Concentrator Location in Telecommunication Networks [1 ed.] 0387235310, 9780387235318

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Concentrator Location in Telecommunications Networks

COMBINATORIAL OPTIMIZATION VOLUME 16 Through monographs and contributed works the objective of the series is to publish state of the art expository research covering all topics in the field of combinatorial optimization. In addition, the series will include books, which are suitable for graduate level courses in computer science, engineering, business, applied mathematics, and operations research. Combinatorial (or discrete) optimization problems arise in various applications, including communications network design, VLSI design, machine vision, airline crew scheduling, corporate planning, computer-aided design and manufacturing, database query design, cellular telephone frequency assignment, constraint directed reasoning, and computational biology. The topics of the books will cover complexity analysis and algorithm design (parallel and serial), computational experiments and application in science and engineering. Series Editors Ding-Zhu Du, University of Minnesota Panos M. Pardalos, University of Florida Advisory Editorial Board Alfonso Ferreira, CNRS-LIP ENS London Jun Gu, University of Calgary David S. Johnson, AT&T Research James B.Orlin, M.I.T. Christos H. Papadimitriou, University of California at Berkeley Fred S. Roberts, Rutgers University Paul Spirakis, Computer Tech Institute (CTI)

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

by

HANDE YAMAN Department of Industrial Engineering Bilkent University, Ankara, Turkey

Springer

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0-387-23532-9 0-387-23531-0

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Contents

List of Figures List of Tables

ix xiii

List of Algorithms Acknowledgments

xv xvii

Part I

Problem Definition and Survey

1. INTRODUCTION 2.

A SURVEY ON LOCATION PROBLEMS WITH APPLICATIONS IN TELECOMMUNICATIONS 2.1 Facility Location Problems Problems 2.2 2.3 Hub Location Problems with Single Assignment 2.4 Conclusion

3. QCL-C: RELAXATIONS AND SPECIAL CASES 3.1 Problem Definition and Formulation for the QCL-C 3.2 Relaxations Based on the Capacity Constraints 3.3 Special Cases Based on the Backbone Network and the Routing Cost Structure 3.4 The Relationship between the Problems

3 9 10 18 20 22 25 26 28 28 29

vi

Part II

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Polyhedral Analysis

4. UNCAPACITATED PROBLEMS 4.1 Uncapacitated Concentrator Location Problem 4.2 Uncapacitated Concentrator Location Problem with Star Routing 4.3 Uncapacitated Concentrator Location Problem with Complete Routing 4.4 Conclusion

35 36 63 80 96

5. PROBLEMS WITH LINEAR CAPACITY CONSTRAINTS 5.1 Linear Capacitated Concentrator Location Problem 5.2 Linear Capacitated Concentrator Location Problem with Star Routing 5.3 Linear Capacitated Concentrator Location Problem with Complete Routing 5.4 Conclusion

99 100

6. PROBLEMS WITH QUADRATIC CAPACITY CONSTRAINTS 6.1 Quadratic Knapsack Constraint 6.2 Quadratic Capacitated Concentrator Location Problem 6.3 Quadratic Capacitated Concentrator Location Problem with Star Routing 6.4 Quadratic Capacitated Concentrator Location Problem with Complete Routing 6.5 Conclusion

145 146 156

165 171

Summary of Results of Part II

173

Part III

124 127 142

163

Solving QCL-C with Branch and Cut

7. QCL-C: FORMULATIONS AND PROJECTION INEQUALITIES 7.1 Comparing Formulations 7.2 Projection Inequalities 7.3 Semi-extended Formulations

179 180 186 187

8. BRANCH AND CUT ALGORITHM FOR QCL-C 8.1 Preprocessing and Basic Strengthening 8.2 Other Valid Inequalities and Separation Algorithms 8.3 Branch and Cut Algorithm and Computational Results

197 198 200 206

Contents

vii

9. CONCLUSION

221

Appendices A Facets of for and B Detailed Results of Tests to Determine Useful Cuts B.1 Results of Test 1 B.2 Results of Test 2

225 231 232 244

References

251

Index

257

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List of Figures

1.1 3.1 4.1 4.2 4.3 4.4 4.5

Complete backbone and star access networks The relaxations and projections Star backbone and star access networks A W-2 inequality A k-triangle configuration The solution to show that The solution to show that for

4.6

The solution to show that such that is even and The solution to show that such that is odd and The solution to show that such that and are odd and The solution to show that such that is odd, is even and The solution to show that such that is odd, is even and The solution to show that such that and are even and The solution to show that such that is even, is odd and The solution to show that such that and are even and The solution to show that such that is even, is odd and Odd holes in the conflict graph

4.7 4.8 4.9

4.10 4.11 4.12 4.13 4.14 4.15

5 31 36 40 42 43 44

for 45

for 45

for 46

for 46

for 47

for 47

for 47

for 48

for 48 49

x

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25

Removing a triangle From a linear ordering to a Removing node Removing nodes and Removing nodes and A k-leaf configuration Solutions and to show that and Solutions and to show that and Solutions and to show that Solutions and to show that

configuration

51 52 53 54 54 56

for 56

for 57 57

for all for

58

4.26 4.27 4.28 4.29 4.30 4.31 4.32 5.1

Solutions

and

to show that

for

Arcs with coefficient 1 in a k-leaf configuration The 2-cycle inequality (4.24) The 2-cycle inequality (4.25) Routing on a star backbone network Routing on a complete backbone network The separation problem as a mincut affinely independent points in that satisfy

58 59 61 61 64 81 94 102

5.2 5.3 5.4 5.5 5.6 5.7

5.8

Solutions and to show that for and Solutions and to show that for and Solutions and to show that for and Solutions and to show that for and Solutions and to show that for Solutions and to show that for and if there exists a such that and Solutions and

and

to show that if there exists a

for such that

105 105 105 106 106

107 107

xi

List of Figures

5.9

5.10 5.11 5.12

Solutions and

and

to show that for all

for and

Solutions and to show that and if for all Solution if Solution if

for and

if

108 108 110 110

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List of Tables

3.1 3.2 3.3 6.1 6.2 6.3 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

Concentrator Location Problems Concentrator Location Problems with Star Routing Concentrator Location Problems with Complete Routing Summary of polyhedral results for concentrator location problems with no routing cost Summary of polyhedral results for concentrator location problems with star routing Summary of polyhedral results for concentrator location problems with complete routing Different formulations - Duality gaps Different formulations - CPU times Projection inequalities Semi extended formulation - Duality gaps Semi extended formulation - CPU times Second semi extended formulation - Duality gaps Second semi extended formulation - CPU times Preprocessing and basic strengthening Branching and enumeration strategies Cuts Cuts-2 Problems with 17 nodes-Branch and Cut Problems with 17 nodes-CPLEX CPLEX with strengthening Problems with 22 nodes Hub location problems-1

30 30 30 174 175 176 184 185 188 191 192 195 195 201 210 211 212 215 215 216 218 219

xiv

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

8.10 B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 B.9 B.10 B.11 B.12 B.13 B.14 B.15 B.16 B.17 B.18

Hub location problems-2 Preprocessing Cover inequalities Strengthened projection inequalities Step inequalities Quadratic binpacking inequalities Binpacking and residual capacity inequalities Effective capacity inequalities W-2 inequalities k-triangle inequalities k-leaf inequalities 2-cycle inequalities Odd hole inequalities Binpacking and residual capacity inequalities-2 Effective capacity inequalities-2 W-2 inequalities-2 k-triangle inequalities-2 k-leaf inequalities-2 2-cycle inequalities-2

220 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249

List of Algorithms

6.1 6.2 7.1 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Separation of projection inequalities Preprocessing and strengthening Computation of lower bounds for Separation of W – 2 inequalities Separation of k-triangle inequalities Separation of 2-cycle inequalities Separation of effective capacity inequalities Rounding heuristic

154 155 187 199 200 202 203 203 207 209

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Acknowledgments

I started studying the polyhedra of concentrator location problems for my PhD thesis, at Service d’Optimisation, Université Libre de Bruxelles. It was a great chance and pleasure to meet my supervisor Martine Labbé and to work in her team. I owe much to her for introducing me to this domain and for her guidance and advise. Of course, I am indebted to several other people for their suggestions, feedback and support. Hoping not to forget anyone, I would like to thank Jean-Paul Doignon, Eric Gourdin, Pierre Hansen, Thomas M. Liebling, Ridha Mahjoub, Valéry Paternotte, Yves Pochet, Jean-François Raskin and Phillippe Vincke for taking time to read this text and for their valuable comments. Andrea Lodi provided the code to compute the bound. Alberto Caprara proposed the branch and bound algorithm to solve the quadratic knapsack problems. Yves Pochet suggested the semi-extended formulations and Mathieu Van Vyve provided his manuscript on approximate extended formulations. Corinne Feremans and Inmaculada Rodriguez Martin helped me a lot with ABACUS, Gilles Fasbender, Hadrien Mélot and David Huygens with the computer, and Françoise Van Brussel and Véronique Bastin with the administrative work. I would like to thank them all. The problems that make the subject of this book are proposed by France Telecom. Their collaboration and financial support are gratefully acknowledged.

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PART I

PROBLEM DEFINITION AND SURVEY

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Chapter 1 INTRODUCTION

“Telecommunications” comes from the greek “tele” (far) and the latin “communicare” (share). Its meaning has become so clear to us, as we buy books, read newspapers or transfer money from the internet, send pictures, e-mail family or call friends, that we almost forgot how significantly it affected our lives. As a matter of fact, technology has considerably evolved since the inventions of Chappe, Morse and Bell, let alone since our ancestors cleverly thought of using smoke signs and other “tamtam” devices, often to communicate vital information. It seems reasonable to state that for a given technology (smoke signs, telephone lines, ISDN,...), the system’s performance depends on its organization. In other words, on optimization. The significant change in telecommunications networks over the last decade has consequently raised new optimization problems, among which the optimal location of concentrators remains a crucial one, especially when combined with other features of the new telecommunications networks. This book deals with such problems. In a typical telecommunications network, “terminals” are the users that have some traffic demand. The traffic gathered from many terminal nodes is progressively combined in order to fill links of increasing capacity and is finally forwarded to its destination. As a result, telecommunications networks, often, have a multi-layer hierarchical architecture. At a lower level of such an architecture, the traffic is collected to be sent across an upper level. Although the real telecommunications networks are usually structured into many such levels, most design problems considered by the telecommunications companies concern only part of the overall network. Here we are interested in two-level networks.

4

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

A two-level generic telecommunications network consists of backbone and access networks. The backbone is the top level network through which the concentrators are connected to each other. The access networks make the level below and they connect the terminals to concentrators. Different topologies are possible for these networks. Backbone network is complete if there are direct links between all pairs of concentrators. If it is too costly to realize such a high level of connection, the concentrators can be connected by a ring, a path, a tree or they may be connected to some central unit by direct links forming a star. Similar structures are possible for the access networks. As the issues of quality and survavibility are more critical on the top level of the network, usually access networks are less dense compared to the backbone network. This book is devoted to the study of concentrator location problems in telecommunications networks where access networks are stars. We consider concentrator location problems with different cost structures and capacity constraints. We started studying these problems for a project funded by France Telecom. The principal problem of this project and this book is as follows. We are given a set of terminal nodes and a traffic matrix whose entries represent the amount of traffic to be routed between pairs of terminals. We determine a subset of the terminal nodes to be the concentrator locations. Each node that is not a concentrator is connected to exactly one concentrator node by direct links. Concentrators are connected to each other by a complete backbone network. Figure 1.1 depicts a network with a complete backbone network and star access networks. Nodes 1 to 5 are the concentrator nodes and they are connected by a complete network. Each of the remaining nodes is connected to a single concentrator. It is also possible to have concentrators to which no other node is connected. The traffic between two nodes is routed on the shortest path between these nodes. We assume that the cost of routing the traffic is a function of the distance and it satisfies the triangle inequality. If two nodes are connected to the same concentrator, the traffic between them does not enter the backbone network. So the traffic routed on the backbone links is the traffic of nodes that are connected to different concentrators. More specifically, the traffic on a backbone link from node to node is the sum of the traffic between all nodes connected to node and all nodes connected to node

5

Introduction

Figure 1.1. Complete backbone and star access networks

A concentrator has a fixed capacity in terms of the traffic that passes through it. This traffic is the sum of the traffic on the links between this concentrator and the nodes that are connected to it and the traffic on the backbone links between this concentrator and the other concentrators. There is a fixed cost for installing a concentrator at a certain node and a cost per unit of traffic routed between two given nodes. The aim is to locate the concentrators and to connect the remaining nodes to the concentrator nodes to minimize the total cost for installing concentrators and the cost for routing the traffic in the network. We call this problem “Quadratic Capacitated Concentrator Location Problem with Complete Routing” and abbreviate it by “QCL-C”. The aim of this book is to investigate the polyhedral properties of the QCL-C and to develop a branch and cut algorithm to solve it. When the project started, we first reviewed the literature on concentrator location problems in telecommunications networks. We did not encounter any study on the QCL-C. We saw that exact methods exist usually for the concentrator location problems with no routing cost. The concentrator location problems with routing cost are usually solved by heuristic approaches.

6

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Moreover, very little is known about their polyhedral properties. Then, we started to study the polyhedron of the QCL-C and to develop a branch and cut algorithm to solve it. As the problem seemed rather complicated, we found it appropriate to study some relaxations and special cases which could hopefully give some insight to solve QCL-C. We consider two relaxations of the QCL-C based on the capacities. The first relaxation is obtained by changing the capacity requirements to be as follows. Each concentrator has a fixed capacity in terms of the traffic adjacent at nodes (the sum of the traffic of commodities having this node either as origin or destination) that are connected to it (and not on the traffic between this concentrator and the other concentrators). This version of the problem is called “Linear Capacitated Concentrator Location Problem with Complete Routing”. The second relaxation is obtained by removing the capacity constraints and it is called “Uncapacitated Concentrator Location Problem with Complete Routing”. We also study two special cases of the problem. The first case occurs when the routing cost on the backbone network is negligible. In this case we have a “Quadratic Capacitated Concentrator Location Problem”. The second case corresponds to solving the location and routing problem on a network with a star backbone. In this case we are given a central unit and each concentrator node is connected to this node by direct links. The resulting backbone is a star. In such a network there is a single path connecting two concentrator nodes and this path passes through the central node. The corresponding problem is called “Quadratic Capacitated Concentrator Location Problem with Star Routing”. For both special cases, we also consider the relaxations with respect to the capacity structures. This makes nine different problems which are closely related. We study the polyhedra corresponding to these nine problems. We start with uncapacitated problems. Then we consider linear capacitated and quadratic capacitated problems. Once the capacity structure is fixed, the problems are considered in the following order: no routing cost, routing cost defined on a star network and routing cost defined on a complete network. The idea for this ordering is to start with easier problems (problems with fewer or easier constraints or problems with fewer variables) and try to generalize the results to more complicated cases. This sometimes leads to repetitions, which we nonetheless keep for the sake of completeness.

Introduction

7

Most of the results presented in this book are from Labbé and Yaman [50], [51], [52], and Labbé et al. [53]. The book is organized as follows. In Chapter 2, we survey the literature on location problems in telecommunications. In Chapter 3, we formally define the QCL-C. We give a formulation of this problem and then discuss its relaxations and special cases. The aim is to show how these problems are related and why it can be interesting to consider the other eight problems. In Chapter 4, we investigate the polyhedra of the uncapacitated problems. We modify some of these results in Chapter 5 for the polyhedra related with the problems with linear capacity constraints. We also give some results based on the capacities. In Chapter 6, we investigate the structure of the polyhedra related with the problems with quadratic capacity constraints. We use these results to develop a branch and cut algorithm to solve the QCL-C. The formulations, algorithm and computational results are presented in Chapters 7 and 8. Finally, we give some conclusions and future research directions in Chapter 9.

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Chapter 2 A SURVEY ON LOCATION PROBLEMS WITH APPLICATIONS IN TELECOMMUNICATIONS

The aim of this survey is to gather some significant examples of location problems that arise in the design of telecommunications networks and review the polyhedral properties of these problems. The survey in Gourdin et al. [37] gives a more detailed analysis of the models and solution methods for most of the problems presented here. For earlier surveys, one can also refer to Boffey [10] and Klincewicz [46]. There is a very recent survey by Campbell et al. [13] on hub location problems. Here, we review only the hub location models for the single assignment version. The basic questions related with the design of a telecommunications network are where should the concentrators be located, what type of machinery should be used, how should the terminals be connected (assigned) to the concentrators and how should the concentrators be connected among themselves or to some central node, how should the traffic be routed and what type of links and how many of these links should be installed on the edges of the network. It is quite hard to come up with answers to all these questions simultaneously. So, in most methods, the design is done in an iterative manner. Initially, the concentrator locations and assignments of terminals to the concentrators are determined. Then the design of the access networks and the design of the backbone network become independent and can be handled separately. Thus the location of concentrators is a crucial issue.

10

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

2.1

Facility Location Problems

The facility location models for telecommunications networks are concerned with the location of concentrators and the assignment of terminals to these concentrators. The basic problems in most of these models are the Uncapacitated Facility Location Problem (UFLP), the Capacitated Facility Location Problem (CFLP) and the Capacitated Facility Location Problem with Single Assignment (CFLPS).

2.1.1

Uncapacitated Facility Location Problem

In this section, we formulate the UFLP and present some of its variants that have applications in telecommunications. The UFLP is as follows. Given the set of clients N and the set of possible locations for the facilities M, determine the location of facilities and assign the clients to these facilities in order to minimize the sum of the cost of installing facilities and the cost of serving clients via the installed facilities. Let be 1 if a facility is installed at location Define to be the fraction of demand of client facility at location

and 0 otherwise. that is served by the

Using these two sets of variables, the UFLP can be formulated as follows:

where and

denotes the cost of serving client by facility at location denotes the cost of installing a facility at location

By constraints (2.2) and (2.4), the demand of each client is served, and by constraints (2.3), it is served by a facility only if this facility is installed. The cost function (2.1) is the sum of the cost of serving clients and the cost of installing facilities.

A Survey on Location Problems with Applications in Telecommunications

11

For more information about the UFLP, one may refer to, e.g., Cornuejols et al. [22], Krarup and Pruzan [48], and Labbé et al. [49]. In the context of telecommunications, clients are terminals and facilities are concentrators. The UFLP can be solved to decide about the locations of concentrators and the assignments of terminals. Then the design of the backbone and access networks can be done separately. It can also be solved to design a network where both the backbone and access networks are stars (see Chapter4). For polyhedral properties of UFLP, see, e.g., Canovas et al. [14], Cho et al. [18, 19], Cornuejols and Thizy [24], and Guignard [40]. Next, we present examples of network design problems where UFLP is either generalized or appears as a subproblem. These examples are about the design of networks where at least one component is a star or a tree. Helme and Magnanti [43] consider the design of a satellite communications network where each terminal is directly assigned to a concentrator and concentrators are directly assigned to a central unit. Thus the backbone and access networks are stars. There is an operating cost for the concentrators, which is the cost of using the capacity of concentrators. To compute this cost, it is necessary to differentiate the traffic between terminals assigned to the same concentrator and the traffic between terminals assigned to different concentrators. This results in a quadratic cost function. The authors present a linearization for the problem and a branch and bound algorithm to solve it. Chardaire et al. [17] consider the design of a two level network where backbone and access networks are stars. Each terminal is connected to a first level concentrator which is connected to a second level concentrator. All second level concentrators are connected to a central unit. They present two integer programming formulations, a simulated annealing algorithm and a family of cuts. Chung et al. [21] consider the design of a network where the backbone is fully connected and the access networks are stars. The authors develop a formulation to find such a network that minimizes the cost of installing concentrators, cost of assigning terminals to concentrators and the cost of interconnecting concentrators. The total cost function is quadratic due to the last component. They linearize the formulation and present a dual-based solution procedure.

12

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Mateus et al. [60] design a network, where the backbone and access networks are trees, in three phases. In the location phase, they solve a UFLP where the number of concentrators to be installed is bounded from above. Then, for each concentrator and the set of terminals assigned to it, they find a minimum cost tree network with the concentrator as the root. In the third phase, the backbone network is designed. Current and Pirkul [26] consider the problem where the concentrators are connected to each other by a path of primary arcs and the terminals are connected to the concentrators by paths of secondary arcs. Two heuristics based on Lagrangian Relaxation are given. Pirkul and Nagarajan [70] and Lee et al. [55] consider the design of a network where the backbone is a tree and access networks are stars. Pirkul and Nagarajan (1992) use a two-phase algorithm where in the first phase, they divide the set of nodes into regions and in the second phase, they determine, for each region, a path from the furthest node of the region to the central node such that the nodes of this path are concentrators. Lee et al. [55] present a formulation and apply Lagrangian Relaxation. Gavish [35] formulates the problem of designing a network where the backbone is a star and access networks are trees. The objective function involves the cost of establishing the links and installing the concentrators. The model chooses among different types of links with different costs and capacities. Gavish [36] discusses the evolution of the network topologies and the network design process and gives a model to design a network with no a priori topology. The author presents a Lagrangian Relaxation based solution procedure.

2.1.2

Capacitated Facility Location Problems

Capacitated location problems in telecommunications, relax the assumption that a concentrator can serve all terminals. These models are mostly variants of CFLP and CFLPS. The CFLP is defined as follows. Given a set of clients N with known demands for each client and a set of possible locations for facilities M with a given capacity for each the aim is to install facilities and serve the demands of the clients respecting the capacities of the facilities and

A Survey on Location Problems with Applications in Telecommunications

13

to minimize the total cost of installing facilities and the cost of serving clients. The CFLP can be formulated as follows:

Constraints (2.6) ensure that client is served by facility only if a facility is installed at location and if the capacity of that facility is not exceeded. Sridharan [77] gives a survey of the CFLP. Cornuejols et al. [23] compare relaxations and heuristics. Aardal [1] gives a branch and cut algorithm. The CFLPS (also known as the CFLP with single sourcing or capacitated concentrator location problem) has an additional constraint that asks the demand of each client to be satisfied by exactly one facility. The CFLPS can be formulated as follows:

Neebe and Rao [64] formulate the CFLPS as a set partitioning problem and solve it by a branch and price algorithm. The pricing problem decomposes into knapsack problems. Heuristics based on Lagrangian relaxation can be found in, e.g., Beasley [9], Cortinhal and Captivo [25], Darby-Dowman and Lewis [27], Klincewicz and Luss [47], Mirzaian [63], Pirkul [69], Sridharan [76]. Variants of CFLP and CFLPS exist. In the first variant, multitype facilities are available. Each type of facility has a different cost and capacity. For this problem, Lee [54] proposes an algorithm based on cross decomposition, which combines Benders Decomposition and Lagrangian Relaxation and presents computational results. Amiri [3] studies a variant with delay costs. He models the system as a M/M/1 queuing system and gives two Lagrangian Relaxation based heuristics to solve this problem.

14

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

If reliability is an important issue, it is possible to serve one client by several facilities. Tang et al. [78] consider the problem where it is necessary to connect client to exactly facilities. This variation is called CFLP with multiple homing. Pirkul et al. [71] study the case where every client is served by two facilities, once for the primary coverage and then for the secondary or backup coverage. In the remainder of this section, we review the polyhedral results on CFLP and CFLPS. The CFLP is a relaxation of CFLPS. So, the valid inequalities for the CFLP are also valid for the CFLPS. Moreover, under some conditions, the facet defining inequalities of the polytope associated to CFLP are also facet defining for the polytope associated to CFLPS. We first survey the valid and facet defining inequalities for the CFLP polytope.

CFLP: Valid Inequalities and Facets Leung and Magnanti [56] study the valid inequalities and facets of the polytope associated to the CFLP. They consider the case where all facilities have the same capacity Q. Let F be the set of such that satisfies constraints (2.6) and

and let Leung and Magnanti [56] prove that PF is full-dimensional and give facet defining inequalities. For

define Let and (mod Q) with if is a multiple of Q. If facilities serve the demands of clients in with full capacity, then the residual demand to be satisfied by the last facility is

T HEOREM 2.1 (Leung and Magnanti [56]) Residual capacity inequality

is valid and defines a facet of PF when

and

A Survey on Location Problems with Applications in Telecommunications

15

Aardal et al. [2] study the valid inequalities and facets of the polytope associated to the CFLP. Define to be the flow between client and facility In other words, Consider the set X defined by the following constraints:

Let The authors assume that for all Then results due to Aardal et al. [2].

Below we review

Let be such that Set is called a cover with respect to N and M and a minimal cover if, in addition, for all we have As the demand of all clients cannot be satisfied through facilities in at least one of the facilities in has to be installed. So valid inequality. THEOREM 2.2 (Aardal et al. [2]) If M and N, and

is a

is a minimal cover with respect to then

defines a facet of

A subset such that cover set with respect to M and N. If facility at node the maximum amount of flow between facilities in change if and decreases by otherwise.

is called a flow is closed, then and clients does not

THEOREM 2.3 (Aardal et al. [2]) Let be a flow cover with respect to M and N such that and for all Then the flow cover inequality

defines a facet of PX.

16

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Flow cover inequalities can be generalized as follows. Let For each let Define is a flow cover with respect to M and if

and Set

THEOREM 2.4 (Aardal et al. [2]) Let be a flow cover with respect to M and Let be the set of facilities for which Assume for all The effective capacity inequality

defines a facet of PX if and only if for all there exists with

for all for all

and if

then

Another family of valid inequalities by Aardal et al. [2] is the family of submodular inequalities. A set function on is submodular if for all Let and for all The function

is submodular on M. Define THEOREM 2.5 (Aardal et al. [2]) Let all The submodular inequality

for and

for

is valid for PX. Conditions under which submodular inequalities define facets of PX are given in Aardal et al. [2].

A Survey on Location Problems with Applications in Telecommunications

17

CFLPS: Valid Inequalities and Facets Let is the polytope associated to the CFLPS. Assume that all facilities have the same capacity Q. As CFLP is a relaxation of the CFLPS, the residual capacity inequalities are also valid for this polytope. But they are not facet defining in general. Consider the case where the demands of the clients are equal. Then we can assume that each client has unit demand and the capacity of a facility is in terms of the number of clients assigned to it. THEOREM 2.6 (Leung and Magnanti [56]) Assume that the demands of all clients are equal. Let Q be the maximum number of clients that can be served by a facility. Let and and define with is a multiple of Q. The residual capacity inequality

is facet defining for

when

and

Leung and Magnanti [56] develop an alternative formulation for the CFLPS with unit demands and different capacities. The capacity of a facility is viewed as a collection of unit capacity facilities. Define to be 1 if client is assigned to unit of facility and 0 otherwise for all and and define also for all Then the feasible set of CFLPS is defined by the following set of constraints:

Constraints (2.10) imply that a client can be assigned to at most one unit of capacity of a facility. Constraints (2.11) ensure that if a client is assigned to the unit of capacity of facility then there has to be a facility at node and that no other client can be assigned to this capacity unit. If client is

18

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

served by some unit of capacity of facility node by constraints (2.12).

then there has to be a facility at

Leung and Magnanti [56] use the set packing structure of the problem to derive facet defining inequalities. They show that the cliques of the conflict graph associated to this set packing give rise to families of facets which are constraints (2.10), (2.11) and (2.12). They also investigate the odd holes in the conflict graph and derive valid inequalities. Deng and Simchi-Levi [28] study the general case where clients can have different demands and they give the following valid inequalities: THEOREM 2.7 (Deng and Simchi-Levi [28]) Let and Define to be the minimum number of facilities needed to serve all clients in The binpacking inequality

is valid for Under some conditions, these inequalities define facets of the convex hull of the feasible set when the values of some of the variables are fixed. Other valid and facet defining inequalities are presented by Deng and Simchi-Levi [28].

2.

Problems

Given a set of demand points, the problem asks to locate facilities on a subset of these nodes to minimize the total distance from each demand point to the closest facility. In the problem, different from the UFLP, the number of facilities to be located is given. So to obtain a formulation for the problem, we add the constraint to the formulation of UFLP. Refer to Bozkaya et al. [11], Labbé et al. [49] and Mirchandani [62] for surveys on the problem. Consider the case where N = M. Let

be the distance between nodes

and So

If a facility is located at node for all

then node

is assigned to itself as

A Survey on Location Problems with Applications in Telecommunications

19

Avella and Sassano [4] eliminate variables by substituting Then the problem can be formulated as follows:

For if node is assigned to node then by constraints (2.15), node cannot be assigned to any other node and so there is a facility at node If no node is assigned to node then constraint (2.15) implies that node can be assigned to at most one node. As we require facilities to be located, the number of nodes that are assigned to other nodes is and this is given in constraint (2.16). If we remove constraint (2.16), we obtain a special stable set problem. This means that valid inequalities known for the stable set polytope are also valid for the polytope. Let be the convex hull of vectors that satisfy (2.15)-(2.17). Avella and Sassano [4] study the properties of THEOREM 2.8 (Avella and Sassano [4]) For dimension

the polytope

has

Avella and Sassano [4] present two families of valid inequalities for the polytope namely W – 2 inequalities and I* -cover inequalities. Here we review the results on the W – 2 inequalities. Define THEOREM 2.9 (Avella and Sassano [4]) Let with and such that for each there exists exactly one such that Let there is no such that The W – 2 inequality

20

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

is valid for only if

The inequality defines a facet of the

poly tope if and

W – 2 inequalities include families of well known inequalities like clique, lifted odd hole, lifted odd anti-hole, etc. Avella and Sassano [4] give separation algorithms for the two families of inequalities and present computational results. The capacitated p-median problem is not studied much. Lorena and Senne [57] apply column generation and Maniezzo et al. [58] propose a heuristic method to solve this problem.

2.3

Hub Location Problems with Single Assignment

The hub location problems are different from the facility location problems in the sense that they consider the communication between pairs of nodes. There is a traffic between any pairs of nodes. The traffic from node to node goes from node to the facility (hub) to which node is assigned, then to the facility to which node is assigned and finally to node There is a cost for routing this traffic in the network. We review the models for hub location problems with single assignment.

2.3.1

Uncapacitated Hub Location Problem with Single Assignment

Let denote the traffic from node to node Let denote the cost of routing the traffic between demand point and facility and let denote the cost of routing the traffic between two facilities and Then Uncapacitated Hub Location Problem with Single Assignment (UHLP) can be formulated as follows (O’Kelly [66]):

s.t. (2.2), (2.3), (2.5), and (2.7). If constraints (2.7) are replaced by constraints (2.4), then one obtains a formulation of the Uncapacitated Hub Location Problem with Multiple Assignment. Even though for the UFLP, there exists always an optimal solution which satisfies (2.7), this is no longer true for the Uncapacitated Hub Location Problem with Multiple Assignment.

A Survey on Location Problems with Applications in Telecommunications

21

Notice that the objective function (2.18) is quadratic. Several linearizations exist in the literature (see Campbell [12], Ebery [29], Ernst and Krishnamoorthy [31], Labbé et al. [53], and Skorin-Kapov et al. [75]). For and in N and and in M, define et al. [75] give the following linear formulation:

Skorin-Kapov

s.t. (2.2), (2.3), (2.5), and (2.7)

For the case where the routing costs satisfy the triangle inequality, Ernst and Krishnamoorthy [31] propose the following formulation. For and define to be the flow of traffic originating at node and traveling on the direct link from to

s.t. (2.2), (2.3), (2.5), and (2.7)

For and define Ebery [29] derives the following formulation:

s.t. (2.2), (2.3), (2.5), and (2.7)

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

The formulation by Skorin-Kapov et al. [75] uses variables. Ernst and Krishnamoorthy [31] decrease the size of variables by an order of by modeling the traffic as multicommodity flow aggregated by origins. But the LP relaxation of this formulation is weaker than the one of the first formulation. The formulation by Ebery [29] uses variables but the author mentions that solving this formulation takes longer than solving the formulation of Ernst and Krishnamoorthy [31]. A branch and bound algorithm and an exact method based on shortest paths for the case where the number of hubs is fixed can be found in Ernst and Krishnamoorthy [31] and [32], respectively. A Lagrangian Relaxation heuristic is given in Pirkul and Schilling [72].

2.3.2

Capacitated Hub Location Problem with Single Assignment

The Capacitated Hub Location Problem with Single Assignment (CHLP) generalizes the UHLP by introducing capacity constraints. Ernst and Krishnamoorthy [30] study the CHLP. They derive a formulation by adding capacity constraints to the formulation by Ernst and Krishnamoorthy [31]. They develop heuristics based on simulated annealing and random descent.

2.4

Conclusion

Having reviewed the literature on location problems for telecommunications networks, we did not encounter any work on the QCL-C. Chung et al. (1992) consider a simpler version where there are no capacity constraints and the cost of connecting two concentrator nodes is independent of the terminals assigned to them and thus is independent of the traffic. Hub location problems are closer to QCL-C. But, the type of capacity in QCL-C is not considered in this literature. Polyhedral approaches are rarely used to solve concentrator location problems. In fact, exact methods are rare in general. There are polyhedral results on location problems (UFLP, CFLP, CFLPS, and p-median) but not for versions where routing decisions are involved or where there is a cost

A Survey on Location Problems with Applications in Telecommunications

23

associated to traffic. This is also the case for hub location problems. Campbell et al. (2002) mention that little is known about the polyhedral structure of hub location problems. Chapters 4, 5 and 6 of this book attempt to fill this gap by presenting polyhedral properties of the QCL-C and its variants.

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Chapter 3 QCL-C: RELAXATIONS AND SPECIAL CASES

In this chapter, we formally define the “Quadratic Capacitated Concentrator Location Problem with Complete Routing” and present a formulation. Then we give formulations of the eight other problems and discuss how these problems are related.

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

3.1

Problem Definition and Formulation for the QCL-C

Let I denote the set of terminal nodes with and K the set of commodities. Commodity has origin destination and traffic Origins and destinations of commodities are terminal nodes and any pair of terminal nodes defines a commodity. The values are defined to be 0 for all Each terminal is either a concentrator or is connected to another concentrator node. Let be the total traffic adjacent at node We can compute as

The results remain valid if for all If node is connected to node then the traffic on the link between nodes and is The cost of routing traffic on the link between node and node is denoted by Any node that becomes a concentrator is assigned to itself. In this case, is the amount of demand that should be processed by node The cost of installing a concentrator at node and processing demand is denoted by We define the arc set We denote by the cost of routing a unit traffic on arc if it becomes a backbone arc, i.e., if both nodes and are concentrators. We assume that the cost vector B satisfies the triangle inequality and for all If nodes and are concentrators, then the amount of flow on arc is given by where F* is the set of commodities such that is assigned to and is assigned to If node becomes a concentrator node, then the total amount of traffic transiting through node cannot be larger than its capacity M. This total amount of traffic is equal to the sum of the amounts of traffic adjacent to nodes assigned to concentrator (including and the amounts of traffic on the backbone arcs leaving and the backbone arcs entering Our mixed integer programming formulation of QCL-C uses two types of variables. We have assignment variables:

for all If node is a concentrator then takes value one and node is assigned to itself. Further, we define to be the total traffic on the arc

27

QCL-C: Relaxations and Special Cases

for all

Now, we can present our formulation: (QCL-C)

Constraints (3.2) and (3.7) ensure that each terminal either becomes a concentrator location or is assigned to exactly one other node. By constraints (3.3), if a terminal is assigned to a node then this node is a concentrator location. Constraints (3.4) relate the traffic vector to the assignment vector Consider arc Given that and (3.7), constraint (3.4) is equivalent to

where and In fact, F* is the set of all commodities whose traffic travels along arc So constraint (3.4) defined by F* computes the traffic on arc correctly, where the constraints defined by other subsets F are redundant. The capacity constraints (3.5) ensure that if we install a concentrator at node then the traffic adjacent to node does not exceed the capacity M. This traffic is equal to the sum of the traffic adjacent at nodes that are assigned to node and the traffic between the nodes that are assigned to and the nodes that are not. If we do not install a concentrator at node then we cannot assign any terminal to node Let

denote the set of pairs

that satisfy constraints (3.2)-(3.7).

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

3.2

Relaxations Based on the Capacity Constraints

Uncapacitated Case: If it is possible to assign all nodes to a single concentrator, i.e., if then the problem is uncapacitated. In this case we can remove the capacity constraints from the formulation. Define to be the set of pairs that satisfy constraints (3.2)-(3.4), (3.6), and (3.7). Linear Capacitated Case: If the capacity of a concentrator is defined in terms of the traffic adjacent to nodes assigned to it, then the capacity constraints (3.5) should be replaced by

Let be the set of pairs (3.7), and (3.8).

that satisfy constraints (3.2)-(3.4), (3.6),

The proposition below gives the relationship between the feasible sets of the three problems. The proof is omitted. PROPOSITION 3.1

3.3

if and only if

Special Cases Based on the Backbone Network and the Routing Cost Structure

Concentrator Location Problem: If the routing cost on the backbone network B is zero, then we have a pure location problem. In this case we can remove the variables and the related constraints from the formulation QCL-C. We end up with the following formulation:

Let be the set of vectors that satisfy constraints (3.2), (3.3), (3.5), and (3.7). We also consider the two relaxations based on the capacities. Define and be the set of vectors that satisfy constraints (3.2), (3.3), (3.7), and (3.8) and constraints (3.2), (3.3), and (3.7) respectively. Concentrator Location Problem with Star Routing: If the backbone network is a star then each concentrator is connected directly to a central node. In this case, the routing cost B satisfies for all where denotes the cost of routing a unit traffic on the link between node and the central node. We can simplify the formulation QCL-C by

QCL-C: Relaxations and Special Cases

29

defining aggregate traffic variables and removing the variables Define to be the traffic on the link between node and the central node for all Clearly, The formulation simplifies to:

Define and be the set of feasible for the uncapacitated, linear capacitated and quadratic capacitated cases respectively.

3.4

The Relationship between the Problems

Tables 3.1, 3.2, and 3.3 give the names, abbreviations, and formulations of the nine problems with reference to their capacity structures and routing costs. We modify the result in Proposition 3.1 for the two special cases as follows. PROPOSITION 3.2 and

and if and only if

Define

to be the projection of the set F on the space, i.e., (or depending on the problem)}. The following propositions are about how the location problems with routing are related to the pure location problems. P ROPOSITION 3.3 and PROPOSITION 3.4 and These projection properties are later used to show that some of the facets of the polyhedra of problems with routing costs project onto facets of the polyhedra of pure location problems. Figure 3.1 depicts Propositions 3.1, 3.2, and 3.3.

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

QCL-C: Relaxations and Special Cases

Figure 3.1. The relaxations and projections

31

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PART II

POLYHEDRAL ANALYSIS

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Chapter 4 UNCAPACITATED PROBLEMS

In this chapter, we study the polyhedra corresponding to the uncapacitated concentrator location problem with no routing cost (UCL), with routing on a star backbone (UCL-S) and with routing on a complete backbone (UCL-C). We start with the polytope of UCL. We present several families of facet defining inequalities and investigate the complexity of the related separation problems. These inequalities are sufficient to describe the polytope when the problem is defined on three or four nodes. Then we study the polyhedra of the uncapacitated concentrator location problems with routing cost. We relate some of the facet defining inequalities of these polyhedra with those of the UCL polytope. For the star routing case, we present a family of facet defining inequalities.

36

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

4.1

Uncapacitated Concentrator Location Problem

This section is devoted to the study of the Uncapacitated Concentrator Location Problem (UCL) polytope. Remember that the UCL is defined as follows. Given a set of nodes I with we choose a subset of nodes to locate concentrators. Each node in set I is either a concentrator or it is assigned to a concentrator node. The cost of locating a concentrator at node is denoted by and the cost of assigning node to node is denoted by The aim is to minimize the total cost of concentrator location and assignment. The UCL can be used to design an uncapacitated network with a star backbone and star access networks that we call a star/star network. In such a network, there is a central node to which all concentrators are connected. If we define to be the cost of installing a concentrator at node and connecting it to the central node and to be the cost of connecting the terminal node to concentrator node then we can see the design problem as a UCL. In fact, in a network design problem where the local access networks are stars, we can expect to have the UCL as a subproblem. In Figure 4.1, we see a star/star network with node 0 as the central node. Nodes 1 to 5 are concentrators and they are connected to the central node with direct links. Each of the remaining nodes is connected to exactly one of these four concentrator nodes.

Figure 4.1. Star backbone and star access networks

Uncapacitated Problems

37

We now define the variables and present a formulation of the UCL. Define

for all If node is a concentrator, then takes value 1 and node is assigned to itself. We can formulate the UCL as follows:

Constraints (4.1) and (4.3) ensure that each node is either a concentrator or it is assigned to exactly one other node. By constraints (4.2), if node is assigned to node then node is a concentrator. The UCL is very close to the uncapacitated facility location problem (UFLP) which is reviewed in Chapter 2. The UFLP is defined as follows. Given a set of clients N and a set of possible locations for facilities M, locate facilities on a subset of M and assign each node in N to a facility in order to minimize the cost of facility location and assignment. Let denote the cost of locating a facility at node and let denote the cost of assigning client to the facility at node Define to be 1 if a facility is located at node and 0 otherwise and to be 1 if client is assigned to the facility at node and 0 otherwise. Then the UFLP can be formulated as follows:

Given an instance of UCL, we can define N = M = I, for all and for all such that and for all Then the UFLP has an optimal solution where for all So we can substitute in the formulation. Then we obtain the UCL. This shows that UCL is a special case of UFLP.

38

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

The UFLP is an NP-hard problem (see Cornuejols et al. [22]). The theorem below establishes the complexity status of UCL. THEOREM 4.1 (Labbé and Yaman [51]) The UCL is NP-hard. Proof: We show that the recognition version of the UCL is NP-complete by a reduction from the recognition version of the UFLP. The recognition version of the UFLP (RUFLP) is as follows. Given sets M and N, vector F, matrix D and a constant B, does there exist a solution to UFLP with cost less than or equal to B? Similarly, the recognition version of the UCL (RUCL) is, given set I, cost matrix C and a constant B, does there exist a solution to UCL with cost less than or equal to B? The problem RUCL is in NP since given a solution we can verify in polynomial time that it has cost less than or equal to B. Given an instance of the RUFLP, we reduce it to an instance of RUCL. Define and set where node is a dummy node. Let for all and Set for each node in M and set for each node Define for all and for all and such that For nodes in N, set if and and set if and We show that there exists a solution to RUFLP if and only if there exists a solution to RUCL. Assume that the RUCL has a solution with cost less than or equal to B. Clearly, this solution does not use any arcs with cost So in node is assigned to itself and each node in set M is either assigned to itself or it is assigned to node Moreover, each node in N is assigned to a node in M. Now define as follows. Let if for and 0 otherwise and if for and and 0 otherwise. The cost of such a solution is and is therefore less than or equal to B. Equivalently, any solution of the RUFLP can be transformed to a solution of RUCL as follows. Set if and otherwise for each and set if for each and It can be shown easily that the two solutions have the same cost. So we can conclude that there exists a solution to RUFLP if and only if there exists a solution to RUCL. Hence, RUCL is NP-complete. To our knowledge, the polyhedral structure of the UCL has not been studied before. However, Avella and Sassano [4] investigate the structure of the p-median polytope and they derive a family of facet defining inequalities

39

Uncapacitated Problems

which are also facet defining for the UCL polytope. Define Following what Avella and Sassano [4] have done for the p-median problem, we substitute for all in the formulation of UCL to obtain:

Let satisfies (4.7)} and The polytope is a special stable set polytope (see Balas and Padberg [7] for a review of results on the stable set polytope). It can be easily shown that is full dimensional and constraints (4.7) which are clique inequalities and the nonnegativity constraints for all define facets of the polytope

4.1.1

W-2 Inequalities

Now, we discuss the family of facet defining inequalities that are introduced by Avella and Sassano [4] for the p-median polytope. THEOREM 4.2 (Avella and Sassano [4]) Let exists exactly one such that there is no such that Let Then the W – 2 inequality

is valid for

with such that for each Define also

The inequality defines a facet of

and there

if and only if

Figure 4.2 depicts a W – 2 inequality where the set W = {1,2,3,4}. The arcs of H are the arcs with dashed lines. We see that node 3 has all its outgoing arcs to nodes in W, hence U = {3}. Two easy examples of W – 2 inequalities are the clique and triangle inequalities. The clique inequality (4.7) for a given arc is a W – 2 inequality where for some and

40

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Figure 4.2. A W-2 inequality

The triangle inequalities have the following form:

and are W –2 inequalities where

and

The triangle inequalities (4.10) and the clique inequalities (4.7) correspond to cliques in the conflict graph. The conflict graph has the arcs of set A as its vertices. There is an edge between two vertices of the conflict graph if and only if there exists a constraint (4.7) where the two arcs defining the two vertices appear together. We show that the sets of arcs who have coefficient 1 in triangle inequalities and clique inequalities (4.7) are the only maximal cliques in the conflict graph. PROPOSITION 4.1 (Labbé and Yaman [50]) The maximal cliques in the conflict graph are either of the form for or of the form Proof: In the conflict graph, there is an edge between vertices and if and only if (i) and or (ii) or (iii) If we look at cliques that contain and for then these cliques can contain at most one vertex and any number of vertices of the form A maximal such clique is of the second form described in the proposition. Now we can look at cliques where no node is repeated as the tail, i.e., if is in the clique, no other vertex of the form can be in the clique.

Uncapacitated Problems

41

Then such a clique contains two vertices and with In this case, the only vertex that can appear in such a clique (without having any vertex of the form or is and this is the first form given in the proposition. Avella and Sassano [4] observe that the W – 2 inequalities include lifted versions of known classes of facets for the stable set polytope like lifted odd holes, lifted odd anti-holes and lifted anti-webs. When

the clique inequalities (4.7), the nonnegativity constraints for all and the two triangle inequalities give the full description of the UCL polytope Though for it is not the case anymore. We can verify these using PORTA (see Christof [20]) and the facet defining inequalities for and can be found in Appendix A. In the remaining part of this section, we present three families of facet defining inequalities for the UCL polytope that are not W – 2 inequalities.

4.1.2

k-triangle Inequalities

In this subsection, we present a family of facet defining inequalities called “k-triangle inequalities”. We see that when is 1, the k-triangle inequalities are the triangle inequalities (4.10) which are also W – 2 inequalities. But for the k-triangle inequalities are not W – 2 inequalities. PROPOSITION 4.2 Consider the graph where with for some Without loss of generality, we number the nodes in from 1 to The arc set consists of all arcs for and all arcs of the form where and are both odd and Then the k-triangle inequality

is valid for Figure 4.3 depicts a k-triangle configuration. Proof: We prove that the k-triangle inequality (4.11) is valid for by induction on Notice that when we have the triangle inequality which is valid for Now assume that inequality (4.11) is valid for for all We show that it is also valid for

42

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Figure 4.3. A k-triangle configuration

Assume to the contrary that there exists an

Then we should have This is possible only if

such that

and

since and

because of constraints (4.7). This also shows that So to have a vector

which satisfies

(4.12), we should have Constraints (4.7) imply that Moreover since the following:

and we should have

. If we repeat the same argument, we can show that for and

satisfies

But all odd nodes such that are assigned to some node, so node cannot be assigned to any of these nodes. Therefore such a point cannot be in This proves that the k-triangle inequality (4.11) is valid for T HEOREM 4.3 The k-triangle inequality (4.11) is facet defining for all Proof: We prove that the k-triangle inequality (4.11) is facet defining for for all by sequential lifting. For a given define The odd hole inequality

for

43

Uncapacitated Problems

is facet defining for the polytope (see Padberg [68]). Now, starting from the odd hole inequality, we lift the variables that are fixed to 0 sequentially. As we take the maximum lifting coefficients for each variable, we obtain a facet defining inequality for of the form

where is the optimal lifting coefficient of variable with Let be the set of variables that are lifted before Then the optimal lifting coefficient of is:

Let

denote the set of

that satisfy (4.14)-(4.16).

We first lift the variables such that Notice that these are all such that and are odd and We do the lifting for if all such that and and are already lifted. We show that for all by induction on the order of the lifting variables. The first variable to lift is When we can show that The clique inequalities (4.7) imply that for all and So for all Consider the solution such that for all and for all other (see Figure 4.4). It is easy to verify that is in and This shows that

Figure 4.4. The solution

all

Now, assume that we are lifting We show that

to show that

where As

and we have that

for

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

If node nodes imply that

then the k-triangle inequality (4.11) for the first So we get

As we have that inequalities (4.7) imply that and So

For

Assume that node for all

The clique

the clique inequalities (4.7) imply that

If we sum up the inequalities, (4.17), (4.18) and (4.19) for we obtain

for all

Now, if we consider the solution such that for all and for all other easily show that is in and

we can So

In Figure 4.5, we see the solution

Figure 4.5. The solution

to show that

Next we should show that for all k-triangle inequality (4.11) is valid, we have that

for

Since the for all

45

Uncapacitated Problems

and and

As the first variable to lift, consider Consider the solution such that As is in and

such that and are not in for all we can conclude that

Now consider a node such that and the solution such that for all for all and Let and set The vector (see Figure 4.6) is in Since we can conclude that We can also show that in a similar way.

Figure 4.6. The solution

Let for all So

to show that

for

such that is even and

such that and the solution such that and for all Pick The vector (see Figure 4,7) is in and satisfies Similarly,

Figure 4.7. The solution

to show that

for

and set

such that is odd and

Now consider such that and are in and and are odd. Let and Consider the solution such that for all for all and for all other Clearly, (see Figure 4.8) is in and

46

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

This shows that

Figure 4.8. and

The solution

to show that

for

such that and

are odd

Now, we lift an

and

Figure 4.9. even and

If all and

all

such that is odd, is even and Assume that and Consider the solution such that for all and for all for all other Clearly, (see Figure 4.9) is in As we have that

The solution

to show that

for

such that is odd,

then consider the solution such that for all for all other Since (see Figure 4.10) is in we have that So we can conclude that such that is odd, is even and

The remaining variables are the such that is even and and for some then the solution 4.11) such that for all all and for all other and it satisfies So

is

for and for

If (see Figure for is in

47

Uncapacitated Problems

Figure 4.10. even and

The solution

to show that

Figure 4.11. even and

The solution

to show that

If solution

Figure 4.12. odd and

for some (see Figure 4.12). So

The solution

If and such that Figure 4.13) is in

for

for

such that is odd,

is

such that and

are

then we can again consider the same

to show that

for some for all and and it satisfies

for

such that is even,

is

then we consider the solution for all for all other Again, (see showing that

If for some then we again consider solution (see Figure 4.14). This shows that Therefore we can conclude that for all such that is even and

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Figure 4.13. The solution even and

to show that

Figure 4.14. The solution odd and

to show that

for

for

such that and

are

such that is even,

is

As all variables are lifted and the resulting inequality is the k-triangle inequality is facet defining for It is clear that the triangle inequality (4.10) is a W – 2 inequality. In fact, is the only for which the k-triangle inequality is a W – 2 inequality. Notice that for each node there is at least one arc This means that is the same as W. The right hand side of the k-triangle inequality, is equal to if and only if So the W – 2 and k-triangle inequalities coincide only for the case The k-triangle inequalities are lifted odd hole (in the conflict graph) inequalities. But clearly there are odd holes in the conflict graph such that the corresponding lifted odd hole inequality is not a k-triangle inequality. Assume that On the left hand side of Figure 4.15, we see the subgraph of the conflict graph induced by the arcs of a 2-triangle configuration. On the right hand side of Figure 4.15, there is another odd hole in the conflict graph. Assume that the corresponding odd hole inequality can be lifted to a 2-triangle inequality. Notice that in a 2-triangle configuration, any even node has exactly one incoming and one outgoing arc. The first odd node has two incoming arcs and one outgoing arc. The last odd node has two outgoing arcs and one incoming arc and the middle odd node has exactly two incoming arcs and two outgoing arcs. As node 1 has two outgoing arcs it should be an odd node. Node

49

Uncapacitated Problems

4 should also be an odd node since it has two incoming arcs. As no nodes can have more than two outgoing arcs, we cannot have arc (1,4). So node 4 should come after node 1. Then as node 1 has two outgoing nodes, it cannot be the first odd node. Hence it should be the middle odd node. This implies that node 4 has to be the last odd node. It has two incoming arcs. But the last node can have only one incoming arc. So it is impossible to order these nodes to have a 2-triangle configuration. Therefore this odd hole inequality cannot be lifted to a 2-triangle inequality.

Figure 4.15.

Odd holes in the conflict graph

Below we investigate the complexity of the separation problem for the ktriangle inequalities. First, we give two properties that can help us to reduce the domain of the separation problem. PROPOSITION 4.3 (Labbé and Yaman [50]) If a fractional solution violates a k-triangle inequality, then there exists a violated k-triangle inequality where for all Proof: Assume that the fractional solution violates a k-triangle inequality. If this k-triangle configuration satisfies that for all then we are done. Otherwise, let be the smallest index such that We can remove the triangles and still have a violated inequality since

The first all

triangles give a violated inequality that satisfies

for

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

PROPOSITION 4.4 If a fractional solution violates a k-triangle inequality, then there exists a violated k-triangle inequality where for all

Proof:

Consider a for all

arcs going out of node arcs going out of node

violated

k-triangle As

inequality

which

satisfies and for all The remaining arcs are the and the arcs for The cannot have value 1 since and for all

Now assume that there exists at least one arc such that Let be the arc set of the (k-l)-triangle obtained by removing the nodes and and connecting node to (see Figure 4.16). We have

which proves that the (k-1)-triangle inequality corresponding to is also violated. Repeating this argument we obtain a violated k-triangle inequality where for all THEOREM 4.4 (Labbé and Yaman [50]) The separation problem for the ktriangle inequalities is NP-complete. Proof: The proof is by reduction from the linear ordering problem.

51

Uncapacitated Problems

Figure 4.16.

Removing a triangle

Suppose we are given a set of items N with and a set of arcs Let denote the weight of arc The recognition version of the linear ordering problem is as follows. Does there exist a linear ordering L on N such that (equivalently, for a small enough where if comes after in L? This problem is NP-complete (see Garey and Johnson [34] [GT8]). Now given an instance of the linear ordering problem, divide all the values and B by a very large number so that The problem obtained by this scaling is equivalent to the original problem in the sense that the ordering that solves one solves also the other. Now, to avoid the trivial cases, we assume that for each and that for each arc (these are useful to transform the instance to a fractional solution of the UCL). The separation problem for the k-triangle inequalities is as follows: Given a fractional solution of UCL does there exist a violated k-triangle inequality? Given any violated k-triangle inequality, we can verify in polynomial time that it is violated. So the problem is in NP. Given an instance of the linear ordering problem, we construct the following instance for the k-triangle separation problem: for each node create a dummy node and set Create also three nodes, and set and for all Set also for all Define and All other arcs have This can be a fractional solution of the UCL instance defined on the node set We claim that there exists a k-triangle inequality violated by if there exists a linear ordering L on N with

if and only

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Given a k-triangle configuration with nodes, we renumber the nodes from 1 to We call the nodes for “odd nodes” and the remaining nodes “even nodes”. If node for some then node is called its even node. Suppose that we are given a linear ordering L on N with We construct a configuration which yields a violated inequality. Let the odd nodes be the nodes in N ordered as in L and for each odd node let the even node be its dummy node. At the end, add the triangle Then

So the

inequality is violated. In Figure 4.17 we see a configuration obtained from a linear ordering where nodes

are ordered as

Figure 4.17. From a linear ordering to a

configuration

Now assume that we have a violated k-triangle inequality. By Proposition 4.3 we can assume that for This implies that a dummy node can be an odd node only if it is the last node since all of its outgoing arcs have value 0. Assume that the last node is a dummy node. As all its outgoing arcs have value 0, we can then remove the last triangle and still have a violated inequality. So we look at the case where none of the odd nodes is a dummy node. Suppose node is an odd node. Its even node is either or some node Assume that the latter is the case. If the next odd node is then changing node to node cannot decrease the violation since for all as for all and If the next odd node is one of the nodes and then

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Uncapacitated Problems

as violation.

we can change node

by node

without decreasing the

Now we show that removing nodes and from where they are and putting them as the last triangle does not decrease the violation. If none of these nodes are used in the k-triangle configuration, then

As all odd nodes are in N, we already have a linear ordering on a subset of N with nodes which has cost more than 2Bk. This is impossible since we assumed that Assume that node is the last node. Then the most advantageous case would be to have as the last triangle since is the only outgoing arc of with nonzero value and all arcs with have value B and the remaining outgoing arcs of have value 0. Even in this case we can remove the last triangle since So we can assume that is not the last node. Assume that only one of these three nodes and is used. Then it is either or since all arcs incident at node have value 0 except and so it can only be the last node and can be removed in this case as explained above. The node which is or must be an odd node, with the even node being a node As (since and for all removing this node and its even node cannot decrease the violation. We see how we can remove node in Figure 4.18. Node can be removed in a similar way.

Figure 4.18. Removing node

Assume now two of these nodes are used. If these two nodes are and then is an odd node and is its even node. In this case we can remove these

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

two nodes since at most we can have Figure 4.19 shows how we can remove nodes

Figure 4.19. Removing nodes

on the triangle. and

and

If the nodes are and then has to be the odd node and has to be its even node. In this case So we can again remove these two nodes as shown in Figure 4.20.

Figure 4.20. Removing nodes

and

If we have the nodes and then as the arcs connecting them have value 0, we should have them both as odd nodes. As this is the same as having them separately, they can be removed. If all three nodes are used and if they are not together, then node has to be used with or So we are in one of the cases above. If these nodes are used together, it is either the triangle or the triangle since and In the latter case, changing the triangle to does not decrease the violation. Finally moving this triangle to the end of the sequence cannot decrease the violation. If these nodes are removed from the k-triangle configuration, then we append this triangle at the end since it has value when

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Uncapacitated Problems

This implies that we obtain a k-triangle configuration where the odd nodes except and are in N. If we have triangles, we get a linear ordering on nodes that we call

This partial linear ordering can be completed by adding the remaining nodes to the end of the order and it gives a linear ordering L with

4.1.3

k-leaf Inequalities

Now, we present a family of facet defining inequalities for “k-leaf inequalities”. PROPOSITION 4.5 Given arc the k-leaf inequality

is valid for

and a subset

called the with

for all

In Figure 4.21 we see a k-leaf configuration. Proof: Let If then as for all and for all inequality (4.20) is valid. If then for all and for all inequality (4.20) is satisfied.

for some As

The only case that remains is when Then as for all we can conclude that the k-leaf inequality is valid for THEOREM 4.5 The k-leaf inequality (4.20) is facet defining for

for all

Proof: We prove that the k-leaf inequality (4.20) is facet defining for for by the indirect method. Define Consider an inequality of the form

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Figure 4.21.

A k-leaf configuration

We show that if all points in satisfy the inequality (4.21) at equality, then (4.21) is a positive multiple of (4.20). Consider a node For a given consider the solution and for all other pairs The solution Now consider the solution such that and for all other pairs (see Figure 4.22). The vector is also in This shows that for all and

Figure 4.22.

Solutions

and

to show that

for

and

Inequality (4.21) becomes

Now let such that

for all

and and

Consider the solution for all other pairs The second

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Uncapacitated Problems

solution is the same as except that both and are in This proves that and

Figure 4.23. Solutions

and

to show that

(see Figure 4.23). Clearly, for all

for

and

Now we can rewrite inequality (4.22) as:

Next, we show that the solution where for all and we have that

Figure 4.24.

for all Given consider for all and the solution where (see Figure 4.24). As both and are in for all

Solutions

Consider also the solution we can show that

and

to show that

such that

for all

Since both

and

are in

Now, consider a node and the solutions defined as follows: for all (see Figure 4.25). As and are all in we have that for all This implies that for all and for all

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Figure 4.25. Solutions

and

to show that

for

We should show that for all It is enough to consider the solution where and for some (see Figure 4.26). As and are both in we have that So for all

Figure 4.26. Solutions

and

to show that

for

The inequality (4.23) becomes:

If we plug in the solution inequality (4.21) is a (4.20) is facet defining for

we can show that This proves that of the k-leaf inequality (4.20). So inequality

The k-leaf inequalities coincide with the W – 2 and the k-triangle inequalities only for since whenever the coefficients of the variables are not only 0 or 1. Now we would like to see if k-leaf inequalities can be obtained by lifting some rank inequality for which is facet defining for In Figure 4.27 we see the conflict graph induced by arcs that have coefficient 1 in a k-leaf inequality

Uncapacitated Problems

59

for For the corresponding inequality is which can be obtained by summing inequalities for all This shows that this inequality is not facet defining for where and and Moreover, for any set the inequality does not define a facet of Therefore a k-leaf inequality with cannot be obtained by lifting a rank inequality which is facet defining for

Figure 4.27. Arcs with coefficient 1 in a k-leaf configuration

Now we investigate the complexity of the separation problem for the k-leaf inequalities. PROPOSITION 4.6 (Labbé and Yaman [50]) The separation problem for the k-leaf inequalities can be solved in time. Proof: Let be a fractional solution of UCL and choose an arc would like to find a set such that where

We

If then we have a violated k-leaf inequality. Otherwise we conclude that there is no violated k-leaf inequality defined by arc

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Let be written as:

Then for any

the value

can

It is easy to show that the maximizing set This gives a very simple algorithm which has a complexity of

4.1.4

2-cycle Inequalities

The last family of inequalities we introduce in this section are given by PORTA as facet defining inequalities for when (see Appendix A). We generalized these inequalities to be facet defining for any Each one of these inequalities is defined by a subset of four nodes. We do not know any generalization to any subset of nodes. PROPOSITION 4.7 Consider a subset with and a node Let C be a directed cycle on the nodes of D. Renumber the nodes such that D = {1,2,3}, the cycle is 1, 2, 3, 1, the node and

1 The 2-cycle inequality

is valid for

2 Let and be the node in D such that Renumber the nodes such that and Then the 2-cycle inequality

is valid for Inequalities (4.24) and (4.25) are depicted in Figures 4.28 and 4.29 respectively. Proof: We prove the first statement and leave the second which can be done in

Uncapacitated Problems

Figure 4.28. The 2-cycle inequality (4.24)

Figure 4.29. The 2-cycle inequality (4.25)

a similar way. Consider the following inequalities:

61

62

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

where the first inequality is a triangle inequality, the following four inequalities are implied by the clique inequalities and the last one is a W – 2 inequality where W = {1,2,3,4} and U = {4}. If we sum up all the inequalities, we get

To get inequality (4.24), it is enough to divide the inequality (4.26) by 2 and round down the right hand side. THEOREM 4.6 The 2-cycle inequalities (4.24) and (4.25) are facet defining for if Proof: The PORTA output in Appendix A for shows that both inequalities are facet defining for when We prove that inequality (4.24) is facet defining for for any by lifting. The proof for (4.25) can be done in a similar way. Now let

and consider the polytope if or ). Inequality (4.24) is facet defining for Let L be the set of indices of variables fixed to 0 and be the set of variables that are lifted before with Denote by the optimal lifting coefficient of We can compute as follows:

Suppose, first we are lifting a variable consider the solution such that other then we can show that

such that

and and

If we for all

Now consider such that and solution such that and for all other

It is enough to consider the where and and to show that If we are

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Uncapacitated Problems

lifting other

with we take such that Then we can show that

and

for all

Suppose that we are lifting such that solution where where for all other This proves that

and and

The only remaining variables are then becomes

Suppose that The lifting coefficient

where

The triangle (4.10) inequality implies that Consider the solution where This proves that

and

Consider the and

for any for all other

The trivial facet defining inequalities, the k-leaf inequalities, the W – 2 inequalities and the 2-cycle inequalities are sufficient to have the full description of when (see Appendix A).

4.2

Uncapacitated Concentrator Location Problem with Star Routing

In this section we consider the Uncapacitated concentrator location problem where we are given a central node and each concentrator node is connected to this central node. This results in a star topology for the backbone network. In addition to the concentrator location cost, there is also a cost associated with routing the traffic in the network. The traffic between two nodes is routed on the shortest path connecting these two nodes. In a network with a star backbone network and star access networks, there is a single path connecting any pair of nodes. The aim is to locate the concentrators and to assign the remaining nodes to the concentrators in order to minimize the cost of concentrator location and the cost of routing the traffic. This problem is called “Uncapacitated Concentrator Location Problem with Star Routing” and is abbreviated by UCL-S. Let I denote the set of nodes and denote the cost of installing a concentrator at node We denote by K the set of directed pairs of nodes. The traffic from node to node is We can compute the total traffic

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

between nodes and as If node is assigned to node then the traffic on the link connecting nodes and is Let denote the cost of routing units of traffic on the link between nodes and If two nodes and are assigned to the same concentrator, say the traffic from node to node follows the path So this traffic does not enter the backbone network. However, if node is assigned to node and node is assigned to node then the traffic from node to node follows the path where node 0 stands for the central node. The total traffic on the link between node and node 0 is the sum of the traffic between all nodes that are assigned to and all nodes that are not assigned to We denote by the cost of routing a unit traffic between nodes and 0. In Figure 4.30, we see a network with 18 nodes where nodes 1 to 5 are concentrators. The traffic from node 8 to node 12 follows the path since node 8 is assigned to node 1 and node 12 is assigned to node 3. The traffic from node 9 to node 10 follows the path

Figure 4.30. Routing on a star backbone network

The UCL-S is studied in Labbé and Yaman [51]. Two formulations whose constraints are facet defining are presented. A branch and cut algorithm and a Lagrangian Relaxation heuristic are compared in this paper.

Uncapacitated Problems

65

A similar problem has been studied by Helme and Magnanti [43]. Their work is summarized in Chapter 2. To formulate the UCL-S, we use the assignment variables defined for UCL and we define traffic variables. Let denote the total traffic between node and the central node for Now we can formulate the UCL-S as follows:

Constraints (4.27), (4.28) and (4.30) are the same as in UCL. Constraints (4.29) define the traffic variables in terms of the assignment variables. Suppose we are given an assignment vector The traffic on the link between node and the central node is As we have The maximizing set is Then So constraint (4.29) defined by F* computes the real value of We can also give a formulation without using the variables

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Let and

satisfies (4.32)-(4.35)}

The remaining part of this section is about properties and the facets of

4.2.1

The UCL-S Polyhedron

Here we investigate the dimension of the polyhedron PROPOSITION 4.8 (Labbé and Yaman [51]) The polyhedron is full dimensional, i.e., Proof: We show that if all points satisfy an equality then and For and consider the pair which is the same as except that for some As both and are in we have and This implies that for all Now consider a pair where enough N for all This pair For for all Thus

4.2.2

and is in

for some large So

consider the solution where except This pair is in and it satisfies for all

and

Facet Defining Inequalities Involving Only the Assignment Variables

In Chapter 3, we saw that the UCL polytope is the projection of on the We now discuss the relationship between the facets of and the ones of THEOREM 4.7 (Labbé and Yaman [51]) The inequality defines a facet of if and only if it defines a facet of Proof: Assume that defines a facet of We use the indirect method to prove that it also defines a facet of Assume that all points Consider such such that also satisfy except For a point define to be the same as and it satisfies for some Clearly that for So we should have This implies that and as are full dimensional, and As all for some and Assume that also satisfies

defines a facet of and that any such that Since is full dimensional we should show

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Uncapacitated Problems

that and for some to prove that defines a facet of As any such that also satisfies This proves that and for some since is full dimensional. Thus the inequality defines a facet of

4.2.3

Facet Defining Inequalities Involving Only the Traffic Variables

Now we would like to characterize facet defining inequalities of on the traffic variables. T HEOREM 4.8 No inequality of the form unless it is a positive multiple of for some

based

defines a facet of

Proof: We first show that if defines a facet of then Assume that there exists a such that Let Consider to be the same as be such that but that for some Then As cannot be valid. This shows that inequality we should have

and except So and

Consider the inequality with and Consider a node and the pair such that for all and This pair is in but Therefore the inequality is not valid and thus cannot be facet defining. If defines a facet of then Let

be such that Then any pair should satisfy at equality. So if it is a positive multiple of

PROPOSITION 4.9 The inequality and only if for all Proof: Let

defines a facet of

and assume that there exists a node implies that inequality is not a multiple of cannot be facet defining. Then

If

such that defines a facet of for

if

such that As the the latter

for all and if there exist two nodes and such that Then all points satisfy This implies that has dimension at most So cannot define a facet of

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

If a facet of

for all

it is easy to see that inequality

defines

COROLLARY 4.1 (Labbé and Yaman [51]) If there exists at least one such that then has no facet defining inequality of the form In fact, the case where all for all does not make sense since we can remove the variables which are zero in an optimal solution. Then the problem reduces to a UCL.

4.2.4

Facet Defining Inequalities Involving Assignment and Traffic Variables

In this section we show that the inequalities (4.33) define facets of under some conditions. The theorem below gives these conditions. THEOREM 4.9 (Labbé and Yaman [51]) For a given and the inequality (4.33) defines a facet of exists an ordering on the nodes of such that

if there

The proof of the theorem proceeds as follows. First for a given node we fix the values of all assignment variables related to node i.e., the values of variables and for all Then the traffic on the link between node and the central node is fixed and yields a lower bound for This corresponds to a facet defining inequality for the polyhedron obtained by fixing the values of variables. We apply sequential lifting to variables with fixed values to obtain a facet defining inequality for Finally, we show that these inequalities form a subset of the inequalities (4.33). An alternative proof is given in Labbé and Yaman [51]. Below we present the three lemmas describing these steps. The proof of Theorem 4.9 is given after the lemmas. LEMMA 4.1 Let and The inequality

be such that

is facet defining for the polyhedron

where

Proof: It is easy to see that (4.36) is a valid inequality for has dimension where

and

The polyhedron (we can see it like

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Uncapacitated Problems

removing nodes in for the assignment part). As the points in that satisfy inequality (4.36) only have the value fixed (except the assignment variables related with nodes in the inequality defines a facet of Now we lift the variables whose values are fixed to 0 or 1 to obtain a facet defining inequality for The resulting inequality is of the form:

Suppose we are lifting with Define to be the set of variables with indices in and that are lifted before L EMMA 4.2 For

and for

the optimal lifting coefficient of

the optimal lifting coefficient of

and respectively

is

is

Proof: The proof is by induction on the lifting order. It is easy to see that if the first variable to lift is with then which is the change in the traffic when we assign node to node Similarly, if

Assume that we are lifting and all the coefficients for are computed as described in the Lemma 4.2. If the optimal lifting coefficient can be computed as:

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

As appears with a positive coefficient in the objective function, it takes the minimum possible value, so inequality (4.38) is tight at the optimal solution. If we plug this in (4.37), the objective function becomes:

The objective function has a fixed, a linear (in and and a quadratic component. We try to simplify the fixed and linear components. The fixed component is:

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Uncapacitated Problems

Let denote the coefficient of if linear part of the objective value. For

and of

if

For

For a given

satisfying (4.39), define The objective value for

and is:

in the

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Define Then

As

and

we have

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Uncapacitated Problems

As all items in have nonnegative coefficients, the minimizing set for any Then The minimizing set

If

As as:

Therefore

the optimal lifting coefficient

can be computed as:

takes the smallest possible value, we can rewrite the objective function

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

The fixed term is :

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Uncapacitated Problems

if Let denote the coefficient of linear part of the objective value. For

and of

if

For

For a given

satisfying (4.42), define The objective value for

and is:

in the

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Define Then

As

and we have

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Uncapacitated Problems

As all items in are and

have nonpositive coefficients, the maximizing sets Then

Lemma 4.2 implies that the lifting coefficient of with is independent of the lifting coefficients of with and that the lifting coefficient of with is independent of the lifting coefficients of with LEMMA 4.3 For

Proof: We set

As

the optimal lifting coefficient of

Then

for all

and

So

Proof of Theorem 4.9: The above three lemmas imply that

is

So

we have

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

defines a facet of

where and are subsets of such that and and and are orderings defined on and respectively. By rearranging the terms and using we can obtain:

The right hand side of inequality (4.43) is:

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Uncapacitated Problems

Define the order to be followed by the reverse ordering of right hand side of (4.43) is equal to:

Then the

for So for inequality (4.33) defined by F for which there exists an order on that defines a facet of

an such

We call the inequalities (4.33) that satisfy the requirement of Theorem 4.9 “ordering inequalities”. Now we show that separating the ordering inequalities is the same as separating inequalities (4.33). PROPOSITION 4.10 (Labbé and Yaman [51]) Ordering inequalities can be separated in time. Proof: We can separate the inequalities (4.33) in time as follows. For a given with we take If the corresponding inequality is violated, then this is the most violated inequality (4.33) for Otherwise, there is no violated inequality (4.33) for For a given the order obtained by ordering the nodes in in decreasing order of leads to the set which gives the most violated inequality (4.33). So the most violated inequality (4.33) satisfies the condition of Theorem 4.9 and can be found in time. The proof of Proposition 4.10 implies that the inequalities of the form (4.33) which are not ordering inequalities are never violated alone, i.e., if there exists a violated inequality (4.33), then there exists also a violated ordering inequality. This shows that the remaining inequalities (4.33) are not necessary to describe the polyhedron so they cannot be facet defining. COROLLARY 4.2 (Labbé and Yaman [51]) Inequality (4.33) is facet defining for if and only if it is an ordering inequality.

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

4.3

Uncapacitated Concentrator Location Problem with Complete Routing

In this section, we study the uncapacitated concentrator location problem where the concentrator nodes are connected by a complete backbone network. Similar to the UCL-S, we have a cost for locating concentrators and a cost for routing the traffic. However, in this case, we may have several paths connecting two concentrator nodes. The routing is done on the shortest path. This problem is called “Uncapacitated Concentrator Location Problem with Complete Routing” and is abbreviated by UCL-C. Most of the input data are the same as for the UCL-S. Here, we discuss the routing in detail. If two nodes and are assigned to the same concentrator, say the traffic from node to node follows the path However, if node is assigned to node and node is assigned to node then the traffic from node to node follows the path Therefore the total traffic on arc is the sum of the traffic of commodities whose origins are assigned to node and whose destinations are assigned to node We denote by the cost of routing a unit traffic from node to node In Figure 4.31, we see a network with 18 nodes where the first five are concentrators. The traffic from node 8 to node 12 follows the path since node 8 is assigned to node 1 and node 12 is assigned to node 3. The traffic from node 9 to node 10 follows the path and does not use any backbone link since both nodes 9 and 10 are assigned to node 2. In the literature, this problem is called “the Uncapacitated Hub Location Problem with Single Assignment”. For a recent survey on hub location problems, see Campbell et al. [13]. The polyhedral structure of this problem has been studied by Labbé and Yaman [52] and Labbé et al. [53]. Hamacher et al. [41] present polyhedral results for the Uncapacitated Hub Location Problem with Multiple Assignment. The polytope they consider is defined on a higher dimensional space than the one considered here To formulate the UCL-C, we use the assignment variables for all defined for UCL. We also define traffic variables; is the total traffic on the arc from node to node We can formulate the UCL-C as follows:

Uncapacitated Problems

81

Figure 4.31. Routing on a complete backbone network

Constraints (4.44), (4.45) and (4.47) are the same as in UCL. Constraints (4.46) define the traffic variables in terms of the assignment variables as explained in Chapter 3. We can also give a formulation without using the variables

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Let satisfies (4.49)-(4.52)} and In the remaining part of this section, we discuss the properties, valid and facet defining inequalities of

4.3.1

The UCL-C Polyhedron

We start by investigating the dimension of the polyhedron PROPOSITION 4.11 (Labbé et al. [53]) The polyhedron sional, i.e.,

is full dimen-

Proof: See the proof of Proposition 4.8.

4.3.2

Facet Defining Inequalities Involving Only the Assignment Variables

We know that the UCL polytope is the projection of on the and that the facet defining inequalities of were also facet defining for A similar result can be proved for T HEOREM 4.10 (Labbé et al. [53]) The inequality of if and only if it defines a facet of

defines a facet

Proof: See the proof of Theorem 4.7.

4.3.3

Facet Defining Inequalities Involving Only the Traffic Variables

The following proposition gives a necessary condition for an inequality to define a facet of T HEOREM 4.11 (Labbé et al. [53]) No inequality of the form defines a facet of unless it is a positive multiple of for some

83

Uncapacitated Problems

Proof: See the proof of Theorem 4.8. PROPOSITION 4.12 (Labbé et al. [53]) For

inequality

if

then the

defines a facet of

Proof: Let N denote a very large number. Consider the following points:

for each except

if

if for all in addition we have

if

for each

then

and if then

in addition we have points in

We have

such that

and

for all

if

and

for all

such that

and if

then except

such that except

and

then

that satisfy

Now we show that these points are affinely independent, i.e., any that

such

and the row for

For each arc

For

we have

For

For

This implies that This

we have

implies that

implies that

Then with

we have that which implies that

So

4.3.4

Valid Inequalities Involving Assignment and Traffic Variables

Here we discuss the link between valid inequalities for for both involving traffic and assignment variables. PROPOSITION 4.13 The inequality

and the ones

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

is a valid inequality for

if and only if

is a valid inequality for Proof: Assume that inequality (4.54) is not a valid inequality for Then there exists a pair which does not satisfy the inequality. Consider where for all The pair and does not satisfy (4.53). So inequality (4.53) is not a valid inequality for Consider Assume that inequality (4.53) is not a valid inequality for which does not satisfy the inequality (4.53). Define Then for each we have for all then If (4.54) is a valid inequality for So we have

Therefore inequality (4.54) is not valid for

4.3.5

More Valid Inequalities - Projection

Here, we present two other ways of formulating the UCL-C in higher dimensions and we discuss how we can obtain valid inequalities from the projections of these higher dimensional polyhedra. Most of the results are due to Labbé and Yaman [52]. We use the following variables in both formulations:

for all variables easily.

and We present the results without removing the for ease of presentation but they can be modified for very

Multicommodity Flow Formulation and its Projections As the routing cost B satisfies the triangle inequality, we can formulate the UCL-C using multicommodity flows. To obtain the multicommodity

Uncapacitated Problems

85

flow formulation, we replace constraints (4.46) with the following set of inequalities:

Notice that the flow conservation constraints are replaced by the inequality forms (4.55) (see Mirchandani [61]). If the origin of commodity is assigned to concentrator but not the destination, there is a net flow of one unit that goes out of concentrator On the contrary if the destination is assigned to but not the origin, there is a net flow of one unit that comes into node If both the origin and destination are assigned to or neither one is assigned to the flow on the arcs incoming to node is equal to the flow on the arcs outgoing from node concerning this commodity. Constraints (4.56) imply that the traffic on arc should be at least the sum of the traffic of commodities whose origins are assigned to node and whose destinations are assigned to node We sometimes refer to the traffic on the backbone links as capacity to be able to follow the terminology of flows and cuts. To our knowledge, there are two ways of projecting out the flow variables in this system. The first method used by Mirchandani [61] is doing a direct projection. This method leads to inequalities known as the metric inequalities (see Iri [44] and Onaga and Kakusho [67]). Mirchandani [61] studies the extreme rays of the resulting cone for the single commodity and multi commodity cases. For the single commodity case the projection inequalities are the well known cut inequalities that state that with fixed values of the capacities on the arcs of a network, the amount of flow from an origin to a destination is bounded by the capacity of the minimum capacity cut separating the origin and the destination. However for the multicommodity case, we do not know any characterization of all the extreme rays of the resulting cone. Mirchandani [61] presents necessary conditions for a ray to be extreme. The second method used by Rardin and Wolsey [74] is to replace the flow conservation constraints by the corresponding cut constraints and do the projection afterwards. The authors call the projection inequalities “dicut inequalities” and discuss some special cases for specific problems.

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

We do the projection for the multicommodity flow formulation using both methods and compare the results. The comparison gives us a necessary condition for the dicut inequalities not to be dominated. Projection 1: Direct Method For the first projection, we follow the method used by Mirchandani [61]. If we associate dual variables to constraints (4.55) and to constraints (4.56), by Farkas’ Lemma, we have the following result: given a solution and there exits a vector X satisfying (4.55) and (4.57) if and only if

for all

such that

As

for all it is enough to consider only Moreover, as all the data are rational, we can just consider integer vectors So we can conclude that for a given there exists a vector X which satisfies (4.55)-(4.57) if and only if

for all integer Projection 2: Indirect Method Now, we discuss the projection using the method described in Rardin and Wolsey [74]. Let denote the capacity of arc used for commodity We know that for integer (otherwise the origin and destination of the commodity are not properly defined), there exists a feasible flow for commodity if and only if

where can be replaced by the following:

So constraints (4.55) and (4.57)

Uncapacitated Problems

87

If we associate dual variables to inequalities (4.61) and to inequalities (4.62) we get that, given a solution and there exits a vector X satisfying (4.55) and (4.57) if and only if

for all

such that

Clearly, we need to consider only and integer vectors So for a given there exists a vector X which satisfies (4.55)(4.57) if and only if

for all integer The are interpreted in Rardin and Wolsey [74] as follows. If we let denote a collection of cuts for commodity and define we can interpret as the number of times cut S is repeated in the collection for a given Comparison of Projections 1 and 2 It is interesting to check if any of the projections give stronger inequalities. PROPOSITION 4.14 (Labbé and Yaman [52]) For a given define for each Then for a given there exists a vector X which satisfies (4.55)-(4.57) if and only if

for all

and integer such that we can sort the cut sets in each where

as

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Proof: Suppose we have an inequality of type (4.64) for a pair such that for all If we define then is the number of times node is repeated in the cut sets in collection The right hand side of inequality (4.64) is

which is equal to the right hand side of inequality (4.58) for this Now we compare the left hand sides. For inequality (4.58), we take for all In fact,

Suppose

Then

So we have for all This proves that the left hand side of inequality (4.64) is greater than or equal to the left hand side of inequality (4.58). As the right hand sides are equal, we can conclude that inequality (4.58) is at least as strong as inequality (4.64). Next, we show how we can construct a pair inequality as a given pair. For each commodity We define to be the collection of sets for so that we have We let for all and Moreover, if

that gives the same let where for all Then we have

then whenever

we have

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Uncapacitated Problems

Therefore

and if

for all

then there is no

such that This proves that

So

and

Hence for inequalities of type (4.64), we only need to consider where we can sort the cut sets in each such that where is the number of cut sets in The inequalities given by other are dominated by these. Hub Location Formulation and its Projection Here we use the hub location structure to formulate UCL-C. We replace the constraints (4.46) by

to obtain the hub location formulation. This formulation is given in SkorinKapov et al. [75]. Note that we replaced constraints (4.66) and (4.67) which are originally equalities by their inequality forms. PROPOSITION 4.15 (Labbé and Yaman [52]) Given that satisfies (4.66)-(4.69) if and only if

there exists a X

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

such that

for all

Proof: If we associate dual variables to constraints (4.66), to constraints (4.67) and to constraints (4.68), by Farkas’ Lemma, we get the result. We consider first the case for a single commodity and try to characterize all nondominated inequalities. Single Commodity Case Suppose above. Given if and only if

and

We drop the index from the variables we defined Proposition 4.15 implies that there exists a feasible X

such that

for all

Let C1 be the cone of that satisfy inequalities (4.74) to (4.75). We would like to characterize all nonzero extreme rays of C1. PROPOSITION 4.16 (Labbé and Yaman [52]) The ray is an extreme ray of the cone C1 if and only if it belongs to one of the classes:

1

for one

2

for one

3

for all entries are 0.

where

for all

where if

4 that

and the rest of the entries are 0. and the rest of the variables are 0.

for all

and the rest of the

such for all where and the rest of the entries are 0.

Proof: The proof is very similar to the proof of Proposition 3.5 given by Mirchandani [61] (the proof of the proposition which characterizes the

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Uncapacitated Problems

extreme rays of the projection cone for a single commodity flow). We give it here for the sake of completeness. (Sufficiency) Classes 1 and 2 are trivial. For classes 3 and 4, consider Let a because Notice that and is not an extreme ray of C1. Then of constraint (4.75). Suppose that in C1 which are not multiples of there exist distinct and This implies that and if and if if In Class 3, and and

and T = I. As for all and we have that Then and

for all

As and are multiples of

in A, and for all

In Class 4, B, S and T are all nonempty. If and then By the above So we have and for all and for all and discussion, and If and

then

and

and

we get are multiples of

and

As and

So

For classes 3 and 4, we showed that a ray satisfying the requirements of one of these cases cannot be written as a linear combination of two distinct rays of C1. Thus such a ray is an extreme ray. (Necessity) Given an extreme ray of C1, define the sets B, S and T. Assume that It is easy to show that if then should belong to class 1 and if then should belong to class 2. If both T and B are not empty, then this can be written as a linear combination of rays of classes 1 and 2, so it cannot be extreme. Now, assume that rays

and for all for all

Let By feasibility, we have Clearly, Consider the two defined as follows: for all for all for all for all for all or and the rest of entries are 0.

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

We can show easily that both extreme ray implies that So to be extreme for all entries are 0.

and

are in C1 and that But the fact that is an and should be multiples of for all should have the form: for all and the rest of the

If by feasibility we have T = I. Then Otherwise, it is in Class 4. PROPOSITION 4.17 (Labbé and Yaman [52]) Given that satisfies (4.66)-(4.69) if and only if

is in Class 3. there exists an X

for all Proof: The inequalities defined by the extreme rays are as follows: 1

for all

2

for all

3

for all

4

for all

The first three families of inequalities are already implied by the constraints in the formulation. The only nonredundant inequalities are the inequalities of the fourth form. These inequalities are quite similar to cut inequalities for the single commodity flow. In fact when we take a cut in the case of hub location, we choose two disjoint subsets of the set I, S and and consider all the arcs going from S to Given a solution not.

and

we can easily check if there exits a feasible X or

PROPOSITION 4.18 (Labbé and Yaman [52]) Given we can check if there exists an X that satisfies (4.66)-(4.69) and if not, we can find a violated inequality of the form (4.76) by solving a mincut problem. Proof: We should find

such that is minimized. Let denote this minimum value.

Uncapacitated Problems

93

There exists a feasible X if and only if To compute we transform the problem into a classical mincut problem. Consider the layered graph where V includes the nodes and the set I and a duplicate of set I. The arc set is defined as Let denote the capacity of arc Define the arc capacities as follows:

A cut C is a subset of the set I such that and Define and and Notice that if S is not a subset of T, the cut has an infinite capacity. Suppose that C is a cut with a finite capacity. We compute the capacity of the cut C.

Consider the cut This cut has a capacity equal to As a result of this, the min capacity cut has always a finite capacity, that is the mincut problem cannot be unbounded on graph So We can conclude that there exists a feasible X for a given pair if and only if the min capacity cut on the graph has a capacity not less than In Figure 4.32, the set and the corresponding inequality is:

So the sets are S = T = {1, 2}

Multi Commodity Case For a single commodity, we are able to characterize all extreme rays of the projection cone. However for the multicommodity case, we do not know such a characterization. Still, we are able to investigate some special cases. Let C be the cone of (4.72). Consider an

that satisfy inequalities (4.71) and in C. Define the following sets: for all

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Figure 4.32. The separation problem as a mincut

for all

and

PROPOSITION 4.19 If

for all

extreme ray if and only if it has the form of the entries are 0.

then

for one

is an

and the rest

An extreme ray of this kind leads to an inequality for some A which is already implied by the constraints of the formulation. So this family of extreme rays give redundant inequalities. PROPOSITION 4.20 If

for all

an extreme ray if and only if it has the form and the rest of the entries are 0.

and

then

for one

is

and for one

Like the previous class of extreme rays, this class also leads to redundant inequalities of the form for all and PROPOSITION 4.21 If have for all only if

and In this case,

for all

then we should is an extreme ray if and

The inequalities defined by this class are for all and These inequalities are dominated by the assignment constraints (4.44).

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Uncapacitated Problems

For the remaining extreme rays, we should have and In this case, we have a sufficient condition for a special class of such rays to be extreme.

PROPOSITION 4.22 (Labbé et al. [53]) A ray

of the if there and all the rest of where

form if if and such that exists a the entries are 0 is an extreme ray if the graph and and such that and there exists a where is connected, the bipartite graphs are connected for all or for all

and

Proof: Assume that the conditions are satisfied. Then we show that if there exist distinct and in C1 such that then they are multiples of

For

if

we have

if

and If

and As and

conclude that

As for all and as

such that for all

and

and

for all

Let

Pick

if

we have

and As the graph and

such that and and

and we have that is connected, we can for for all

and and Then and we have As the graph is connected So both for all and

are multiples of

A special class of these extreme rays are the following:

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

and the rest of the entries are 0. Such an inequality:

defines the following

which is equivalent to:

This inequality says that the capacity of links going from the set S to the set should be sufficient to carry the traffic of the commodities whose origin is assigned to a node in set S and destination is assigned to a node in

COROLLARY 4.3 (Labbé et al. [53]) The inequality

where S and T are nonempty disjoint subsets of I and is a valid inequality for and it is not dominated by other projection inequalities. Constraints (4.51) are special cases of the projection inequalities (4.77) where and Labbé and Yaman [52] identify some facet defining inequalities that involve both the assignment and traffic variables.

4.4

Conclusion

In this chapter, we studied the polyhedra of UCL, UCL-S and UCL-C. We started with the polytope of UCL. We considered all facet defining inequalities for and We generalized these inequalities to any We tried to characterize these inequalities in terms of the known facet defining inequalities for the stable set polytope. For these family of inequalities, we investigated the complexity of the separation problems. Then we studied the polyhedron of UCL-S. We showed that the facet defining inequalities of UCL-S polyhedron involving only assignment

Uncapacitated Problems

97

variables are the facet defining inequalities of the UCL polytope. If there exists at least one commodity with positive traffic demand (which is in fact necessary for the problem to make sense), then we showed that there were no facet defining inequalities involving only the traffic variables. We also presented a family of facet defining inequalities involving both assignment and traffic variables and we showed that they can be separated in polynomial time. In the last section of this chapter, we investigated the structure of the UCL-C polyhedron. Similar to the UCL-S, we could characterize the facet defining inequalities of the UCL-C polyhedron involving only the assignment variables in terms of the facet defining inequalities of the UCL polytope. Then we presented a necessary condition for inequalities involving only the traffic variables to be facet defining. Finally, we derived some valid inequalities involving both assignment and traffic variables projecting out the extra variables coming from higher dimensional formulations. See the end of this part for a summary of the polyhedral results.

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Chapter 5 PROBLEMS WITH LINEAR CAPACITY CONSTRAINTS

This chapter is about the concentrator location problems with linear capacity constraints. Each concentrator has a fixed capacity in terms of the demands of nodes that are assigned to it. We consider the problem with no routing cost (LCL), with routing on a star backbone network (LCL-S) and with routing on a complete backbone network (LCL-C). As the uncapacitated version of each problem is a relaxation of the linear capacitated version, the valid inequalities for the uncapacitated case remain valid for the capacitated case. We investigate when these inequalities remain facet defining and derive new families of valid and facet defining inequalities for the problems with linear capacity constraints. We define to be the demand of node and M to be the capacity of a concentrator. In this chapter, we assume that all nodes cannot be assigned to a single concentrator, i.e., and that any two nodes can be assigned together, i.e., for all and

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

5.1

Linear Capacitated Concentrator Location Problem

In this section, we study the polytope corresponding to the linear capacitated concentrator location problem (LCL). In addition to the UCL, we have the capacity constraints saying that the sum of the demands of nodes assigned to a certain concentrator cannot exceed its capacity. We first discuss when the facet defining inequalities of the knapsack and the UCL polytopes are facet defining for the LCL polytope. Then we modify the valid inequalities of the capacitated facility location problems, so that they are valid for the LCL polytope.

5.1.1

Formulation and the LCL polytope

In this section, we present the formulation for the LCL. We use the assignment variables as in UCL. We can formulate the LCL as follows:

We can also give a formulation without using the variables project out these variables, we obtain the following formulation:

Let Define called an open node, if

:

If we

satisfies (5.5) and (5.6)} and for all Node is If in addition

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Problems with Linear Capacity Constraints

then node

is called free.

We first investigate the dimension and the trivial facets of the polytope PROPOSITION 5.1 The LCL polytope Proof: Consider the vectors for each We have so PROPOSITION 5.2 defines a facet of

is full dimensional, i.e., and for all affinely independent points in

the nonnegativity constraint

Proof: Consider the vectors described in the proof of Proposition 5.1 except the vector where We have affinely independent points in that satisfy If

and

then the inequality and the nonnegativity constraints

describe the polytope In the sequel, we assume that PROPOSITION 5.3 For of if and only if

the clique inequality (5.5) defines a facet for all

Proof: Assume that following points:

for all

for some

1

Consider the

such that

and

for all 2

for some

and

for all

3

for some

and

for all

4

for some

and

for all

5

and

for all

6

and

for all

7

for some

and

for all

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Figure 5.1.

affinely independent points in

that satisfy

The list of these points is given in Figure 5.1, where I, 0, and 1 denote the identity matrix, matrix of 0’s and 1 ’s of the corresponding sizes, respectively. The columns correspond to points. The number of points of each type is indicated in the first row. It is easy to verify that in total we have affinely independent points in that satisfy inequality (5.5) at equality. If there exists a node such that then inequality (5.5) is dominated by the valid inequality (which is a lifted cover inequality, see the following section):

So in this case, inequality (5.5) cannot be facet defining.

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Problems with Linear Capacity Constraints

The remaining part of this section is devoted to the study of nontrivial facets of and some valid inequalities.

5.1.2

Knapsack Based Inequalities

We lift the facet defining inequalities of the knapsack polytope to be facet defining for A similar result is given in Deng and Simchi-Levi [28] for the polytope of the capacitated facility location problem with single assignment. PROPOSITION 5.4 If the inequality for the knapsack polytope then the inequality

is a valid inequality

is valid for Proof: If for all then as inequality is equivalent to the capacity constraint (5.6) for node inequality (5.8) is a valid inequality for If for some then for all and for all So, inequality (5.8) is valid for

THEOREM 5.1 If the inequality is a facet defining inequality for the polytope then inequality (5.8) is facet defining for

Proof: If the inequality the polytope then the inequality

is a facet defining inequality for is facet defining for

We lift all variables fixed to 0 sequentially. We start with variables Suppose we fix then for all and for all Hence the coefficient of should be When all such variables are lifted, we obtain inequality (5.8). Now we should lift the variables where and Suppose we fix We can take for some This shows that the lifting coefficient of should be 0. So inequality (5.8) is facet defining for

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5.1.3

UCL Based Inequalities W – 2 Inequalities

THEOREM 5.2 Let with and such that for each there exists exactly one Define and there is no such that Let Then the W – 2 inequality

defines a facet of

such that

where

and

if

and

for all Proof: Assume that and for all We use the indirect method to prove that in this case the W – 2 inequality (5.9) is facet defining for Let Consider an inequality of the form

We show that if all points in that satisfy the inequality (5.9) at equality also satisfy the inequality (5.10) at equality, then (5.10) is a positive multiple of (5.9). We define for all Assume that if and satisfies inequality (5.9) at equality, then it also satisfies Let such that and Consider a point such that for all for some and for all other arcs Consider also the solution that is the same as except that (see Figure 5.2). As both and are in and they satisfy inequality (5.9) at equality, we have that Consider a node and a node There exists a node such that Consider the solution such that for all and for all other arcs Define and to be the same as except and (see Figure 5.3).The vectors and are in and they satisfy inequality (5.9) at equality. So

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Problems with Linear Capacity Constraints

Figure 5.2.

Solutions

and

to show that

for

Figure 5.3.

Solutions

and

to show that

for

and

and

The inequality (5.10) becomes

Consider a node a node and the solution such that for all and for all other arcs Define to be the same as except that for some (see Figure 5.4). As both and are in and they satisfy inequality (5.9) at equality, we have that for all This shows that for all

Figure 5.4.

Solutions

and

to show that

for

and

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Let such that and There exists a node such that Consider the solution such that for all and for all other arcs Consider that is the same as except that and (see Figure 5.5). Both vectors and are in and they satisfy inequality (5.9) at equality. So for all

Figure 5.5.

Let where for all

Solutions

and

to show that

for

and

be two different nodes in U. Consider the solutions and for all and for all other arcs and and for all other arcs (see Figure 5.6).

Figure 5.6.

Since both we have that

and

and

Solutions

are in

and

to show that

for

and they satisfy inequality (5.9) at equality,

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Problems with Linear Capacity Constraints

So

for all

The inequality (5.11) becomes

Consider arc such that and If there exists a node such that and then consider the solutions and such that for all and for all other arcs and is the same as except that and (see Figure 5.7). As both and are in and they satisfy inequality (5.9) at equality, we have that

Figure 5.7. exists a

Solutions and such that

to show that and

If there exists a node such that such that for all is the same as except that that

for

and and

and

if there

then consider the solutions and for all other arcs and (see Figure 5.8). We have

So

Figure 5.8. exists a

Solutions and such that

to show that

for

and

if there

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

If

for all nodes consider the node If then Now consider the solutions and such that for all and for all other arcs and is the same as except that and (see Figure 5.9). The vectors and are in and they satisfy inequality (5.9) at equality. So we have that

Figure 5.9. Solutions and for all and

to show that

for

If

then for all and Consider the solutions and such that for all other arcs and for all all other arcs (see Figure 5.10). The vectors and satisfy inequality (5.9) at equality. So we have that There fore

Figure 5.10. Solutions and for all and

to show that

for

and

if

for all for all and for are in and they

and

if

Consider any solution such that for all for some and for all other arcs Clearly, the right hand side is This shows that inequality (5.10) is an of inequality (5.9). Starting from the W – 2 inequality (5.9) and applying sequential lifting, we can obtain a facet defining inequality for of the form:

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Problems with Linear Capacity Constraints

We first show that independently of the lifting order, the optimal lifting coefficients are either 0 or 1. For define and to be the set of arcs in and in H, respectively, such that is lifted before Define also PROPOSITION 5.5 The coefficient Proof: The coefficient

for all

can be computed as:

Consider the solution where for all The vector is in and it satisfies So As and it takes only integer values, Now we prove that the coefficients

with

PROPOSITION 5.6 The coefficient Proof: Given

for some

are either 0 or 1. for all

in H, we can compute the coefficient

as:

If

or then there exist two nodes and in As there exists at least one node such that we have that Similarly we can show that There exists at least one node in such that or If then consider the solution such that and for all (see Figure 5.11). If then consider the solution such that and for all In both cases, the vector is in and satisfies So, we have that If solution

and such that and for all have that

then for all

is either or Consider the and (see Figure 5.12). Since So we can conclude that As takes only integer values and as we

We show how we can compute the lifting coefficients of the variables with for sequences where these variables are lifted before the variables with

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Figure 5.11. Solution

Figure 5.12. Solution

if

if

PROPOSITION 5.7 Assume that the variables with are not yet lifted or the ones that are lifted have coefficient 0. Then for a given the coefficient is 1 if and only if there does not exist a set such that and the nodes in can be assigned to nodes in I \ W. Proof: If there exists a set such that and the nodes in can be assigned to nodes in I \ W, then clearly there exists an with So Assume that there does not exist a set such that and the nodes in can be assigned to nodes in I \ W. Assume to the contrary that there exists an with If nodes are open in then So and node is the only open node. As there does not a set such that and the nodes in can be assigned to nodes in I \ W, So Notice that the coefficients do not depend on the lifting order on the set H. Moreover for all with and if then for all In general, computing amounts to solving

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Problems with Linear Capacity Constraints

the feasibility version of a generalized assignment problem, i.e., only if the following system has no solution:

is 1 if and

Now, we compute the lifting coefficients of the variables with for sequences where these variables are lifted before the variables with PROPOSITION 5.8 Assume that the variables with are not yet lifted or the ones that are lifted have coefficient 0. Then for the coefficient is 1 if and only if all the fallowing conditions hold: 1

for all

2

for all

such that

Proof: If there exists a node such that then consider the solution such that all where is such that we have that If there exists a node and and for all

such that then the solution such that where So

and

and As

for and

is in

and

Assume that the two conditions listed in the proposition are satisfied for arc We prove that by induction on the lifting order. We first prove that condition 1 implies

for all that

Assume to the contrary that there exists a vector If there are nodes that are open in

such then

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

So we should have node, condition 1 implies that nodes in So

If node is the only open cannot be assigned to node

Now assume that and let denote the other open node. As cannot be assigned to node we have (5.12). This proves that if condition 1 is satisfied the first variable to lift has coefficient 1. Now suppose we are lifting and all variables with are lifted as described in the proposition. Assume to the contrary that the two conditions of the proposition are satisfied and that there exists a vector such that If there are nodes open in then So we should have Conditions 1 and 2 imply that if for some and with no other node in I \ U whose corresponding assignment variable has coefficient 1 can be assigned to node Therefore we have at least nodes open in and either the nodes that are not open in W \ U are assigned to the remaining open node, say node or some node in is assigned to node As there are at least two nodes not open in W \ U, we should have all these nodes assigned to node But as cannot be assigned to node and to node we have that So Notice that the lifting coefficient for depends on the lifting coefficients for but is independent of the with and The lifting coefficients can be computed in time. k-triangle Inequalities THEOREM 5.3 Consider the graph where with for some We number the nodes in from 1 to The arc set consists of all for and all arcs of the form where and are both odd and Then the inequality:

is facet defining for 1

if for all

2

for all

such that

3

for all

such that

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Problems with Linear Capacity Constraints

4

for all

5

such that

for all

Proof: This theorem can be proved as Theorem 4.3. We only say briefly where the conditions of the theorem are used in the proof. The solutions that are used in the proof of Theorem 4.3 are depicted in Figures 4.4-4.14. We refer to these figures. Condition 4 for

ensures that solution

in Figure 4.5 is feasible.

Condition 4 for

ensures that solution

in Figure 4.8 is feasible.

Condition 5 for

ensures that solution

in Figure 4.9 is feasible.

Condition 1 for ensures that solution

in Figure 4.10 is feasible.

Condition 2 for

ensures that solution

in Figure 4.11 is feasible.

Condition 3 for

ensures that solution

in Figure 4.12 is feasible.

Condition 2 for

ensures that solution

in Figure 4.13 is feasible.

Condition 3 for

ensures that solution

in Figure 4.14 is feasible.

k-leaf Inequalities THEOREM 5.4 Given an arc the k-leaf inequality

and a subset

is facet defining for if and only if Proof: If

and

then the following cover inequality is valid:

The clique inequality (5.5) for

is

with

for all

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

If we sum these two inequalities, we get

which dominates the k-leaf inequality. So in this case, the k-leaf inequality cannot be facet defining. If there exists a node the inequality

If

such that

then we consider

this is the same as the k-leaf inequality and is valid. Assume that Then As we also have The inequality becomes:

and it is valid. So inequality (5.15) is valid. As it dominates the k-leaf inequality, the latter cannot be facet defining. Assume that and for all We use the indirect method to prove that in this case, the k-leaf inequality (5.14) is facet defining for The proof is very similar to the proof of Theorem 4.5. We give it for the sake of completeness. Define and Define also Consider an inequality of the form

We show that if all points in satisfy the inequality (5.16) at equality, then (5.16) is a positive multiple of (5.14). Consider a node the solution and solution such that both and are in

For a given consider for all other arcs Now consider the and for all other arcs As for all and

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Problems with Linear Capacity Constraints

Let as

Consider the solution where for all and for all other arcs Now consider the solution that is the same except Both and are in So for all

For consider the solutions and such that and for all other arcs and and for all other arcs As both and are in Then inequality (5.16) becomes

For node and as except that for all

consider the solution such that for all for all other arcs The second solution is the same Clearly, both and are in This proves that Now we can rewrite inequality (5.17) as:

Next, we show that the solution where for all and for all Consider also the solution we can show that Now, consider a node for all and

for all Given for all and the solution As both and are in

such that

where we have that

and

are in

Consider the solutions defined as follows: As and are all in we have that for all This implies that for all for all

We should show that for all consider the solution where and and are both in we have that The inequality (5.18) becomes:

If we plug in the solution inequality (5.16) is a

Since both

consider

for some So

It is enough to As for all

we can show that This proves that of the k-leaf inequality (5.14). So inequality

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(5.14) is facet defining for Now we can lift the variables whose values are fixed to 0. Let optimal lifting coefficient for with

denote the

PROPOSITION 5.9 Define variables

and such that

1 The coefficients 2 The coefficient

then

Suppose that we are first lifting the and then the variables such that

for all where

3 For the variable with define to be the set of indices of variables that are lifted before The lifting coefficient of is where

and

Proof: We first prove the first statement. For let Let for all The vector is in and it satisfies So for

be such that and

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Problems with Linear Capacity Constraints

Now we consider the second statement. The value

can be computed as:

which is equivalent to:

s.t. (5.19) and (5.20). So is the minimum number of nodes that should be removed from the set so that the remaining nodes can be assigned to node with node Now, we prove the third statement. For coefficient can be computed as:

We investigate two cases: 1 If

the problem becomes:

The maximum value of this problem is

with

the lifting

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2 If then as there is at least on node in that cannot be assigned to node when the other nodes in and node are assigned to node Node can be assigned to this node. So the problem becomes:

Then

is equal to the optimal value of this problem.

Thus

This proves the third statement.

The maximization problems to compute values and are both knapsack problems. If is lifted after for then we solve the second knapsack problem to compute i.e., We can lift again by solving the second problem, but we do not add 1 to the optimal value of this maximization problem, i.e., 2-cycle Inequalities THEOREM 5.5 Consider a subset with and a node Let C be a directed cycle on the nodes of D. Renumber the nodes such that D = {1,2,3}, the cycle is 1, 2, 3, 1, the node and If and for all such that and for all then 1 the 2-cycle inequality

defines a facet of 2 Let and be the node in D such that Renumber the nodes such that and Then the 2-cycle inequality

defines a facet of

Proof: See the proof of Theorem 4.6.

Problems with Linear Capacity Constraints

5.1.4

119

CFLP Based Inequalities

The capacitated facility location problem with single assignment (CFLPS), also called the capacitated concentrator location problem, is defined on two sets I and J, the set of terminals and the set of possible locations for concentrators, respectively. Each terminal node should be assigned to exactly one location where a concentrator is installed. For more information, see Chapter 2. The feasible set of the CFLPS is the set of pairs following constraints:

which satisfy the

Let denote the convex hull of this set of points. We first show that we can derive a valid inequality for from a valid inequality for PROPOSITION 5.10 Assume I = J. If inequality

is a valid inequality for

then the inequality

is a valid inequality for Proof: Assume that inequality (5.26) is not valid for Then there exists a point which does not satisfy inequality (5.26). Define as follows: for all and for all As and as it does not satisfy inequality (5.25), this inequality cannot be valid for Now we discuss three families of valid inequalities for

that come from

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Binpacking Inequalities Binpacking inequalities are introduced by Deng and Simchi Levi [28] for the poly tope of CFLPS. Choose subsets and Define to be the minimum number of concentrators to be installed to assign all nodes in The binpacking inequality

is valid for and defines a facet for a smaller dimensional polytope under some conditions. We define binpacking inequalities for the LCL and discuss when they are facet defining for COROLLARY 5.1 Let

and

The binpacking inequality

is valid for Modified Binpacking Inequalities The computation of assigned to itself. Define

PROPOSITION 5.11 Let

does not take into account that a concentrator is as follows:

The modified binpacking inequality

is valid for Proof: If the variables

are set to 0 for

and

then in any

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Problems with Linear Capacity Constraints

solution This implies that inequality (5.29) is valid in this case. As relaxing for some and cannot increase the left hand side of inequality (5.29), it is valid for One particular case is when the form

In this case we obtain an inequality of

which gives a lower bound on the number of concentrators to be installed. The modified binpacking inequality (5.29) remains valid if we compute a lower bound to The value is a lower bound for Still it requires the solution of a binpacking problem. A computationally less demanding solution would be to consider the bound of Martello and Toth [59]. THEOREM 5.6 Let be such that for all Then the modified binpacking inequality (5.29) is facet defining for Proof: Assume, without loss of generality, that Assume also that Consider an inequality of the form

and for all

Assume that all points in that satisfy the modified binpacking inequality (5.29) at equality also satisfy inequality (5.31) at equality. We show that inequality (5.31) is a positive multiple of (5.29). Consider a solution where node is free, nodes are open so that the remaining nodes are assigned to these nodes. Such a solution satisfies the modified binpacking inequality (5.29) at equality. Let denote such a solution. Choose two nodes and not in Let be the same as except that As both and are in and they both satisfy the modified binpacking inequality (5.29) at equality, we have Similarly we can show that since node is free in Then inequality (5.31) becomes:

For each node. Define

and

there exists an where node is assigned to some As to be the same as except that and

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both points are in and they both satisfy the modified binpacking inequality (5.29) at equality, we have for all and Now consider some and let denote the set of the open node and the nodes assigned to it. As and as any two nodes can be assigned together, each has at least two nodes. Assume that is the open node in Let be another node in which is assigned to Replace node by node 1. This is feasible since As the resulting point is in and as it satisfies (5.29) at equality, we have for all As any node in can be the open node, we have that for any two nodes and in This shows that if Repeating the same argument for nodes we can show that if and As we conclude that if Consider some Choose and Let and denote the smallest nodes in and respectively. Assume, without loss of generality, that nodes and are not open in and that Define to be the same as except that node is free, all nodes in are assigned to node and node 1 is assigned to the open node in Define also to be the same as except that node 1 is assigned to node Both vectors and are in and satisfy (5.29) at equality. So This shows that Therefore for all and Consider some As nodes are open, node 1 is free and the remaining nodes are assigned to these open nodes, So inequality (5.31) is an of the modified binpacking inequality (5.29). Residual Capacity Inequalities Residual capacity inequalities are introduced by Leung and Magnanti [56] for the capacitated facility location polytope. The capacitated facility location problem is similar to the CFLPS except that we do not force each terminal node to be assigned to exactly one concentrator node. So the variables are continuous. This problem is a relaxation of the CFLPS (see Chapter 2). The residual capacity inequality is defined as follows. Let Define

and

and The residual

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Problems with Linear Capacity Constraints

capacity inequality

is valid for COROLLARY 5.2 Let residual capacity inequality

and

The

is valid for

Effective Capacity Inequalities The capacitated concentrator location problem with different capacity values has been studied by Aardal et al. [2] (see Chapter 2). The flow cover inequalities for this version of the problem simplify to the residual capacity inequalities in case of equal capacities. This is not the case for the effective capacity inequalities. In case of equal capacities, the effective capacity inequality is defined as follows. Let For each choose and define Define also for each inequality

and

If

then the effective capacity

is valid for COROLLARY 5.3 Let Define also If

is valid for

For each

let for each

and define and

then the effective capacity inequality

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5.2

Linear Capacitated Concentrator Location L Problem with Star Routing

In this section we consider the capacitated version of the UCL-S. We call this problem “Linear Capacitated Concentrator Location Problem with Star Routing” and abbreviate it by LCL-S. We can formulate the LCL-S as follows:

Different from the UCL-S, we have the capacity constraints (5.37). We removed constraints (4.28) as they are implied by constraints (5.37). Now, we give the formulation without using the variables

Let and

satisfies (5.41)-(5.45)}

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Problems with Linear Capacity Constraints

5.2.1

The LCL-S polyhedron

We first investigate the dimension of the polyhedron PROPOSITION 5.12 The polyhedron

has full dimension,

i.e.,

Proof: See the proof of Proposition 4.8. As the LCL polytope the following result:

is the projection of

THEOREM 5.7 The inequality it defines a facet of

on the

we have

defines a facet of

if and only if

Proof: See the proof of Theorem 4.7. This theorem characterizes all facet defining inequalities of of the form Now we investigate facet defining inequalities of the form PROPOSITION 5.13 Let Then,

1

be a facet defining inequality for

and

2 Let

If there exists an

such that

then

3 If of 4 For

then for some

i.e., the inequality

the inequality for all

is a positive multiple

defines a facet of

if and only if

Proof: For part 1, see the first paragraph of the proof of Theorem 4.8. For part 2, if there exists an such that then there exists a such that and for all So As all pairs that satisfy satisfy also for all if is facet defining then it is a positive multiple of for some Hence is a singleton. This proves part 3. Part 4 can be proved as Proposition 4.9. COROLLARY 5.4 If there exists an no facet defining inequality of the form

with

then

has

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5.2.2

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Computing Lower Bounds on the Traffic

As the concentrators have capacities and as all nodes cannot be assigned to the same concentrator. Using the capacity restrictions, we can compute some lower bounds on the traffic which flows on the backbone links. Assume that node is a concentrator, i.e., it is assigned to itself. Let denote the minimum amount of traffic adjacent at node that travels on the link We can compute as follows:

where is 1 if node is assigned to node and 0 otherwise. We try to minimize the traffic between node and the nodes that are not assigned to node PROPOSITION 5.14 For

is valid for

where

the traffic bound inequality

is computed as described above for all

We can obtain stronger inequalities by computing PROPOSITION 5.15 For

is valid for

for each

where

for each

the traffic bound inequality

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Problems with Linear Capacity Constraints

5.3

Linear Capacitated Concentrator Location Problem with Complete Routing

We now study the linear capacitated concentrator location problem with complete routing which is denoted by LCL-C. Different from the UCL-C, we have capacity constraints for the concentrators. The LCL-C can be formulated as follows:

The formulation without the variables

Let satisfies (5.53)-(5.57)} and

is as follows:

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5.3.1

The LCL-C polyhedron

The below propositions are about the polyhedral structure of PROPOSITION 5.16 (Labbé et al. [53]) The polyhedron dimension, i.e.,

has full

Proof: See the proof of Proposition (4.8). THEOREM 5.8 (Labbé et al. [53]) The inequality if and only if it defines a facet of

defines a facet of

Proof: See the proof of Theorem (4.10). PROPOSITION 5.17 (Labbé et al. [53]) Let inequality for Then,

1

be a facet defining

and

2 Let either

3 If

such that or

If there exists an and for every

and an

then

then

4 If and defines a facet of

for all

then

Proof: The first three statements can be proved as the first three statements of Proposition 5.13. For the fourth statement, see the proof of Proposition 4.12. PROPOSITION 5.18 The inequality

is a valid inequality for

if and only if

is a valid inequality for Proof: See the proof of Proposition 4.13.

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Problems with Linear Capacity Constraints

We can compute lower bounds on the traffic variables using the idea in Proposition 5.14. PROPOSITION 5.19 For

are valid for

where

the traffic bound inequalities

and

are computed as follows:

and

Proof: See Proposition 5.14. PROPOSITION 5.20

are valid for

For

where

the traffic bound inequalities

and

are computed as follows:

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

and

Proof: See Proposition 5.15.

5.3.2

Strengthened Projection Inequalities

In this subsection, we present a family of valid inequalities which give lower bounds on the amount of traffic on the backbone links. We then show that these inequalities dominate constraints (5.50). In the remaining part of this subsection, we use the variables for ease of presentation. To derive the strengthened projection inequalities, we consider some substructures. More precisely, for a given arc and a node we consider the following structure:

where on arc

is the amount of traffic that originates at node and that travels given the assignment vector Let be the set of points that satisfy constraints (5.64) to (5.68) and If we have an inequality which is valid for for each then the inequality is a valid inequality for since PROPOSITION 5.21 The inequality

Problems with Linear Capacity Constraints

131

where

is valid for Proof: If node is assigned to node is not assigned to node then

then

If node

The value of can be improved if we fix the values of some of the variables. Let and be two disjoint subsets of I, define and compute

Then

is valid for when for all and for all Since the values of some of the variables are fixed, By lifting the variables whose values are fixed to 0 or 1, we can obtain a valid inequality of the form

The following proposition tells us how to compute the lifting coefficients. Let

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

THEOREM 5.9 The inequality (5.71) is valid for for all where

and and

and

for any

such that

is the set of indices of the variables that have been lifted before for all where

is the set of indices of the variables that have been lifted before

Proof: We are first lifting the variables with indices in and then the variables with indices in Suppose we are lifting with We have an inequality of the form

which is valid for the inequality

We would like to find an

is valid for inequality (5.72) is valid for

When

such that

for any value of the So we consider the

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Problems with Linear Capacity Constraints

case

To have inequality (5.72) valid, we should have:

We have two cases to consider: Case1: and we get

Let

Then

be the following:

In this case if we pick any

Case 2: we have

We plug this in inequality (5.73)

then inequality (5.72) is valid for

So

Then

It is easy to show by induction that for all case, for any inequality (5.72) is valid for We can choose an which is not more than strongest inequality we choose

and

So in this

To have the

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Now we do the lifting for inequality

We would like to choose

is valid for When valid for any We consider the case where choose an such that

We need to consider two cases again: Case1:

Let

When Case 2:

Then

So

be the following:

we can choose any Then

such that the

the inequality (5.74) is So we would like a

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Problems with Linear Capacity Constraints

So

For all and all

we know that The solution with for all for all is a feasible solution since we could fix for So by induction we can show that for all Therefore

For inequality (5.74) to be valid for should choose which is not less than

and

we The best choice is

The above proposition shows that the computation of each lifting coefficient amounts to solving a knapsack problem. Here we compare the strength of these inequalities with the constraints (5.50). We should note that if we fix some variables to 0 while computing the values we can obtain a valid inequality even without lifting. PROPOSITION 5.22 For any dominated by the inequality

where

and

for some

the inequality (5.50) is

with

and

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Proof: The inequality (5.50) is

Clearly for all dominates inequality (5.50).

This shows that inequality (5.76)

We can prove a similar result for projection inequalities defined by two sets S and T which are not necessarily singletons. The proof can be done similarly. PROPOSITION 5.23 For any disjoint subsets S and T of the node set I and the inequality

is dominated by the inequality

where for some

with

with for all

and

for all As the set T becomes large, the gain over the projection inequalities will decrease. If all nodes in can be assigned to the nodes in T, then If this is the case for all then the inequalities (5.77) and (5.78) are the same.

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Problems with Linear Capacity Constraints

5.3.3

Step Inequalities

Here we try to find lower bounds on the traffic on arc using the total number of terminals assigned to concentrators and Let denote the number of terminals assigned to and If we know that we cannot assign terminals to only one of the concentrators and then we can compute a lower bound on the traffic traveling on the links and PROPOSITION 5.24 (Labbé et al. [53]) Let denote a lower bound on the traffic from concentrator to when terminals are assigned to and for Define for all If for all then the step inequality

is valid inequality for Proof: Let

If

for any Then by definition Since for all So

then we can show that

If

then we have

Hence

Now, we present an easy way to compute lower bounds Let be the maximum number of terminals that can be assigned to a given concentrator Because of the assumption that a given node is assigned to itself if it is a concentrator, and can be different. Let If is larger than B, then the configuration that minimizes the number of commodities that travel on the arc is to assign B terminals to one of the concentrators and the remaining to the other concentrator. Then we have commodities that travel from to Let denote the minimum traffic. We define the lower bounds as follows:

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

It is easy to verify that the bounds satisfy So they lead to valid inequalities. If we consider a subset

for all

we obtain the inequality:

where is computed as follows. Let be the maximum of and which are the maximum number of terminals in set that can be assigned to concentrators and respectively. Define also to be the smallest traffic with origin and destination in Then

for all

on link for any

and for all Inequality (5.80) is valid when for all However, for any if we have that the traffic cannot decrease. So the step inequality (5.80) is valid in general

We can also set some variables at 1 to obtain better bounds. However in this case we cannot guarantee that the resulting inequality remains valid in general. So we should lift down these variables. In the remaining part of this section we discuss the lifting problem. Let

and

The following inequality:

is a valid inequality if for all for all and is computed accordingly, that is after subtracting the demands of the items in sets and from the capacities of concentrators and Now assume that we are lifting the variables with indices in the sets

with

and that we already lifted and

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Problems with Linear Capacity Constraints

PROPOSITION 5.25 The lifting coefficient of computed as:

where

and

and

Proof: The inequality

with

can be

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

is valid when for all and would like to compute an such that the inequality

is valid when is no longer fixed to 1. When valid for any value of So we consider the case satisfy

for any feasible and inequality, the variable value which is:

for all

We

inequality (5.83) is Then should

As we try to maximize the right hand side of this which has a negative coefficient takes its minimum

Plugging this into inequality (5.84), we get that

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Problems with Linear Capacity Constraints

Let

and define

and

Now we define

to be

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

Notice that the cost for all and for all Besides the corresponding assignment variables appear in the quadratic cost term with a negative coefficient. So there exists an optimal solution to the above problem where for all and for all Therefore we can remove these variables from the problem. The simplified formulation is as follows:

We do not know if this problem can be simplified further. For any for all

5.4

inequality (5.83) is valid when is no longer fixed to 1, and for all

Conclusion

In this chapter, we studied the linear capacitated versions of concentrator location problems studied in Chapter 4. The capacity constraints impose that the sum of demands of nodes assigned to a certain concentrator cannot exceed its capacity. For the concentrator location problem with no routing cost LCL, we showed that facet defining inequalities of the knapsack polytope can be lifted to define facets of the LCL polytope. Then we investigated the conditions under which the facet defining inequalities of the UCL polytope remain facet defining for the LCL polytope. There are cases where these inequalities define facets of the LCL polytope when the values of some of the variables are fixed to

Problems with Linear Capacity Constraints

143

0. For those inequalities, we also discussed how we can lift the variables whose values are fixed. Finally, we modified the valid inequalities of the capacitated facility location problem polytope to be valid for the LCL polytope. For the two problems with routing costs, LCL-S and LCL-C, we characterized the facet defining inequalities involving only the assignment variables in terms of the facet defining inequalities of the LCL. We discussed the properties of facet defining inequalities involving only traffic variables. Then we presented several families of valid inequalities which give lower bounds on the traffic variables. These lower bounds are computed using the capacity restrictions. A summary of the polyhedral results is given at the end of this part.

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Chapter 6 PROBLEMS WITH QUADRATIC CAPACITY CONSTRAINTS

In this chapter we study the concentrator location problems with quadratic capacity constraints. The capacity constraints impose that the sum of the traffic on the links adjacent to a concentrator cannot exceed its capacity. This traffic is equal to the sum of the traffic adjacent at nodes assigned to the concentrator and the sum of the traffic between the nodes that are assigned to the concentrator and the nodes that are not. The results given here are often simple modifications of the results in the previous chapter and the proofs given there also remain valid for the quadratic case. So this chapter is mostly a collection of these modified results given without proofs. Recall that K is the set of commodities (or the set of directed pairs of nodes), i.e., for all and for all In this chapter, we assume that for all We also assume that all nodes cannot be assigned to a single concentrator and that any two nodes can be assigned together. We first investigate the polytope defined by a single knapsack constraint which is quadratic. This gives some insight about how the results for the linear case can be modified for the quadratic case. The results in this chapter are from Labbé and Yaman [50] and Labbé et al. [53].

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6.1

Quadratic Knapsack Constraint

In the concentrator location problems with quadratic capacity constraints (see Chapter 3), the capacity constraint is of the following form:

Different from linear knapsack, this quadratic knapsack inequality involves also the items that are not put in the knapsack. We first present five linearizations of the quadratic knapsack constraint (6.1). Then we identify a family of facet defining inequalities for the corresponding polytope.

6.1.1

Linearizations

Define satisfies (6.1)}. linearizations are based on linear knapsack constraints.

The first three

Linearizations by Knapsack Constraints PROPOSITION 6.1 (Labbé and Yaman [50]) The point if and only if it satisfies

is in

for all Proof: Given in if and only if

define

Then the point

is

The inequality (6.2) defined by I* is equivalent to the above inequality. So to prove the proposition we should show that I* maximizes the left hand side of inequality (6.2). Let The left hand side of inequality (6.2) for reads:

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Problems with Quadratic Capacity Constraints

So set I* maximizes the left hand side of inequality (6.2). Therefore if satisfies inequality (6.2) for I* it satisfies it for all PROPOSITION 6.2 (Labbé and Yaman [50]) The point if and only if it satisfies

is in

for all Proof: Given

define and The left hand side of inequality (6.3) for F* is:

Clearly if and only if it satisfies inequality (6.3) for F*. As F* maximizes the left hand side of inequality (6.3), the other sets define redundant inequalities. So we can say that if and only if it satisfies inequality (6.3) for each The third linearization is based on the results in Balas and Mazzola [6]. Define for all PROPOSITION 6.3 (Labbé and Yaman [50]) The point if and only if it satisfies

for all

such that

Proof: Given define and otherwise for all pairs

is in

for all and Then

if

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So if and only if it satisfies inequality (6.4) for As maximizes the left hand side of inequality (6.4), if and only if it satisfies inequality (6.4) for all Let denote the set of feasible solutions for the LP relaxation of the linearization. We have the following result. PROPOSITION 6.4 (Labbé and Yaman [50]) Proof: We first prove that Then

Given

let

So for a given there exists a set which defines the same inequality as For a given F, we can take to be if and to be otherwise. Following the equations above in the reverse order, we can show that defines the same inequality as F. So Now we show that Given Inequality (6.2) defined by defined by F. This shows that It is likely that

define is the same as inequality (6.3)

Consider the following example:

EXAMPLE 6.1 Consider I = {1,2,3}, M = 39, for all and for So we have the following constraint:-

Consider the fractional solution This solution is not in F={(3,l),(3,2),(2,l)}as

where and since inequality (6.3) is violated for

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Problems with Quadratic Capacity Constraints

However

satisfies inequalities (6.2) for all

The second and third linearizations are also advantageous compared to the first one in terms of separation. The separation problem for inequalities (6.2) is a specific max cut problem defined on a complete graph on the node set I, where the profit of arc is However, inequalities (6.3) and (6.4) can be separated in time by examining all pairs Linearizations by Cover Inequalities The next two linearizations are based on cover inequalities. define a “quadratic cover”.

We first

DEFINITION 6.1 (Labbé and Yaman [50]) A subset such that is called a quadratic cover. A quadratic cover C is called a minimal quadratic cover if no proper subset of C is a quadratic cover. PROPOSITION 6.5 (Labbé and Yaman [50]) If then the quadratic cover inequality

is a quadratic cover,

is valid for Proof: For a vector i.e.,

let

and assume that

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So set

is also a quadratic cover and vector

is not feasible.

PROPOSITION 6.6 (Labbé and Yaman [50]) The point if and only if it satisfies

for all

such that C is a minimal quadratic cover.

PROPOSITION 6.7 (Labbé and Yaman [50]) The point if and only if it satisfies

for all

is in

is in

such that C is a minimal cover for a knapsack constraint

for some Let and denote the set of feasible solutions of the LP relaxations of the last two linearizations. PROPOSITION 6.8 (Labbé and Yaman [50]) Proof: Assume we are given a vector and we would like to check if there exists a cover C for a knapsack constraint (6.7) defined by some for which the cover inequality is violated. Then we solve the following problem:

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Problems with Quadratic Capacity Constraints

where 1 if as

since to

if and

and

and 0 otherwise. If is given, we take to be and 0 otherwise. Then the constraint can be rewritten

for all

Then the problem is equivalent

This is exactly the problem we would solve to find a violated quadratic cover inequality. This shows that if and only if So We cannot compare and It is well known that there are fractional solutions satisfied by the knapsack constraint which violate cover inequalities. So is not included in In general is not included in Consider the following example: EXAMPLE 6.2 (Labbé and Yaman [50]) Let I = {1,.., 5}, M = 80, for all and for Consider the fractional solution where and for This solution is not in since inequality (6.3) is violated for F = {(1,2), (1,3), (1,4), (1,5)} as

If we consider the separation problem discussed in the proof of Proposition 6.8, we see that no cover inequality is violated since for any two items and we have and no one item subset is a cover. So satisfies all inequalities (6.6).

6.1.2

Quadratic Knapsack Polytope

Define The set is called the quadratic knapsack polytope. We assume that for all Then the polytope has dimension and the inequalities for are the trivial facets of As for all we have

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an independence system. This implies that any non trivial facet defining inequality for has the form with and (see Nemhauser and Wolsey [65]). The cover inequalities define facets of the linear knapsack polytope when some of the variables are fixed to zero (see Balas [5], Hammer et al. [42] and Wolsey [80]). Here we have a very similar result. PROPOSITION 6.9 (Labbé and Yaman[50]) If cover, then the quadratic cover inequality

is a minimal quadratic

is facet defining for In the following section we discuss the separation and lifting problems.

6.1.3

Separation and Lifting for Quadratic Cover Inequalities

To find a quadratic cover inequality that is violated by a fractional solution we solve the following problem:

If then gives a violated quadratic cover inequality. The above problem can be written as a maximization problem. Define for all

We propose a branch and bound algorithm (Algorithm 6.1) to solve this maximization problem. The idea is due to Caprara [15] and the algorithm is a

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Problems with Quadratic Capacity Constraints

modification of the algorithm given in Caprara et al. [16]. Let N be the set of items and the problem:

where

for all

Suppose we would like to solve

and

for all

In Algorithm 6.1, denotes the cost of the current solution and is the amount of capacity already used by The lower bound on the optimal value is denoted by the upper bound on the optimal value at a given node is denoted by and the best known solution is denoted by We initialize all entries of and to be 0. We call this recursive algorithm by quadsep(0, 0, 1) (which implies that and are initialized to be 0). Given a minimal quadratic cover C, the quadratic cover inequality is facet defining for We apply sequential lifting to the variables with Suppose that we are lifting the variable and let be the set of variables lifted before Coefficient denotes the lifting coefficient of We solve the following problem to compute the lifting coefficient:

We propose a branch and bound algorithm (Algorithm 6.2) to solve this problem. This algorithm is very similar to the algorithm “quadsep”. However, in this case we are able to compute better upper bounds since the linear terms

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

are dominant. Suppose the problem has this form:

where

for all for all

for all

and

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Problems with Quadratic Capacity Constraints

We initialize to be 0 and all entries of algorithm by quadlift(0, 0, 1).

and

to be 0. We call the

The inequality

is valid for for all

if

for all

and is facet defining for

if

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6.2

Quadratic Capacitated Concentrator Location Problem

In this section, we study the polytope corresponding to the quadratic capacitated concentrator location problem (QCL). We can formulate the QCL as follows:

We can also give a formulation without using the variables project out these variables, we obtain:

Let Define

If we

satisfies (6.11) and (6.12)} and for all

PROPOSITION 6.10 (Labbé and Yaman [50]) The QCL polytope dimensional.

is full

Proof: See the proof of Proposition 5.1. PROPOSITION 6.11 (Labbé and Yaman [50]) The nonnegativity constraint defines a facet of for each Proof: See the proof of Proposition 5.2.

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Problems with Quadratic Capacity Constraints

If and then similar to the case of LCL poly tope, the polytope is defined by the inequality and the nonnegativity constraints. So in the remaining part of this section, we assume that PROPOSITION 6.12 (Labbé and Yaman [50]) The inequality (6.11) defines a facet of if and only if is not a quadratic cover for all Proof: See the proof of Proposition 5.3. PROPOSITION 6.13

and Yaman [50]) If the inequality is a valid inequality for the quadratic knapsack

(Labbé

polytope then the inequality

is valid for Proof: See the proof of Proposition 5.4. THEOREM 6.2 (Labbé and Yaman [50]) If the inequality is a facet defining inequality for the polytope then inequality (6.14) is facet defining for Proof: See the proof of Theorem 5.1. THEOREM 6.3 (Labbé and Yaman [50]) Let

with such that for each Define also

there exists exactly one such that there is no such that Let Then the W – 2 inequality

defines a facet of

where

and is not a quadratic cover for all Proof: See the proof of Theorem 5.2.

if

and

and

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If we lift first the variables result:

with

in H, then we have the following

PROPOSITION 6.14 (Labbé and Yaman [50]) Assume that the variables with are not yet lifted or the ones that are lifted have coefficient 0. Then for a given the coefficient is 1 if there does not exist a set such that is not a quadratic cover and the nodes in can be assigned to nodes in I \ W and is 0 otherwise. Proof: See the proof of Proposition 5.7. If we lift first the variables following result:

with

in

then we can prove the

PROPOSITION 6.15 (Labbé and Yaman [50]) Assume that the variables with are not yet lifted or the ones that are lifted have coefficient 0. Then given the coefficient is 1 if is a quadratic cover for all and is a quadratic cover for all such that and and is 0 otherwise. Proof: See the proof of Proposition 5.8. THEOREM 6.4 (Labbé and Yaman [50]) Consider the graph where with for some We number the nodes in from 1 to The arc set consists of all arcs for and all arcs of the form where and are both odd and Then the inequality:

is facet defining for

1

if is not a quadratic cover for all

2

is not a quadratic cover for all such that is not a quadratic cover for all

3 such that

4

is not a quadratic cover for all such that

5

is not a quadratic cover for all

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Problems with Quadratic Capacity Constraints

Proof: See the proof of Theorem 5.3. THEOREM 6.5 (Labbé and Yaman[50])Given an arc with the k-leaf inequality

is facet defining for if and only if quadratic covers.

and

and a subset

for all

are not

Proof: See the proof of Theorem 5.4. PROPOSITION 6.16 (Labbé and Yaman [50]) Define is not a quadratic cover} and Suppose that we are first lifting the variables such that then and then the variables such that

1 The coefficients

for all

2 The coefficient is not a quadratic cover}.

where

with define to be the set of indices 3 For the variable of variables that are lifted before The lifting coefficient of is where

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

and

Proof: See the proof of Proposition 5.9. THEOREM 6.6 (Labbé and Yaman [50]) Consider a subset with and a node Let C be a directed cycle on the nodes of D. Renumber the nodes such that D = {1,2,3}, the cycle is 1, 2, 3, 1, the node and If {1,.., 4} is not a quadratic cover and is not a quadratic cover for all such that and for all then

1 the 2-cycle inequality

defines a facet of

2 Let and be the node in D such that Renumber the nodes such that and Then the 2-cycle inequality

defines a facet of

Proof: See the proof of Theorem 4.6. Now we present a family of facet defining inequalities which forms a subset of modified binpacking inequalities presented for

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Problems with Quadratic Capacity Constraints

PROPOSITION 6.17 (Labbé and Yaman [50]) Let If is a quadratic cover for all with then the quadratic binpacking inequality

is valid for PQ . Proof: The following quadratic cover inequalities are valid:

If we sum these inequalities over all

we get

which simplifies to

If we divide both sides by obtain inequality (6.15).

and round down the right hand side, we

THEOREM 6.7 (Labbé and Yaman [50]) Let Define If is a quadratic cover for all with is not a quadratic cover for all with and then the quadratic binpacking inequality (6.15) is facet defining for Proof: Assume that is a quadratic cover for all is not a quadratic cover for all with Inequality (6.15) becomes:

and

with and

We use the indirect method to show that inequality (6.16) is facet defining for Consider an inequality

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

with to be the set of open Let Choose a subset into sets where nodes. We partition the set open node. Such a solution for all Assign the nodes in to the and it satisfies the inequality (6.16) at equality. Moreover, node is is in free in Let be the set of such solutions. Consider two nodes and that are not in and solution that is the same as except that As it satisfies inequality (6.16) at equality, we have that and

Consider the is also in and for all

Consider and Consider also a node Consider the two solutions and that are the same as except that and Both solutions are in and satisfy inequality (6.16) at equality. So for all and The inequality (6.17) becomes

For any that

consider an arc such that both and are in As nodes can be assigned together, there exists a solution such Consider the solution which is the same as except that and The solution is also in and it satisfies inequality (6.16) at equality. So As we can repeat the same argument for other nodes we can show that for all

Consider two nodes and in Consider a solution where node is open. Define to be the same as x except that we change node by node We have that for all Now, inequality (6.17) becomes

Then for some This proves that inequality (6.17) is an of the inequality (6.16). So inequality (6.16) is facet defining for

Consider a solution

Problems with Quadratic Capacity Constraints

6.3

163

Quadratic Capacitated Concentrator Location Problem with Star Routing

In this section we consider the concentrator location problem with routing on a star backbone network where the capacities are quadratic. We call this problem “Quadratic Capacitated Concentrator Location Problem with Star Routing” and abbreviate it by QCL-S. We can formulate the QCL-S as follows:

Now we give the formulation without using the variables

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

satisfies (6.24)-(6.28)}

Let

and P ROPOSITION 6.18 The polyhedron

has full dimension, i.e.,

Proof: See the proof of Proposition 4.8. THEOREM 6.8 The inequality it defines a facet of

defines a facet of

if and only if

Proof: See the proof of Theorem 4.7.

PROPOSITION 6.19 Let Then,

1

be a facet defining inequality for

and

2 Let

If there exists an

such that

then 3 If

4 For

then the inequality for all

defines a facet of

Proof: See the proof of Proposition 5.13. PROPOSITION 6.20 For

is valid for

where

the traffic bound inequality

is computed as

if and only if

Problems with Quadratic Capacity Constraints

165

Proof: See Proposition 5.14. PROPOSITION 6.21 For

is valid for

the traffic bound inequality

where

for each Proof: See Proposition 5.15.

6.4

Quadratic Capacitated Concentrator Location Problem with Complete Routing

This section is on the Quadratic Capacitated Concentrator Location Problem with Complete Routing (QCL-C). We basically modify the results presented for LCL-C. This problem can be formulated as:

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

The formulation without the variables

is as follows:

Let satisfies (6.36)-(6.40)} and PROPOSITION 6.22 (Labbé et al. [53]) The polyhedron dimension.

has full

Proof: See the proof of Proposition 4.8. THEOREM 6.9 (Labbé et al. [53]) The inequality if and only if it defines a facet of

defines a facet of

Proof: See the proof of Theorem 4.7. PROPOSITION 6.23 (Labbé et al. [53]) Let inequality for Then,

1

be a facet defining

and

2 Let either

such that or

If there exists an and for every then

and an

Problems with Quadratic Capacity Constraints

3 If

4 If

167

then

and

is not a quadratic cover for all defines a facet of

Proof: See the proof of Proposition 5.17. PROPOSITION 6.24 The inequality

is a valid inequality for

if and only if

is a valid inequality for Proof: See the proof of Proposition 4.13. PROPOSITION 6.25 For

are valid for

where

the traffic bound inequalities

and

are computed as follows:

then

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

and

Proof: See Proposition 5.14. PROPOSITION 6.26 (Labbé et al. [53]) For inequalities

are valid for

and

where

and

the traffic bound

are computed as follows:

Problems with Quadratic Capacity Constraints

169

Proof: See Proposition 5.15. Now we modify the strengthened projection inequalities for the case with quadratic capacities. Again, we use the variables for ease of presentation. Similar to the linear capacitated case, we consider the following structure:

where is the amount of traffic that originates at node and that travels on arc given the assignment vector Let be the set of points and that satisfy constraints (6.47) to (6.51) and If the inequality is valid for for each then the inequality is a valid inequality for THEOREM 6.10 (Labbé et al. [53]) The inequality

is valid for

where

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

and for any

such that

and

is the set of indices of the variables that have been lifted before for all where

and

and

for all

where

is the set of indices of the variables that have been lifted before

Proof: See the proof of Theorem 5.9. PROPOSITION 6.27 (Labbé et al. [53]) For any inequality (6.33) is dominated by the inequality

and

the

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Problems with Quadratic Capacity Constraints

where

for some

with

and

Proof: See the proof of Proposition 5.22.

6.5

Conclusion

In this chapter, we studied the concentrator location problems with quadratic capacity constraints. We first considered the polytope defined by a quadratic knapsack inequality. We modified the definition of a cover for the quadratic case. We presented two branch and bound algorithms, one to solve the separation problem and the other one to compute the lifting coefficients for quadratic cover inequalities. With this new definition of a cover, most of the results for the linear capacitated case remained valid for the quadratic case. This chapter gave a list of these results without proofs. The summary of polyhedral results presented in Chapters 4, 5 and 6 is provided next.

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Summary of Results of Part II

Here we summarize the polyhedral results presented in Chapters 4, 5 and 6. In Table 6.1, we see which families of inequalities are facet defining for the polytopes of concentrator location problems with no routing cost. We give the reference of the results stating the conditions under which inequalities define facets when necessary. In Tables 6.2 and 6.3, we summarize the polyhedral results for concentrator location problems with star and complete routing respectively.

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Summary of Results of Part II

175

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PART III

SOLVING QCL-C WITH BRANCH AND CUT

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Chapter 7 QCL-C: FORMULATIONS AND PROJECTION INEQUALITIES

In the following two chapters, we present a branch and cut algorithm for the QCL-C. In this chapter, we first compare several formulations to choose the one to use in the algorithm. We strengthen this formulation using projection inequalities. Then we also try to obtain stronger formulations keeping a subset of multicommodity flow variables. The results show that the formulation with the least number of variables is the most advantageous. Most of the results given in this chapter and the next one are from Labbé et al. [53].

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7.1

Comparing Formulations

This section is devoted to the comparison of our formulation with hub location, multicommodity flow and aggregated multicommodity flow formulations. We first present the formulation we are using in our branch and cut algorithm. This formulation is called QCL-C1 and it uses the assignment variables:

for all

and the traffic variables

which is the total traffic on arc

(QCL-C1)

We changed the quadratic capacity constraints to linear constraints using the traffic variables As for all in an optimal solution takes the minimum value implied. The inequalities (7.2) imply that In an optimal solution, the inequality is tight, implying the quadratic capacity constraints (6.32). Formulation QCL-C1 has variables and exponentially many constraints. When we start the branch and cut algorithm, we do not include constraints (7.2) in the LP relaxation. We add these inequalities whenever we find them to be violated by the optimal solution of the current LP relaxation. They can be separated as follows. For a given arc and a solution let If the set yields the most violated inequality.

QCL-C: Formulations and Projection Inequalities

181

Now we present the other formulations and show how they relate to QCLC1. For more information, refer to Chapter 2. We define the variables:

for all and Constraints (7.2) in formulation QCL-C1 can be obtained by projecting out the variables from the system:

In fact these are the only nonredundant projection inequalities of this system. The second formulation we consider is the hub location formulation. O’Kelly [67] gives a quadratic integer programming formulation for the uncapacitated single assignment hub location problem which can be easily modified for our problem. Campbell [12] and Skorin-Kapov et al. [75] present linearizations of O’Kelly’s formulation. We use the linearization of SkorinKapov et al. [75] to obtain the second formulation. This formulation is denoted by QCL-C2. To obtain QCL-C2, we replace constraints (7.2) in QCL-C1 by the following set of constraints:

We can easily verify that takes value 1 if and only if both and are equal to 1 and it takes value 0 otherwise. Formulation QCL-C2 has variables and constraints.

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

We can also obtain a multicommodity flow formulation using the variables as follows:

Let QCL-C2m denote this formulation. This formulation is valid since the cost for routing the traffic in the backbone network satisfies the triangle inequality. So the traffic travels from one concentrator to the other directly without visiting other nodes. This also implies that the LP relaxations of QCL-C2 and QCL-C2m have the same value. Ernst and Krishnamoorthy [32] suggest a formulation with a smaller number of variables for the capacitated single assignment hub location problem. They aggregate the commodities by their origins to decrease the number of flow variables. To give this aggregated multicommodity flow formulation that we call QCL-C3, we define to be the flow of commodities originating at node and traveling along the arc for all and Formulation QCLC3 can be obtained by replacing constraints (7.2) in QCL-C1 by the following set of constraints:

Formulation QCL-C3 has variables and constraints. Obviously the linear programming relaxation of QCL-C2 is stronger than the one of QCL-C1 and the one of QCL-C3. However, the comparison between QCLC1 and QCL-C3 is not so clear. One is tempted to conclude that the LP relaxation of formulation QCL-C1 is weaker than the one of QCL-C3. Though the following example shows that it is not the case. EXAMPLE 7.1 (Labbé et al. [53]) Suppose we have five nodes and the only commodities with nonzero traffic are from node 1 to nodes 2 and 3 with one unit of traffic. Consider the vector where

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QCL-C: Formulations and Projection Inequalities

As there is no flow on the arcs adjacent at nodes 1,2 and 3. The flow conservation constraints for nodes 4 and 5 are:

So the vector QCL-C3.

where

is feasible for the LP relaxation of

However is not feasible for the LP relaxation of QCL-C1 since constraint (7.2) for arc (4, 5) and F = {(1, 2)} implies that

and constraint (7.2) for arc (5, 4) and F = {(1, 3)} implies that

Now consider the vector

where

As and for all where is feasible for the LP relaxation of QCL-C1. However the flow conservation constraint for node 1 is:

which implies that any solution relaxation of QCL-C3.

with

is infeasible for the LP

If we use the formulations with the strongest LP relaxation, i.e., QCL-C2 or QCL-C2m, we spend a lot of time to solve the LP’s since they have variables and constraints. The question is whether we gain enough from the strengthening of the formulation. In Tables 7.1 and 7.2, we present the results obtained from the four formulations for a small sample of test problems. We have seven problems with 10 nodes, six problems with 12 nodes and five problems with 15 nodes generated from the same cost and traffic data. We keep M = 10 and multiply the traffic values by a coefficient Q to have different types of problems with respect to the tightness of capacities and the effect of the routing cost. We take Q to be 1, 2/3, 1/2, 1/3, 1/4, 1/5 and 1/6. This set of problems is used in the remainder of this chapter and in the next chapter to decide about different components of the branch and cut algorithm.

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To compare the formulations presented above, we use a branch and cut algorithm which is described in detail in Section 8.3. We should note that for formulation QCL-C1, the only valid inequalities separated are constraints (7.2). For the other formulations, we have a pure branch and bound algorithm. The problems with Q = 1 were infeasible for and and the problem with Q = 2/3 was infeasible for so we do not include these problems in the results. Table 7.1 gives the duality gap at the root node of the branch and bound tree (computed using the optimal solution). In Table 7.2, the columns “total” and “first LP” give the CPU time in seconds to solve the problem and the CPU time to solve the first LP relaxation. We set a time limit of one hour. If we cannot solve the problem to optimality within one hour of CPU time, then we write “timelimit” in the column time. If we run out of memory, then we write “memory”. We observe the following in Tables 7.1 and 7.2. The duality gap decreases as Q decreases, independently of which formulation we choose. This is because the QCL-C approaches the uncapacitated concentrator location problem as Q tends to 0.

QCL-C: Formulations and Projection Inequalities

185

The duality gap is very large when we use formulation QCL-C1. It goes up to 72 percent for the problem with and Q = 1. As increases, it becomes harder to close the gap. As a result, we were unable to solve any problem with 15 nodes using formulation QCL-C1. Formulations QCL-C2 and QCL-C2m have the same duality gap. However, there is a big difference (a factor of two) between these formulations when we look at the CPU time. These formulations have the disadvantage that it takes much more time to solve the LP’s. We report the time to solve the first LP for QCL-C2m. We see that it goes from 4 seconds to 3 minutes when we go from 10 nodes to 15 nodes. For QCL-C2m we also tried to solve the first LP using the barrier method with and without crossover as Fortz [33] suggested. The results improved considerably (it became around one minute for problems with 15 nodes). However, when we used the barrier methods to solve all LP’s, the total CPU time increased. The problem with and Q = 1/4 was not solved in one hour. If we only solve the first LP by barrier and then switch to the dual

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

simplex, the comparison would not change in general. Formulation QCL-C3 performs better in the average. Still there are six problems that are not solved to optimality because we ran out of memory. This is basically because the tree grows faster compared to the two other formulations since we do not add any cuts and the time to solve the LP’s is shorter compared to QCL-C2m. Still, for the problems that are solved to optimality, this formulation is often the fastest. To sum up we can say that formulations QCL-C2 and QCL-C2m do not seem promising to solve big problems as the time to solve the LP relaxations grows very fast. However, the other two formulations do not perform much better as they have larger duality gaps and it takes a lot of time to close this gap. It is clear that if we would like to use one of these two formulations, we should strengthen it. Below, we discuss how we can do this.

7.2

Projection Inequalities

In Chapter 4, we obtained valid inequalities for the UCL-C by projecting a higher dimensional polyhedron and we called these inequalities “projection inequalities”. Given a solution to check whether there exists a violated projection inequality (4.70) amounts to solving a linear programming problem with variables and constraints. If we solve the separation problem exactly, then we can obtain the same duality gap as QCL-C2 using the formulation QCL-C1 or QCL-C3 and adding the projection inequalities. However, this does not make much sense since the size of the corresponding LP is about the same as that of the LP relaxation of QCL-C2. In consequence, we use heuristics to separate some specific families of projection inequalities. We focus on the projection inequalities that have the following form:

where S and T are disjoint subsets of the node set I and Even though we do not know how to separate exactly this specific family of projection inequalities, we can show that the separation is easy if we are given the sets S and T. For given sets S and T and a solution we choose

Let be the current fractional solution. separation algorithm for the projection inequalities.

Algorithm 7.1 is the

QCL-C: Formulations and Projection Inequalities

187

In Table 7.3, we present the results obtained by using the projection inequalities with formulations QCL-C1 and QCL-C3. For the remainder, we use the following abbreviations: gap: the duality gap at the root node, i.e., where is the lower bound at the root node and is the best upper bound known for that problem. time: the total CPU time in seconds Here we see that formulation QCL-C1 with the projection inequalities outperforms the other formulations. The duality gap is usually close to the one of formulation QCL-C3 with the projection inequalities and we gain a lot from the CPU time by using QCL-C1 rather than QCL-C3. We also observe that we do not have the problem of memory with formulation QCL-C3 when we use the projection inequalities. However, the use of projection inequalities slows down the program and it performs worse than before for the problems that could be solved in both cases.

7.3

Semi-extended Formulations

The hub location formulation can be seen as an extended formulation for QCL-C. This formulation is stronger than the other formulations presented in this chapter but it has a large number of variables and constraints. The idea of a semi-extended formulation is to keep some of these variables and constraints and have a relaxation of the extended formulation which has a dual bound

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

close to the one of the extended formulation (Pochet [73], Van Vyve [79]). We present two ways of obtaining semi-extended formulations. The results are based on ideas by Pochet [73] and Van Vyve [79]. To derive the first semi-extended formulation, we present a new formulation which has a structure that allows us to remove some of the flow variables. Consider the following set of constraints:

QCL-C: Formulations and Projection Inequalities

189

If we replace the inequalities (7.2) in QCL-C1 by the above set of constraints then we obtain a valid formulation for QCL-C. Let QCL-C4 denote this formulation. PROPOSITION 7.1 The LP relaxation of formulation QCL-C4 has the same optimal value as the LP relaxation of formulation QCL-C2. Proof: Let QCL-C2. Clearly, (7.21). For a given

be feasible for the LP relaxation of formulation satisfies constraints (7.17), (7.18), (7.20) and as for all we have So also satisfies constraints (7.19) and is feasible for the LP relaxation of formulation QCL-C4. Let be feasible for the LP relaxation of formulation QCL-C4. Assume that there exists a and such that Then So cannot be feasible for the LP relaxation of formulation QCL-C4. This shows that for all and We can similarly show that for all and So satisfies constraints (7.6) and (7.7). As constraints (7.8) and (7.9) are also satisfied by we can conclude that is feasible for the LP relaxation of formulation QCL-C2. Let and be subsets of the node set I for each Then by replacing the inequalities (7.17)-(7.21) by the following system of constraints, we get a valid formulation of the QCL-C:

The LP relaxation of the new formulation is a relaxation of the LP relaxation of QCL-C4. When for all we obtain formulation QCL-C4 and when for all we obtain formulation QCL-C1. If we choose a fixed upper bound U on the size of the sets and

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so that variables.

and

for all

then the formulation has

The semi-extended formulation can also be strengthened by projection inequalities. In this case, we consider formulation QCL-C4 and project out the variables for or for all We call these inequalities “semi-projection inequalities”. PROPOSITION 7.2 The inequality

is valid for

such that

Proof: Associate dual variables and to constraints (7.17)-(7.20) respectively. By Farkas’ Lemma we obtain the result. If we try to find semi-projection inequalities of the form (7.16), then we can consider the following Choose two disjoint subsets S and T of I and a subset F of the commodity set K. Let if and if and if if and and set the rest of the components at 0. This choice of satisfies inequalities (7.28) and (7.29). So inequality (7.27) defined by is a valid inequality. COROLLARY 7.1 Let S and T be disjoint subsets of I and F be a subset of K. Then the semi-projection inequality

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and Q = 2/3 and

and Q = 1/2 respectively.

The demands can be used to obtain a relaxation of the QCL-C which is a LCL-C. As we can expect so that such a relaxation is stronger than a relaxation obtained using demands This relaxation is used often to avoid solving quadratic knapsack problems in separation and lifting of various inequalities.

8.2

Other Valid Inequalities and Separation Algorithms

This section is about the valid inequalities used in the branch and cut algorithm and the corresponding separation procedures. Let denote the current solution.

Branch and Cut Algorithm for QCL-C

8.2.1

201

W-2 Inequalities

We separate the W-2 inequalities using an algorithm similar to the one described in Avella and Sassano [4]. We do not lift the variables with since the computation of the lifting coefficients amounts to solving generalized assignment problems. However, the computation of lifting coefficients of with and is equivalent to checking conditions of Proposition 6.15. As this can be done in time, we lift the variables with and For a given we do the lifting in decreasing order of because of the second condition. The separation algorithm is given in Algorithm 8.3.

8.2.2

k-leaf Inequalities

The k-leaf inequalities are separated exactly in time as described in the proof of Proposition 4.6. Then the variables with are lifted by solving linear knapsack problems (using demands We add the most violated inequality for each

202

8.2.3

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

k-triangle Inequalities

As the separation problem for the k-triangle inequalities is NP-complete, to separate these inequalities, we developed a simple greedy heuristic (Algorithm 8.4).

8.2.4

2-cycle Inequalities

We separate the 2-cycle inequalities of the form (4.24). The separation algorithm is as in Algorithm 8.5.

8.2.5

Odd Hole Inequalities

As the UCL is a stable set problem, the inequalities valid for the stable set polytope are valid for the UCL polytope and therefore valid for QCL-C. We use the odd hole inequalities in our branch and cut algorithm. If O is the vertex set of an odd hole in the conflict graph, the odd hole inequality is valid. These inequalities can be separated in polynomial

Branch and Cut Algorithm for QCL-C

203

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CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

time by solving a series of shortest path problems on a bipartite graph (see Grötschel et al [38]). The bipartite graph B has A as one side of the node set and a copy of A that we call as the other side. Let be the node set of B. The arc set and or or and or or If then its weight is which is nonnegative. A path from in B is a cycle of odd length passing through node on the conflict graph. Let P be the shortest over all of the shortest paths. Let denote the nodes on P. Then the length of P is equal to There exists a violated odd hole inequality if and only if The procedure to separate the odd hole inequalities consists of defining the bipartite graph B and then solving a shortest path problem for each Then we take the shortest of these paths to be P. If then we have the most violated odd hole inequality.

8.2.6

Cover Inequalities

To separate the quadratic cover inequalities, we use the algorithm given in Section 6.1. For a given we define and We fix for all and for all So usually the separation problem has a size smaller than the size of the original problem. If we find a cover C, we also fix for We lift first the variables whose values are fixed to 0 and who have If the resulting inequality is violated, then we lift the variables whose values are fixed to 1 except After, we lift the remaining variables whose values are fixed to 0. The last variable we lift is This is similar to the order given by Gu et al. [39]. We can also separate cover inequalities on linear knapsack constraints which give relaxations of the quadratic constraint. One such linear constraint comes from the second linearization of the quadratic knapsack constraint (see Proposition 6.2). In this case, we choose a set F to obtain a linear knapsack constraint. We take Then we separate and lift the cover inequality on the knapsack inequality (6.3) defined by F. Another possibility is to consider the inequality where are computed during preprocessing. In general these inequalities are different from the first set of linear knapsack constraints. However we are not able to compare their strengths. Still, we can say that as the first set of linear

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knapsack inequalities have demands that are based on the actual assignment vector and the associated flow in the backbone network, these inequalities are likely to be more useful for problems which do not have very tight capacities. To find violated lifted cover inequalities, we use the algorithm given by Gu et al. [39]. For each node we first look for a violated lifted cover inequality based on the first linear knapsack. If we find one, we add it to the formulation and pass to Otherwise, we try to find one with the demands We do it sequentially to avoid adding the same inequality twice.

8.2.7

Binpacking and Residual Capacity Inequalities

The separation problems for binpacking and residual capacity inequalities involve the choice of two subsets, and J. We developed a heuristic to separate a subset of these inequalities. This heuristic is based on the following observations. If the set is given then the binpacking inequality is

The set of the inequality. Notice that

maximizes the left hand side

If we are given the set J, then we can rewrite the binpacking inequality as:

Finding the set which maximizes the left hand side of the inequality is not easy. A heuristic approach is to take The separation algorithm for the binpacking and residual capacity inequalities is as follows. We consider all sets of the form for some such that For a given the binpacking inequality becomes

We take We evaluate the binpacking inequality for this choice of and J and add it to the formulation if it is violated. Otherwise we check whether the residual capacity inequality defined by the same sets and J is violated.

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8.2.8

Quadratic Binpacking Inequalities

We separate quadratic binpacking inequalities on three nodes by enumeration. We consider each triple If the triple forms a quadratic cover, we check if the inequality is violated. We add all such violated inequalities to the formulation.

8.2.9

Effective Capacity Inequalities

We separate these inequalities as described in Aardal [1], though we do not look for the P-depots structures (see Algorithm 8.6).

8.2.10

Strengthened Projection Inequalities

We separate the strengthened projection inequalities as follows. For a given for each node we take and We also pick the sets and We fix the values of for all and for all We compute for each node and then lift the variables in the following order: and Then we sum up the values of to compute a bound on the total traffic. If the bound exceeds the current value of the traffic, we introduce the violated inequality to the formulation.

8.2.11

Step Inequalities

We do not know any algorithm to solve the separation problem exactly for step inequalities. So we adopted the following simple heuristic. For a given arc we take and evaluate the inequality for this choice of We add the violated inequalities to the formulation.

8.3

Branch and Cut Algorithm and Computational Results

In this section, we present the basic parts of our branch and cut algorithm and discuss the computational results. The branch and cut algorithm is implemented in C++ using ABACUS 2.3 (see and Thienel [45]) and the LP solver CPLEX 7.0. The runs are taken on an Intel Pentium III, 1 GHz, 1 GB RAM running under Suse 7.2. We start with the LP relaxation that contains constraints (7.1), (7.3) and for all and for all We run the preprocessing and strengthening algorithm. We delete the variables whose values are fixed to 0 or 1 from the formulation and add the valid inequalities generated by Algorithm 8.1. Then we introduce the formulation to ABACUS.

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8.3.1

207

Primal Heuristic

The solution of the current LP relaxation is feasible if and satisfies constraints (7.2). If is integer but satisfy constraints (7.2), we compute a feasible as follows:

for all an integer

is integer does not

If is not integer, we apply a rounding heuristic to obtain that satisfies capacity constraints (7.3). If we can find such an

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we compute Algorithm 8.7.

as described above. The rounding heuristic is given in

If we obtain a feasible solution whose cost is less than the current upper bound, then we update the upper bound and the best solution.

8.3.2

Branching and Enumeration Strategies

We consider two branching strategies. The first strategy is to branch on the most fractional variable, i.e., we branch on if If all are integer then we branch on the most fractional variable The reason for giving a priority to variables is the following. Fixing means to fix for all and fixing means to fix for all So the two branches have the same number of variables fixed. However for when we fix we have for all and for all and when we fix we cannot fix the values of other variables. So on one branch we have variables whose values are fixed and on the other branch we have only one variable whose value is fixed. This is why we can expect to have a more balanced tree by branching on the variables first. The second strategy is to branch on the assignment constraints (7.1). We take the inequality for node which has the minimum value. Then in one branch we fix to 1, and in the other branch we fix to 1. Three enumeration strategies are tested: breadth first, depth first and best first. Now we present the results for the two branching strategies and three enumeration strategies. We are separating the quadratic cover inequalities, the strengthened projection inequalities and the step inequalities. For this analysis, we excluded the problems with 10 nodes, since all these problems were solved in less than 5 seconds after preprocessing. We added problems with 16 nodes with Q = 1/2,1/3,1/4,1/5,1/6. The results using the first and second branching strategies coupled with the three enumeration strategies are presented in Table 8.2. The column nodes gives the average number of nodes in the branch and cut tree and the column time gives the average CPU time in seconds. The first row marked by “var” stands for the first branching strategy and the second one marked by “sos”

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stands for the second branching strategy. We observe that for the first branching strategy, there is not a big difference between the results obtained using different enumeration strategies though the best first strategy seems to be slightly better. For the second branching strategy, depth first search performs worse than the other ones. The difference between the two branching strategies is also clear. The CPU time for the second strategy is three to four times the CPU time for the first strategy. Based on these results, we decided to use the first branching strategy and the best first enumeration strategy in our branch and cut algorithm.

8.3.3

Useful Cuts

Now we would like to see which cuts are useful to close the duality gap and to decrease the CPU time. To do this, we conduct the following test. Initially we separate only the projection inequalities. Then we add each family of cuts to the existing cuts in the following order: 1 Lifted quadratic cover inequalities (cover) 2 Lifted strengthened projection inequalities (spro) 3 Step inequalities (step) 4 Quadratic binpacking inequalities on three nodes (quadbin) 5 Binpacking and residual capacity inequalities (binres) 6 Effective capacity inequalities (effcap) 7 Lifted W – 2 inequalities (w-2) 8 k-triangle inequalities (k-tri) 9 Lifted k-leaf inequalities (k-leaf) 10 2-cycle inequalities (2-cycle) 11 Odd hole inequalities on the conflict graph (odd)

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For each family of cut, the name in the parenthesis refers to the name of its row in the table. We excluded the problems with 12 nodes and the problem with 16 nodes and Q = 1/2 as they were solved in a short time. In Table 8.3, we report the average number of violated inequalities (column no. of violated ineq.s), average percentage improvement in the duality gap (column % imp. in gap), and in the CPU time (column % imp. in time) for each family of cuts (detailed results are given in Appendix B.1). In parenthesis, we report also the average gap and the average CPU time. Note that the percentage improvements are the averages of the percentage improvements taken over all problems, not the percentage improvement computed on the average values. For odd hole inequalities, we did not use the problems with 16 nodes as it takes too long to solve them. We can see that the cover inequalities and the strengthened projection inequalities are the most useful cuts. The step inequalities are also useful for problems with large Q values. We also observe that even if they seem to be useful to reduce the duality gap, the odd hole inequalities take too long to separate. The separation time can be decreased by changing the algorithm to look for a violated odd hole inequality rather than the most violated one. However as these inequalities are not violated often, the results do not improve significantly. So we remove the odd hole inequalities from the branch and cut algorithm.

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It is clear that the cover inequalities and the strengthened projection inequalities are useful and should be kept in the branch and cut algorithm. We decided to keep also the step inequalities as they are mostly violated for problems with large Q values and as it does not take much time to separate them. To decide for the rest of the inequalities, we conduct the following test. The branch and cut algorithm contains the first three families of inequalities. Then we add each family separately to see if it improves the results. The results are presented in Table 8.4 (see Appendix B.2 for detailed results). We observe that none of the families of inequalities seem very useful to improve the results. The k-triangle inequalities improve the CPU time but mainly for problems with small Q values. The detailed results show that the W – 2 inequalities improve the results for some of the problems and behave very badly for some others. They are also mostly useful for problems with small Q values. As the hard problems are the ones with large Q values and as the separation for k-triangle and W – 2 inequalities are based on enumeration of triples, so they can become time consuming for large we do not keep these inequalities in the branch and cut algorithm. So we remove the inequalities except cover, strengthened projection and step inequalities from the branch and cut algorithm. We should still be careful about the cover inequalities. As the separation and lifting procedures for cover inequalities require the solution of quadratic knapsack problems, it can be useful to switch to procedures which approximate the quadratic knapsacks by linear knapsacks, as the problem size grows bigger. We discuss this further when we present the computational results.

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8.3.4

213

Tailing Off

We observed that the improvement in the lower bound decreases as the number of LP’s solved at a certain node increases. As the number of LP’s solved increases, the violated inequalities are usually projection inequalities and the violation is not big. So it can be more efficient to branch rather than adding new violated cuts and reoptimizing. We apply the following rule. If the improvement over the five consecutive LP’s is less than 0.05% then we branch.

8.3.5

Test Instances

We have two sets of data from France Telecom, the first one has 17 nodes and the second one has 22 nodes. Using the traffic and cost data of these two sets and changing the concentrator capacities M and the traffic demand level Q, we obtained different problems. Each problem name has the form f t, M, Q where is the number of nodes, M is the capacity of a concentrator and Q is the demand level. We only consider the feasible problems, since infeasibility is reported during preprocessing. In these problems, the traffic matrix T and the routing cost vector B are symmetric. So we remove half of the commodities and the links. We take the fixed demand of each terminal to be In the problems with 17 nodes, all commodities have positive traffic demand. This is not the case for the problems with 22 nodes. As a result, we see that the problems with 22 nodes are easier than the problems with 17 nodes in general. We also generated several problems using the AP data set for hub location problems from the OR Library (see Beasley [8]). These problems have 10, 20, 25, 40 and 50 nodes. There are four problems of each kind, differing in the fixed cost for installing concentrators and the capacity of concentrators. The letter “L” stands for loose and “T” stands for tight. We modified the data to fit to our program (we took M to be the average of the concentrator capacities, we made the traffic and cost data symmetric and removed the commodities from a node to itself) and we created new problems changing the value Q. We take Q = 1,1/2,1/3,1/4,1/5. Each one of these problems is called Q where C and K are either L or T, denoting the type of cost and capacity, respectively.

8.3.6

Computational Results

In Tables 8.5-8.9, we present the results for the test instances described above. We report the duality gap at the root node (column gap), the number of nodes in the branch and cut tree (column nodes) and the CPU time for each

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problem (column time). We have a time limit of four hours. If the problem is not solved to optimality because of the time limit then we write “time”. If we run out of memory then we write “memory”. In both cases we also report the final gap if a feasible solution is found. For each problem we give the number of concentrators installed in the optimal solution to give an idea about the tightness of capacities. We also report the number of violated inequalities of each family. For the remainder, we use the following abbreviations: conc: number of concentrators installed in the optimal solution cover: number of violated cover inequalities spro: number of violated strengthened projection inequalities step: number of violated step inequalities When some of the nodes are assigned to themselves and removed from the problem during preprocessing, we add the number of such nodes to the number of concentrators installed. For instance for the problem f t, 17,10,1/2, five nodes were fixed to become concentrator nodes and were removed from the problem during preprocessing and seven of the remaining nodes became concentrator locations, thus the 7+5. We observe that the step inequalities are violated when Q is large. After a certain value of Q there are no more violated step inequalities. The number of violated quadratic cover and strengthened projection inequalities also decreases as Q decreases since the capacities become loose. For problems with 17 nodes, we compare the performance of our branch and cut algorithm with the one of CPLEX 7.0 MIP solver. The formulation given to CPLEX is the aggregated multicommodity flow formulation QCL-C3 with the additional constraints for all The variables whose values are fixed to 0 or 1 during the preprocessing are removed from this formulation. We report the duality gap, the number of nodes in the branch and cut tree and the CPU time in Table 8.6. The duality gap is computed using the best upper bound found by any of the two methods. When the program stops because of the time limit then we report the final duality gap computed using the best upper bound of the corresponding method. For problems with large Q values, our branch and cut algorithm is faster than CPLEX 7.0. The biggest difference occurs for problem f t,17,20,1. For this problem CPLEX takes around 125 times more CPU time. The branch and

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cut algorithm is not able solve the problems ft, 17,15,1/2 and ft,17, 20, 2/3 to optimality and stops with a gap of 0.032% and 1.59% respectively. The duality gaps at the root node are 17.33% and 19.56% which are very big. However for CPLEX, the gaps were 30.83% and 37.37% which shows that these were hard problems. The other problems that are not solved to optimality by CPLEX in four hours are solved by the branch and cut algorithm. The longest one ft,17, 10, 1/3, takes less than one and a half hour. As Q decreases, the duality gap we obtain using the aggregated flow formulation with CPLEX is smaller than the duality gap we obtain using the branch and cut algorithm. For these problems, the two solution methods do not differ much for the CPU time. There are two problems where CPLEX is faster than the branch and cut algorithm but the biggest difference is around one minute. The cuts used by the MIP solver of CPLEX are GUB cover, clique, flow cover and Gomory fractional cuts. It is possible to use the preprocessing and strengthening algorithm to strengthen the formulation before giving it to CPLEX MIP solver. We would like to see if this can improve the results. We add all inequalities that are generated by Algorithm 8.1 to the formulation. Moreover, we also add knapsack inequalities of the form for each where are computed by Algorithm 8.1. This enables CPLEX to generate cover inequalities. In Table 8.7, we give the results with these inequalities for the first five problems with 17 nodes. We see that strengthening the formulation decreases the duality gap and the number of nodes in the tree for all problems. However, the CPU time decreases for problems with large Q values and increases for problems with small Q values. We can conclude that for the hard problems, the strengthening improves the results considerably. Still, the branch and cut algorithm performs better than CPLEX on these problems. So, we do not give a comparison for

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217

the remaining test instances. Before moving to other test instances, we would like to see if we can improve the performance of the branch and cut algorithm for the two unsolved problems, namely ft, 17,15,1/2 and ft, 17,20,2/3. We try the following strategies: 1 Use Algorithm 8.2 during preprocessing and separate the cover inequalities on linear knapsacks as discussed in Section 8.2.6. These two changes avoid solving quadratic knapsack problems. 2 Separate also the inequalities that are removed from the branch and cut algorithm except the odd hole inequalities, i.e., quadratic binpacking inequalities on three nodes, binpacking, residual capacity, effective capacity, lifted W – 2, k-triangle, lifted k-leaf and 2-cycle inequalities. 3 Give the objective value of the best solution found as the starting upper bound. 4 Use depth first search.

We did one change at a time. The problem ft, 17, 15, 1/2 was solved to optimality in 12613.79 seconds of CPU time when we solved linear knapsacks rather than quadratic knapsacks. The other changes did not improve the results. With depth first search, the algorithm stopped because of the timelimit, reporting a gap of 8.91% for the problem ft, 17, 15, 1/2 and 21.18% for the problem ft, 17, 20, 2/3. We also solved the other problems applying the first strategy. The results did not change significantly. Based on this result and some preliminary tests, we decided to use Algorithm 8.2 during preprocessing and to separate the cover inequalities on linear knapsacks for the remaining test instances as they have bigger sizes than the problems we solved up to now. We also updated the separation of projection inequalities so that we only consider the commodities with nonzero traffic demands. In Table 8.8, we report the results for problems with 22 nodes. There is one problem that is not solved to optimality as we ran out of memory; the problem ft, 22, 20, 1 where the program stopped with a gap of 1.38%. The remaining instances were solved rather fast except the problem ft, 22, 15, 2/3 which took around 15 minutes.

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We see that there is only one step inequality violated. For these problems, we can remove the step inequalities from the branch and cut algorithm. But, as the separation of step inequalities does not take much time, the total CPU time would not change significantly. In Tables 8.9, we report the results for the hub location problems from the OR library. We omit the results for problems with 10 nodes as all of these problems were solved at the root node in less than one second. We also remove the problems with Q = 1/4 and Q = 1/5 when the capacities are loose (problems of type LL, TL) since in these problems the capacity constraints were redundant. The problems with tight capacities (problems of type TT, LT) were infeasible with Q= 1 and Q = 1/2 and are not considered. So we report the results for 48 problems. Among 48 problems, 34 of them are solved to optimality. For the 14 problems not solved, we ran out of memory. All problems with 20 and 25 nodes are solved to optimality. We could solve the problems with 40 and 50 nodes only for small values of Q. Once again, we see that the difficulty of the problem depends a lot on the demand and capacity level of the problem.

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For two problems, the program stopped before finding a feasible solution. In fact, it is trivial to find a feasible solution, i.e., we can assign each node to itself. In most cases, such a solution has a very big cost compared to the one of the optimal solution (the number of concentrators installed is not more than five for the problems solved), which probably does not improve the performance of the algorithm. There are also two problems where the algorithm stopped with a gap of more than 20 %. Looking at the gaps of the other problems, we can say that this may be due to the bad quality of the upper bound rather than the lower bound. In fact, for bigger size problems, we believe that a better heuristic can improve the performance of the branch and cut algorithm.

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To conclude, we can say that the problems with small values of Q can be solved by the CPLEX MIP solver for reasonable sizes. The hard problems are the ones where Q is large, so that the traffic demand is high and capacities are tight. The branch and cut algorithm is able to solve such problems of small size. But for hard problems of big size, it cannot prove optimality in general.

Chapter 9 CONCLUSION

The principal problem studied in this book is the concentrator location problem with routing cost on a complete backbone network where concentrators are subject to quadratic capacity constraints. As relaxations, we considered the uncapacitated and linear capacitated cases. We also considered problems with no routing cost and routing cost on a star backbone network. In total, we studied the polyhedra of nine problems closely related to each other. The nice relationship between the problems with routing cost and problems with no routing cost, (i.e., the projections of the polyhedra of the problems with routing cost on the assignment variables correspond to the polytopes of the problems with no routing cost), helped us to characterize all facet defining inequalities involving the assignment variables of the polyhedra of the problems with routing cost in terms of the facet defining inequalities of the polytopes of the problems with no routing cost. In some cases, we also characterized the facet defining inequalities of the polyhedra of problems with routing cost involving only traffic variables. We introduced a family of facet defining inequalities involving both assignment and traffic variables for the polyhedron of the uncapacitated concentrator location problem with star routing. It was easier to study the polyhedra of problems with no capacity restrictions. We started with these problems. We identified several families of facet defining inequalities for the corresponding polyhedra. Then we investigated when the facet defining inequalities of the polyhedra of uncapacitated problems remain facet defining for the polyhedra of problems with linear and quadratic capacity restrictions.

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The quadratic capacity constraints also led us to investigate the polytope defined by a quadratic knapsack constraint. We modified the definition of a cover, which we used to develop cover inequalities for this polytope. Branch and bound algorithms to solve two variants of optimization problems defined on a quadratic knapsack constraint have been developed, to solve the separation problem for cover inequalities and to compute the lifting coefficients. The results are used to develop a branch and cut algorithm to solve the quadratic capacitated concentrator location problem with routing on a complete backbone network. We compared several formulations, their strength and the time it took to solve the problems by the branch and cut algorithm. The results showed that it was more advantageous to use the formulation with the least number of variables (the formulation on natural variables) and to strengthen it using inequalities obtained by projecting out the extra variables (multicommodity flow variables) of a higher dimensional and stronger formulation. The initial tests showed that the problems where the traffic demand was high were the hardest problems and they had a huge duality gap even with the strongest formulation. The preprocessing and strengthening algorithm developed based on the capacity constraints proved to be very effective to decrease the gap and the computation time. We tried to see which families of cuts were useful to close the remaining gap and to further decrease the computation time. The cover inequalities and the strengthened projection inequalities turned out to be the most effective. Then we tested the branch and cut algorithm on test instances supplied by France Telecom and some instances from the OR Library. The results showed that the branch and cut algorithm is able to close a big part of the duality gap and to reduce the computation time considerably. However there are still some problems of reasonable size that we cannot solve to optimality. For these problems, the algorithm stopped usually because we ran out of memory. Computational results for UCL-S and QCL are reported in Labbé and Yaman [51] and [50], respectively. The branch and cut algorithm for UCL-S uses the ordering inequalities. Problems generated from the AP data are used. The algorithm solves all problems with up to 100 nodes and some problems with up to 150 nodes in 4 hours of CPU time. For QCL, cover inequalities, quadratic binpacking inequalities on three nodes, triangle inequalities and W – 2 inequalities are separated. Again, problems are created using the AP data. For loose capacities, problems with

Conclusion

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up to 190 nodes are solved in 4 hours of CPU time. For tight capacities, 100 node problems are solved to optimality. It would be interesting to see the performance of the branch and cut algorithm for the other problems. In fact, we need to choose good parameters for each problem and identify the useful cuts. The results can be interesting to see if the inequalities like W – 2, k-triangle, k-leaf, 2-cycle, bin packing, effective capacity etc.. are useful for the other problems.

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Appendix A Facets of for

and

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Here we give the facet defining inequalities given by PORTA for we have 14 inequalities:

and

For

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(10) (11) (12) (13) (14)

Inequalities (1) to (6) are nonnegativity constraints, (10) and (11) are triangle inequalities and the remaining are clique inequalities. For

the number of facet defining inequalities becomes 133: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

(11) (12) (13) (14) (15) (16) (17) (18)

APPENDIX A: Facets of

(19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57)

for

and

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(58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) (76) (77) (78) (79) (80) (81) (82) (83) (84) (85) (86) (87) (88) (89) (90) (91) (92) (93) (94) (95) (96)

APPENDIX A: Facets of

(97) (98) (99) (100) (101) (102) (103) (104) (105) (106) (107) (108) (109) (110) (111) (112) (113) (114) (115) (116) (117) (118) (119) (120) (121) (122) (123) (124) (125) (126) (127) (128) (129) (130) (131) (132) (133)

for

and

229

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Inequalities (1) to (12) are nonnegativity constraints, (13) to (20) are triangle inequalities and (21) to (32) are the clique inequalities. The 2-leaf inequalities are from (33) to (44). Inequalities (45) to (101) are W – 2 inequalities. For inequalities (55), (59), (60), (71), (73), (74), (81), (84) and (85) the set U is empty. For the remaining ones, U is a singleton. The remaining inequalities (102)-(133) are the miscellaneous inequalities.

Appendix B Detailed Results of Tests to Determine Useful Cuts

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Results of Test 1

Tables B. 1- B. 12 give the detailed results for the first test to determine useful cuts. For each table, the improvements are computed using the results of the preceding table. The following abbreviations are used: LP’s: the total number of LP’s solved no. of ineq.s: number of violated inequalities % imp. in gap: percentage improvement in gap % imp. in LP: percentage improvement in the number of LP’s solved % imp. in time: percantage improvement in total CPU time

APPENDIX B: Detailed Results of Tests to Determine Useful Cuts

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B.2

Results of Test 2

Tables B.13-B.18 give the results of the second test. The improvements are computed with respect to results of Table B.4.

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References

[1] K. Aardal. Capacitated facility location: Separation algorithms and computational experience. Mathematical Programming, 81:149–175, 1998. [2] K. Aardal, Y. Pochet, and L.A. Wolsey. Capacitated facility location: Valid inequalities and facets. Mathematics of Operations Research, 20:562–582, 1995.

[3] A. Amiri. Solution procedures for the service system design problem. Computers & Operations Research, 24:49–60, 1997. [4] P. Avella and A. Sassano. On the p-median polytope. Mathematical Programming, 89:395–411,2001. [5] E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146–164, 1975. [6] E. Balas and J.B. Mazzola. Nonlinear 0-1 programming: I. linearization techniques. Mathematical Programming, 30, 1984. [7] E. Balas and M:W. Padberg. Set partitioning: A survey. SIAM Review, 18:710–760, 1976.

[8] J.E. Beasley. Or-library: Distributing test problems by electronic mail. Journal of the Operational Research Society, 41:1069–1072, 1990.

[9] J.E. Beasley.

Lagrangean heuristics for location problems. Operational Research, 65:383–399, 1993.

European Journal of

[10] T.B. Boffey. Location problems arising in computer networks. Journal of the Operational Research Society, 40:347–354, 1989. [11] B. Bozkaya, J. Zhang, and E. Erkut. An Efficient Genetic Algorithm for the p-Median Problem, chapter in Facility Location: Applications and Theory, Z.Drezner and H.W. Hamacher (eds). Springer, 2002. [12] J.F. Campbell. Integer programming formulations of discrete hub location problems. European Journal of Operational Research, 72:387–405, 1994.

252

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

[13] J.F. Campbell, A.T. Ernst, and M. Krishnamoorthy. Hub Location Problems, chapter in Facility Location: Applications and Theory, Z.Drezner and H.W. Hamacher (eds). Springer, 2002. [14] L. Canovas, M. Landete, and A. Marin. On the facets of the simple plant location packing polytope. Discrete Applied Mathematics, 124, 2002. [15] A. Caprara. Personal communication. 2002. [16] A. Caprara, D. Pisinger, and P. Toth. Exact solution of the quadratic knapsack problem. INFORMS Journal on Computing, 11:125–137, 1999.

[17] P. Chardaire, J.L. Lutton, and A. Sutter. Upper and lower bounds for the two-level simple plant location problem. Annals of Operations Research, 86, 1999. [18] D.C. Cho, E.L. Johnson, M. Padberg, and M.R. Rao. On the uncapacitated plant location problem i: Valid inequalities and facets. Mathematics of Operations Research, 8, 1983. [19] D.C. Cho, E.L. Johnson, M. Padberg, and M.R. Rao. On the uncapacitated plant location problem ii: Facets and lifting theorems. Mathematics of Operations Research, 8, 1983. [20] Th. Christof. Ein verfharen zur transformation zwischen polyederdarstellungen. Diplomarbeit, Universität Augsburg, 1991. [21] S. Chung, Y. Myung, and D. Tcha. Optimal design of a distributed network with a twolevel hierarchical structure. European Journal of Operational Research, 1992:105–115, 62. [22] G. Cornuejols, G.L. Nemhauser, and L.A. Wolsey. The Uncapacitated Facility Location Problem, chapter in Discrete Location Theory, P.B. Mirchandani and R.L. Francis (eds). Wiley, New York, 1990. [23] G. Cornuejols, R. Sridharan, and J.M. Thizy. A comparison of heuristics and relaxations for the capacitated plant location problem. European Journal of Operational Research, 50, 1991. [24] G. Cornuejols and J.M. Thizy. Some facets of the simple plant location polytope. Mathematical Programming, 23, 1982. [25] M.J. Cortinhal and M.E. Captivo. Upper and lower bounds for the single source capacitated location problem. European Journal of Operational Research, 151, 2003. [26] J. Current and H. Pirkul. The hierarchical network design problem with transshipment facilities. European Journal of Operational Research, 52:338–347, 1991. [27] K. Darby-Dowman and H.S. Lewis. Lagrangian relaxation and the single source capacitated facility location problem. Journal of the Operational Research Society, 39, 1988. [28] Q. Deng and D. Simchi-Levi. Valid inequalities, facets and computational results for the capacitated concentrator location problem. Technical report, Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027-6699, 1992.

REFERENCES

253

[29] J. Ebery. Solving large single allocation p-hub problems with two or three hubs. European Journal of Operational Research, 128, 2001. [30] A.T. Ernst and M. Krishnamoorthy. Efficient algorithms for the uncapacitated single allocation p-hub median problem. Location Science, 4:139–154, 1996. [31] A.T. Ernst and M. Krishnamoorthy. Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem. European Journal of Operational Research, 104:100–112, 1998.

[32] A.T. Ernst and M. Krishnamoorthy. Solution algorithms for the capacitated single allocation hub location problem. Annals of Operations Research, 86:141–159, 1999. [33] B. Fortz. Personal communication. 2002. [34] M.R. Garey and D.S. Johnson. Computers and Intractibility: A Guide to the Theory of NP-Completeness. Freeman, New York, 1979.

[35] B. Gavish. Topological design of centralized computer networks: Formulations and algorithms. Networks, 12:355–377, 1982. [36] B. Gavish. Topological design of computer communication networks- the overall design problem. European Journal of Operational Research, 58, 1992. [37] E. Gourdin, M. Labbé, and H. Yaman. Telecommunication and Location, chapter in Facility Location: Applications and Theory, Z.Drezner and H.W. Hamacher (eds). Springer, 2002. [38] M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, 1988.

[39] Z. Gu, G.L. Nemhauser, and M.W.P. Savelsbergh. Lifted cover inequalities for 0-1 integer programs: Computation. INFORMS Journal on Computing, 10, 1998. [40] M. Guignard. Fractional vertices, cuts and facets of the simple plant location problem. Mathematical Programming, 12, 1980.

[41] H. W. Hamacher, M. Labbé, S. Nickel, and T. Sonneborn. Adapting polyhedral properties from facility to hub location problems. Discrete Applied Mathematics, in press. [42] P.L. Hammer, E.L. Johnson, and U.N. Peled. Mathematical Programming, 8:179–206, 1975.

Facets of regular 0-1 polytopes.

[43] M.P. Helme and T.L.Magnanti. Designing satellite communication networks by zero-one quadratic programming. Networks, 19:427–450, 1989. [44] M. Iri. On an extension of the max-flow min-cut theorem to multicommodity flows. Journal of Operations Research Society of Japan, 13:129–135, 1971. [45] M. Jünger and S. Thienel. The abacus system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization. Software Practice and Experience, 30, 2000.

[46] J.G. Klincewicz. Hub location in bacbone/tributary network design: a review. Location Science, 6:307–335, 1998.

254

CONCENTRATOR LOCATION IN TELECOMMUNICATIONS NETWORKS

[47] J.G. Klincewicz and H. Luss. A lagrangian relaxation heuristic for capacitated facility location with single source constraints. Journal of the Operational Research Society, 37, 1986. [48] J. Krarup and P.M. Pruzan. The simple plant location problem: Survey and synthesis. European Journal of Operational Research, 12:36–81, 1983.

[49] M. Labbé, D. Peeters, and J.F. Thisse. Location on Networks, chapter in Handbooks in Operations Research and Management Sciences, Vol. 8, M.O. Ball, T.L. Magnanti, C.L. Monma and G.L. Nemhauser (eds). North-Holland, Amsterdam, 1995. [50] M. Labbé and H. Yaman. Polyhedral analysis for concentrator location problems. Technical report, ISRO/OR Preprint 2003/13, Université Libre de Bruxelles, Brussels, 2003. [51] M. Labbé and H. Yaman. Solving the hub location problem in a star-star network. Technical report, revised version of ISRO/OR Preprint 2003/15, Université Libre de Bruxelles, Brussels, 2004.

[52] M. Labbé and H. Yaman. Networks, to appear.

Projecting the flow variables for hub location problems.

[53] M. Labbé, H. Yaman, and E. Gourdin. A branch and cut algorithm for hub location problems with single assignment. Mathematical Programming, to appear.

[54] C.Y. Lee. An algorithm for the design of multitype concentrator networks. Journal of the Operational Research Society, 44, 1993. [55] B.H. Lim Lee Y. and J.S. Park. A hub location problem in designing digital data service networks: Lagrangian relaxation approach. Location Science, 4:185–194, 1996. [56] J.M.Y. Leung and T.L. Magnanti. Valid inequalities and facets of the capacitated plant location problem. Mathematical Programming, 44:271–291, 1989.

[57] L.A.N. Lorena and E.L.F. Senne. A column generation approach to capacitated p-median problems. Computers & Operations Research, 31, 2004. [58] V. Maniezzo, A. Mingozzi, and R. Baldaci.

[59] S. Martello and P. Toth. Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York, 1990. [60] G.R. Mateus, F.R.B. Cruz, and H.P.L. Luna. An algorithm for hierarchical network design. Location Science, 2, 1994.

[61] P. Mirchandani. Projections of the capacitated network loading problem. European Journal of Operational Research, 122:534–560, 2000. [62] P.B. Mirchandani. The p-Median Problem and Generalizations, chapter in Discrete Location Theory, P.B. Mirchandani and R.L Francis (eds). Wiley, New York, 1990. [63] A. Mirzaian. Lagrangian relaxation for the star-star concentrator location problem: Approximation algorithm and bounds. Networks, 15, 1985.

REFERENCES

255

[64] A.W. Neebe and M.R. Rao. An algorithm for the fixed-charge assigning users to sources problem. Journal of the Operational Research Society, 34, 1983. [65] G.L. Nemhauser and L.A. Wolsey. Integer and Combinatorial Optimization. Wiley, New York, 1988. [66] M.E. O’Kelly. A quadratic integer program for the location of interacting hub facilities. European Journal of Operational Research, 32, 1987. [67] M.E. O’Kelly. Hub facility location with fixed costs. Papers in Regional Science The Journal of RSAI, 71:293–306, 1992. [68] M.W. Padberg. On the facial structure of the set packing polyhedra. Mathematical Programming, 5:199–215, 1973. [69] H. Pirkul. Efficient algorithms for the capacitated concentrator location problem. Computers & Operations Research, 14, 1987. [70] H. Pirkul and V. Nagarajan. Locating concentrators in centralized computer networks. Annals of Operations Research, 36, 1992.

[71] H. Pirkul, S. Narasimhan, and P. De. Locating concentrators for primary and secondary coverage in a computer communications network. IEEE Transactions on Communications, 36, 1988. [72] H. Pirkul and D. Schilling. An efficient procedure for designing single allocation hub and spoke systems. Management Science, 44, 1998. [73] Y. Pochet. Personal communication. 2002 [74] R.L. Rardin and L.A. Wolsey. Valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems. European Journal of Operational Research, 71:95–109, 1993. [75] D. Skorin-Kapov, J. Skorin-Kapov, and M. O’Kelly. Tight linear programming relaxations of uncapacitated p-hub median problem. European Journal of Operational Research, 94:582–593, 1996.

[76] R. Sridharan. A lagrangian heuristic for the capacitated plant location problem with single source constraints. European Journal of Operational Research, 66, 1993. [77] R. Sridharan. The capacitated plant location problem. European Journal of Operational Research, 87, 1995.

[78] D.T. Tang, L.S. Woo, and L.R. Bahl. Optimization of teleprocessing networks with concentrators and multiconnected terminals. IEEE Transactions on Computers, 27, 1978. [79] M. Van Vyve. Approximate extended formulations, 2002. [80] L. Wolsey. Faces for a linear inequality in 0-1 variables. Mathematical Programming, 8, 1975.

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Index

I*-cover inequalities, 19 2-cycle inequalities, 60–63, 118, 160, 174–176, 202, 203, 210, 217, 223, 242, 249 ABACUS, 206 access networks, 4, 5, 9, 11, 12, 36, 63 anti-web, 41 assignment constraints, 94, 208 backbone, 4–6, 9, 11, 12, 26, 28, 35, 36, 63, 64, 80, 81, 85, 99, 126, 130, 163, 182, 205, 221, 222 best first search, 208, 210 binpacking inequalities, 18, 120, 174–176, 205, 210, 217, 237, 244 modified, 120–122, 160, 174–176 quadratic, 161, 174–176, 206, 210, 217, 236 problem, 121 branch and bound, 11, 22, 152, 153, 171, 184, 222 branch and cut, 5–7, 13, 179, 180, 183, 184, 191, 194, 196, 197, 200, 202, 206, 208, 210, 212–220, 222, 223 branch and price, 13 breadth first search, 208, 210 capacity constraints, 4, 6, 7, 22, 27, 28, 99, 100, 124, 127, 142, 145, 146, 171, 180, 198, 207, 218, 221, 222 quadratic, 145, 146, 180, 222 clique, 18, 20, 39–41, 43, 44, 62, 113, 174–176, 216, 226, 230 concentrator location problem, 4–6, 22, 25, 28, 35, 36, 63, 80, 99, 100, 119, 123, 124, 127, 142, 145, 146, 156, 163, 171 conflict graph, 18, 40, 48, 49, 58, 202, 204, 210

convex hull, 14, 15, 17–19, 39, 43, 58, 59, 62, 66–68, 82, 100, 103, 104, 113, 119, 124, 127, 130, 131, 151–153, 156, 157, 159, 164, 166, 169 cover, 15, 150, 151, 171, 204, 222 flow, 15, 16, 123, 216 GUB, 216 inequalities, 102, 113, 149–152, 174–176, 198, 204, 205, 211, 212, 214, 216, 217, 222, 233 quadratic, 149, 151–153, 161, 171, 174–176, 204, 208, 210, 214 quadratic, 149, 150, 152, 153, 157–161, 167, 198, 206 CPLEX, 206, 214–216, 220 depth first search, 208, 210, 217 dicut inequalities, 85, 86 dimension, 14, 15, 39, 66–68, 80, 82, 84, 97, 101, 120, 125, 128, 151, 156, 164, 166, 186 effective capacity inequalities, 16, 123, 174– 176, 206, 207, 210, 217, 223, 238, 245 facet, 14–18, 20, 29, 35, 38, 39, 41–43, 48, 55, 58–60, 62, 63, 66–69, 78, 79, 82, 83, 96, 97, 99–104, 108, 112–114, 116, 118, 120, 121, 125, 128, 142, 143, 146, 151–153, 155–162, 164, 166, 167, 173–176, 221, 226 facility location problem, 12, 37, 100, 103, 119, 143 flow conservation, 85, 183, 193 flow cover, see cover generalized assignment problem, 111, 201 GUB cover, see cover heuristic, 5, 12, 13, 22, 186, 202, 205–208, 219

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hub location problem, 9, 20, 22, 23, 80, 89, 92, 181, 182, 187, 193, 194, 213, 218– 220 k-leaf inequalities, 55, 58, 59, 63, 113–115, 159, 174–176, 201, 210, 217, 223, 241, 248 k-triangle inequalities, 39–42, 44, 48–52, 58, 62, 84, 112, 158, 174–176, 202, 203, 210, 212, 217, 223, 240, 247 knapsack, 13, 100, 103, 118, 135,142, 145, 146, 150–152, 157, 171, 198, 200, 201, 204, 205, 212, 216, 217, 222 Lagrangian relaxation, see relaxation lifting, 43, 58, 59, 62, 63, 69, 112, 116, 131, 132, 134, 135, 138, 152, 153, 159, 201, 204–206, 210, 212, 217 coefficient, 43, 62, 63, 69, 73, 77, 103, 109, 111, 112, 116, 117, 131, 135, 139, 153, 159, 171, 201 order, 43, 69, 109–111 sequential, 42, 68, 108, 153 linear programming relaxation, see relaxation, 182 linearization, 11, 21, 146–150, 181, 204 metric inequalities, 85 mincut, 92–94 modified binpacking inequalities, see binpacking multicommodity flow , 84–86, 179, 180, 182, 192–194, 214 NP-complete, 38, 50, 51, 202 NP-hard, 38 odd anti-hole, 20, 41 odd hole, 18, 20, 41–43, 48, 49, 202, 204, 210, 211, 217, 243 ordering inequalities, 79, 175 polyhedron, 6, 7, 29, 35, 38, 66, 68, 79, 80, 82, 84, 96, 97, 125, 128, 164, 166, 186

polytope, 14, 15, 17, 19, 20, 35, 36, 38, 39, 41, 43, 62, 66, 80, 82, 96, 97, 100, 101, 103, 120, 122, 125, 142, 143, 145, 146, 151, 152, 156, 157, 171, 202 PORTA, 41, 60, 62 preprocessing, 197, 198, 201, 204, 206, 208, 213, 214, 216, 217, 232 projection, 29, 31, 66, 82, 84–87, 89, 91, 93, 125 inequalities, 85, 96, 136, 169, 176, 179, 181, 186, 187, 190–192,217 quadratic binpacking inequalities, see binpacking quadratic capacity constraints, see capacity constraints quadratic cover, see cover relaxation, 6, 7, 28, 31, 99, 122 Lagrangian, 12, 13, 22 linear programming, 148, 150 residual capacity inequalities, 14, 17, 122, 123, 174–176, 205, 210, 217, 237, 244 semi projection inequalities, 190, 191, 193, 194 separation, 20, 35, 49–51, 59, 94, 96, 149, 151, 152, 171, 186, 187, 191, 197, 200– 207, 211, 212, 217, 218 sequential lifting, see lifting set partitioning, 13 stable set, 19, 39, 41, 96, 202 step inequalities, 137, 138, 176, 206, 208, 210– 212, 214, 218, 235 strengthened projection inequalities, 130, 176, 206, 208, 210–212, 214, 234 tailing off, 213 traffic bound inequalities, 126, 129, 164, 165, 167, 168, 175, 176 triangle inequality, 4, 21, 26 W-2 inequalities, 19, 20, 39–41, 48, 58, 62, 63, 104, 108, 157, 174–176, 201, 202, 210, 212, 217, 223, 230, 239, 246