The linear ordering problem: Exact and heuristic methods in combinatorial optimization PDF
Preview The linear ordering problem: Exact and heuristic methods in combinatorial optimization
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 Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermitted onlyundertheprovisions oftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Coverdesign:deblik,Berlin Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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
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