test sets (19,092 new friend pairs C 200,000 non-friend pairs) and sort them in decreasing order of ranking values. If two pairs have the same ranking value, their order is random. Going through from top to bottom in the sorted list, draw points in the ROC figure from original point (0,0). If the current ranked user pair is indeed a friend, move the cursor along the y-axis with the normalized distance y D 1=19; 092 and draw a point. If the current ranked user pair is not a friendship, move along the x-axis with distance x D 1=200; 000 and draw a point. After drawing a point, check the next user pair and repeat the process until no more data is left. Viewing the ranking result as friendship prediction, the x-axis denotes the false positives and y-axis denotes the true positives. For perfect friendship ranking, all the true friend pairs should aggregate at the top of the ranking values, the ROC curve thus yields the line from original point (0,0) to the upper left corner (0,1) and to the upper right corner (1,1). A completely random ranking, however, would give a diagonal line from the left bottom to the top right corner.

Optimization Problems in Online Social Networks Fig. 17 ROC curves of friendship ranking

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The non-diagonal solid line is the ROC curve for the proposed model. It can be seen that there were more than 80 % true friend pairs against about 10 % false positives. In other words, by using the friendship ranking, about 80 % friend pairs aggregate at the top 10 % of the data set. The performance of the three-layer friendship model is superior to the classical data mining algorithms. In particular, the standard probabilistic naive Bayes classification algorithm and the C4.5 revision 8 decision tree learner algorithm (J48) were carried out on the same data sets used by the three-layer friendship model. Figure 17 also depicts the performance of these two algorithms, where the dashed line denotes the performance of naive Bayes algorithm and the dotted line denotes the performance of the J48 decision tree algorithm. The ROC area was calculated to quantify the overall performance, which is the area below the ROC curve. Of course, the ROC area of the perfect ranking is 1 and the random ranking is 0.5. Table 1 shows the comparison results of the three algorithms. Though the results are close, the three-layer model has a larger ROC area than the other two models, indicating better performance. For a friend recommendation application, it is more interesting to compare user pairs for high ranking values. This is because when a recommended list of friends is provided to a user, they would probably only browse the top, say 100, suggestions and ignore the rest. Considering more than 70,000 users in Brightkite, the top 1 % or even top 0.1 % recommended users are the most important. To show the performance of the three algorithms in the part of high ranking values, it suffices to show a sub-figure of magnifying ROC curves in Fig. 17. This figure includes the three ROC curves of the x-axis from 0 to 0.05. The three-layer friendship model shows a superior performance over the other two algorithms in most cases, where J48 is better than naive Bayes. Table 2 shows the detailed numbers on the friend-pair prediction against different non-friend pairs (false positive rate). Each row indicates prediction results of the three algorithms in a given top user pairs. For example, the second row of Table 2

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Table 1 The ROC area of three algorithms

Multilayered

Naive Bayes

J48 tree

0.906

0.882

0.869

Table 2 Predication performance comparisons Top 10,000 5,000 1,000 500 100 50

Friend pairs number and percentage Multilayer Naive Bayes 8; 672 86:7 % 6; 498 4; 804 96:1 % 3; 277 982 98:2 % 639 487 97:4 % 324 98 98:0 % 68 49 98:0 % 37

65:0 % 65:5 % 63:9 % 64:8 % 68:0 % 74:0 %

J48 tree 8; 380 4; 406 905 449 90 48

83.8 % 88.1 % 90.5 % 89.8 % 90.0 % 96.0 %

indicates that in the top 100 user pairs with high ranking values, the three-layer model successfully predicted 98 friend pairs, which means that in those ranked list, there are 98 true friend pairs in the top 100 ranked pairs and the true friend pairs percentage is 98 %. At the same time, the J48 tree found 90.0 % true friend pairs and the naive Bayes found only 68.0 %. In the third row from bottom of Table 2, the three-layer model successfully predicted 487 true friend pairs in the top 500 ranked pairs. Though the three algorithms have close overall prediction performance, Table 1 shows that the three-layer friendship model achieved better prediction accuracy than the other two classical algorithms, especially for the part of high ranking values.

5.3.2 Location Metric To explore the relationship between location-based metric and user friendship for location-based OSNs, the ROC area of the proposed model without the location layer was calculated. That is, the model has only two layers: the social graph and tag graph. The ROC area is 0.758, which incurs substantial performance reduction compared with the three-layer model. Figure 18 shows the comparison of two ROC curves. Additionally, one can use the information gain to quantify the impact of different attributes to friendship prediction. In addition to the three attributes of distance, tags, and locations, one may also calculate the information gain of gender difference and age difference in friendship prediction. Table 3 shows the results. It shows that the location-based metric provided comparable information gain as the social graph distance in friendship prediction. The tag metric has smaller information gain than the other two metrics, possibly for the reason that a large number of users used tags out of the top 1,000 popular ones, and so their tag metrics become zero. The gender and age attributes, however, provide few information gain

Optimization Problems in Online Social Networks Fig. 18 Comparison of ROC curves with/without location metric

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and so do not help predict friendship. One of a possible reason is that a large number of users hide such information for privacy consideration.

6

Conclusion

This chapter introduces and describes four major issues in the application research of online social networks. They are influence maximization, community detection, data collection, and multilayer friendship modeling. The first two are combinatorial optimization problems. The last two study how to collect data and determine friendships, which are needed for studying the first two issues from a holistic point of view. For each issue, this chapter provides theoretical results and approximation algorithms. Online social networks provide a new platform of rich applications for fruitful research of combinatorial optimizations. Acknowledgements The author “Jie Wang” was supported in part by the NSF under grants CNS-0958477 and CNS-1018422. The author “You Li” was supported in part by the NSF under grant CNS-1018422. The author “Jun-Hong Cui” was supported in part by the US Department of Education under Grants P200A100141 and P200A090342. The author “Benyuan Liu” was supported in part by the NSF under grant CNS-0953620. The author “Guanling Chen” was supported in part by the NSF under grant IIS-0917112.

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Cross-References Greedy Approximation Algorithms Network Optimization Social Structure Detection

Recommended Reading 1. Y.-Y. Ahn, S. Han, H. Kwak, S. Moon, H. Jeong, Analysis of topological characteristics of huge online social networking services, in Proceedings of the International World Wide Web Conference (WWW), Banff, 2007 2. L. Becchetti, C. Castillo, D. Donato, A. Fazzone, A comparison of sampling techniques for web graph characterization, in Workshop on Link Analysis: Dynamics and Static of Large Networks (LinkKDD), Philadelphia, 2006 3. F. Benevenuto, T. Rodrigues, M. Cha, V. Almeida, Characterizing user behavior in online social networks, in Proceedings of the ACM SIGCOMM Internet Measurement Conference (IMC), Chicago, 2009 4. R.S. Burt, Positions in networks. Soc. Forces 55, 93–122 (1976) 5. J. Caverlee, S. Webb, A large-scale study of MySpace: observations and implications for online social networks, in Proceeding of International Conference on Weblogs and Social Media, Seattle, 2008 6. W. Chen, Y. Wang, S. Yang, Efficient influence maximization in social networks, in KDD, Paris, 2009 7. W. Chen, C. Wang, Y. Wang, Scalable influence maximization for prevalent viral marketing in large scale social networks, in KDD, Washington, DC, 2010 8. W. Chen, Y. Yuan, L. Zhang, Scalable influence maximization in social networks under the linear threshold model, in ICDM, Sydney, 2010 9. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms (MIT, Cambridge, 2009) 10. P. Domingos, M. Richardson, Mining the network value of customers, in KDD, San Diego, 2001 11. W. Gauvin, B. Ribeiro, B. Liu, D. Towsley, J. Wang, Measurement and gender-specific analysis of user publishing characteristics on MySpace. IEEE Netw. 24(5), 38–43 (2010) 12. W. Gauvin, Providing anonymity for online social network users. Ph.D. Dissertation, Department of Computer Science, University of Massachusetts Lowell, 2011 13. M. Girvan, M.E.J. Newman, Community structure in social and biological networks. PNAS 99(12), 7821–7826 (2002) 14. M. Gjoka, M. Kurant, C.T. Butts, A. Markopoulou, Walking in Facebook: a case study of unbiased sampling of OSNs, in Proceedings of the IEEE INFOCOM, San Diego, 2010 15. C. Gkantsidis, M. Mihail, A. Saberi, Random walks in peer-to-peer networks, in Proceedings of the IEEE INFOCOM, Hong Kong, 2004 16. J. Goldenberg, B. Libai, E. Muller. Talk of the network: a complex systems look at the underlying process of word-of-mouth. Mark. Lett. 12(3), 211–223 (2001) 17. J. Goldenberg, B. Libai, E. Muller, Using complex systems analysis to advance marketing theory development. Acad. Mark. Sci. Rev. 9, 1–19 (2001) 18. W.K. Hastings, Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970) 19. M.R. Henzinger, A. Heydon, M. Mitzenmacher, M. Najork, On near-uniform URL sampling, in Proceedings of the International World Wide Web Conference (WWW), Amsterdam, 2000 20. D. Kempe, J.M. Kleinberg, E. Tardos, Maximizing the spread of influence through a social network, in KDD, Washington, DC, 2003

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21. D. Kempe, J.M. Kleinberg, E. Tardos, Influential nodes in a diffusion model for social networks, in ICALP, Lisbon, 2005 22. B. Krishnamurthy, P. Gill, M. Arlitt, A few chirps about Twitter, in Proceedings of the First Workshop on Online Social Networks (WOSN), Seattle, 2008 23. M. Kumura, K. Saito, Tractable models for information diffusion in social networks, in ECML PKDD, Berlin, 2006 24. M. Kurant, A. Markopoulou, P. Thiran, On the bias of BFS (Breadth-First-Search) sampling, in Proceedings of the 22nd International Teletraffic Congress (ITC’22), Amsterdam, 2010 25. S.H. Lee, P.-J. Kim, H. Jeong, Statistical properties of sampled networks. Phys. Rev. E 73, 102–109 (2006) 26. J. Leskovec, A. Krause, C. Guestrin, C. Faloustos, J. VanBriesen, N.S. Glance, Cost-effective outbreak detection in networks, in KDD, San Jose, 2007 27. N. Li, G. Chen, Multi-layer friendship modeling for location-based mobile social networks, in MobiQuitous, Toronto, 2009 28. Y. Li, Z. Fang, J. Wang, An efficient random-walk algorithm for finding network communities. Technical report, Department of Computer Science, University of Massachusetts, Lowell, 2012 29. L. Lovasz, Random walks on graphs, a survey. Combinatorics 2, 1–46 (1993) 30. N. Metropolis, M. Rosenbluth, A. Rosenbluth, A. Teller, E. Teller, Equations of state Calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953) 31. A. Mislove, M. Marcon, K.P. Gummadi, P. Druschel, B. Bhattacharjee, Measurement and analysis of online social networks, in Proceedings of the 5th ACM/USENIX Internet Measurement Conference (IMC’07), San Diego, 2007 32. R. Narayanm, Y. Narahari, A shapley value based approach to discover influential nodes in social networks. IEEE Trans. Autom. Sci. Eng. 8,(1), 130–147 (2011) 33. G. Nemhauser, L. Wolsey, M. Fisher, An analysis of the approximations for maximizing submodular set functions. Math. Program. 14, 265–294 (1978) 34. M.E.J. Newman, Detecting community structure in networks. Eur. Phys. J. B 38, 321–330 (2004) 35. M.E.J. Newman, Modularity and community structure in networks. Technical report, physics/0602124, 2006 36. A. Rasti, M. Torkjazi, R. Rejaie, N. Duffield, W. Willinger, D. Stutzbach, Respondent-driven sampling for characterizing unstructured overlays, in Proceedings of the IEEE Infocom MiniConference, Rio de Janeiro, 2009 37. B. Ribeiro, D. Towsley, Estimating and sampling graphs with multidimensional random walks, in Proceeding of ACM SIGCOMM Internet Measurement Conference (IMC), Melbourne 2010 38. M. Richardson, P. Domingos, Mining knowledge-sharing sites for viral marketing, in Proceedings of the 8th International Conference on Knowledge Discovery and Data Mining, Edmonton, 2002 39. D. Stutzbach, R. Rejaie, N. Duffield, S. Sen, W. Willinger, On unbiased sampling for unstructured peer-to-peer networks, in Proceeding of ACM SIGCOMM Internet Measurement Conference (IMC), Rio de Janeriro, 2006 40. L.G. Valiant, The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979) 41. C. Wilson, B. Boe, A. Sala, K. Puttaswamy, B. Zhao, User interactions in social networks and their implications, in Proceedings of the EuroSys, Nuremberg, 2009

Optimizing Data Collection Capacity in Wireless Networks Shouling Ji, Jing (Selena) He and Yingshu Li

Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problems, Motivations, and Main Contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Capacity Issues in Wireless Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Capacity for Single-Radio Single-Channel Wireless Networks. . . . . . . . . . . . . . . . . . . . . 3.2 Capacity for Multichannel Wireless Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Network Model and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Interference Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Network Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Routing Tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Vertex Coloring Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Capacity of Snapshot Data Collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Scheduling Algorithm for Snapshot Data Collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Capacity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Capacity of Continuous Data Collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Compressive Data Gathering (CDG). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Pipeline Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Capacity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Capacity of the Pipeline Scheduling Algorithm in Single-Radio Multichannel WSNs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Simulations and Results Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Performance of MPS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Performance of PS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Impacts of N and M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S. Ji () • Y. Li Department of Computer Science, Georgia State University, Atlanta, GA, USA e-mail: [email protected]; [email protected] J.S. He Department of Computer Science, Kennesaw State University, Kennesaw, GA, USA e-mail: [email protected]

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8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2542 Cross-References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2544 Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2544

Abstract

Data collection is an important operation of wireless sensor networks (WSNs). The performance of data collection can be measured by its achievable network capacity. Most existing works focus on the capacity of unicast, multicast, or/and snapshot data collection (SDC) in single-radio single-channel wireless networks, and no works dedicatedly consider the continuous data collection (CDC) capacity for WSNs under the protocol interference model. In this chapter, firstly, a multipath scheduling algorithm for SDC in single-radio multichannel WSNs is proposed. Theoretical analysis of the multipath scheduling algorithm W shows that its achievable network capacity is at least 2 .1:812 Cc , where d 1 Cc2 /=H e W is the channel bandwidth, H is the number of available orthogonal channels,  is the ratio of the interference radius over the transmission radius of a node, 2 c1 D p C 2 C1, and c2 D p3 C 2 C2, which is a tighter lower bound compared 3

W with the previously best result which is 8 2 (Chen et al. (2010) Capacity of data collection in arbitrary wireless sensor networks. In: IEEE INFOCOM, San Diego, USA). For CDC, an intuitive method is to pipeline existing SDC methods. However, such an idea cannot actually improve the network capacity. The reason for this failure is discussed and a novel pipeline scheduling algorithm for CDC in dual-radio multichannel WSNs is proposed. This pipeline scheduling algorithm significantly speeds up the data collection process and achieves a capacity of

8 nW ˆ ˆ  ; if e  12  ˆ ˆ 3:632 C c3  C c4 ˆ ˆ ˆ 12M ˆ < H ˆ ˆ nW ˆ ˆ ; if e > 12  ˆ 2 Cc Cc  ˆ 3:63 ˆ 3 4 ˆ : Me H

;

where n is the number of the sensors, M is a constant value and usually M  n, e is the maximum number of the leaf nodes having a same parent in the 8 8 routing tree (i.e., data collection tree), c3 D p C  C 2, and c4 D p C 3 3 2 C6. Furthermore, for completeness, the performance of the proposed pipeline scheduling algorithm in single-radio multichannel WSNs is also analyzed, which shows that for a long-run CDC, the lower bound of the achievable asymptotic network capacity is 8 nW ˆ ˆ  ; if e  12  ˆ 2 Cc Cc ˆ 3:63 ˆ 3 4 ˆ ˆ 16M ˆ < H ˆ ˆ nW ˆ ˆ  ; if e > 12  ˆ ˆ 3:632 C c3  C c4 ˆ ˆ : M.e C 4/ H

:

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Extensive simulation results indicate that the proposed algorithms improve the network capacity significantly compared with existing works.

1

Introduction

Wireless sensor networks (WSNs) as shown in Fig. 1, consisting of small nodes with sensing, computation, and wireless communication capabilities, attract many interests from the research community recently. Generally speaking, WSNs are deployed for monitoring and controlling systems where human intervention is not desirable or feasible. Therefore, WSNs are widely used in many applications [2, 15, 34]: – Disaster relief applications. A typical scenario is wildfire detection, where a WSN consisting of sensor nodes that are equipped with thermometers and can determine their own locations (relative to each other or in absolute coordinates) is depoyed. In such an application, sensor nodes should be cheap enough to be considered disposable since a large number is necessary. – Environment monitoring. WSNs can be used to monitoring environments. For instance, with respect to monitor chemical pollutants, a possible application is monitoring garbage dump sites. The main advantage of WSNs here is the longterm and unattended monitoring. – Intelligent buildings. Buildings waste vast amounts of energy because of inefficient monitoring of humidity, ventilation, and/or air conditioning usage. A better, real-time, and high-resolution monitoring of temperature, airflow, humidity, and other physical parameters in a building by means of WSNs can considerably increase the comfort level of inhabitants and reduce the energy consumption. – Facility management. In the management of facilities larger than a single building, WSNs also have a wide range of possible applications. Simple examples include keyless entry applications where people wear badges that allow a WSN to check which person is allowed to enter which areas of a larger company site. This example can be extended to the detection of intruders. – Machine surveillance and structural health monitoring (SHM). One idea is to fix sensor nodes to difficult-to-reach areas of machinery where they can detect vibration patterns that indicate the need for maintenance. – Precision agriculture. Applying WSNs to agriculture allows precise irrigation and fertilizing by placing humidity/soil composition sensors into the fields. – Medicine and health care. The use of WSNs in healthcare applications is potentially very beneficial. Possibilities range from postoperative and intensive care, where sensors are directly attached to patients. – Logistics. In several different logistics applications, it is conceivable to equip goods with simple sensors that allow a simple tracking of these objects during transportation or facilitate inventory tracking in stores or warehouses. – Traffic control. Sensors embedded in the streets or road sides can gather information about traffic conditions at a much finer-grained resolution. Such deployed sensors could also interact with the cars to exchange danger warnings about road conditions or traffic jams ahead.

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Fig. 1 A wireless sensor network (WSN)

2

Problems, Motivations, and Main Contributions

As mentioned in the previous section, WSNs are mainly used for gathering data from the physical world. Data gathering can be categorized as data aggregation [1, 28, 33, 44, 57], which obtains aggregated values from WSNs, e.g., maximum, minimum, or/and average value of all the data, and data collection [11, 12, 31, 32, 46, 65, 66], which gathers all the data from a network without any data aggregation. For data collection, the union of all the sensing values from all the sensors at a particular time instance is called a snapshot [12, 31]. The problem of collecting all the data of one snapshot is called snapshot data collection (SDC). Similarly, the problem of collecting multiple continuous snapshots is called continuous data collection (CDC). Taking the WSN shown in Fig. 2 as an example, this WSN consists of one sink node denoted by s0 and nine sensors denoted by si .1  i  9/. Assume the data value (packet) produced at sensor si .1  i  9/ at a particular time instance is di .1  i  9/ as shown in Fig. 2. Then, the set fdi j 1  i  9g is called a snapshot. In a data aggregation task, if the aggregation function is max./,1 then

1 Under the max./ function, the maximize value of a snapshot will be collected by the sink, i.e., max.fd1 ; d2 ;    ; dn g/ D dmax such that dmax 2 fd1 ; d2 ;    ; dn g and for 8di 2 fd1 ; d2 ;    ; dn g, dmax  di .

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Fig. 2 Explanation of data aggregation and data collection

s0

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only the maximize value of set fdi j 1  i  9g will be collected by the sink s0 . On the other hand, in a data collection task, all the nine values of set fdi j 1  i  9g will be collected by the sink s0 . Different from wired networks, WSNs suffer from the interference problem, which degrades the network performance. Consequently, network capacity, which can reflect the achievable data transmission rate, is usually used as an important measurement to evaluate network performance. Particularly, for a data collection WSN, the average data receiving rate at the sink during the data collection process, referred to as data collection capacity [12, 31] (formally defined in Sect. 4.2), is used to measure its achievable network capacity, i.e., data collection capacity reflects how fast data have been collected by the sink. In this chapter, the snapshot and continuous data collection problems (formally defined in Sect. 4.2), as well as their achievable capacities in single/dual-radio multichannel WSNs, are studied. After the first work [24], extensive works emerged to study the network capacity issue for variety of network scenarios, e.g., multicast capacity [40, 41, 48], unicast capacity [52, 53], broadcast capacity [42], and SDC capacity [11, 12, 46, 66]. Most of the previous studies on network capacity are for single-radio single-channel WSNs [4, 8–12, 20, 22, 24, 40, 41, 46, 48, 51–53, 55, 59, 60, 63, 66], where a network consists of a number of nodes which have only one radio and all the nodes communicate over a common single channel. Because of the inherent limitations of such networks, transmissions suffer from the radio confliction problem [13, 23, 37, 47, 56] and the channel interference problem [3, 6, 7, 14, 16–19, 21, 25, 29, 29, 35–37, 43, 43, 45, 49, 54, 61, 64] seriously. This degrades network performance significantly. The radio confliction problem is caused by the fact that each node is equipped with only one radio, which means a node can only work on a half-duplex mode, i.e., this node cannot receive and transmit data simultaneously. The channel

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Fig. 3 A dual-radio sensor node [30, 38]

interference problem is caused by all the nodes working over a common channel. When one node transmits data, all the other nodes within its interference radius cannot receive any other data and all the other transmissions that interfere with this transmission cannot be carried out simultaneously. Fortunately, many current off-the-shelf sensor nodes are capable of working over multiple orthogonal channels, e.g., IEEE 802.11 b/g standard supports 3 orthogonal channels and IEEE 802.11a standard supports 13 orthogonal channels [5, 13], respectively, which can greatly mitigate the channel interference problem. Furthermore, with the development of hardware technologies and the decreasing of hardware cost, a sensor node can be equipped with multiple radios. As shown in Fig. 3, it is a dual-radio sensor node developed by HLJU and HIT [30, 38]. This helps with solving the radio confliction problem. Therefore, multi-radio multichannel WSNs currently begin to attract more and more interests from researchers [13, 23, 37, 47, 56]. Different from the previous works which investigate the capacity issues for single-radio single-channel WSNs, this chapter studies the network capacity problem for both CDC and SDC in dual-radio multichannel WSNs under the protocol interference model. Similarly as [12], in this chapter, the data collection capacity is defined as the average data rate at the sink to continuously receive data from sensor nodes. Two channel scheduling algorithms for both CDC and SDC are proposed, respectively. Meanwhile, the results obtained in this chapter can be extended to WSNs under the physical interference model [42]. The motivation of the work in this chapter lies on the fact that dual-radio multichannel WSNs can make nodes work in a full-duplex manner without incurring high hardware cost, while the channel interference problem can be mitigated significantly. Most of the previous works focus on addressing the SDC capacity problem, while the work in this chapter is the dedicated one investigating the CDC capacity problem in detail under the protocol interference model. Besides, the work in this chapter is suitable for dualradio multichannel WSNs. The main contributions of this chapter are as follows: – For the SDC problem in single-radio multichannel WSNs, a new multipath scheduling algorithm is proposed. Furthermore, theoretical analysis of the multipath scheduling algorithm shows that this algorithm can achieve the order-optimal network capacity ‚.W / and has a tighter lower bound W compared with the previously best result in [12], which is 2d.1:812 Cc1 Cc2 /=H e W , where W is the channel bandwidth, H is the number of available orthogonal 82 channels in a WSN,  is the ratio of the interference radius over the transmission 2 C 2 C 1, and c2 D p3 C 2 C 2. radius of a node, c1 D p 3

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– Currently, there are no solutions dedicated for the CDC capacity problem. Therefore, a novel pipeline scheduling algorithm that combines compressive data gathering (CDG) [46] and pipeline together is proposed, which significantly improves the CDC capacity for dual-radio multichannel WSNs. Theoretical analysis shows that the achievable asymptotic network capacity of this pipeline scheduling algorithm in a long-run is 8 nW ˆ ˆ  ; if e  12  ˆ 2 ˆ C c3  C c4 3:63 ˆ ˆ ˆ 12M < H ; nW ˆ ˆ ˆ > 12 ; if    e ˆ ˆ 3:632 C c3  C c4 ˆ ˆ : Me H where n is the number of the sensors in a WSN, M is a constant value determined by the correlations of the data in a snapshot and usually M  n, e is the maximum number of the leaf nodes having a same parent in the routing tree (i.e., data 8 8 collection tree), c3 D p C C2, and c4 D p C2 C6. A straightforward upper 3 3 bound of data collection of a dual-radio WSN is 2W , since a dual-radio sink can simultaneously receive two data packets at most. Whereas, thanks to the benefit brought by the pipeline technique and CDG, analysis shows that the proposed pipeline scheduling algorithm can even achieve a capacity higher than 2W . – For completeness, the performance of the proposed pipeline scheduling algorithm in single-radio multi-channel WSNs is also examined, denoted by the single-radio-based pipeline scheduling algorithm. Theoretical analysis shows that for a long-run CDC, the lower bound of the achievable asymptotic network capacity of the single-radio-based pipeline scheduling algorithm for single-radio multi-channel WSNs is 8 nW ˆ ˆ  ; if e  12  ˆ 2 ˆ C c3  C c4 3:63 ˆ ˆ ˆ < 16M H nW ˆ ˆ ˆ  ; if e > 12  ˆ ˆ 3:632 C c3  C c4 ˆ ˆ : M.e C 4/ H

:

– The simulation results indicate that the proposed algorithms have a better SDC capacity compared with the previously best works. Particularly, when  D 2 and H D 3, for SDC in a WSN with 4,000 nodes, the improvements of the capacity of the proposed multipath scheduling algorithm are 74.3 and 29 % compared with BFS [12] and SLR [6], respectively. For CDC in a WSN with 10,000 nodes, the proposed pipeline scheduling algorithm achieves a capacity 7.6 times of that of CDG [46], 22.8 times of that of BFS [12], and 19.4 times of that of SLR [6], respectively.

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The rest of this chapter is organized as follows: Sect. 3 summarizes the advances in studying network capacity issues for wireless networks. Section 4 introduces the interference model, the network model, and the preliminaries. The multi-channel scheduling algorithm for SDC in single-radio multichannel WSNs is proposed and analyzed in Sect. 5. Section 6 presents a novel pipeline scheduling algorithm for CDC and its theoretical achievable asymptotic network capacity. For completeness, the performance of the pipeline scheduling algorithm in single-radio multichannel WSNs is also analyzed in Sect. 6. The simulations to validate the performance of the proposed algorithms are shown in Sect. 7. Section 8 concludes this chapter and points out possible future research directions.

3

Capacity Issues in Wireless Networks

In this section, the advances in studying network capacity issues for wireless networks are surveyed and summarized. At the end of this section, the differences between the work in this chapter and existing works are also compared.

3.1

Capacity for Single-Radio Single-Channel Wireless Networks

Following the seminal work [24] by Gupta and Kumar, extensive works emerged to study the network capacity issue. The works in [4, 8, 9, 22] focus more on the MAC layer to improve the network capacity. In [8], the network capacity with randomaccess scheduling is investigated. In this work, each link is assigned a channel access probability. Based on which some simple and distributed channel access strategies are proposed. Another similar work is [9], in which the authors studied the capacity of CSMA wireless networks. The authors formulated the models of a series of CSMA protocols and study the capacity of CSMA scheduling versus TDMA scheduling. They also proposed a CSMA scheme which combines a backboneperipheral routing scheme and a dual carrier-sensing and dual-channel scheme. In [22], the authors considered the scheduling problem where all the communication requests are single-hop and all the nodes transmit at a fixed power level. They proposed an algorithm to maximize the number of links in one time slot. Unlike [22], the authors in [4] considered the power-control problem. A family of approximation algorithms were presented to maximize the capacity of arbitrary wireless networks. The works in [40, 41, 48, 52, 53, 59] study the multicast and/or unicast capacity of wireless networks. The multicast capacity for wireless ad hoc networks under the protocol interference model and the Gaussian channel model is investigated in [40, 41], respectively. q In [40], the authors  showed that the network multicast   n W n n capacity is ‚ when k D O and is ‚.W / when k D  ,  log n k log n log n where W is the bandwidth of a wireless channel, n is the number of the nodes in a network, and k is the number of the nodes involved in one multicast session. In [41], the authors showed that when k  1 .log n/n2˛C6 and ns  2 n1=2Cˇ , the

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capacity that each multicast session can achieve is at least c8 pn , where k is the ns k number of the receivers in one multicast session; n is the number of the nodes in the network; ns is the number of the multicast sessions; 1 , 2 , and c8 are constants; and ˇ is any positive real number. Another similar work [48] studies the upper and lower bounds of multicast capacity for hybrid wireless networks consisting of ordinary wireless nodes and multiple base stations connected by a high-bandwidth wired network. Considering the problem of characterizing the unicast capacity scaling in arbitrary wireless networks, the authors proposed a general cooperative communication scheme in [52]. The authors also presented a family of schemes that address the issues between multi-hop and cooperative communication when the path loss exponent is greater than 3. In [53], the authors studied the balanced unicast and multicast capacity of a wireless network consisting of n randomly placed nodes and obtained the characterization of the scaling of the n2 -dimensional balanced unicast and n2n -dimensional balanced multicast capacity regions under the Gaussian fading channel model. A more general .n; m; k/-casting capacity problem was investigated in [59], where n, m, and k denote the total number of the nodes in the network, the number of destinations for each communication group, and the actual number of communication group members that receive information, respectively. In [59], the upper and lower bounds for the .n; m; k/-cast capacity were obtained for random wireless networks. In [29], the authors investigated the network capacity scaling in mobile wireless ad hoc networks under the protocol interference model with infrastructure support. In [64], the authors studied the network capacity of hybrid wireless networks with directional antenna and delay constraints. Unlike previous works, the authors in [36] studied the capacity of multi-unicast for wireless networks from the algorithmic aspects, and they designed provably good algorithms for arbitrary instances. The broadcast capacity of wireless networks under the protocol interference model is investigated in [35], where the authors derived the upper and lower bounds of the broadcast capacity in arbitrary connected networks. When the authors in [45] studied the data gathering capacity of wireless networks under the protocol interference model, they concerned the per source node throughput in a network where a subset of nodes send data to some designated destinations while other nodes serve as relays. To gather data from WSNs, a multi-query processing technology is proposed in [14]. In that work, the authors considered how to obtain data efficiently with data aggregation and query scheduling. Under different communication organizations, the authors in [17] derived the many-to-one capacity bound under the protocol interference model. Another work that studied the manyto-one capacity issue for WSNs is [49], where the authors considered to use data compression to improve the data gathering efficiency. They also studied the relation between a data compression scheme and the data gathering quality. In [19], the authors studied the scaling laws of WSNs based on an antenna-sharing idea. In that work, the authors derived the many-to-one capacity bounds under different power constraints. In [61], the authors studied the multicast capacity of MANETs under the physical interference model, called MotionCast. They considered the network capacity of MANETs in two particular situations, which are the LSRM (local-based speed-restricted) model and the GSRM (global-based speed-restricted) model.

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The multi-unicast capacity of wireless networks is studied in [18] via percolation theory. By applying percolation theory, the authors obtained a tighter capacity bound for arbitrary wireless networks. The SDC capacity of WSNs is studied in [10–12, 31, 46, 51, 66]. In [66], the authors considered the collision-free delay-efficient data-gathering problem. Furthermore, they proposed a family of path scheduling algorithms to collect all the data to the sink and obtained the network capacity through theoretical analysis. The authors of [12, 31] extended the work of [66]. They derived tighter upper and lower bounds of the capacity of data collection for arbitrary WSNs. Luo et al. [46] is a work studying how to distribute the data collection task to the entire network to achieve load balancing. In that work, all the sensors transmit the same number of data packets during the data collection process. In [10, 11, 32], the authors investigated the capacity of data collection for WSNs under the protocol interference model and the physical interference model, respectively. They proposed a grid partition method which divides the network into small grids to collect data and then derived the network capacity. The worst-case capacity of data collection of a WSN is studied in [51] under the physical and protocol interference models. The capacity and energy efficiency of wireless ad hoc networks with multipacket reception under the physical interference model are investigated in [60]. With the multi-packet reception scheme, a tight bound of the network capacity is obtained. Furthermore, the authors showed that a trade-off can be made between increasing the transport capacity and decreasing the energy efficiency. In [63], a scheduling partition method for large-scale wireless networks is proposed. This method decomposes a large network into many small zones, and then localized scheduling algorithms which can achieve the order optimal capacity as a global scheduling strategy are executed in each zone independently. A general framework to characterize the capacity of wireless ad hoc networks with arbitrary mobility patterns is studied in [20]. By relaxing the “homogeneous mixing” assumption in most existing works, the capacity of a heterogeneous network is analyzed. Another work [55] studies the relationship between the capacity and the delay of mobile wireless ad hoc networks, where the authors studied how much delay must be tolerated under a certain mobile pattern to achieve an improvement of the network capacity.

3.2

Capacity for Multichannel Wireless Networks

Since wireless nodes can be equipped with multiple radios, and each radio can work over multiple orthogonal channels, multi-radio multichannel wireless networks attract many research interests recently [3, 21, 25, 43]. In [21], the authors studied the data aggregation issue in multichannel WSNs under the protocol interference model. Particularly, they designed a constant factor approximation scheme for data aggregation in multi-channel WSNs modeled by unit disk graphs (UDGs). Unlike [21], this chapter studies the snapshot/continuous data collection capacity issue for WSNs. In [3, 25, 43], the authors investigated the joint channel assignment and routing problem for multi-radio wireless mesh networks, software-defined radio

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networks, and multichannel ad hoc wireless networks, respectively. They focused on the channel assignment and routing issues, while in data collection, especially continuous data collection, one focuses on how to solve the data accumulation problem at the sensors near the sink to improve the achievable network capacity. The issue of the capacity of multichannel wireless networks also attracts a lot of attention [6, 7, 16, 37, 54]. In [6, 7], the authors studied the connectivity and capacity problem of multichannel wireless networks. They considered a multichannel wireless network under constraints on channel switching, proposed some routing and channel assignment strategies for multiple unicast communications and derived the per-flow capacity. The multicast capacity of multi-channel wireless networks is studied in [54]. In this work, the authors represented the upper bound capacity of per multicast as a function of the number of the sources, the number of the destinations per multicast, the number of the interfaces per node, and the number of the available channels. Subsequently, an order-optimal scheduling method is proposed under certain circumstances. In [16], the authors first proposed a multichannel network architecture, called MC-MDA, where each node is equipped with multiple directional antennas, and then obtained the capacity of multiple unicast communications, under arbitrary and random network models. The impact of the number of the channels, the number of the interfaces, and the interface switching delay on the capacity of multichannel wireless networks is investigated in [37]. In this work, the authors derived the network capacity under different situations for arbitrary and random networks.

3.3

Remarks

Unlike the above-mentioned works, this chapter has the following two main characteristics. First, most of the above-mentioned works are specifically for singleradio single-channel wireless networks, while the work in this chapter considers the network capacity for dual-radio multichannel WSNs. Second, the work in this chapter is the dedicated one that investigates the network capacity for CDC in detail under the protocol interference model, whereas most of the previous works study the network capacity for multicast or/and unicast and etc. For the works that study the data collection capacity of wireless networks, they focus on the SDC problem which is a special case of CDC. Compared with them, the methods proposed in this chapter are more universal.

4

Network Model and Preliminaries

In this section, the interference model, the network model, and the assumptions are described; the routing tree used for data collection is constructed; and some necessary preliminaries are made. For the frequently used notations in this chapter, they are listed in Table 1.

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Table 1 Notations used in this chapter Notation

Description

G.V; E/

The network topology graph, V is the set of all the nodes, E is the set of all the possible links The number of sensors in a WSN The interference radius The H available orthogonal channels The bandwidth of a channel The size of a data packet A time slot The time consumption of snapshot/continuous data collection The number of snapshots in a continuous data collection task The snapshot/continuous data collection capacity The set of dominators/connectors The graph constructed by the nodes in D The radius of G 0 The data collection tree The conflicting graph of A The maximum/minimum degree of a graph The maximum number of leaf nodes having a same parent in T The inductivity of a graph The number of the dominators within a half-disk with radius r Snapshot data collection Continuous data collection Compressive data gathering Super time slot Connected dominating set

n  1 ;    ; H W b t  N D/C G0 L0 T R.A/ .//ı./ e ı  ./ ˇr SDC CDC CDG STS CDS

4.1

Interference Model

When model the interference in wireless networks, three interference models are frequently used: the protocol interference model [12, 31], the physical interference model [11], and the generalized physical interference model [1, 41]. Assume u and v are two nodes in a wireless network and u is transmitting some data to v. Furthermore, assume T is set of all the nodes that transmit data to some nodes concurrently with u. Then, the transmission of u and v under these three interference models will be introduced respectively in the following.

4.1.1 Protocol Interference Model Under the protocol interference model, u can successfully transmit data to v only if 8w 2 T ; w ¤ u; kw  vk  .1 C /  r;

(1)

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where w is any node transmitting data concurrently with u, kk is the Euclidian distance between two nodes,   0 is a constant value, and r is the communication radius of the nodes in a wireless network.

4.1.2 Physical Interference Model Under the physical interference model, u can successfully transmit data to v only if the signal-to-interference-plus-noise ratio (SINR) value at v associated with u is no less than a threshold value as follows: ƒ.u; v/ D SI NRu;v D

Pu  ku  vk

P  ; N0 C Pw  kw  vk

(2)

w2T ;w¤u

where Pu (respectively, Pw ) is the transmission power of u (respectively, w), > 2 is the path loss exponent, N0 represents the ambient noise, and  is the threshold SINR value.

4.1.3 Generalized Physical Interference Model Under the generalized physical interference model, the data receiving rate at v from u is given by Ru;v D W  log.1 C ƒ.u; v//;

(3)

where W is the bandwidth of the channel on which u transmits data to v. In this chapter, the snapshot and continuous data collection, as well as their achievable capacities, in single/dual-radio multichannel WSNs are investigated under the protocol interference model.

4.2

Network Model

In this chapter, a WSN consisting of n sensors and one sink is considered and represented by a connected undirected graph G D .V; E/, where V is the set of all the nodes in the network and E is the set of all the possible links among the nodes in V . Every sensor in the WSN produces one data packet in a snapshot. Each sensor has two radios and each radio has a fixed transmission radius normalized to one and a fixed interference radius, denoted by ,   1, as shown in Fig. 4. Since the protocol interference model is used, for any receiving node v, v can receive a data packet successfully from a transmitting node u if ku  vk  1 and there is no other node s satisfying ks  vk   and trying to transmit a data packet simultaneously over the same channel with u. Furthermore, two links are interfering links if at least one transmission over them will fail if they transmit data simultaneously. Each radio can work over H orthogonal channels, denoted by 1 ; 2 ; : : : ; H , respectively. A fixed data-rate channel model [12] is adopted in this chapter, which means each sensor can transmit at a rate of W bits/second over a wireless channel. The size of every

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Fig. 4 Communication radius vs. interference radius Sensor Node Communication Radius Interference Radius

data packet transmitted in the network is set to be b bits. All the transmissions are assumed to be synchronized and the size of a time slot is t D b=W seconds. The problems studied in this chapter are further formally defined as follows. For a WSN consisting of n sensors and one sink, every sensor produces a data packet with b bits at a particular time instant. The union of all the n data packets produced by the n sensors at a particular time instant is called a snapshot. The process to collect all the data of a snapshot to the sink is called snapshot data collection (SDC). The SDC capacity is defined as D nb  , where  is the time used to collect all the data of a snapshot to the sink, i.e., SDC capacity reflects the average data receiving rate at the sink during SDC. Similarly, the process to collect all the data of N continuous snapshots is called continuous data collection (CDC). The CDC capacity is defined as D Nnb , where  now is the time consumption to collect all the data of these N snapshots to the sink, i.e., CDC capacity reflects the average data receiving rate at the sink during CDC. In this chapter, the SDC and CDC problems for WSNs, as well as their achievable network capacities, are studied.2

4.3

Routing Tree

G is a unitdisk graph representing a WSN. The sink s0 is defined as the center of G. The radius of G with respect to s0 is the maximum depth of the breadth-first search (BFS) tree rooted at s0 . For a subset U of V , U is a dominating set (DS) of G if every node in V is either an element of U or adjacent3 to at least one node in U . If the subgraph of G induced by U is connected, then U is called a connected dominating set (CDS) of G. Since a CDS can serve as a virtual backbone of a WSN, it receives a lot of attention [26, 27, 39, 57, 58, 62] recently.

2 In the following of this chapter, the snapshot/continuous data collection capacity and network capacity are used interchangeably without confusion. 3 In this chapter, if two nodes u and v are said adjacent/connected, that means u and v are within the communication range of each other, i.e., ku  vk  1.

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a

c

s0

s0

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s0

s0

Fig. 5 The construction of a CDS based routing tree. s0 is the sink. The black nodes in (b) are dominators, and the blue nodes in (c) are connectors

Taking the WSN shown in Fig. 5a as an example, a CDS-based routing tree T is built as shown in Fig. 5d using the method proposed in [57]. T is rooted at the sink and can be built according to the following steps. First, construct a BFS tree on G beginning at the sink and obtain a maximal independent set (MIS) D according to the search sequence. As shown in Fig. 5b, the set of all the black nodes is an MIS of the network shown in Fig. 5a. Note that D is also a DS of G and an element in D is called a dominator. Clearly, every dominator is out of the communication range of any other dominators. Let G 0 be a graph on D in which two nodes in D linked by an edge if and only if these two nodes have a common neighbor in G. Obviously, sink s0 is in G 0 and also s0 is defined as the center of G 0 . Suppose that the radius of G 0 with respect to s0 is L0 and the union of dominators at level l .0  l  L0 / is denoted as set Dl . Note that D0 D fs0 g. Second, choose nodes, also called connectors, to connect all the nodes in D to form a CDS. Let Sl .0  l  L0 / be the set of the nodes adjacent to at least one node in Dl and at least one node in DlC1 and compute 0 a minimal cover Cl  Sl for DlC1 . Let C D [0L 1 Cl and therefore D [ C is a CDS of G. As shown in Fig. 5c, the blue nodes are connectors chosen to connect the dominators in D. Meanwhile, the union of the dominators and connectors in Fig. 5c forms a CDS of the network shown in Fig. 5a. Finally, for any other node u, also called a dominatee, not belonging to D [ C , choose the nearest dominator as u’s parent node. Thus, the routing tree T is obtained. For instance, the tree shown in Fig. 5d is the obtained routing tree for the network shown in Fig. 5a. For each link in T , assign it a direction from the child node to the parent node along the data transmission flow to the sink as shown in Fig. 5d. Furthermore,

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Fig. 6 (a) A link set and (b) its corresponding conflicting graph

a s1

b

s5

s2

s6 s3 s4

v2

v1

v4

s8 s7

v3

the receiving node, i.e., the parent node, of a link is called a head. Similarly, the transmitting node, i.e., the child node, of a link is called a tail. Suppose that A is a set of links of T . The corresponding conflicting graph of A is denoted by ” in Lemmas 1 and 2. With these modified lemmas, the proof of Folkman-Graham inequality can be modified to a proof of Theorem 2. t u Corollary 3 Let X be a compact convex region on the Euclidean plane. Suppose X contains n independent points. Then 2 1 n < p A.X / C P .X / C 1 2 3 where A.X / and P .X / denote the area and the perimeter of X , respectively.

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Corollary 4 If a circle with radius r contains n independent points, then 2 n < p  r 2 C  r C 1: 3 As one can see from the following sections, Corollary 4 has a lot of applications. But in some other applications, Corollary 4 is far from tight. For example, in the study of wireless sensor networks, the following problem is often considered: Let T be a tree lying in the Euclidean plane with n vertices, satisfying the condition that each edge has length at most one. Let Q.T / be the union of the n disks with radius one and centers at the vertices of T . How many independent points could Q.T / contain? Let mip.n/ denote the maximum number of independent points lying in Q.T / for T over all the trees with n vertices whose maximum p edge lengths are at most one. For n D 1, Corollary 4 results in mip.1/  b.2= 3 C 1/ C 1c D 7. However, the exact value of mip.1/ is five, determined by Wan et al. in [53]. Lemma 3 (Wan et al. [53]) mip.1/ D 5. Proof Let Dv denote the disk with center v and radius one. Let a1 ; a2 ; : : : ; ak be the independent points lying in Dv such that va1 ; va2 ; : : : ; vak are in counterclockwise ordering. Since kvai k  1 for i D 1; 2; : : : ; k and kai aj k > 1 for i ¤ j , one has †a1 va2 > =3; †a2 va3 > =3; : : : ; †ak va1 > =3. Thus, 2 D †a1 va2 C †a2 va3 C    C †ak va1 > k=3. Hence, k < 6. This means mip.1/  5. On the other hand, it is easy to arrange five independent points on the boundary of Dv , which means mip.1/  5. t u Since the upper bound of mip.n/ plays an important role in analyzing approximation algorithms for the virtual backbone of wireless networks, a lot of works focus on the improvement of mip.n/. Unlike wired network, there is no fixed infrastructure in a wireless network. In such a circumstance, if all the sensors participate in the transmission, it is not only a waste of energy, but also can cause too much interference such that nodes cannot decode the transmitted information correctly (which is known as broadcast storm). To solve this problem, virtual backbone was proposed [3, 8, 49], which corresponds to a connected dominating set in a graph. A dominating set (DS) of a graph G D .V; E/ is a vertex set D such that every vertex in V nD has at least one neighbor in D. It is a connected dominating set (CDS) if GŒD , the subgraph of G induced by D, is connected. Using a CDS as the virtual backbone, computations and transmissions are mainly carried out in the CDS, and thus energy is saved; interference is alleviated. Clearly, one expects the size of the CDS to be as small as possible, resulting in the minimum connected dominating set problem (MCDS). A widely used model of homogeneous wireless sensor network is the unit disk graph (UDG), for which every vertex corresponds to a sensor on the plane; two vertices are adjacent in the graph if and only if the Euclidean distance between their corresponding sensors is at most a constant which is scaled to one.

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Consider the MCDS problem in a unit disk graph (UDG) G. A lot of approximation algorithms follow a two-phase strategy [3, 5, 33, 50, 53]. The first phase constructs a maximal independent set (MIS) I (i.e., a set of independent points such that adding any point into it breaks the independency, notice that any MIS is also a DS). The second phase selects a set C of connectors such that D D I [ C is a CDS. The differences of these works lie in how I and C are selected. To analyze the approximation ratio, consider an optimal solution D  . If jI j  ˇ1 jD  j C ˇ2 and jC j  1 jD  j C 2 , then jDj  .ˇ1 C 1 /jD  j C .ˇ2 C 2 /, and thus D is a .ˇ1 C 1 /-approximation solution. Implementation of Lemma 3 for mip.1/ already shows that jI j  5jD  j. A lot of improvements on the upper bound were made continually by Funke et al. [16], Gao et al. [17], Wan et al. [53, 54], Wu et al. [58], and Li et al. [33], which will be introduced in the following. For the ease of statement, some notations should be fixed first. For a positive real number r, use Dwr to denote Sthe disk centered at node w with radius r. For a node r set W , denote by DW D w2W Dwr . Call them as the r-neighbor area of w and r W , respectively. For an MIS I , denote by Iwr D I \ Dwr and IWr D I \ DW the set of MIS nodes in the r-neighbor area of w and W , respectively. When r D 1, the superscript r is omitted and the notations DW ; IW , and terminology neighbor area are used. Thus, Q.T / in the above is exactly DV .T / , and mip.n/ is exactly maxfjIV .T / j W I is an MIS and T is a tree on n nodes in a UDGg. What one is interested in is the spanning tree T of GŒD  where D  is an optimal CDS. Theorem 3 jI j  4jD  j C 1. Proof The proof makes use of the following observation: Suppose W is a connected node set and w is another node which is connected to W , then jIW [fwg n IW j  4

(1)

(refer to Fig. 2 for an illustration), that is, connecting a node to an already connected node set increases the number of MIS nodes in the neighbor area by at most four. For an optimal solution D  , order the nodes as v1 ; v2 ; : : : ; vjD  j such that for any i D 1; 2; : : : ; jD  j, the subgraph of GŒD  induced by Wi D fv1 ; v2 ; : : : ; vi g is connected. Then

jI j D jIv1 j C

 jD Xj



jIvi n IWi j  5 C

i D2

jI j X

4 D 5 C 4.jD  j  1/ D 4jD  j C 1:

i D2

The upper bound is proved. For n D 2, the exact value of mip.2/ has been determined in [58]. Lemma 4 (Wu et al. [58]) mip.2/ D 8.

t u

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u

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v

Fig. 2 Adding node v increases the region Du by the shaded area which contains at most four MIS nodes

Proof Let u and v be two points with distance at most one. First, it was shown that mip.2/  8. Suppose that there exists an independent set I of more than eight points lying in Du [ Dv . A contradiction will be found through a proof consisting of two parts, each part showing a fact. Fact 1. The intersection area A D Du \ Dv contains exactly one vertex in I . Indeed, suppose A contains k vertices in I . By Lemma 3, Du  A contains at most 5  k vertices in I and Dv  A contains at most 5  k vertices in I . Thus, Du [ Dv contains at most 10  k vertices in I . Hence, by the assumption jI j  9, one has k  1. To show k  1, let x and y be the two intersection points of the boundaries of Du and Dv . Since d.u; v/  1, one has †xvy D †yux  2=3. Thus, Du  A contains at most four vertices in I , so does Dv  A. This means that I contains at most 8 C k vertices. Since jI j  9, one has k  1. Let a0 denote the unique point of I lying in Du \ Dv . As a consequence, each of Du  A and Dv  A contains exactly four vertices in I and jI j D 9. Assume I D fa0 ; a1 ; : : : ; a8 g, where a0 ; a1 ; ::; a4 lie counterclockwise in Du and a0 ; a5 ; : : : ; a8 lie counterclockwise in Dv . Denote by ubi the radius through ai for i D 2; : : : ; 4 and by vbi the radius through ai for i D 5; : : : ; 8. Using radius one, draw four arc-triangles ub2 c2 , ub3 c3 , vb6 c6 , and vb7 c7 as shown in Fig. 3. Their boundaries intersect the boundary of Du \ Dv at d2 ; d3 ; d6 ; d7 , respectively. Note that none of a1 ; a4 ; a5 ; a8 can lie in the four arc-triangles ub2 c2 , ub3 c3 , vb6 c6 , and vb7 c7 and Du \ Dv . Therefore, a1 ; a4 ; a5 ; a8 must lie in the four small dark areas xc2 d2 , yc3 d3 , yc6 d6 , and xc7 d7 , respectively, as shown in Fig. 3. Fact 2. There exist two small dark areas which are too close to contain two independent vertices. (Therefore, this fact results in a contradiction.) To show this fact, notice that †b2 ub3 > =3 and †c2 ub2 D †b3 uc3 D =3. Hence, †c2 uc3 >  and †c3 uc2 <  (note that †c3 uc2 is the one obtained by sweeping c3 u counterclockwise to c2 u). Similarly, †c7 vc6 < . Therefore,

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c2

Fig. 3 An illustration of the four small dark areas

b2

c7

d2 d 7

a2

u

b3

x

a3

a7

v

a0 d3

b7

a6

d6

b6

y c3

c6

†uc2 c7 C †c2 c7 v C †vc6 c3 C †c6 c3 u > 2. This means that either †uc2 c7 C †c2 c7 v >  or †vc6 c3 C†c6 c3 u > . Without loss of generality, assume the former occurs. Then, it can be shown that the dark area xc2 d2 [ xc7 d7 cannot contain two independent points. To do so, enlarge the area xc7 d7 by turning the arc-triangle vb7 c7 around v until vc7 is parallel to uc2 (Fig. 4). At this position, quadrilateral c2 uvc7 becomes a parallelogram so that kc2 c7 k D kuvk  1. It follows that the distance between two points in areas xc2 d2 and xc7 d7 cannot exceed max.kc2 c7 k; kc2 d7 k; kd2 c7 k; kd2 d7 k/. Moreover, it can be proved that kd2 d7 k  max.kc2 d7 k; kd2 c7 k/. In fact, by noticing that †c7 d7 d2 C †d7 d2 c2 > , one has either †c7 d7 d2 > =2 or †d7 d2 c2 > =2. Therefore, either kd2 c7 k > kd2 d7 k or kc2 d7 k > kd2 d7 k. Now, to complete the proof of Fact 2, it suffices to prove that kc2 d7 k  1 and kd2 c7 k  1. To see kc2 d7 k  1, make kc2 d7 k longer by moving v away from u until kuvk D 1 (Fig. 5). At this limit position, kuvk D kvb7 k D kb7 d7 k D kd7 uk D 1. Therefore, uvb7 d7 is a parallelogram. It follows that kd7 b7 k D kc2 c7 k D 1 and d7 b7 is parallel to uv and hence parallel to c2 c7 . Thus, c2 d7 b7 c7 is a parallelogram. Therefore, kc2 d7 k D kc7 b7 k D 1. Similarly, one can show kd2 c7 k  1. Fact 2 induces a contradiction, which implies that mip.2/  8. On the other hand, eight independent points can be easily placed in Du [ Dv when kuvk D 1. This means that mip.2/  8. t u Making use of Lemma 4, Wu et al. [58] improved the upper bound of jI j as follows: Theorem 4 jI j  3:8jD  j C 1:2. Proof In [58], the authors first showed that any unit disk graph has a minimum spanning tree in which every vertex has degree at most five. Let T be such a spanning tree of GŒD  . The upper bound is established by induction on jT j, the number

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c2

Fig. 4 Turn unit arc-triangle vb7 c7 until vc7 parallels uc2

c7

x d2 d7

u

v

c2 d2

c7

x

d7 u

v

b7

Fig. 5 Move u until kuvk D 1

of nodes in T . The validity for jT j D 1 or 2 follows from Lemmas 3 and 4. For jT j  3, it is easy to see that T has a non-leaf node v which is adjacent with at most one non-leaf node u (if all the neighbors of v in T are leaves, let u be an arbitrary neighbor of v). Let the remaining neighbors of v be x1 ; x2 ; : : : ; xk . Then k  4. By the induction hypothesis, the neighbor area of T 0 D T  fv; x1 ; : : : ; xk g contains at most 3:8.jT j  k  1/ C 1:2 MIS nodes. Denote T0 D T 0 [ fvxk g and Ti D Ti 1 [ fvxi g for i D 1; 2; : : : ; k  1. By Lemma 4, it can be seen that jIfv;xk g n Iu j  7. By Lemma 3, jIxi n Iv j  4. Thus jIT0 n IT 0 j  7 and jITi n ITi 1 j  4 .i D P 1; 2; : : : ; k  1/. Then, jI j D jIT 0 j C jIT0 n IT 0 j C k1 i D1 jITi n ITi 1 j is upper bounded by jI j  3:8.jT j  k  1/ C 1:2 C 7C4.k  1/D3:8jD  j C 1:2 C 0:2.k  4/  3:8jD  j C 1:2:

t u

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From the above analysis, one obtains the feeling that if some nodes in D  are more compact, then the number of MIS nodes in their neighbor area will be smaller. This intuition is further explored in [54]. Theorem 5 (Wan et al. [54]) mip.n/ 

11 n 3

C 43 .

Proof A set W of nodes is called a star if there exists a node w 2 W such that W  Dw . A star containing k nodes is called a k-star. Thus a star is a compact set of nodes, and the larger k is, the more compact the nodes are. It was proved in [54] that any k-star W satisfies 11k C 1: (2) jIW j  3 This inequality is obtained by showing that 

3k C 2; for k  2; minf3k C 3; 21g; for k  3;

jIW j  k D

and k  11k 3 C 1. The intuition for the expression of k is that adding a new node into a star increases the number of MIS nodes in the neighbor area by at most 3, and when k is large enough (in fact, when k > 6), adding new nodes into the star will no longer increase the number of MIS nodes in the neighbor area. A nontrivial star is a k-star with k  2. Decompose D  into nontrivial stars S ; S ; : : : ; Sp such that for any i D 1; 2; : : : ; p, the subgraph of GŒD  induced by S1i 2 j D1 Sj is connected (this can be done as proved in [54]). By (2), jIS1 j 

11jS1 j C 1: 3

S For i D 2; 3; : : : ; p, let ui be a node in ij1 D1 Sj which is adjacent with Si . Then   Si 1 ISi n j D1 Sj [ fui g is independent. Again by (2),

jISi n

i 1 [

IS j j 

j D1

11 jSi j: 3

It follows that jI j D jIS1 j C

p X i D2

jISi n

i 1 [ j D1

IS j j 

11  jD j C 1: 3

Funke et al. [16] presented a new approach and outlined a proof for mip.n/  3:453n C 8:291:

t u

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However, Wan et al. [54] thought that the proof for a key geometric extreme property underlying this claim is missing, and thus the proof is incomplete. Gao et al. [17] proved the key geometric extreme property in detail and found that the proof of Funke et al. actually leads to a better bound. Lemma 5 (Gao et al. [17]) mip.n/  3:453n C 4:839. To understand the idea in deducing this bound, it is useful to first look at the deduction of a looser bound: jI j  3:748jD j C 5:252:

(3)

To prove (3), notice that the disks fDu0:5 j u 2 I g are mutually disjoint and they are 1:5 all contained in DD  . Hence jI j 

1:5 Area.DD / : 2   0:5

(4)

1:5 1:5 to a disk To obtain an upper bound for Area.DD  /, notice that adding a disk Dv  p  1 1:5 2 2 Du increases the 1:5-neighbor area by at most   1:5  2 1:5 arccos 3  22  2:9435 (this can be illustrated also by Fig. 2; in this setting, the disks have radius 1:5, and the upper bound is achieved when the distance between nodes u and v is exactly one). Then by an analysis similar to Theorem 3, it can be seen that 1:5 2   Area.DD  /    1:5 C 2:9435.jD j  1/  2:9435jD j C 4:1251:

(5)

Then inequality (3) follows from (4). The above deduction is not tight, because the set of disks fDu0:5 j u 2 I g cannot 1:5 overspread the whole area of DD  ; there are a lot of “wasted” fragments the areas S 1:5 0:5 of which are counted in Area.DD  / but have no contribution to Area u2I Du . To overcome this shortcoming, Funke et al. [16] proposed using Voronoi cells instead of disks of radius 0:5 to aid the analysis. For a node u 2 I , the Voronoi cell of u, denoted by V .u/, is the set of points on the plane which are closer to u than 1:5 to the other nodes of IS. Call T V .u/ D V .u/ \ DD  as the truncated Voronoi cell 1:5 of u. Clearly, DD  D u2I T V .u/. The advantage is that there is no waste in using truncated Voronoi cell. The key step in the paper [17] is to establish a lower bound: Area.T V .u//  0:8525 .8u 2 I /:

(6)

Replacing the denominator of (4) by 0:8525 and using (5), the upper bound in Lemma 5 is obtained. 1:5 In the case that the region DD  has no holes, Gao et al. [17] proved an upper bound: jI j  3:399jD j C 4:874:

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It was obtained by combining the lower bounds for the truncated Voronoi cells with 1:5 the Euler’s formula. In the case when DD  has holes, they proved jI j  3:473jD j C 4:874: Li et al. [33] improved the approach of Funke et al. [16] and proved the following: Lemma 6 (Li et al. [33]) mip.n/  3:4306n C 4:81854. Proof This bound is obtained by further exploring the lower bounds for the areas of the truncated Voronoi cells. The authors partition I into two subsets: I1 D fu 2 p 1= 3

I j Du

1:5  DD  g and I2 D I n I1 . It was proved that

p 3=2; for u 2 I1 ; Area.T V .u//  ; for u 2 I2 ; where   0:855. Hence p

p 3 3 1:5 jI1 j C jI2 j D jI j  Area.DD  /  2 2

! p 3   jI2 j: 2

(7)

Then the authors proved that p   1:51= 3 Peri DD  jI2 j  p ; 2.1  1= 3/

(8)

where Peri represents the perimeter of the region. By induction on jD  j, it can be proved that    p    2 1 1:51= 3  .jD j  1/ arcsin p C 2 3 p : Peri DD  2 3 3  2= 3 Then, the desired upper bound follows from (5), (7), (8), and (9).

(9) t u

Vahdatpour et al. [52] claimed that they proved mip.n/  3n C 3:

(10)

If their proof is correct, then this is best possible because Wan et al. [54] have constructed an example showing that mip.n/  3n C 3:

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Unfortunately, an important part of the proof was missing from [52]. Therefore, their result cannot be accepted by many researchers in the literature. What is the important part which was missing from the proof? To explain, the terminology of semi-exclusive neighboring set is needed. Let T be an arbitrary spanning tree of GŒD  and assume v1 ; v2 ; : : : ; vjT j is an arbitrary traversal of T and v1 is not a leaf vertex. For any i , 2  i  jT j, the set Ui D Ivi n Ifv1 ;:::;vi 1 g (the subset of MIS nodes which are adjacent with vi but not adjacent to any of the previous nodes v1 ; v2 ; : : : ; vi 1 ) is called the semi-exclusive neighboring set of node vi . By (1), one has jUi j  4 for i D 2; : : : ; jT j. Vahdatpour et al. tried to prove (10) by using induction on jT j. If there exists a leaf vertex vj with  jUj j  3, then the induction hypothesis can be applied on T  vj to yield jID  j  3.jT j  1/ C 3 C 3 D 3jT j C 3. Hence one may assume that every leaf vertex vj has jUj j D 4. For each leaf vertex vj , denote by f .vj / the first fork vertex (i.e., a vertex of degree at least three in T ) that is met when one starts from vj and goes along the path on T toward the root v1 . If there exists a vertex vi between vj and f .vj / such that jUi j  2, denote the path on T between vj and vi as P . Notice that T  V .P / is still a tree.  Hence one can apply the induction hypothesis to T  V .P / to obtain jID  j  3.jT j  t/ C 3 C 2 C 3.t  2/ C 4 D 3jT j C 3, where t is the number of nodes on P . What remains is the case that for every leaf vertex vj , every vertex vi between vj and f .vj / has jUi j  3, and the focus is on the distribution of MIS nodes near a fork vertex. This is the most complicated case, but Vahdatpour et al. did not provide a sufficient argument to deal with it. Due to the above reason, one can conclude that up to now, the best known upper bound for mip.n/ for general n is the one in Lemma 6 given by Li et al. [33]. In all the above works, geometry is explored deeper and deeper to obtain a tighter bound for the packing of disks with radius 0:5 in the neighbor area of a minimum CDS. The interested readers are referred to the mentioned literatures for the detailed implementation of geometry.

4

Sink Placement in Underwater Sensor Networks

Underwater Sensor Network has a lot of differences from the Terrestrial Sensor Network. It uses acoustic channels instead of radio-frequency signals, which cannot travel far in water. Thus a message which is forwarded through more intermediate nodes will experience a significantly longer data latency. To solve this problem, multiple UnderWater-Sink (UW-Sink) architecture was proposed [1]. Since UWSinks are very expensive, it is desirable to use as less UW-Sinks as possible while the number of relays for each sensor can be controlled within a given constant d . This consideration leads to the model of minimum d -hop sink placement (MdHSP) problem, the goal of which is to place a minimum number of sinks such that each sensor is at most d -hops away from at least one sink. Even in UDG, MdHSP problem is NP-hard [29]. However, this problem admits a PTAS in UDG [57]. In the following, the ideas in [29] which gave a constant approximation algorithm for MdHSP are introduced.

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A node set Id of a graph G D .V; E/ is a d -hop independent set (d IS) if any pair of nodes in Id have hop distance greater than d , it is a maximal d -hop independent set (d MIS) if any super-set of Id is no longer a d -IS, and it is a d -hop dominating set (dDS) if every node v 2 V n D is at most d -hops away from some node u 2 D (v is said to be d -hop dominated by u). Clearly, a d MIS is also a dDS. Let G be the UDG for the sensors. The algorithm just finds a d MIS Id in G and place a sink at the location of each node in Id . Clearly, Id is a solution to MdHSP. The authors in [29] proved that Id approximates the optimal solution within a factor of O.d /. First, they proved that optdDS  5optM dHSP ; (11) where optdDS and optM dHSP are the sizes of a minimum dDS and an optimal MdHSP, respectively. To show (11), suppose OPTM dHSP is an optimal solution to MdHSP. For each c 2 OPTM dHSP , find an MIS Ic in N.c/, whereSN.c/ is the set of sensors at Euclidean distance at most one from c. Let I 0 D c2OPTM dHSP Ic . For each node v 2 V , it is at most d -hops away from some c 2 OPTM dHSP . Let .v; w1 ; : : : ; wk ; c/ be a shortest path from v to c, where k  d  1. Then wk is either in Ic or adjacent with a node in Ic . In any case, v is at most d -hops away from some node in Ic . Hence I 0 is a dDS of G, and thus optdDS  jI 0 j. By Lemma 3, one has jI 0 j  5optM dHSP . In view of inequality (11), what remains to show is that jId j  O.d /optdDS

(12)

holds for any d MIS Id . Since a d MIS must be an MIS, an upper bound of factor O.d 2 / follows easily from Corollary 4: Each node in Id is d -hop dominated by some node in OPTdDS ; nodes which are d -hop dominated by u 2 OPTdDS are in the disk centered at u with radius d ; since the nodes in Id have mutual distances greater than one, Corollary 4 gives   2 2 jId j  p d C d C 1 optdDS D O.d 2 /optdDS : 3 To reduce the order of the factor, an intuitive idea might be that two nodes which are d -hop independent are far from each other. Unfortunately, this intuition is not true, as can be seen from Fig. 6. Thus a more elaborated analysis is in need. For a node u, denote by Id .u/ the set of nodes of Id which are d -hop dominated .1/ by u. Divide Id .u/ into two parts: Id .u/ D fv 2 Id .u/ j v is dd=2e-hop .2/ .1/ dominated by ug and Id .u/ D Id .u/ n Id .u/. It was proved in [29] that for any node u, .1/

(13)

.2/

(14)

jId .u/j  5: jId .u/j  O.d /:

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u2

u1

u3

u

ud+1 ud

Fig. 6 The node ud C1 is d -hop independent with node u; their Euclidean distance is only a little larger than one

.1/

To show (13), for each node v 2 Id .u/, let nPv be a shortest path o from v to u, and .1/

let wv be the last node on Pv . Then W .u/ D wv j v 2 Id .u/ is an independent set. In fact, if this is not true, suppose wv1 is adjacent with wv2 , then there is a path from v1 to v2 through edge wv1 wv2 with hop distance at most 2.dd=2e  1/ C 1  d , contradicting that v1 and v2 are d -hop independent. Thus it follows from Lemma 3 .1/ that jId .u/j  jW .u/j  5. .2/ To show (14), for each node v 2 Id .u/, let Pv be a shortest path from v to u, say, Pv D w0 w1 w2 : : : wp u, where w0 D v. Notice that p  dd=2e. For each i D 0; 1; : : : ; dd=2e  1, draw a disk Dw0:5i with center wi and radius 0:5. Denote Sdd=2e1 0:5 A.v/ D i D1 Dwi . First, notice that A.v1 / \ A.v2 / D ; for any pair of nodes .2/ v1 ; v2 2 Id .u/; otherwise, similarly to the above paragraph, one might obtain a contradiction that there is a path from v1 to v2 with hop distance at most d . Second, observe that the disks Dw0:50 ; Dw0:52 ; Dw0:54 : : : are mutually disjoint since Pv is a shortest path. Thus

 .d  3/ 1 d    .0:5/2  : Area.A.v//  2 2 16 S Since v2I .2/ .u/ A.v/ is contained in the disk centered at u with radius d C 0:5, one d has .d C 0:5/2 .2/ D O.d /: jId .u/j  .d  3/=16 Then, (12) follows from (13) and (14).

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Theorem 6 The MdHSP problem in unit disk graph admits an approximation of factor O.d /. The above analysis also shows that the minimum d -Hop dominating set problem can be approximated within factor O.d /, which provides an approximation guarantee for the previous works such as [2], and can also be a step stone for some other more constraint problems such as minimum connected d -hop dominating set [43], minimum d -hop connected d -hop dominating set [30, 48], and minimum connected k-fold d -hop dominating set [32, 60].

5

Data Aggregation Scheduling

In data aggregation problem, a sink node s collects an information from the other nodes. During the aggregation, each node other than the sink combines all the information it receives with its own and sends the combined information only once. For example, to detect a burst of fire, the base station is interested in the highest temperature in a sensed field. Each node in the field detects its local temperature. When a node receives a data sent from another node, it compares the received temperature with the highest temperature stored in it, chooses the higher one, and puts it into the current data as well as the corresponding location. When it has aggregated all the data that are planned to be sent to it, it forwards the aggregated data. It can be seen that the topology of an aggregation can be modeled as an inward arborescence rooted at the sink node, call it as an aggregation arborescence. Because of interference, the aggregation might be delayed. In the protocol interference model, node u can receive data from node v only when u is in the transmission range of v and no other nodes are in the interference range of u. Suppose each node has a fixed transmission radius which is scaled to one and an interference radius  > 1. Then two links u1 v1 and u2 v2 interfere with each other if either u1 v2 or u2 v1 has length at most . A set of links are interference-free if they are mutually interference-free. In an aggregation schedule, one has to find an aggregation arborescence and assign time-slots to its links such that (i) a node sends the combined data to its parent only after it has collected all the data from its children and (ii) all the links assigned to a common time-slot are interference-free. The latency of the aggregation schedule is the number of time-slots used. The goal of the minimum-latency aggregation schedule problem (MLAS) is to find an aggregation schedule with the minimum latency. Even for the case that  D 1, the MLAS problem is NP-hard [6]. For a set of nodes V in the plane, let G be the unit disk graph on V . The maximum hop distance from the sink node s to the other nodes is the radius of G with respect to s, denoted by R. The approximation algorithms given in [6] and [27] achieve latencies at most .  1/R and 23R C   18, respectively, where  is the maximum degree of G. In [55], Wan et al. presented three approximation algorithms called SAS, PAS, and EPAS, which produce latencies at most 15R C   4, 2R C  C O.log R/, and

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 p  1 C ‚ log R= 3 R R C . In Sects. 5.1 and 5.2, the main ideas of PAS and EPAS are introduced, respectively. An aggregation schedule for interference radius  D 1 is said to be well separated if at each time-slot of the schedule, either all the transmitting nodes have mutual distances greater than one or all the receiving nodes have mutual distances greater than one. Using well-separated schedule, Wan et al. [55] introduced a generic method for the aggregation scheduling problem with  > 1. This method is presented in Sect. 5.3. Parts of the above works were generalized to many-to-one data aggregation scheduling problem in multichannel multi-hop wireless networks [34].

5.1

Pipelined Aggregation Scheduling (PAS)

A basic step for PAS is the establishment a single-hop aggregation schedule from X to Y , where X; Y are two disjoint node sets. The schedule is described in Algorithm 1 called iterative minimal covering (IMC), in which data are sent through the links specified by S . In each iteration, a minimal cover C of the remaining node set X 0 is formed. By the minimality of C , every node y 2 C has a private neighbor x in X 0 , that is, y is the only neighbor of x in C . Then xy is the link through which x is set to report its data. The integer `.xy/ indicates the time-slot assigned to the link xy. It can be seen that `.xy/ is no more than the number of neighbors of y in X .

(15)

By choosing x to be a private neighbor of y, the links with the same label are interference-free. Algorithm PAS first constructs a connected dominating set (CDS) D of the unit disk graph G. For each node u in V nD, assign an arbitrary dominator v as its parent, indicating that u reports data through link uv. In the first phase of the aggregation, data are sent from V n D to D using the single-hop aggregation schedule IMC. By the property in (15), this phase takes at most   1 time-slots. In the second phase of the aggregation, transmissions are scheduled within the CDS D. In the following, the focus is put on the construction of D and the scheduling within D. Though the problem of computing a minimum CDS has constant approximations, the CDS D in [55] is constructed in a specific way so that it has stronger properties to be used in analyzing the latency. It starts from constructing a breadthfirst-search tree T of G and finds a maximal independent set (MIS) U induced by T in a first-fit manner: order the nodes of T layer by layer as v1 ; v2 ; : : : ; vn , where v1 D s; set U D fv1 g initially; for i D 2 to n, add vi to U if vi is not adjacent with any node in U . Notice that any MIS of G is also a dominating set of G. After that, more nodes W called connectors are added in the following way such that D D U \ W is a CDS of G: Let H be the graph with node set U ; two nodes in U are adjacent in H if they have a common neighbor in G. Suppose the radius of H

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Algorithm 1 IMC 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

S D ;, ` D 0, X 0 D X, Y 0 D Y . while X 0 ¤ ; do ` D ` C 1. Let C  Y 0 be a minimal cover of X 0 . for each y 2 C do Let x 2 X 0 be a private neighbor of y. S D S [ fxyg. `.xy/ D `. X 0 D X 0 n fxg. Y 0 D C. end for end while Output S and `.

with respect to s is R0 . It can be seen that R0  R 1. For i D 0; 1; : : : ; R0 , let Ui be the set of nodes at distance i from s in H , and let Pi be the set of nodes which are adjacent with some node in Ui and some node in Ui C1 . Compute a minimal cover S 0 Wi  Pi for Ui C1 . The connector set W D iRD11 Wi . Since all the nodes in U are contained in the disk centered at s with radius R, it follows from Corollary 4 that 2 jU j  p R2 C R C 1: 3

(16)

Furthermore, since Wi is a minimal cover of Ui C1 , one has jWi j  jUi C1 j, and thus jW j  jU j  1. Hence 4 jU [ W j  p R2 C 2R C 1: 3

(17)

Let D D U \ W . Denote the subgraph of G induced by D as GŒD . Clearly, GŒD is connected. Let RD be the radius of GŒD with respect to s. For i D 0; 1; : : : RD , let Vi be the set of nodes at distance i from s in GŒD . The aggregation arborescence TD within D is constructed together with a link labeling and a node ranking rank as in algorithm CBFS. For 0  i  RD and 0  j  rank.s/, let tij D .RD  i / C 44j . Each link uv of TD is scheduled to the time-slot tij C 4.`  1/, where i is the depth of u in TD (i.e., u 2 Vi ), j D rank.u/, and ` is the label of the link uv. It was proved in [55] that the above method gives an aggregation schedule with latency at most t0;rank.s/ . Notice that the node ranking produced by CBFS has the property that rank.s/  log jDj. Combining this with (17), rank.s/  O.log R/. Hence, by observing RD  2.R  1/, one has r0;rank.s/ D RD C 44  rank.s/  2R C O.log R/:

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Algorithm 2 CBFS 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:

TD D .D; ;/. For each u 2 VRD , rank.u/ D 0. for i D RD down to 1 do Let J D frank.v/ W v 2 Vi g. for each j 2 J do Let Vij D fv 2 Vi W rank.v/ D j g. Augment TD by applying Algorithm IMC to the sets Vij and Vi1 . end for for each u 2 Vi1 do if u has no child then rank.u/ D 0 elseLet r be the maximum rank of the children of u. if u has only one child with rank r then rank.u/ D r else rank.u/ D r C 1. end if end if end for end for Output T and rank.

Taking the first phase of aggregation into account, the latency of the scheduling produced by the algorithm PAS is at most 2R C O.log R/ C .

5.2

Enhanced Pipelined Aggregation Scheduling (EPAS)

EPAS uses the same outline as PAS but different set of connectors and different function of time-slot assignment. The set W of connectors is constructed based on the same MIS U as in PAS, which is induced by the breadth-first-search tree T . For each node v, denote by p i .v/ the i th ancestor of v, that is, p i .v/ is i hops away from i on the path of T from v to s. Fix a positive integer k. Set W 0 D ; initially. For each node u 2 U n fsg, suppose i is the smallest positive integer such that p i .u/ is in U , add fp j .u/ W 1  j  minfi  1; kgg into W 0 . Notice that minfi  1; kg  1, and thus by the construction of U , the subgraph GŒU [ W 0 is connected. In GŒU [ W 0 , construct a shortest path tree T 0 from s to all the nodes in U n fsg. Let W be the subset of W 0 which is contained in V .T 0 /. Since for each node u 2 U n fsg, at most k nodes are added into W 0 , one has jW j  jW 0 j  k.jU j  1/: Combining this with (16),   2 jDj D jU [ W j  .k C 1/ p R2 C R C 1: 3

(18)

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Furthermore, it was proved in [55] that   1 RD  1 C R; k

(19)

where RD is the radius of the subgraph p GŒD with respect to s. The EPAS uses parameter k D ‚. 3 R= log R/ to find a connector set W . Let TD be the aggregation arborescence of GŒU [ W constructed by CBFS. Suppose `0 is the maximum label on the links of TD . For 0  i  RD and 0  j  rank.s/, let tij D .RD i /C3`0 j . Each link uv of TD is scheduled to the time-slot tij C3.`1/, where i is the depth of u in TD , j D rank.u/, and ` is the label of the link uv. Similar to PAS, the number of time-slots used by EPAS is at most t0;rank.s/ . By (18), (19), and the observation rank.s/  O.log R/, t0;rank.s/

   log R R: D RD C 3`0 rank.s/  1 C ‚ p 3 R

Taking the first phase ofaggregation into account, EPAS produces a schedule with p latency at most 1 C ‚ log R= 3 R R C .

5.3

A Generic Method for  > 1

Let S D fSi W 1  i  `g be the output of a well-separated schedule using interference radius  D 1, where ` is the latency of the schedule, and Si is the set of links scheduled at time-slot i . For each 1  i  `, links in Si might have interference with respect to  > 1 and thus need to be further scheduled. For a link set L, the conflict graph of L is the undirected graph C G.L/ with vertex set L; two vertices are adjacent in C G.L/ if the corresponding two links interfere with each other. It can be seen that scheduling Si into interference-free subsets is equivalent to properly coloring the vertices of C G.Si /, with each color class corresponding to a set of interference-free links. Suppose H is a graph to be colored, and V .H / is ordered as v1 ; v2 ; : : : ; vn . The first-fit coloring assigns colors to the vertices sequentially. When vertex vi is to be colored, it is assigned the first available color, namely, the color with the smallest index which has not been used by any of its previous neighbors. Such a coloring strategy uses at most ı  C 1 colors, where ı  D max1i n fdprev .vi /g and dprev .vi / is the number of previous neighbors of vi in the ordering (also see [39]). By symmetry, it suffices to consider the case that the transmitting nodes of Si have mutual distances greater than one.

(20)

Order the vertices of C GŒSi as u1 v1 ; u2 v2 ; : : : ; up vp such that the tails u1 ; u2 ; : : : ; up of the corresponding links are located from left to right. When

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vertex ui vi is to be colored in C GŒSi , all those links corresponding to the previous neighbors of vertex ui vi w