The linear ordering problem: Exact and heuristic methods in combinatorial optimization

184 Pages·2011·1.66 MB·English

by  Rafael Martí

Applied Mathematical Sciences
Volume 175
Editors
S.S. Antman J.E. Marsden L. Sirovich
Advisors
P. Holmes J. Keener J. Keller R. Laubenbacher
B.J. Matkowsky A. Mielke C.S. Peskin
K.R.S. Sreenivasan A. Stevens A. Stuart
Forfurthervolumes:
http://www.springer.com/series/34
Rafael Martí (cid:129) Gerhard Reinelt
The Linear Ordering Problem
Exact and Heuristic Methods
in Combinatorial Optimization
123
Prof.Dr.RafaelMartí Prof.Dr.GerhardReinelt
DptoEstadísticaeInv.Operativa UniversitätHeidelberg
UniversitatdeValència Inst.Informatik
Dr.Moliner50 ImNeuenheimerFeld326
46100Valencia 69120Heidelberg
Spain Germany
[email protected] [email protected]
Editors:
S.S.Antman J.E.Marsden
DepartmentofMathematics ControlandDynamicalSystems,107-81
and CaliforniaInstituteofTechnology
InstituteforPhysical Pasadena,CA91125
ScienceandTechnology USA
UniversityofMaryland [email protected]
CollegePark,MD20742-4015
USA
[email protected]
L.Sirovich
LaboratoryofAppliedMathematics
DepartmentofBiomathematicalSciences
MountSinaiSchoolofMedicine
NewYork,NY10029-6574
[email protected]
ISSN0066-5452
ISBN978-3-642-16728-7 e-ISBN978-3-642-16729-4
DOI10.1007/978-3-642-16729-4
SpringerHeidelbergDordrechtLondonNewYork
MathematicsSubjectClassification(2010):90Cxx,90C57,90C59,0C90
(cid:2)c Springer-VerlagBerlinHeidelberg2011
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To AmparoRico andXimoSero´, el padrino,
forshowingmetheway.
RafaMart´ı
.
Preface
The idea for writing thisbookcame up when the authorsmet atthe Universityof
Valenciain2005.Whilecomparingourexperienceswithregardtovariousaspects
of the linear ordering problem (LOP), we realized that most of the optimization
technologieshadbeensuccessfullyappliedtosolvethisproblem.Wealsofoundthat
therewereonlyasmallnumberofbookscoveringallstate-of-the-artoptimization
methodsforhardoptimizationproblems(especiallyconsideringbothexactmethods
andheuristicstogether).WethoughtthattheLOPwouldmakeanidealexampleto
surveythesemethodsappliedtooneproblemandfeltthetimewasripetoembark
ontheprojectofwritingthismonograph.
Facedwith thechallengeof solvinghardoptimizationproblemsthataboundin
therealworld,classicalmethodsoftenencounterseriousdifficulties.Importantap-
plicationsinbusiness,engineeringoreconomicscannotbetackledbythesolution
methodsthathavebeenthepredominantfocusofacademicresearchthroughoutthe
past three decades. Exact and heuristic approachesare dramatically changing our
abilitytosolveproblemsofpracticalsignificanceandareextendingthefrontierof
problems that can be handled effectively. In this text we describe state-of-the-art
optimization methods, both exact and heuristic, for the LOP. We actually employ
theLOPtoillustratecurrentoptimizationtechnologiesandthedesignofsuccessful
implementationsof exactand heuristic procedures.Therefore,we do notlimitthe
scopeofthisbooktotheLOPbut,onthecontrary,weprovidethereaderwiththe
background and strategies in optimization to tackle different combinatorial prob-
lems.
This monographis devoted to the LOP, its origins, applications, instances and
especiallytomethodsforitseffectiveapproximateorexactsolution.Ourintention
istoprovidebasicprinciplesandfundamentalideasandreflectthestate-of-the-art
of heuristic and exact methods, thus allowing the reader to create his or her per-
sonal successfulapplicationsof the solution methods.The bookis meantto be of
interestforresearchersandpractitionersincomputerscience,mathematics,opera-
tionsresearch,managementscience,industrialengineering,and economics.Itcan
beusedasatextbookonissuesofpracticaloptimizationinamaster’scourseoras
areferenceresourceforengineeringoptimizationalgorithms.
vii
viii Preface
To make the book accessible to a wider audience, it is to a large extent self-
contained,providingthereaderwiththebasicdefinitionsandconceptsinoptimiza-
tion. However, in order to limit the size of this monographwe have not included
extensiveintroductions.Readersinterestedin furtherdetailsarereferredto appro-
priatetextbookssuchas[4,84,102,117,118,124].
Thestructureofthisbookisasfollows.Chapter1providesanintroductiontothe
problemanditsapplicationsanddescribesthesetofbenchmarkinstanceswhichwe
areusingforourcomputationalexperimentsandwhichhavebeenmadepublically
available. Chapter 2 describes such basic heuristic methods such as construction
and localsearches. Chapter 3 expandson Chapter2 and coversmeta-heuristicsin
whichthesimplemethodsarenowembeddedincomplexsolutionalgorithmsbased
ondifferentparadigms,suchasevolutionorlearningstrategies.Chapter4discusses
branch-and-bound,theprincipalapproachforsolvingdifficultproblemstooptimal-
ity. A special version based on polyhedral combinatorics, branch-and-cut, is pre-
sentedinChapter5.Chapter6dealsinmoredetailwiththelinearorderingpolytope
whichisatthecoreofbranch-and-cutalgorithms.ThebookconcludeswithChap-
ter7,whereanumberoffurtheraspectsoftheLOPandpotentialissuesforfurther
researcharedescribed.
Rafael Mart´ı’s research was partially supported by the Ministerio de Ciencia e
Innovacio´nofSpain(GrantRefs.TIN2006-02696andTIN2009-07516).
We areindebttomanypeople,butinparticulartosomeverygoodfriendsand
colleagues who helped us to gain a deeper understanding of the linear ordering
problem:VicenteCampos,Thomas,Christof,AngelCorbera´n,CarlosGarc´ıa,Fred
Glover,MartinGro¨tschel,MichaelJu¨nger,ManuelLagunaandDionisioPe´rez.The
proofreading by Cara Cocking, Elena Ferna´ndez, He´ctor Fraire, Marcus Oswald,
RodolfoPazos,HannaSeitz,MarkusSpethandPeiWangisparticularlyacknowl-
edged. Finally, our special thanks go to Abraham Duarte, who implementedmost
ofthemethodsdescribedinChapters2and3andperformedsomeoftheassociated
experiments.
Valencia,Heidelberg RafaelMart´ı
October2010 GerhardReinelt
Contents
1 Introduction................................................... 1
1.1 Basicdefinitions ........................................... 1
1.2 ApplicationsoftheLinearOrderingProblem ................... 3
1.2.1 EquivalentGraphProblems............................ 3
1.2.2 RelatedGraphProblems .............................. 4
1.2.3 AggregationofIndividualPreferences................... 4
1.2.4 BinaryChoiceProbabilities............................ 5
1.2.5 TriangulationofInput-OutputTables.................... 5
1.2.6 OptimalWeightedAncestryRelationships ............... 6
1.2.7 RankinginSportsTournaments ........................ 6
1.2.8 CorruptionPerception ................................ 7
1.2.9 CrossingMinimization ............................... 8
1.2.10 LinearOrderingwithQuadraticObjectiveFunction ....... 8
1.2.11 SchedulingwithPrecedences .......................... 9
1.2.12 LinearOrderingwithCumulativeCosts ................. 9
1.2.13 CoupledTaskProblem................................ 9
1.2.14 TargetVisitationProblem ............................. 10
1.3 BenchmarkProblems ....................................... 10
1.3.1 DataFormat ........................................ 10
1.3.2 Input-OutputMatrices ................................ 12
1.3.3 RandomlyGeneratedInstancesA(Type1)............... 13
1.3.4 RandomlyGeneratedInstancesA(Type2)............... 13
1.3.5 RandomlyGeneratedInstancesB....................... 13
1.3.6 SGBInstances....................................... 13
1.3.7 InstancesofSchiavinottoandStu¨tzle.................... 14
1.3.8 InstancesofMitchellandBorchers ..................... 14
1.3.9 FurtherSpecialInstances.............................. 14
2 HeuristicMethods.............................................. 17
2.1 Introduction ............................................... 17
2.1.1 AssessingtheQualityofHeuristics ..................... 19
ix