handbhandbook of modeling for discert optimizationook of modeling for discert optimization

8/10/2019 Handbhandbook of modeling for discert optimizationook of Modeling for Discert Optimization

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HANDBOOK ON MODELLING

FOR DISCRETE OPTIMIZATION


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Recent titlesin the

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*A list of the early publications in the series is at the end of the book *


3/441

HANDBOOK ON MODELLING

FOR DISCRETE OPTIMIZATION

Edited by

GAUTAM APPA

Operational Research Department

London School of Economics

LEONIDAS PITSOULIS

Department of Mathematical and Physical Sciences

Aristotle University of Thessaloniki

H.PAUL

W ILLIAMS

Operational Research Department


Sprin

ger


4/441

Gautam Appa Leonidas Pitsoulis

London School of Econ om ics Aristotle Unive rsity of Thessa loniki

United Kingdom Greece

H. Paul W illiams


United Kingdom

Library of Congress Control Number: 2006921850

ISBN -10: 0-387-32941-2 (HB) ISBN-10: 0-387-32942-0 (e-book)

ISBN-13:

978-0387-32941-3 (HB) ISBN -13: 978-0387-32942-0 (e-book)

Printed on acid-free paper.

2006 by Springer Science+Business Media, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without

the written permission of the publisher (Springer Science + Business Media, Inc., 233

Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with

reviews or scholarly analysis. Use in connection with any form of information storage

and retrieval, electronic adaptation, computer software, or by similar or dissimilar

methodology now know or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks and similar terms,

even if the are not identified as such, is not to be taken as an expression of opinion as to

whether or not they are subject to proprietary rights.

Printed in the United States of America.

9 8 7 6 5 4 3 2 1

springer.com


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Contents

List of Figures ix

List of Tables xiii

Con tributing Authors xv

Preface xix

Acknowledgments xxii

Part I Me thods

1

The Form ulation and Solution of Discrete Optimisation Models 3

H. Paul Williams

1.

The App licability of Discrete Optimisation 3

2.

Integer Programming 4

3.

The Uses of Integer Variables 5

4.

The Modelling of Comm on Conditions 9

5.

Reformulation Techniques 11

6. Solution Methods 22

References 36

2

Con tinuous App roaches for Solving Discrete Optimization Problem s 39

Panos M Pardalos,

Oleg

A Prokopyev and Stanislav B usy gin

1.

Introduction 39

2.

Equivalence of Mixed Integer and Com plementarity Problem s 40

3.

Continuous Formulations for 0-1 Programming Problems 42

4.

The Maximum Clique and Related Problems 43

5. Th e Satisfiability Problem 48

6. The Steiner Problem in Graphs 51

7.

Semidefinite Program ming Approaches 52

8. Minimax Approaches 54

References 55

3

Logic-Based Modeling 61

John N H ooker

1.

Solvers for Logic-Based Constraints 63

2.

Good Formulations 64


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vi HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION

3.

Prepositional Logic 69

4. Cardinality Formulas 77

5.

0-1 Linear Inequalities 83

6. Cardinality Rules 85

7.

Mixing Logical and Con tinuous Variables 87

8. Add itional Global Constraints 92

9. Conclusion 97

References 99

4

Mo delling for Feasibility - the case of Mutually Orthogonal Latin Squares 103

Problem

Gautam Appa, Dimitris Mag os, loannis M ourtos and Leonidas P itsoulis

1.

Introduction 104

2.

Definitions and notation 106

3.

Formulations of the fcMOL S problem 108

4.

Discussion 122

References 125

129

129

130

133

134

137

139

141

144

146

148

6

Modeling and Optimization of Vehicle Routing Problem s 151

Jean-Francois Cordeau an d Gilbert Laporte

1. Introduction 151

2.

The Vehicle Routing Problem 152

3.

The Chinese Postman Problem 163

4.

Constrained Arc Rou ting Problem s 168

5.

Conc lusions 181

References 181

Part II Applications

7

Radio Resource Managem ent 195

Katerina Papad aki and

Vasilis

Friderikos

1. Introduction 196

2. Problem Definition 199

;wc

ugl

1.

2.

3.

4.

5.

)rk Modelling

as

R.

Shier

Introduction

Transit Networks

Amplifier Location

Site Selection

Team Elimination ir

1

Sports

6. Reasoning in Artificial Intelligence

7.

Ratio Com parisons in Decision Analysis

8. DNA Sequencing

9. Computer Memory Managem ent

References


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Contents

vii

3.

Myopic Problem Formulations 203

4.

The dynamic downlink problem 208

5.

Concluding Remarks 222

References 224

Strategic and tactical planning models for supply chain: an application of 227

stochastic mixed mteger programming

Gautam M itra, C handra Poojari and Suvrajeet Sen

1.

Introduction and Background 228

2.

Algorithms for stochastic mixed integer program s 234

3.

Supply chain planning and management 237

4.

Strategic supply chain planning : a case study 244

5.

Discussion and conclusions 259

References 260

9

Log ic Inference and a Decomposition Algorithm for the Resource-C onstrained 265

Scheduling of Testing Tasks in the Development of New Pharmaceu

tical and Agrochemical Products

Christos

T,

Maravelias and Ignacio E. Grossmann

266

266

268

271

277

281

281

282

283

284

284

285

10

A Mixed-integer Non linear Program ming Approach to the Optimal Plan- 291

ning of Ons ho re Oilfield Infrastructures

Susara

A,

van den Heever and Ignacio E. Grossmann

291

294

295

301

306

309

311

312

314

11

Radiation Treatment Plann ing: Mixed Integer Program ming Form ula- 317

tions and Approaches

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Introduction

Motivating E xample

Model

Logic Cuts

Decomposition Heuristic

Computational Results

Example

Conclusions

Nomenclature

Acknowledgment

References

Appendix: Example Data

1.

2.

3.

4.

5.

6.

7.

8.

Introduction

Problem Statement

Model

Solution Strategy

Example

Conclusions and Future Work

Acknowledgment

Nomenclature

References


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viii HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION

Michael

C.

Ferris, Robert

R .

Meyer and

Warren D 'Souza

1.

Introduction 318

2. Gamma Knife Radiosurgery 321

3.

Brachytherapy Treatment Planning 327

4.

IMRT 331

5.

Conclusions and Directions for Future Research 336

References 336

12

Multiple Hypothesis Correlation in Track-to-Track Fusion Management 341

Aubrey B

Poore,

Sabino M Gad aleta and Benjamin J Slocumb

1.

Track Fusion Architectures 344

2.

The Frame-to-Frame Matching Problem 347

3.

Assignment Problems for Frame-to-Frame Matching 350

4.

Com putation of Cost Coefficients using a Batch Methodology. 360

5.

Summary 368

References 369

13

Com putational Molecular Biology 373

Giuseppe Lancia

1. Introduction 373

2.

Elemen tary Molecular Biology Concepts 377

3.

Alignment Problems 381

4. Single Nucleotide Polymorphisms 401

5.

Genom e Rearrangements 406

6. Genomic Mapping and the TSP 412

7. Applications of Set Covering 415

8. Conclusions 417

References 418

Index 427


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List of Figures

1.1 A piecewise linear approximation to a non-linear function 8

1.2 The convex hull of a pure IP 12

1.3 The convex hull of a mixed IP 13

1.4 Polytopes with different recession directions 15

1.5 A cutting pattern 20

1.6 Optimal pattern 21

1.7 An integer programm e 23

1.8 An integer programm e with Gom ory cuts 27

1.9 Possib le values of an integer variable 28

1.10 The first branch of a solution tree 29

1.11 Solution space of the first branch 29

1.12 Final solution tree 30

3.1 Conversion ofF to CNF without additional variables. A

formula of the form (if

A

/) V G is regarded as having

t h e f o r m G V ( i J A / ) . 7 2

3.2 Linear-time conversion to CN F (adapted from [21]). The

letter

C

represents any clause. It is assumed that

F

does

not contain variables x i , X 2 ,. .. . 73

3.3 The resolution algorithm applied to clause setS 74

3.4 The cardinality resolution algorithm applied to card inal

ity formula set 5 81

3.5 The 0-1 resolution algorithm applied to set 5 of

0-1

inequalities 85

3.6 An algorithm, adapted from [40], for generating a con

vex hull formulation of the cardinality rule (3.26). It is

assumed that a^,bj G {0,1} is part of the formulation.

The cardinality clause {a^} > 1 is abbreviated a . The

procedure is activated by calling it with (3,26) as the ar

gument. 86

5.1 A transit system Gwith 6 stops 131

5.2 The time-expanded network G 132


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HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION

5.3 Bipartite flow network 136

5.4 Bipartite flow network associated with Team 3 138

5.5 A constraint graph 141

5.6 Network for assessing probabilities 142

5.7 Revised network for assessing probabilities 144

5.8 DNA sequencing network 145

6.1 The Konigsberg bridges problem 165

6.2 Exam ple for the Frederickson's heuristic does not yield

an optimal solution. 169

6.3 Illustration of procedure SHORTEN 171

6.4 Illustration of procedure DR OP 172

6.5 Illustration of procedure AD D 172

6.6 Illustration of procedure 2-OPT 173

6.7 Illustration of procedure PASTE 177

6.8 Illustration of procedure CU T 178

7.1 Feasible region for two users 211

7.2 System events in the time domain for the original state

variable and pre-decision state variable in time periods

t

a n d t + 1 214

7.3 Geom etrical interpretation of the heuristic used for the

embedded IP optimization problem for user

i.

The next

feasible rate tov iis r/" u{si + ai) 218

7.4 Com putational complexity of the LAD P, textbook DP,

and exhaustive search in a scenario where the outcome

space consist of an eight state Markov channel, the ar

rivals have been truncated to less than twelve packets per user 219

7.5 Com putational times of the LA DP algorithm in terms of

CPU-time as a function of the number of mobile users 222

8.1 A scenario tree 231

8.2 Hierarchy of the supply chain planning 238

8.3 A strategic supply chain network 239

8.4 Supply chain systems hierarchy (source- Shapiro, 1998) 242

8.5 SCHUM ANN Models/Data Flow 243

8.6 Influence of time on the strategic decisions 248

8.7 The Lagrangian algorithm 254

8.8 Pseudo Code

1

256

8.9 Best hedged-value of the configuration 258

8.10 The frequency of the configuration selected 258

8.11 The probability weighted objective value of the configuration 259

9.1 Motivating exam ple 267


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Listo f Figures

xi

9.2 Different cycles for four tests 273

9.3 Branch & bound tree of motivating example 274

9.4 Incidence matrix of constraints 278

9.5 Decomposition heuristic 280

10.1 Configuration of fields, well platforms and production

platforms (Iyer

et al

, 1998) 292

10.2 Logic-based OA algorithm 303

10.3 Iterative aggregation/disaggregation algorithm 306

10.4 Results 308

10.5 The fina l configuration 309

10.6 The optimal investement plan 310

10.7 Production profile over six years 310

11.1 The Gam ma Knife Treatment Unit. A focusing helmet is

attached to the frame on the patient's head. The patient

lies on the couch and is moved back into the shielded

treatment area 321

11.2 Underdose of target regions for (a), (c) the pre-treatment

plan and (b), (d) the re-optimized plan, (a) and (b) show

the base plane, while (c) and (d) show the apex plane 332

12.1 Diagram s of the (a) hierarchical architecture without feed

back, (b) hierarchical architecture w ith feedback, and (c)

fully distributed architecture. S-nodes are sensor/tracker

nodes, while F-nodes are system/fusion nodes 344

12.2 Diagram showing the sensor-to-sensor fusion process 346

12.3 Diagram showing the sensor-to-system fusion process 347

12.4 Illustration of source-to-source track correlation 349

12.5 Illustration of frame-to-frame track correlation 350

12.6 Illustration of two-dimensional assignm ent problem for

frame-to-frame matching 352

12.7 Illustration of three-dimensional assignment problem for

frame-to-frame matching 354

12.8 Illustration of multiple hypo thesis, multiple frame, cor

relation approach to frame-to-frame matching 357

12.9 Detailed illustration of multiple hypo thesis, multiple frame,

correlation approach to frame-to-frame matching 358

12.10 Illustration of sliding window for frame-to-frame matching 360

12.11 Illustration of the batch scoring for frame-to-frame matching 365

13.1 Schematic DNA replication 378

13.2 (a) A noncrossing matching (alignment), (b) The di

rected grid. 382


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xii


13.3 Graph of RNA secondary structure 3 91

13.4 (a) An unfolded protein, (b) After folding, (c) The con

tact map graph. 395

13.5 An alignment of value 5 396

13.6 A chromosom e and the two haplotypes 401

13.7 A SNP matrix M and its fragment conflict graph 403

13.8 Evolutionary events 407


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List of Tables

3.1 Prim e implications and convex hull formulations of some

simple propositions. The set of prime implications of a

proposition can serve as a consistent formulation of that

proposition. 78

3.2 Catalog of logic-based constraints 98

3.3 Advantages of consistent and tight formulations 99

4.1 A pair of OLS of order 4 107

5.1 A system of routes and stops 131

5.2 Poss ible locations for amplifiers 134

5.3 Revenues

Vij

and costs Q 135

5.4 Current team rankings 137

5.5 Gam es remaining to be played 138

5.6 History of requests allocated to frames 147

5.7 An optimal assignment of requests 148

5.8 Another optimal assignment of requests 148

6.1 Solution values produced by several TS heuristics on

the fourteen CM T instances. Best known solutions are

shown in boldface. 164

7.1 Analysis on the performance of different algorithm s 223

8.1 App lications of SM IPs 233

8.2 Configuration generation 249

8.3 Dim ension of the strategic supply chain model 251

8.4 The stochastic metrics 257

9.1 Model statistics of test problem s 272

9.2 Addition of cycle-breaking cuts: Num ber of constraints

and LP relaxation 272

9.3 Testing data for the motivating exam ple 273

9.4 Income data for the motivating example. 273

9.5 Preprocessing algorithm (PPRO CA LG) 274


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xiv


9.6 Precedence implications and cuts: No. of constraints

and LP relaxation 276

9.7 Solution statistics 277

9.8 Decomposition algorithm (DECA LG ) 280

9.9 Solution statistics of the exam ples 281

9.10 Solution statistics of the exam ples 282

9.11 Optimal solution 282

9.12 Heuristic solution 282

9.13 Resource assignment of heuristic solution 283

9.A.1 Testing data of product A 286

9.A.2 Testing data of product B 287

9.A.3 Testing data of product C 288

9.A.4 Testing data of product D 288

9.A.5 Incom e data for products 289

11.1 Target coverage of manual pre-plan vs optimized pre

plan vs OR re-optimized plan 331


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Contributing Authors

Gautam Appa

Department of

Operational

Research

London Schoo l of Economics, London

United Kingdom

[email protected]

Vasilis Friderikos

Centre for Telecommunications Research

King'sCollegeLondon

United Kingdom

[email protected]

Stanislav Busygin

Department of Industrial and Systems

Engineering, University of

Florida

303

WeilHall,

Gainesville FL 32611

USA

busygin @uf

I.edu

Sabino M. Gadaleta

Numeric a

PO Box 271246

Fort Collins, CO 805 27-1246

USA

[email protected]

Jean-Fran9ois Cordeau

Canada R esearch Ch air in Distribution

Management and G ERAD, HEC Montreal

3000 chemin de la Cdte-Sainte-Catherine

Montreal, Canada H3T2A7

cordeau @crt.umontreal.ca

Ignacio E. Grossmann

Department of Chemical Engineering

Carnegie Mellon

University,

Pittsburgh

PA 15213,

USA

[email protected]

Warren D'Souza

University of Maryland Scho ol of Medicine

22 South Green Street

Baltimore, MD 21201

USA

[email protected]

Susara A. van den H eever

Department ofChemicalEngineering

Carnegie Mellon University, Pittsburgh

PA 15213,

USA

[email protected]

Michael C. Ferris

Com puter Sciences Department

University of

Wisconsin

1210

West

Dayton

Street,

Madison

Wl53706,

USA

[email protected]

John N. H ooker

Graduate School of Industrial Adm inistration

Carnegie Mellon U niversity, PA 1 5213

USA

[email protected]


16/441

XVI


Giusseppe Lancia

Dipartimento di M atematica e Informatica

Universita di Udine

Viad elle Scienze 206 , 33100 Udine

Italy

[email protected]

Gilbert Laporte

Canada Research Chair in Distribution

Management and GERAD , HEC Montreal

3000 chemin de la Cbte-Sainte-Catherine

Montreal, Canada H3T2A7

[email protected]

Dimitris Magos

Department of Informatics

Technological

Ed ucational Institute of Athens

12210

Athens, Greece

[email protected]

Christos T. Maravelias

Department ofChemicaland B iological

Engineering, U niversity ofWisconsin

1415 Engineering D rive,

Madison, WI53706-1691,

USA

[email protected]

Robert R. Meyer

Com puter Sciences Department

University of

Wisconsin

1210

West

Dayton

Street,

Madison

WI53706,

USA

[email protected]

Gautam Mitra

CARISMA

School of Information Systems,

Computing and Mathematics,

Brunei University, London

United Kingdom

[email protected]

loannis M ourtos

Department of Economics

University ofPatras, 26500

Rion,

Patras

Greece

[email protected]

Katerina Papadaki

Department of

Operational

Research

London School of Economics, London

United Kingdom

[email protected]

Panos Pardalos

Department of Industrial and Systems

Engineering, University of Florida

303

WeilHall,


USA

[email protected]

Leonidas Pitsoulis

Department o f Mathematical and

Physical Sciences, Aristotle U niversity

of Thessaloniki, 54124 Thessaloniki,

Greece

[email protected]

Chandra Poojari

CARISMA

School of Information Systems,

Computing and Mathematics

Brunei University, London

United Kingdom

[email protected]

Aubrey Poore

Department of Mathematics

Colorado State U niversity

Fort Co llins, 80523,

USA

[email protected]


17/441

Contributing Authors

xvn

Oleg A Prokopyev

Department o f Industrial and Systems

Engineering, University of Florida

303 Weil

Hall,


USA

[email protected]

Suvrajeet Sen

Department of Systems and Industrial

Engineering , University of Arizona,

Tuscon,AZ 85721

USA

[email protected]

Douglas R. Shier

Department of Mathem atical Sciences

Clemson University, Clemson,

SC 29634-0975

USA

[email protected]

Benjamin J, Slocumb

Num eric a

PO Box 271246

Fort Collins, CO 80527-12 46

USA

[email protected]

H. Paul Williams

Department of

Operational

Research

London School of Econom ics, London

United Kingdom

[email protected]


18/441

Preface

The primary reason for producing this book is to demonstrate and commu

nicate the pervasive nature of Discrete O ptimisation. It has applications across

a very w ide range of activities. Many of the applications are only known to

specialists. Our aim is to rectify this.

It has long been recognized that ''modelling" is as important, if not more

important, a mathematical activity as designing algorithms for solving these

discrete optimisation problem s. Nevertheless solving the resultant models is

also often far from straightforward. Although in recent years it has become

viable to solve many large scale discrete optimisation problems some problem s

remain a challenge, even as advances in mathematical methods, hardware and

software technology are constantly pushing the frontiers forward.

The subject brings together diverse areas of academic activity as well as di

verse areas of applications. To date the driving force has been Operational R e

search and Integer Programming as the major extention of the well-developed

subject of Linear Program ming. However, the subject also brings results in

Computer Science, Graph Theory, Logic and Combinatorics, all of which are

reflected in this book.

We have divided the chapters in this book into two parts, one dealing with

general methods in the modelling of discrete optimisation problems and one

with specific app lications. The first chapter of this volume, written by Paul

Williams, can be regarded as a basic introduction of how to model discrete

optimisation problems as Mixed Integer Programmes, and outlines the main

methods of solving them.

Chapter 2, written by Pardalos et al., deals with the intriguing relationship

between the continuous versus the discrete approach to optimisation problems.

The authors in chapter 2 illustrate how many well known hard discrete optimi

sation problems can be modelled and solved by continuous methods, thereby

giving rise to the question of whether or not the discrete nature of the problem

is the true cause of its computational complexity or the presence of noncon-

vexity.

Another subject of great relevance to modelling is Logic. This is covered in

chapter 3. The author, John Hooker, describes the relationship with an alter-


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XX HANDBOOK ONMODELLING FORDISCRETE OPTIMIZATION

native solution (and modelling) approach known as Constraint Satisfaction or,

as it is sometimes called, Constraint Logic Programming. This approach has

emerged more from Computer Science than Operational Research. However,

the possibility of "hybrid methods" based on combining the approaches is on

the horizon, and has been realized with some problem specific implementa

tions.

In chapter 4 Appa et al. illustrate how discrete optimisation modelling and

solution methods can be applied to answer questions regarding a problem aris

ing from combinatorial mathematics. Specifically the authors present various

optimisation formulations of the mutually orthogonal latin squares problem,

from constraint programming (which is covered in detail in chapter 3) to mixed

integer programming formulations and matroid intersection, all of which can

be used to answer existence questions for the problem.

It has long been established that Networks can model most of today's com

plex systems such as transportation systems, telecommunication systems, and

computer networks to name a few, and network optimisation has proven to be

a valuable tool in analyzing the behavior of these systems for design purposes.

Chapter 5 by Shier enhances further the applicability of network modelling by

presenting how it can also be applied to less apparent systems ranging from

genomics, sports and artificial intelligence.

Chapter 6 is the last chapter in the methods part of the book, w here Cordeau

and Laporte discuss a class of problems known as vehicle routing problems.

Vehicle routing problems enjoy a plethora of applications in the transportation

and logistics sector, and the authors in chapter 6 present the state of the art with

respect to exact and heuristic methods for solving them.

In the second part of the book various real life applications are presented,

most of them formulated as mixed integer linear or nonlinear programming

problem s. Chapter 7 by Papadaki and Friderikos, is concerned with the so

lution of optimization problems arising in resource management problems in

wireless cellular systems by employing a novel approach, the so called approx

imate dynamic programming.

Most of the discrete optimisation models presented in this book are of de

terministic nature, that is the values of the input data are assumed to be known

with certainty. There are however real life applications where such an assump

tion is inapplicable, and stochastic m odels need to be considered. This is the

subject of chapter 8, by Mitra et al. where stochastic mixed integer program

ming models are discussed for supply chain management problems.

In chapters 9 and 10 Grossmann et al. present how discrete optimisation

modeling can be efficiently applied to two specific application areas. In chap

ter 9 mixed integer linear programming models are presented for the problem

of scheduling regulatory tests of new pharmaceutical and agrochemical prod

ucts, while in chapter 10 a mixed integer nonlinear model is presented for the


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PREFACE xxi

optimal planning of offshore oilfield infrastructures. In both chapters the au

thors also present solution techniques.

Optimization models to radiation therapy for cancer patients is the subject

discussed in chapter 11 by Ferris and Meyer. They show how the problem of

irradiating patients for treatment of cancerous tumors can be formulated as a

discrete optimisation problem and can be solved as such.

In chapter 12 the data association problem that arises in target tracking is

considered by Poore et al. The objective in this chapter is to partition the data

that is created by multiple sensors observing multiple targets into tracks and

false alarms, which can be formulated as a multidimensional assignment prob

lem, a notoriously difficult integer programming problem which generalizes

the well known assignment problem.

Finally chapter 13 is concerned with the life sciences, and Lancia shows

how some challenging problems of Computational Biology can now be solved

as discrete optimisation models.

Assem bling and planning this book has been much m ore ofachallenge than

we at first envisaged. The field is so active and diverse that it has been difficult

covering the whole subject. Moreover the contributors have themselves been so

deeply involved in practical applications that it has taken longer than expected

to complete the volume.

We are aware that, w ithin the limits of space and the time of contributors we

have not been able to cover all topics that we would have liked. For example we

have been unable to obtain a contributor on Com puter Design, an area of great

importance, or similarly on Computational Finance and Air Crew Scheduling.

By way of mitigation we are pleased to have been able to bring together some

relatively new application areas.

We hope this volume proves a valuable work of reference as well as to stim

ulate further successful applications of discrete optimisation.

GAUTAM APPA , LEONIDAS PITSOULIS, H. PAUL WILLIAMS


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xxii


Acknowledgments

We would like to thank the contributing authors of this volume, and the

many anonym ous referees who have helped us review the chapters all of which

have been thoroughly refereed. We are also thankful to the staff of Springer,

in particular Gary Folven and Carolyn Ford, as well as the series editor Fred

Hillier.

Paul Williams acknowledges the help which resulted from Leverhulme Re

search Fellowship RF& G/9/RFG/2000/0174 and EPSR C Grant EP/C530578/1

in preparing this book.


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I

METHODS


23/441

Chapter 1

THEFORMULATIONANDSOLUTIONOFDISCRETE

OPTIMISATION MODELS

H. Paul W illiams

Department of

Operational

Research

London Scho ol of Econom ics, London

United Kingdom

[email protected]

Abstract

This introductory chapter first discusses the applicability of Discrete Optimisa

tion and how Integer Programming is the most satisfactory method of solving

such problems. It then describes a number of modelling techniques, such as

linearisng products of variables, special ordered sets of variables, logical condi

tions,

disaggregating constraints and variables, column generation etc. The main

solution methods are described, i.e. Branch-and-Bound and Cutting Planes. Fi

nally alternative methods such as Lagrangian Relaxation and non-optimising

methods such as Heuristics and Constraint Satisfaction are outlined.

Keywords:

Integer Program ming, Global Optima, Fixed Costs,Convex Hull, Reformu lation,

Presolve, Logic, Constraint Satisfaction.

1.

The Applicability of Discrete Optimisation

The purpose of this introductory chapter is to give an overview of the scope

for discrete optimisation models and how they are solved. Details of the mod

elling necessary is usually problem specific. Many applications are covered in

other chapters of this book. A fuller coverage of this subject is given in [25]

together with many references.

For solving discrete optimization models, when formulated as (linear) In

teger Programmes (IPs), much fuller accounts, together with extensive refer

ences,

can be found in Nemhauser and Wolsey [19] and Williams [24]. Our

purpose, here, is to make this volume as self contained as possible by describ

ing the main methods.


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4


Limiting the coverage to linear IPs should not be seen as too restrictive in

view of the reformulation possibilities described in this chapter.

It is not possible, totally, to separate modelling from the solution meth

ods. Different types of model will be appropriate for different methods. With

some m ethods it is desirable to modify the model in the course of optimisation.

These tw o considerations are addressed in this chapter.

The modelling of many

physical

systems is dominated by

continuous

(as

opposed todiscrete)mathem atics. Such models are often a simplification of

reality, but the discrete nature ofthereal systems is often at a microscopic level

and continuous modelling provides a satisfactory simplification. What's more,

continuous mathematics is more developed and unified than discrete mathe

matics. The calculus is a powerful tool for the optimisation of many contin

uous problems. There are, however, many systems where such models are

inappropriate. These arise with physical systems (e.g. construction problems,

finite element analysis etc) but are much more common in decision making

(operational research) and information systems (computer science). In many

ways,

we now live in a 'discrete world'. Digital systems are tending to replace

analogue systems.

2. Integer Programm ing

The most common type of model used for discrete optimisation is an

Inte

ger Programm e

(IP) although Constraint Logic Programmes (discussed in the

chapter by Hooker) are also applicable (but give more emphasis to obtaining

feasible rather than optimal solutions). An IP model can be written:

Maximise/Minimise c 'x - i - d 'y (1.1)

Subject to: A x + B y 10

Xi + X2 > 10

xi + 2x4 > 23

(1.81)

(1.82)

(1.83)

X1,

X2,

X3,X4 > 0 and integer

is to leave out some of the conditions or constraints on a (difficult) IP model

and solve the resultant (easier) model. By leaving out constraints we may

well obtain a 'better' solution (better objective value), but one which does not

satisfy all the original constraints. This solution is then used to advantage in

the subsequent process.

For the above example the optimal solution of the LP Relaxation is

. 3 o 5 _ , . . . 1

xi = 4- ,X 2 = 3 , Objective =

12'

6

We have obtained the optimal fractional solution at C. The next step is to de

fine acutting plane which will cut-off the fractional solution at C, without


44/441

24


removing any of the feasible integer solutions (represented by bold dots). This

is known as the

separation

problem. Obviously a number of cuts are possi

ble. The facet defining constraints, are possible cuts as are "shallower" cuts

which are not facets. A major issue is to create a

systematic

way of gener

ating those cuts (ideally facet defining) which separate C from the feasible

integer solutions. Before we discuss this, however, we present a major result

due to Chvatal [6]. This is that all the facet defining constraints (for a PIP) can

be obtained by a finite number of repeated applications of the following two

procedures.

(i)

Addtogethercon straints

in suitable multiples (when all in the same form

eg " < " or " > " ) and add or subtract "= " constraints in suitable m ultiples.

(ii) Divide through the coefficients by their greatest common divisor and

round the right-hand-side coefficient

up (in the case of " > " constraints)

or down (in the case of " < " constraints).

We illustrate this by deriving all the facet defining constraints for the exam

ple above. However, we emphasise that our choice of which constraints to add,

and in what multiples, is ad-hoc. It is a valid procedure but not systematic.

This aspect is discussed later.

1. Add - x i + 2 x 2 < 7

-xi < 0

to give 2^1 + 2x2

0, (2.6)

0


64/441

44 HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION

problem asks to find a maximum clique. Its cardinality is called the clique

number

of the graph and is denoted by a;(G).

Along with the maximum clique problem, we can consider the maximum

independent set problem,

A subset / C

V

is called an

independent set

if the

edge set of the subgraph induced by / is empty. The maximum independent

set problem is to find an independent set of maximum cardinality. The inde

pendent number a{G)

(also called the

stability number)

is the cardinality of a

maximum independent set of

G,

It is easy to observe that / is an independent set of G if and only if / is a

clique of G. Therefore, maximum independent set problem for some graph G

is equivalent to solving m aximum clique problem in the complementary graph

G

and vice versa.

Next, we may associate with each vertex i V of the graph a positive

number

wi

called the vertex

weight.

This way, along with the adjacency matrix

AQ,

we consider the vector of vertex weights

w

G M^. The total weight of a

vertex subset S C.V will be denoted by

ies

The

maximum weight clique problem

asks for a clique

Q

of the largest

W{Q)

value. This value is denoted by u;(G,

w).

Similarly, we can define

maximum

weight independentsetproblem.

The maximum cardinality and the maximum weight clique problems along

with maximum cardinality and the maximum weight independent set problems

are A^P-hard [22], so it is considered unlikely that an exact polynomial time

algorithm for them exists. Approximation of large cliques is also hard. It was

shown in [36] that unless NP == ZPP no polynomial time algorithm can

approximate the clique number within a factor of

n^~^

for any e > 0. In [42]

this margin was tightened to

n/2^^^^^^ ~\

The approaches to the problem offered include such comm on combinatorial

optimization techniques as sequential greedy heuristics, local search heuris

tics,

methods based on simulated annealing, neural networks, genetic algo

rithms, tabu search, etc. However, there are also methods utilizing various

formulations of the clique problem as a continuous optimization problem. An

extensive survey on the maximum clique problem can be found in [5].

The simplest integer formulation of the maximum weight independent set

problem is the following so called edge formulation:

n

maxf(x) =Y2 i i

(2-8)

subject to

Xi+ Xj < 1, V(ij) eE, xe {0 ,l}"". (2.9)


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Continuous Approaches

45

In [60] Shor considered an interesting continuous formulation for the max

imum weight independent set by noticing that edge formulation (2.8)-(2.9) is

equivalent to the following multiquadratic problem:

n

m a x / ( x ) ^

^^WiXi,

(2.10)

subject to

XiXj - 0 , V ( i , j ) eE, Xi^ -Xi-^Q, i=- l , 2 , . . . ,n . (2.11)

Applying dual quadratic estimates, Shor reported good computational results

[60]. Lagrangian bounds based approaches for solving some related discrete

optimization problem on graphs is discussed in [61].

Next two continuous polynomial formulations were proposed in [34, 35].

T H E O R E M

6

Ifx'^ is the solution of the following (continuous)quadraticpro

gram

n

m a x / ( x )

= 2_\î ~ y^ î^j = ^^^ 1/2XÂGX

subject to

0 < î 0 (2.13)

iev

is

A recent direct proof of this theorem can be found in [1]. Similar formula

tions with some heuristic algorithms were proposed in [8, 24].

In [7], the formulation was generalized for the maximum weight clique

problem in the following natural way. Let

Wuim

be the smallest vertex weight

existing in the graph and a vector d

G

R^ be such that

W i

Consider the following quadratic program:

m a x / ( x ) =

X^{AG

+ d iag ( ( i i , , . .

^dn))x

(2.15)

subject to

^Xi l,

x>0. (2.16)

T H E O R E M 9

The global optimum value oftheprogram (2.15)-(2.16) is

1 - ^ ^ .

(2.17)

Furthermore, for each maximum weight clique

Q*

of the graph

G(V,

E) there

is a global maximizer

x*

of

the

program (2,15, 2.16) such that

^* _ /

Wi/uj{G,w), ifi e

Q*

"" ~ " \ 0,

ifieV\Q\

^^-^^^

Obviously, when all

Wi=

1, we have the original M otzk in-Straus formulation.

Properties of maximizers of the Motzkin-Straus formulation were investi

gated in [25]. Generally, since the program (2.12) is not concave, its arbitrary

(local) maximum does not give us the clique number. Furtherm ore, if x* is

some maximizer of (2.12), its nonzero components do not necessarily corre

spond to a clique ofthegraph. However, S. Busygin showed in [7] that nonzero

components of any global maximizer of (2.15)-(2.16) correspond to a complete

multipartite subgraph of the original graph and any m aximal clique of this sub

graph is the maximum weight clique of the graph. Therefore, it is not hard to

infer a maximum weight clique once the maximizer is found.


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47

Performing the variable scaling Xi >y/wiXi, one may transform the for

mulation (2.15H2.16) to

mSixf{x) = xÂ^^^x

(2.19)

subject to

z^x

-= 1, X

>

0, (2.20)

whereAQ^ {a\^^)nxn is such that

if = {

V^ ^

'^HhJ)Ê

(2.21)

0, ifzT^ j a n d ( i , j )

^

;,

zeR"" : Zi= y i (2.22)

and

is the vector

of

square roots of the vertex weights. The attractive property

is this rescaled formulation

is

that maximizers corresponding to cliques of

the same weight are located at the same distance from the zero point. Mak

ing use of this, S. Busygin developed a heuristic algortihm for the maximum

weight clique problem, called QUALEX-MS, which is based on the formula

tion (2.19)-(2.20) and shows a great practical efficiency [7]. Its main idea con

sists in approximating the nonnegativity constraint by a spherical constraint,

which leadsto atrust region problem known to be polynomially solvable.

Then, stationary points of the obtained trust region problem show correlation

with maximum weight clique indicators with a high probability.

A good upper bound on uj{G^w) can be obtained using semidefinite pro

gramming. The value

i}{G,w)= m axz Xz, (2.23)

s.tXij O iiJ) eE, tr(X)-l,

where

z

is defined by (2.22), 5 ^ is the cone of positive sem idefinite

nxn

ma

trices, and tr(X)

XlILi

îi

denotes the

trace

of the matrix

X

is known as

the

(weighted) Lovdsz number

(T^-function) of a graph. It bounds from above

the (weighted) independence number of the graph. Hence, considering it for

the complem entary graph, we obtain an upper bound on u;(G,

w).

It was shown

in [9] that unless

P NP,

there exists no polynom ial-time computable upper

bound on the independence number provably better than the Lovasz number

(i.e., such that whenever there wasagap between the independence number

and the Lovasz number, that bound would be closer to the independence num

ber than the Lovasz num ber). It implies that the Lovasz num ber of the comple

mentary graph is most probably the best possible approximation from above

for the clique number that can be achieved in polynomial time in the worst


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48


case.

For a survey on the Lovasz number and its remarkable properties we

refer the reader to [44].

It is worth m entioning that the semidefinite program (2.23) can be also used

for deriving large cliques ofthegraph. Burer, Monteiro, and Zhang employed a

low-rank restriction upon this program in

ih^xx Max-AO

algorithm and obtained

good computational results on a wide range of the maximum clique problem

instances [6].

5. The Satisfiability Problem

Thesatisfiabilityproblem (SATproblem)

is one of the classical hard com bi

natorial optimization problems. This problem is central in mathematical logic,

computing theory, artificial intelligence, and many industrial application prob

lems. It was the first problem proved to be A/^P-complete [12, 22 ]. More

formally, this problem is defined as follows [32]. Let x i , . . . , x ^ be a set of

Boolean variables whose values are either 0 (false), or 1 (true), and let xi de

note the negation of

x^.

A

literal

is either a variable or its negation. A

clause

is an expression that can be constructed using literals and the logical operation

or

(V). Given a set of

n

clauses Ci, . . . ,

Cn,

the problem is to check if there

exists an assignment of values to variables that makes a Boolean formula of

the following

conjunctive normal form (CNF)

satisfiable:

C i

A

C2 A . . .

A

Cn,

where A is a logical and operation.

SAT problem can be treated as a constrained decision prob lem. Another

possible heuristic approach based on optimization of a non-convex quadratic

programming problem is described in [40, 41]. SAT problem was formulated

as an integer programm ing feasibility problem of the form:

B^w < b,

(2.24)

we{-l,ir, (2.25)

whereB e W^"^, b e R'^ md w e R"".It iseasy to observe that this integer

programming feasibility problem is equivalent to the following non-convex

quadratic programming problem:

m ax w^w (2.26)

B^w < b,

(2.27)

-e


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49

A^w


70/441

50


The correspondence between x and y is defined as follows (for

1

< i < m):

r 1 i f ^ i - l

Xi=

0 gap recognition and

a(G)-upper bounds,

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(2003).

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[12] S. Cook. The complexity of theorem-proving procedures, in: Proc. 3rd

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[33] D. Guesfield,

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Dis-

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[35] J. Harant, A. Pruchnewski, and M. Voigt, On dominating sets and inde

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[43] B. Khoury, P.M. Pardalos, D. Heam, Equivalent formulations of the

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Chapter 3

LOGIC-BASED MODELING

John N Hooker

Graduate School of Industrial A dministration

Carnegie Mellon

University,

Pittsburgh, PA 15213

USA

[email protected]

Abstract Logic-based modeling can result in decision mod els that are more natural and

easier to debug. The addition of logical constraints to mixed integer program

ming need not sacrifice computational speed and can even enhance it if the con

straints are processed correctly. They should be written or automatically refor

mulated so as to be as nearly consistent or hyperarc consistent as possible. They

should also be provided with a tight continuous relaxation. This chapter shows

how to accomplish these goals for a number of logic-based cons traints: formu

las of propositional logic, cardinality formulas, 0-1 linear inequalities (viewed as

logical formulas), cardinality rules, and mixed logical/linear constraints. It does

the same for three global constraints that are popular in constraint programming

systems: the all-different, element and cumulative constraints.

Clarity is as important as computational tractability when building scien

tific m odels. In the broadest sense, models are descriptions or graphic rep

resentations of some phenomenon. They are typically written in a formal or

quasi-formal language for a dual purpose: partly to permit computation of the

mathematical or logical consequences, but equally to elucidate the conceptual

structure of the phenomenon by describing it in a precise and limited vocabu

lary. The classical transportation model, for example, allows fast solution with

the transportation simplex method but also displays the problem as a network

that is easy to understand.

Optimization modeling has generally emphasized ease of computation more

heavily than the clarity and explanatory value of the model. (The transportation

model is a happy exception that is strong on both counts.) This is due in part

to the fact that optimization, at least in the context of operations research, is

often more interested in prescription than description. Practitioners who model


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a manufacturing plant, for example, typically want a solution that tells them

how the plant should be run. Yet a succinct and natural model offers several

advantages: it is easier to construct, easier to debug, and more conducive to

understanding how the plant works.

This chapter explores the option of enriching mixed integer programming

(MILP) models with logic-based constraints, in order to provide more natural

and succinct expression of logical conditions. D ue to formulation and solution

techniques developed over the last several years, a modeling enrichment of this

sort need not sacrifice computational tractability and can even enhance it.

One historical reason for the de-emphasis of perspicuous models in opera

tions research has been the enormous influence of linear programming. Even

though it uses a very small number of primitive terms, such as linear inequal

ities and equations, a linear programming model can formulate a remarkably

wide range of problems. The linear format almost always allows fast solution,

unless the model is truly huge. It also provides such analysis tools as reduced

costs, shadow prices and sensitivity ranges. There is therefore a substantial

reward for reducing a problem to linear inequalities and equations, even when

this obscures the structure of the problem.

When one moves beyond linear models, however, there are less compelling

reasons for sacrificing clarity in order to express a problem in a language

with a small number of primitives. There is no framework for discrete or dis

crete/continuous models, for exam ple, that offers the advantages of linear pro

gramming. The mathematical programming community has long used MILP

for this purpose, but MILP solvers are not nearly as robust as linear program

ming solvers, as one would expect because MILP formulates NP-hard prob

lems. Relatively small and innocent-looking problems can exceed the capabil

ities of any existing solver, such as the market sharing problems identified by

Williams [35] and studied by Comuejols and Dawande [16]. Even tractable

problems may be soluble only when carefully formulated to obtain a tight lin

ear relaxation or an effective branching scheme.

In addition MILP often forces logical conditions to be expressed in an un

natural way, perhaps using big-M constraints and the like. The formulation

may be even less natural if one is to obtain a tight linear relaxation. Current

solution technology requires that the traveling salesman p roblem, for exam ple,

be written with exponentially many constraints in order to represent a simple

all-different condition. MILP may provide no practical formulation at all for

important problem classes, including some resource-constrained scheduling

problems.

It is true that MILP has the advantage of a unified solution approach, since

a single branch-and-cut solver can be applied to any MILP model one might

write. Yet the introduction of logic-based and other higher-level constraints no

longer sacrifices this advantage.


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Logic-Based Modeling

63

The discussion begins in Section 3.1 with a brief description of how solvers

can accommodate logic-based constraints: namely, by automatically convert

ing them to MILP constraints, or by designing a solver that integrates MILP

and constraint programm ing. Section 3.2 describes what constitutes a good

formulation for M ILP and for an integrated solver. The rem aining sections de

scribe good formulations for each type of constraint: formulas of propositional

logic, cardinality formulas, 0-1 linear inequalities (viewed as logical formu

las),

cardinality rules, and mixed logical/linear constraints. Section 3.8 briefly

discusses three global constraints that are popular in constraint programming

systems: the all-different, element and cumulative constraints. They are not

purely logical constraints, but they illustrate how logical expressions are a

special case of a more general approach to modeling that offers a variety of

constraint types. The chapter ends with a summary.

1. Solvers for Logic-Based Constraints

1.1 Two Approaches

There are at least two ways to deal with logic-based constraints in a unified

solver.

One possibility is to provide automatic reformulation of logic-based con

straints into MIL P constraints, and then to apply a standard M ILP solver

[26,

28]. The reformulation can be designed to result in a tight linear

relaxation. This is a viable approach, although it obscures some of the

structure of the problem.

A second approach is to design a unified solution method for a diversity

of constraint types, by integrating MILP with constraint programming

(CP). Surveys of the relevant literature on hybrid solvers may be found

in [11, 18 ,21 ,22 ,27 ,3 6] .

Whether one uses automatic translation or a CP/MILP hybrid approach,

constraints must be written or automatically reformulated with the algorith

mic implications in mind. A good formulation for M ILP is one with a tight

linear relaxation. A good formulation for CP is as nearly "consistent" as pos

sible. A consistent constraint set is defined rigorously below, but it is roughly

analogous to a convex hull relaxation in MILP. A good formulation for a hy

brid approach should be good for both MILP and CP whenever possible, but

the strength of a hybrid approach is that it can benefit from a formulation that

is good in either sense.


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1.2 Structured Groups of Constraints

Very often, the structure of a problem is not adequately exploited unless

constraints are considered in groups. A group of constraints can generally

be given a MILP translation that is tighter than the combined translations of

the individual constraints. The consistency-maintenance algorithms of CP are

more effective when applied to a group of constraints whose overall structure

can be exploited.

Structured groups can be identified and processed in three ways.

Automatic detection.

Design solvers to detect special structure and process

it appropriately, as is commonly done for network constraints in MILP

solvers. However, since modelers are generally aware of the problem

structure, it seems more efficient to obtain this information from them

rather than expecting the solver to rediscover it.

Hand

coding.

Ask modelers to exploit structured constraint groups by hand.

They can write a good MILP formulation for a group, or they can write

a consistent formulation.

Structurallabeling.

Let modelers label specially structured constraint groups

so that the solver can process them accordingly. The CP comm unity

implements this approach with the concept of a

global

constraint, which

represents a structured set of more elemen tary constraints.

As an exam ple of the third approach, the global constraint all-different(a,

h,

c)

requires that a,handctake distinct values and thus represents three inequations

a ^ h, a ^ c and h

i=-

c. To take another example, the global constraint

cnf(a V 6, a V -i6) can alert the solver that its argum ents are propositional

formulas in conjunctive normal form (defined below). This allows the solver

to process the constraints with specialized inference algorithms.

2. Good Formulations

2,1 Tight Relaxations

A good formulation of an MILP model should have a tight linear relaxation.

The tightest possible formulation is a

convex hull formulation,

whose continu

ous relaxation describes the convex hull of

the

model's feasible set. T he convex

hull is the intersection of all half planes containing the feasible set. At a min

imum it contains inequalities that define all the facets of the convex hull and

equations that define the affine hull (the smallest dimensional hyperplane that

contains the feasible set). Such a formulation is ideal because it allows one to

find an optimal solution by solving the linear programming relaxation.

Since there may be a large number of facet-defining inequa lities, it is com

mon in practice to generate

separating

inequalities that are violated by the


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65

optimal solution of the current relaxation. This must be done during the so

lution process, however, since at the modeling stage one does not know what

solutions will be obtained from the relaxation. Fortunately, som e comm on con

straint types may have a convex hull relaxation that is simple enough to analyze

and describe in advance. Note, however, that even when one writes a convex

hull formulation of each constraint or each structured subset of constraints, the

model as a whole is generally not a convex hull formulation.

It is unclear how to measure the tightness of a relaxation that does not de

scribe the convex hull. In practice, a "tight" relaxation is simply one that pro

vides a relatively good bound for problems that are commonly solved.

2.2 Consistent Form ulations

A good formulation for CP is

con sistent,

meaning that its constraints explic

itly rule out assignments of values to variables that cannot be part of a feasible

solution. No te that consistency is not the same as satisfiability, as the term

might suggest.

To make the idea more precise, suppose that constraint set S contains vari

a b l e s x i , . . . ,Xn. Each variable Xj has a domain Dj, which is the initial set

of values the variable may take (perhaps the reals, integers, etc.) Let SLpartial

assignment

(known as a

compound label

in the constraints community) specify

values for some subset of the variables. Thus a partial assignment has the form

(^ji ^"'^^3k) = i^h^"'^ ^jk)^ where eachji e D j^ (3.1)

A partial assignment (3.1) is

redundant

for

S

if it cannot be extended to a

feasible solution ofS. That is, every complete assignment

(x i , ...,Xn)='{vi,..., Vn),

where each

j

G

Dj

that is consistent with (3.1) violates some constraint in

S.

By convention, a partial assignment violates a constraint Conly if it assigns

some value to every variable in C Thus if Di = D2 = 71,the assignment

x i ==

1

does not violate the constraint xi + x^ > 0 since

X2

has not been

assigned a value. The assignment is redundant, however, since it cannot be

extended to a feasible solution of

xi + x^ >

0. No value in X2's domain will

work.

A constraint set

S

is

consistent

if every redundant partial assignment vi

olates some constraint in

S.

Thus the constraint set

{xi

+ ^2 > 0} is not

consistent.

It is easy to see that a consistent constraint set

S

can be solved without

backtracking, provided S is satisfiable. First assign x i a value vi GDi that

violates no constraints inS (which is possible because S is satisfiable). Now

assignX2 a valueV2 GD2such that ( x i , 2:2) = ('? i,^ 2)violates no constraints


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67

the smallest finite domains. Bounds consistency is generally maintained by

interval arithmetic and by specialized algorithms for global constraints with

numerical variables.

Both hyperarc and bounds consistency tend to reduce backtracking, because

CP systems typically branch by splitting a domain. If the domain is small or

narrow, less splitting is required to find a feasible solution. Small and narrow

domains also make domain reduction more effective as one descends into the

search tree.

2.3 Prime Implications

There is a weaker property than consistency, namely completeness, that can

be nearly as helpful when solving the problem. Com pleteness can be achieved

for a constraint set by generating its prime im plications, w hich, roughly speak

ing, are the strongest constraints that can be inferred from the constraint set.

Thus when it is impractical to write a consistent formulation, one may be able

to write the prime implications of the constraint set and achieve much the same

effect.

Recall that a constraint set

S

is consistent if any redundant partial assign

ment for

S

violates some constraint in

S. S

is

complete

if any redundant partial

assignment for

S

is

redundant

for some constraint in

S,

Checking whether a

partial assignment is redundant for a constraint is generally harder than check

ing whether it violates the constraint, but in many cases it is practical nonethe

less.

In some contexts a partial assignment is redundant for a constraint only if it

violates the constraint. In such cases com pleteness implies consistency.

Prime implications, roughly speaking, are the strongest constraints that can

be inferred from a constraint set. To develop the concept, suppose constraints

C and D contain variables m x (x i , . . . ,Xn ) . C impliesconstraint D if

all assignments to

x

that satisfy

C

also satisfy

D.

Constraints

C

and

D

are

equivalentif they imply each other.

Let H be some family of constraints, such as logical clauses, cardinality

formulas, or 0-1 inequalities (defined in subsequent sections). We assume H

is semantically finite, meaning that there are a finite number of nonequivalent

constraints in

R.

For example, the set of 0-1 linear inequalities in a given

set of variables is semantically finite, even though there are infinitely many

inequalities.

Let an

H-implication

of a constraint set 5 be a constraint in

H

that

S

im

plies. Constraint C isdi prime H-implication ofS'if C is a iJ-imp lication of 5 ,

and every /^-implication ofS that impliesCis equivalent toC. The following

is easy to show.


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L E M M A 3.1

Given

a sem anticallyfinite

constraint

set H, every H -implication

of a constraint set S is implied by some prime H-implication of S. Thus if

S C H, S is equivalent to the set ofitsprime H-implications.

For example, let

H

be the family of 0-1 linear inequalities in variables

xi, X2,

and let

S

consist of

^ 1 + 3:2 > 1

^ 1 3: 2 ^ 0

S has the single prime ^-im plic atio n x i > 1 (up to equivalence). Any 0-1

inequality implied by S is implied by xi > 1. Since S C H, S isequivalent

to {xi >1}.

Lemma 3.1 may not apply when H is semantically infinite. For instance,

let xi have domain [0,1] and H be the set of constraints of the form xi > a

for a e [0,1 ). Then all constraints in H are implied by 5' =: {xi > 1}

but none are implied by a prime i7-implication of

S,

since

S

has no prime

iJ-implications.

The next section states precisely how Lemma 3.1 allows one to recognize

redundant partial assignments.

2.4 Recognizing Redundant Partial Assignments

Let a clause in variables x i , . . . , x^i with domains D i , . . . , D^ be a con

straint of the form

{xj, ^ ^1) V . . . V{xj^ + v^) (3.2)

where each

2%

^ {I? ?^} and each f

.

G L>j.. A clause with zero disjuncts

is necessarily false, by convention. A constraint set H contains all clauses

for a set of variables if every clause in these variables is equivalent to some

constraint in

H,

L E M M A 3.2

If a semantically finite constraint set H contains all clauses for

the variables in constraint set S, thenanypartial assignm ent that is redundant

for S is redundant for someprime H -implication of S.

This is easy to see. If a partial assignment

{xj^

,...,

Xj^) =

( t ' l , . . . , ^p)

is redundant for

S,

then

S

implies the clause (3.2). By Lemma 3 .1, some

prime if-implication

P of S

implies (3.2). This means the partial assignment

is redundant for P.

As an example, consider the constraint setS consisting of

2x1+ 3x2> 4: (a)

(3 3)

3x1 + 2x2 < 5 (6)


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69

where the domain of eachXj is{0 ,1 ,2} . Since the only solutions of (3.3) are

X

= (1,1 ) and (0 ,2 ), there are two redundant partial assignments: x i

==

2 and

X2

0. Let

H

consist of all linear inequalities in xi,

X2,

plus c lauses in a;i, X2.

The //-prime implications of (3.3), up to equivalence, are (3.3b) and

xi +

2^2 > 3

-x i

+ 2x2 > 1 (3.4)

2 ^1 X2 < 1

It can be checked that each redundant partial assignment is redundant for at

least one (in fact two) of the prime implications. Note that the redundant par

tial assignment X2 == 0 is not redundant for either of the original inequalities

in (3.3); only for the inequalities taken together. Knowledge of the prime im

plications therefore makes it easier to identify this redundancy.

A constraint set

S

is

complete

if every redundant partial assignment for

S

is

redundant for some constraint inS. From Lemm a 3.2 we have:

C O R O L L A R Y

3 .3

If S contains all of

its

prime H-implications for some se-

man tically finite constraint set H that contains all clauses for the variables in

S, then S is complete.

Thus if inequalities (3.4) are added to those in (3.3), the result is a complete

constraint set.

3 . Prepositional Logic

3 ,1 Basic Ideas

A formal logic

is a language in which deductions are made solely on the

basis of the form of statements, without regard to their specific meaning. In

propositional logic,

a statement is made up of

atomic propositions

that can be

true or false. The form of

the

statement is given by how the atomic propositions

are joined or modified by such logical expressions as

not

(-i),

and

(A), and

inclusiveor(V).

A proposition defines a

prop ositional function

that maps the truth values

of its atomic propositions to the truth value of the whole proposition. For

example, the proposition

(a

V

-n6) A (-la

V

h)


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70


contains atomic propositions a,b and defines a function / given by the follow

ing truth table:

a b f{a,b)

0 0 1

0 1 0 (3.5)

1 0 0

1 1 1

Here 0 and 1 denotefalse and true respectively. One can define additional

symbols for implication (>), equivalence (= ) , and so forth. For any pair of

formulas A,

B

A-^

B

=def

-^Ay B

A = B

-def

{A-^B)A{B-^A)

Note that (3.5) is the truth table for equivalence. A

tautology

is a formula

whose propositional function is identically true.

A formula of propositional logic can be viewed as a constraint in which

the variables are atomic propositions and have domains {0,1}, where 0 and

1 signify false and true. The constraint is satisfied when the formula is true.

Many complex logical conditions can be naturally rendered as propositional

formulas, as for exam ple the following.

Alice will go to the party only a -^ c

if Charles goes.

Betty will not go to the party if (aV d) -^-^b

Alice or Diane goes.

Charles will go to the party

c^

{-^d

A

~ie)

unless Diane or Edward goes.

Diane will not go to the party d

>

(c

V

e)

unless Charles or Edward goes.

Betty and Charles never go to the

-^{bAc)

same party.

Betty and Edward are always b = e

seen together.

Charles goes to every party that 6

>

c

Betty goes to.

3 .2 Conversion to Clausal Form

It may be useful to convert formulas to clausal form, particularly since the

well-known resolution algorithm is designed for clauses, and clauses have an

obvious MILP representation.

In propositional logic, clauses take the form of conjunctions of

literals,

which are atomic propositions or their negations. We will refer to a clause

(3.6)


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Logic-Based Modeling 1

1

of this sort as a

propositional clause,

A propositional formula is in

clausal

form, also known asconjunctive norm al form (CNF), if it is a conjunction of

one or more clauses.

Any propositional formula can be converted to clausal form in at least two

ways. The more straightforward conversion requires exponential space in the

worst case. It is accomplished by applying som e elementary logical equiva

lences.

De Morgan slaws

-^{FAG)

= -^Fy-^G

Distribution laws F A{GW H ) = {F AG) W {F A H)

FWIGAH)

- ( F V G ) A ( F V i f )

Dou ble negation -i-nF = F

(Of the two distribution laws, only the second is needed.) For example, the

formula

( a A - n6 ) V - n ( a V - i 6 )

may be converted to CNF by first applying De Morgan's law to the second

disjunct,

(a A -16) V (-la A b)

and then applying distribution,

(a V - la) A (a V 6) A (-16 V - ia) A {-^b V b)

The two tautologous clauses can be dropped. Similarly, the propositions in

(3.6) convert to the following clauses.

- la V c

- laV -16

- i c V - i d

ic

V ie

c V - d V e

^^'^^

-^bW -^c

-16 V e

6 V - i e

1 6 V c

The precise conversion algorithm appears in Fig. 3.1.

Exponential growth for this type of conversion can be seen in propositions

of the form

{aiAbi)V ,.,y{anAbn) (3.8)


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Let

F

be the formula to be converted.

SetA;= and 5 = 0.

R e p l a c e a l l s u b f o r m u l a s o f t h e f or m G~H w i t h {G ^ H) A {H ^ G) .

R e p l a c e a l l s u b f o r m u l a s o f t h e f or m

GDH

w i t h

->G W

H.

P e r f o r m C o n v e r t ( F ) .

T h e CNF f o rm i s t h e c o n j u n c t i o n o f c l a u s e s i n S.

F u n c t i o n C o n v e r t ( F )

If F i s a c l a u s e th e n a d d F to S.

Else if F h a s t h e f o r m - i -i G t he n p e r f o r m C o n v e r t ( G ) .

Else if F h a s t h e f o rm GAH then

p e r f o r m C o n v e r t ( G ) a n d C o n v e r t ( / / ) .

Else if F h a s t h e f o rm -^{G A H) th en p e r f o r m C o n v er t( -i G V - i / / ) .

Else if

F

h a s t h e f o rm

-^{G V H)

th en p e r f o r m C o n v er t( -i G A - i / / ) .

Else if F h a s t h e f o rm GV {H Al) then

P e r f o r m C o n v e rt (C V i f ) a n d C o n ve r t( G V / ) .

Figure 3.1. Convers ion of F to C N F wi thou t addi t ional variables. A form ula of the form

{H A I) V G IS

regarde d as having the form

GV {H A I).

The formula translates to the conjunction of 2^ clauses of the form Li V ... V

Ln,

where eachLj isaj or bj.

By adding new variables, however, conversion to CNF can be accom plished

in linear time. The idea is credited to Tseitin [34] but Wilson's more compact

version [38] simply replaces a disjunction F

V

G with the conjunction

(x i

VX2)

A

{-^xiV

F ) A (^^2

V

G),

where xi,

X2

are new variables and the clauses

-^xi

V

F

and -1x2 V

G

encode

implications

xi -^ F

and

X2> G,

respectively. For example, formula (3.8)

yields the conjunction,

n

(x i V . . .

V

Xn)

A

^

{-^Xj

V

aj

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