Ant Colony Optimization and Constraint Programming

228 Pages·2010·4.911 MB·English

by  Christine Solnon

Ant Colony Optimization and Constraint Programming
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Ant Colony
Optimization and
Constraint Programming

Christine Solnon

Series Editor
Narendra Jussien

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First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Adapted and updated from Optimisation par colonies de fourmis published 2008 in France by Hermes
Science/Lavoisier © LAVOISIER 2008

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
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undermentioned address:
ISTE Ltd John Wiley & Sons, Inc.
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© ISTE Ltd 2010

The rights of Christine Solnon to be identified as the author of this work have been asserted by her in
accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Solnon, Christine.
[Optimisation par colonies de fourmis. English]
Ant colony optimization and constraint programming / Christine Solnon.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-130-8
1. Constraint programming (Computer science) 2. Mathematical optimization. 3. Swarm intelligence.
4. Ant algorithms. I. Title.
QA76.612.S6513 2010
005.1'16--dc22
2009050443

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-130-8
Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne

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Table of Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Overviewofthebook . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1. Constraintprogramming . . . . . . . . . . . . . . . . . . 3
1.1.2. Antcolonyoptimization . . . . . . . . . . . . . . . . . . 4
1.1.3. Constraintprogrammingwithantcolonyoptimization . 4
Chapter2.ComputationalComplexity . . . . . . . . . . . . . . . . 7
2.1. Complexityofanalgorithm . . . . . . . . . . . . . . . . . . . 8
2.2. Complexityofaproblem . . . . . . . . . . . . . . . . . . . . . 10
2.2.1. TheP class . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2. TheNP class . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3. NP-completeproblems . . . . . . . . . . . . . . . . . . 12
2.2.4. NP-hardproblems . . . . . . . . . . . . . . . . . . . . . 14
2.2.5. Undecidableproblems . . . . . . . . . . . . . . . . . . . 14
2.2.6. Complexityofoptimizationproblems . . . . . . . . . . 15
2.3. Wherethemostdifficultinstancescanbefound . . . . . . . . 15
2.3.1. Phasetransition . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2. Searchlandscape . . . . . . . . . . . . . . . . . . . . . . 19
2.4. SolvingNP-hardproblemsinpractice . . . . . . . . . . . . . 21
2.4.1. Exploitationofparticularcases . . . . . . . . . . . . . . 23
2.4.2. Approximationalgorithms . . . . . . . . . . . . . . . . . 23
2.4.3. Heuristicsandmetaheuristics . . . . . . . . . . . . . . . 24
2.4.4. Structuringandfilteringthesearchspace . . . . . . . . 24
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PARTI.CONSTRAINTPROGRAMMING . . . . . . . . . . . . . . . . 27
IntroductiontoPartI . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter3.ConstraintSatisfactionProblems . . . . . . . . . . . . . 31
3.1. Whatisaconstraint? . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1. Definitionofaconstraint. . . . . . . . . . . . . . . . . . 32
3.1.2. Arityofaconstraintandglobalconstraints . . . . . . . 33
3.2. Whatisaconstraintsatisfactionproblem? . . . . . . . . . . . 33
3.2.1. ComplexityofCSPs . . . . . . . . . . . . . . . . . . . . 34
3.3. OptimizationproblemsrelatedtoCSPs . . . . . . . . . . . . . 35
3.3.1. Maximizingconstraintsatisfaction . . . . . . . . . . . . 35
3.3.2. Constrainedoptimization . . . . . . . . . . . . . . . . . 36
3.4. Then-queensproblem . . . . . . . . . . . . . . . . . . . . . . 37
3.4.1. Descriptionoftheproblem . . . . . . . . . . . . . . . . 37
3.4.2. FirstCSPmodel . . . . . . . . . . . . . . . . . . . . . . 38
3.4.3. SecondCSPmodel . . . . . . . . . . . . . . . . . . . . . 39
3.4.4. ThirdCSPmodel . . . . . . . . . . . . . . . . . . . . . . 40
3.4.5. Influenceofthemodelonthesolutionprocess . . . . . 41
3.5. Thestablemarriageproblem . . . . . . . . . . . . . . . . . . . 43
3.5.1. Descriptionoftheproblem . . . . . . . . . . . . . . . . 43
3.5.2. CSPmodel . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6. RandomlygeneratedbinaryCSPs . . . . . . . . . . . . . . . . 46
3.7. Thecarsequencingproblem . . . . . . . . . . . . . . . . . . . 47
3.7.1. Descriptionoftheproblem . . . . . . . . . . . . . . . . 47
3.7.2. CSPmodel . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter4.ExactApproaches . . . . . . . . . . . . . . . . . . . . . . 53
4.1. Constructionofasearchtree . . . . . . . . . . . . . . . . . . . 53
4.2. Constraintpropagation . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1. Forwardchecking. . . . . . . . . . . . . . . . . . . . . . 57
4.2.2. Maintainingarcconsistency . . . . . . . . . . . . . . . . 59
4.3. Orderingheuristics . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1. Heuristicsforchoosingvariables . . . . . . . . . . . . . 61
4.3.2. Heuristicsforchoosingvalues. . . . . . . . . . . . . . . 62
4.3.3. Randomizedrestart . . . . . . . . . . . . . . . . . . . . . 63
4.4. Fromsatisfactiontooptimizationproblems . . . . . . . . . . 63
4.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
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Chapter5.PerturbativeHeuristicApproaches . . . . . . . . . . . . 69
5.1. Geneticalgorithms . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.1. Basicprinciples . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.2. UsingGAstosolveCSPs . . . . . . . . . . . . . . . . . 73
5.2. Localsearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.1. Basicprinciples . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.2. MetaheuristicsbasedonLS . . . . . . . . . . . . . . . . 75
5.2.3. UsingLStosolveCSPs . . . . . . . . . . . . . . . . . . 77
5.3. Particleswarmoptimization . . . . . . . . . . . . . . . . . . . 78
5.3.1. Basicprinciples . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.2. UsingPSOtosolveCSPs . . . . . . . . . . . . . . . . . 80
5.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter6.ConstructiveHeuristicApproaches . . . . . . . . . . . . 85
6.1. Greedyrandomizedapproaches . . . . . . . . . . . . . . . . . 86
6.1.1. Basicprinciples . . . . . . . . . . . . . . . . . . . . . . . 86
6.1.2. UsinggreedyrandomizedalgorithmstosolveCSPs . . 88
6.2. Estimationofdistributionalgorithms . . . . . . . . . . . . . . 88
6.2.1. Basicprinciples . . . . . . . . . . . . . . . . . . . . . . . 88
6.2.2. UsingEDAstosolveCSPs . . . . . . . . . . . . . . . . 90
6.3. Antcolonyoptimization . . . . . . . . . . . . . . . . . . . . . 90
6.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Chapter7.ConstraintProgrammingLanguages . . . . . . . . . . . 93
7.1. Constraintlogicprogramming . . . . . . . . . . . . . . . . . . 94
7.2. Constraintprogramminglibraries . . . . . . . . . . . . . . . . 96
7.3. Constraint-basedlocalsearch . . . . . . . . . . . . . . . . . . 96
7.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
PARTII.ANTCOLONYOPTIMIZATION . . . . . . . . . . . . . . . 101
IntroductiontoPartII . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Chapter8.FromSwarmIntelligencetoAntColonyOptimization 105
8.1. Complexsystemsandswarmintelligence . . . . . . . . . . . 106
8.2. Searchingforshortestpathsbyantcolonies . . . . . . . . . . 108
8.3. Antsystemandthetravelingsalesmanproblem . . . . . . . . 111
8.3.1. Pheromonestructure . . . . . . . . . . . . . . . . . . . . 112
8.3.2. ConstructionofaHamiltoniancyclebyanant . . . . . 114
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8.3.3. Pheromoneupdatingstep . . . . . . . . . . . . . . . . . 115
8.3.4. Artificialversusrealants. . . . . . . . . . . . . . . . . . 115
8.4. GenericACOframework . . . . . . . . . . . . . . . . . . . . . 116
8.4.1. Pheromonestructureandconstructiongraph . . . . . . 116
8.4.2. Constructionofcombinationsbyants . . . . . . . . . . 118
8.4.3. Improvingcombinationswithlocalsearch . . . . . . . . 120
8.4.4. Pheromoneupdatingstep . . . . . . . . . . . . . . . . . 121
8.4.5. ParametersofanACOalgorithm . . . . . . . . . . . . . 122
Chapter9.IntensificationversusDiversification . . . . . . . . . . . 125
9.1. ACOmechanismsforintensifyingthesearch . . . . . . . . . 125
9.2. ACOmechanismsfordiversifyingthesearch . . . . . . . . . 127
9.3. Balancingintensificationanddiversification . . . . . . . . . . 128
9.4. Measuresofdiversification/intensification . . . . . . . . . . . 135
9.4.1. Theλ-branchingfactor. . . . . . . . . . . . . . . . . . . 136
9.4.2. Resamplingratio . . . . . . . . . . . . . . . . . . . . . . 136
9.4.3. Similarityratio . . . . . . . . . . . . . . . . . . . . . . . 137
Chapter10.BeyondStaticCombinatorialProblems . . . . . . . . 141
10.1.Multi-objectiveproblems. . . . . . . . . . . . . . . . . . . . . 141
10.1.1. Definitionofmulti-objectiveproblems . . . . . . . . . . 141
10.1.2. Solvingmulti-objectiveproblemswithACO . . . . . . 143
10.2.Dynamicoptimizationproblems. . . . . . . . . . . . . . . . . 145
10.2.1. Definitionofdynamicoptimizationproblems . . . . . . 145
10.2.2. SolvingdynamicoptimizationproblemswithACO. . . 146
10.3.Optimizationproblemsovercontinuousdomains . . . . . . . 147
10.3.1. Definitionofcontinuousoptimizationproblems. . . . . 147
10.3.2. SolvingcontinuousoptimizationproblemswithACO . 148
Chapter11.ImplementationIssues . . . . . . . . . . . . . . . . . . . 151
11.1.Datastructures. . . . . . . . . . . . . . . . . . . . . . . . . . . 151
11.1.1. Datastructuresassociatedwithpheromonefactors . . . 151
11.1.2. Datastructuresassociatedwithheuristicfactors. . . . . 153
11.1.3. Datastructuresassociatedwithants . . . . . . . . . . . 154
11.2.Selectionofacomponentwithrespecttoprobabilities . . . . 154
11.3.Implementationofalocalsearchprocedure . . . . . . . . . . 157
11.4.Computationofdiversification/intensificationmeasures . . . 157
11.4.1. Resamplingratio . . . . . . . . . . . . . . . . . . . . . . 158
11.4.2. Similarityratio . . . . . . . . . . . . . . . . . . . . . . . 158
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PARTIII.CPWITHACO . . . . . . . . . . . . . . . . . . . . . . . . 161
IntroductiontoPartIII . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Chapter12.SequencingCarswithACO . . . . . . . . . . . . . . . . 165
12.1.Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
12.2.Afirstpheromonestructureforidentifyinggoodcar
sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12.2.1. Pheromonestructure . . . . . . . . . . . . . . . . . . . . 167
12.2.2. Constructionofasequencebyanant . . . . . . . . . . . 168
12.2.3. Pheromonelayingstep . . . . . . . . . . . . . . . . . . . 170
12.3.Asecondpheromonestructureforidentifyingcriticalcars . . 171
12.3.1. Pheromonestructure . . . . . . . . . . . . . . . . . . . . 171
12.3.2. Constructionofasequencebyanant . . . . . . . . . . . 172
12.3.3. Pheromoneupdatingstep . . . . . . . . . . . . . . . . . 172
12.4.Combiningthetwopheromonestructures . . . . . . . . . . . 173
12.4.1. Firstpheromonestructure . . . . . . . . . . . . . . . . . 173
12.4.2. Secondpheromonestructure. . . . . . . . . . . . . . . . 173
12.4.3. Constructionofasequencebyanant . . . . . . . . . . . 173
12.5.ComparisonofthedifferentACOalgorithms . . . . . . . . . 174
12.5.1. Consideredalgorithms . . . . . . . . . . . . . . . . . . . 174
12.5.2. Testsuite. . . . . . . . . . . . . . . . . . . . . . . . . . . 175
12.5.3. Parametersettings . . . . . . . . . . . . . . . . . . . . . 175
12.5.4. Experimentalresults . . . . . . . . . . . . . . . . . . . . 177
12.6.ComparisonofACOwithstate-of-the-artapproaches. . . . . 178
12.6.1. Consideredapproaches. . . . . . . . . . . . . . . . . . . 178
12.6.2. IDWalk . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
12.6.3. VFLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
12.6.4. Experimentalset-up . . . . . . . . . . . . . . . . . . . . 180
12.6.5. Experimentalresults . . . . . . . . . . . . . . . . . . . . 180
12.7.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Chapter13.SubsetSelectionwithACO . . . . . . . . . . . . . . . . 185
13.1.Subsetselectionproblems . . . . . . . . . . . . . . . . . . . . 186
13.1.1. Maximumclique . . . . . . . . . . . . . . . . . . . . . . 186
13.1.2. Multidimensionalknapsack . . . . . . . . . . . . . . . . 187
13.1.3. MaximumBooleansatisfiability . . . . . . . . . . . . . 187
13.1.4. Maximumconstraintsatisfaction . . . . . . . . . . . . . 187
13.1.5. Minimumvertexcover . . . . . . . . . . . . . . . . . . . 188
13.1.6. Maximumcommonsubgraph . . . . . . . . . . . . . . . 188
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13.1.7. Edge-weightedk-cardinalitytree . . . . . . . . . . . . . 189
13.2.DescriptionofAnt-SSP. . . . . . . . . . . . . . . . . . . . . . 189
13.2.1. Constructionofacombinationbyanant . . . . . . . . . 190
13.2.2. Pheromonelayingstep . . . . . . . . . . . . . . . . . . . 192
13.3.InstantiationsofAnt-SSPwithrespecttotwopheromone
strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
13.3.1. Thevertexpheromonestrategy . . . . . . . . . . . . . . 193
13.3.2. Thecliquepheromonestrategy . . . . . . . . . . . . . . 193
13.3.3. Comparisonofthetwostrategies . . . . . . . . . . . . . 194
13.4.InstantiationofAnt-SSPtosolveCSPs . . . . . . . . . . . . . 196
13.4.1. Heuristicfactor . . . . . . . . . . . . . . . . . . . . . . . 196
13.4.2. Localsearch . . . . . . . . . . . . . . . . . . . . . . . . . 197
13.5.Experimentalresults . . . . . . . . . . . . . . . . . . . . . . . 197
13.5.1. Randomlygeneratedbinaryinstances . . . . . . . . . . 197
13.5.2. Resultsoninstancesofthe2006solvercompetition . . 199
13.6.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Chapter14.IntegrationofACOinaCPLanguage . . . . . . . . . 205
14.1.FrameworkforintegratingACOwithinaCPlibrary . . . . . 206
14.1.1. Pheromonestrategy. . . . . . . . . . . . . . . . . . . . . 207
14.1.2. Constructionofassignments. . . . . . . . . . . . . . . . 208
14.1.3. Pheromoneupdatingstep . . . . . . . . . . . . . . . . . 210
14.2.IllustrationofACO-CPonthecarsequencingproblem . . . . 210
14.2.1. CSPmodel . . . . . . . . . . . . . . . . . . . . . . . . . 210
14.2.2. Variableorderingheuristic . . . . . . . . . . . . . . . . . 211
14.2.3. Pheromonestrategies . . . . . . . . . . . . . . . . . . . . 211
14.2.4. Heuristicfactor . . . . . . . . . . . . . . . . . . . . . . . 212
14.2.5. Experimentalresults . . . . . . . . . . . . . . . . . . . . 213
14.3.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Chapter15.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 215
15.1.Towardsconstraint-basedACOsearch . . . . . . . . . . . . . 215
15.2.TowardsareactiveACOsearch . . . . . . . . . . . . . . . . . 216
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
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Foreword
Combinatorial optimization has a very special place in computer science.
Ontheonehand,thisfieldaddressesfundamentalproblemssuchasscheduling,
resourceallocationandvehiculerouting,whicharecentraltooureconomies.
On the other hand, combinatorial optimization problems are extremely hard
from a computational complexity standpoint: it is very unlikely that an
efficient algorithm able to solve all these problems efficiently exists and
that a single approach would outperform all others in this field. Different
combinatorial problems, or even different instances of the same application,
may be solved by very different techniques or by a combination of some of
them. Moreover, whatever the approach considered, solving a combinatorial
optimization problem usually requires a significant amount of programming
andexperimentationwork.
Inthisbook,ChristineSolnonfocusesonAntColonyOptimization(ACO),
a relatively recent approach for solving combinatorial problems. The topic
is relevant: during the last decade, ACO has gradually evolved from an
intellectual curiosity to a metaheuristic that has obtained outstanding results
onsomeapplications.Thisisthecase,forexample,ofschedulinginassembly
lines: a particularly difficult application for which ACO is able to solve a
largeclassofinstanceswithaveryimpressiveefficiencyandsuccessrate.The
scientific article published by the author on this subject was, indeed, a true
revelationformanyresearchers.
However, this book does not introduce ACO in an isolated way, but
providesanoverviewofmanyapproaches.Thefirstpartofthebookprovides
ashortbutexcellentsummaryofthestateoftheart,withafocusonconstraint
satisfaction problems. Not only does this presentation clearly identify ACO
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