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Local Search Algorithms for the Maximum k-Plex Problem

Permanent Link: http://ufdc.ufl.edu/UFE0022261/00001

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Title: Local Search Algorithms for the Maximum k-Plex Problem
Physical Description: 1 online resource (31 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: kplex, local, search, tabu
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The maximum k-plex problem is one of many ways to classify a cohesive subgroup in social network analysis. Considered to be a type of relaxed clique, the maximum k-plex problem is a degree based approach to identifying closely related vertices in a graph. The recent development of applying optimization to the maximum k-plex problem has the capability for growth. Local search algorithms are discussed and applied to benchmark problems for the maximum k-plex problem. Computational experiments are performed, and new lower bounds are successfully attained.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Pardalos, Panagote M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-05-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022261:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022261/00001

Material Information

Title: Local Search Algorithms for the Maximum k-Plex Problem
Physical Description: 1 online resource (31 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: kplex, local, search, tabu
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The maximum k-plex problem is one of many ways to classify a cohesive subgroup in social network analysis. Considered to be a type of relaxed clique, the maximum k-plex problem is a degree based approach to identifying closely related vertices in a graph. The recent development of applying optimization to the maximum k-plex problem has the capability for growth. Local search algorithms are discussed and applied to benchmark problems for the maximum k-plex problem. Computational experiments are performed, and new lower bounds are successfully attained.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Pardalos, Panagote M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022261:00001


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Iwouldliketothankmyadvisor,Dr.PanosM.Pardalos,forallofhisencouragementandsupport.Also,IwouldliketoacknowledgeDr.J.ColeSmithforservingonmycommitteeandforallofhishonestadvice.Lastly,Iwouldliketothankmyfamilyandfriendsfortheirnever-endingmoralsupport. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 2THEMAXIMUMK-PLEXPROBLEM ...................... 10 2.1Foundations ................................... 10 2.2TheMaximumk-PlexProblem ........................ 10 3HEURISTICAPPROACH .............................. 12 3.1Development .................................. 12 3.2Heuristics .................................... 12 3.2.1GreedyAlgorithm ............................ 12 3.2.2TabuSearch ............................... 13 3.2.2.1Method ............................ 13 3.2.2.2Computationalresults .................... 15 3.2.3ExchangeAlgorithm .......................... 15 3.2.3.1Method ............................ 15 3.2.3.2Computationalresults .................... 18 4CONCLUSIONS ................................... 27 REFERENCES ....................................... 29 BIOGRAPHICALSKETCH ................................ 31 5

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Table page 3-1ThetestedDIMACSinstances ............................ 22 3-2Taburesultsfork=1andk=2 .......................... 23 3-3Exchangeresultsfork=1andk=2 ........................ 24 3-4Resultcomparisonfork=1 ............................. 25 3-5Resultcomparisonfork=2 ............................. 26 6

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Figure page 3-1Greedyalgorithmfork-plex ............................. 20 3-2Tabualgorithmfork-plex .............................. 20 3-3Exchangealgorithmfork-plex ............................ 21 7

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Themaximumk-plexproblemisoneofmanywaystoclassifyacohesivesubgroupinsocialnetworkanalysis.Consideredatypeofrelaxedclique,themaximumk-plexproblemisadegreebasedapproachtoidentifyingcloselyrelatedverticesinagraph.Therecentdevelopmentofapplyingoptimizationtothemaximumk-plexproblemhasthecapabilityforgrowth.Localsearchalgorithmsweredevelopedandappliedtobenchmarkproblemsforthemaximumk-plexproblem.Computationalexperimentswereperformed,andnewlowerboundshavebeensuccessfullyattained. 8

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Theapplicationofgraphtheorytothenotionofsocialnetworkshasbeendevelopedformanyyears.Unsurprisingly,theeectsofglobalizationalongwiththeanincreaseofinterestinterrorismhavepropelledanewsurgeofinterestinsocialnetworktheory.Alongwiththestudyofsocialnetworks,certainmathematicalmodelshavebeenrevivedfromthepast.Onesuchexampleisthemaximumk-plexproblem. Themaximumk-plexproblemhasrootsinthestudyofcohesivesubgroupsofanetwork.Accordingto[ 25 ],theconceptofcohesivesubgroupscomesfromthecloserelationshipsbetweenactorsinasocialnetwork.Bystudyingthesegroups,sociologistscanformtheoriesthatexplainthebehaviorofactorsinsocialnetworks.Themostfamousofthesecohesivesubgroupmodelsisknownasthemaximumcliqueproblem.Althoughitiswell-known,soareitsdiculties.Therestrictionsassociatedwithmaximumcliquehavepropelledthestudyandformationofcliquerelaxations.Theserelaxedapproachesgivewaytonewopportunitiesbyfocusingonspecicpropertiesofcohesivesubgroups.Inparticular,themaximumk-plexproblemfocusesonthereachabilityandrobustnessofthegraph,andrelaxesthepropertyoffamiliarity,asstatedin[ 3 ]. Thisthesiswillapplyheuristictechniquestothemaximumk-plexproblem.Chapter2willcoverthebackgroundoftheproblem.Inchapter3,heuristicapproachesandresultsareaddressed.Chapter4discussesconclusionsandthefuturedevelopmentoftechniquesthatcanbeappliedtothemaximumk-plexproblem. 9

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21 ].Manysourceshavepointedoutproblemswiththemaximumcliqueproblem,regardingitsrestrictiveness[ 25 ],modellingdisadvantages[ 23 ][ 10 ],anddicultapproximation[ 5 ]. Inlightoftheseproblems,researchersdevelopedmanydierentrelaxationstothemaximumcliqueproblem.Sincecliquescanbedescribedascohesivesubgroupsofasocialnetwork,certainpropertiescanbeusedtodescribetheirnature:familiarity(eachvertexhasmanyneighbors),reachability(alowdiameter),androbustness(highconnectivity)[ 3 ].Eachofthesepropertieshasitsowndiculties,resultingindierentrelaxationtechniques.Somewell-knownrelaxationmodelsarek-clique[ 18 ],k-club[ 2 ],andk-clan[ 19 ]. Thispaperconsidersanothermodelcalledthemaximumk-plexproblem.ThisparticularrelaxationisdegreeorientedandwasdevelopedbySeidmanandFosterin1978[ 23 ]. 2{1 holds,thesetSinducesak-plex. 10

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3 ],theproblemwasreintroduced,consideringitscomputationalcomplexityanddevelopinganintegerprogrammingformulation.Inregardstocomplexity,whenkisarbitrary,themaximumcliqueproblemexistsasaspecialcaseofmaximumk-plexandisclearlyNP-hard[ 3 ].Inaddition,anNP-completenessproofforndingak-plexofaxedvalueofkinagraphwaspublished[ 3 ]. Also,in[ 3 ]therstknowncomputationalresultsforthemaximumk-plexproblemwaspresented.Startingwithanintegerprogrammingformulation,theyconstructedabranch-and-cutmethodtobeappliedtowell-knowndatasets.Apeelingprocedureusedtoremoveverticeswithlowdegreewasalsousedinpreprocessing.TheirbenchmarksolutionswereobtainedbyimplementingthemethodusingILOGCPLEX9.0r[ILOG]software.Nootherpublishedresultsareknowntoexist. 11

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3 ].Branch-and-cutisconsideredanexactalgorithmthatusescuttingplanemethodstoincreasetheperformanceofabranch-and-boundalgorithm.Althoughthismethodisexact,itsperformanceisveryslow.Therefore,approximationmethodscanbeusedtondsolutionsatamuchfasterrate.Sincethemaximumk-plexproblemisrelativelyunexplored,heuristicmethodshavebeendevelopedtoproducenewbenchmarksolutions.Althoughthebranch-and-cutalgorithmwastestedonseveraldierentinstances,theexperimentsinthisstudyaretestedonlyagainstDIMACSinstances[ 9 ].ThesecanbefoundinTable 3-1 3.2.1GreedyAlgorithm Bydenition,allverticesinthecurrentset,G[S],andthecandidatevertex,vi,havetosatisfyEquation 2{1 toretaink-plexfeasibility.Inordertoimplementthealgorithm,eachvertexisclassiedwhenitisaddedtothecurrentsolution.Verticesareclassiedas 12

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3{1 .Thispropertyisusefulinquicklydeterminingthequalityofcandidatesolutions.Similartomaximumclique,allverticeswithzeropotentialmustbeneighborstoanyvertexselectedtobeaddedtothek-plexinordertoretainfeasibility. Also,thisapproachrequiresthechosenvertextohavethemaximumdegreeamongstallcandidatesoutsideoftheset,degG[S](vi).Inaddition,iftie-breakingisnecessary,theaddedvertexshouldbeselectedatrandom.TheimplementationofthisalgorithmisshowninFigure 3-1 Basicgreedyalgorithmsareknowntoproducegoodresults,buttheyareusuallynotsignicantenoughtoreport.Alone,thegreedyalgorithmhasatendencytoincludeverticesofhighdegree,althoughthoseverticesmaynotbelongtothemaximumk-plex.Formoredependableresults,theuseofmoreadvancedsearchtechniquesareneeded.Therefore,thisgreedyapproachsimplyprovidesaplatformtocontinuedevelopmentofamorein-depthheuristic. 3.2.2.1Method Withintheclassicationoflocalsearchmethods,tabusearchisconsideredtobethemosteective.DevelopedbyFredGlover,tabusearchisamethodthatlimitsthenumber 13

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12 ].Unlikemanyotherlocalsearchtechniques,thismethodprovidesawaytoescapelocaloptima.Withtabusearch,asetofpreviouslychosenmovesarestoredinatabulistthatdisallowsthatmovetobemadeagainuntilaspeciednumberofnewmoveshavebeentaken.Thislimitingfactorreducesthechancethatthealgorithmwillcontinuetorevisitthesamelocalsolution.Consideringitspastperformance,tabusearchwaschosenasstartingpointforapplyinglocalsearchtothemaximumk-plexproblem. Inordertoimplementtabusearch,thequalitiesoftheneighborhoodstructuresneedtobedeterminedforecientresults[ 13 ].Forthismethod,thesolutionsthathavebeenpreviouslychosenusingthegreedyalgorithmwillbeclassiedas"tabu."Thesechoiceswillbeforbiddentobeusedagainuntilaspeciednumberofmoveshavebeenmade.Insuchagreedyalgorithm,thetabusolutionwillbeoverlooked,andthenextbestchoicewillbetaken.Thelengthoftimethatamoveisclassiedastabuiscalleditstabutenure.Althoughtherearemanydierentwaysofdeterminingandadjustingthetabutenure,mostchoicesarebasedonempiricalteststhatbeginwithaninitialsetvalueforallproblems[ 13 ].Fortheresultspresentedinthispaper,thetabutenurewaschosenasasetvalue,withnoadjustmentwithinthealgorithmitself. Toimplementtheneighborhoodsearch,eachlocallymaximalsolutionneedstobechanged.Forthisprocedure,alocalmaximumisadjustedbyremovingavertexfromitscurrentsolutionandaddingnewverticesbasedonthegreedyheuristic.Theselectionofadeletedvertexisbasedonapartiallyrandomapproach.Referencingtothevaluesofpotentialforeachvertexinthegraph,verticeswithzeropotentialarecriticaltothestructureofthegraph.Becausetheseverticesaremorerestrictivetothesolution,arandomvertexfromthissubsetisselectedtoberemovedfromtheset.Insuchacasethatavertexcannotbeselectedwithapotentialofzero,thedeletedvertexmustbechosenfromtherestoftheset.Thisalgorithmrunsinarecursivemanner,continuinguntilastoppingcriteriaismet.ThepseudocodeforthealgorithmcanbefoundinFigure 3-2 14

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tenure=10iterationsandrun time=1000seconds.Althoughbriefexperimentswererunfortabu tenure>10,outputsprovidedinsignicantresults. ThealgorithmwasappliedtoavarietyofgraphsfromthebenchmarkcliqueinstancesoftheSecondDIMACSchallenge[ 9 ].Experimentsonthegraphsweredonewithvaluesk=1(clique)andk=2.Eachexperimentwasperformedfor20independentruns. TheresultsofthetestscanbeseeninTable 3-2 ,showingtheaveragetimetosolution,maximumbestsolution,minimumbestsolution,andaveragebestsolution.Theseresultsarecomparedwithlaterresultsandthebenchmarksolutionsof[ 3 ]inTable 3-4 andTable 3-5 .Comparisonvaluesindicateaveragetimetosolutionandmaximumbestvalues.Valuesinsquarebracketsaretheboundsfornon-optimalterminationsinthebenchmarkinstancesfrom[ 3 ].Valuesinparenthesisarenewbounds.Experimentsthatdidnotobtainoptimum(orcurrentlowerbound)aremarkedwithanasterisk(*). Thetabusearchalgorithmperformedwellonmanyproblems,andnewboundswereobtainedforcertaininstances.Although1-plexesarecliques,thealgorithmprovidedimprovedresultscomparedwithpreviousk-plexresults.AccordingtoDIMACS[ 9 ],the1-plexboundattainedforMANN 27.clqisthepublishedsizeformaximumclique.Additionally,thealgorithmfoundanewlowerboundforhamming10-4.clqwithk=2.Althoughmostresultswerematchedorimproved,somebenchmarkvalueswerenotattained. 3.2.3.1Method 15

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Inthismethod,localsearchisconductedthroughexchangingvertices.Whenalocalmaximumisattained,avertexwithinthesetwillbereplacedbyanexchangepartneroutsideofthesolutionset.Theexchangepartnerisavertexchosenrandomlyandevaluatedforpossibleexchangemoveswithintheset.Inordertodecreasethecomplexityofthealgorithm,therstmovethatsatisesconditionswillbeselected.Oncefeasibilityisestablished,thetwoverticescanbeexchanged.Thealgorithmthencontinuesinagreedyfashion,addingnewverticestotheset. Findinganexchangepairconsistsofevaluatingsimpleconditions.Afteravertexoutsideofthesolutionischosen,itisrandomlycomparedagainsttheverticeswithintheset.Whenchoosinganexchangemove,twoconsiderationsneedtobemade:k-plexfeasibilityofthechosenvertexandstabilityofthecurrentset. Therstconditionisforthechosenvertextohavek-plexfeasibilityafterbeingaddedtotheset.Fromthebasick-plexconditioninEquation 2{1 ,theexchangeconditionscanbeextendedintermsoftherelationshipbetweenthetwochosenvertices.Becausethefeasibilityofthenewvertexisbeingevaluatedbeforeremovingitsexchangepartner,thek-plexconditionsaredependentontherelationshipbetweenthetwo.Iftheverticesareneighbors,theevaluatedconditionsmustbeosetbyone,inordertocompensateforthedegreechangewhenthecurrentvertexisremovedfromtheset.Thissituationoccurs 16

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3{2 .TheconditionsforwhentheyareneighborscanbefoundinEquation 3{3 ,whereSisthecurrentsolutionset,andvnewisthevertexbeingconsideredforadditiontotheset. Oncethevertexisknowntosatisfyk-plexconditions,itmustbecomparedwiththecurrentvertextoensurethatitdoesnotbreakthefeasibilityconditionsfortherestoftheset.Similartotherequirementsofthegreedyalgorithm,thepotentialvaluesofthecurrentsolutionareusedtoensurethestructureofthek-plex.Again,criticalverticesarethosewithpotentialequaltozero,andanynewvertexbeingaddedtothesetmustbeconnectedtothesevertices.Withtheconditionsofexchange,potentialvaluescanonlybeincreasedorremainthesamewhenavertexisremovedfromtheset.Therefore,onlyverticeswithzeropotentialthatarealsoneighborstothevertexbeingconsideredforremovalneedtobecheckedforsetstability.Inordertopreventtheneighboringverticesfromattainingnegativepotential(breakingthek-plex),theyallmustalsobeneighborsoftheexchangepartner.ThisconditioncanbeseeninEquation 3{4 ,whereSisthecurrentsolutionset,SBListhesubsetSwithpotentialequaltozero,vnewisthevertexbeingaddedtotheset,andvoldisthevertexbeingremovedfromtheset. Afterallconditionsaresatised,theexchangecanbemade.Anewsolutionsetisformed,allowingthegreedyalgorithmtocontinuecheckingfornewsolutions.The 17

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3-3 Inadditiontothisexchangealgorithm,anothermethod,usingtabusearch,wasconsidered.Thismethodincorporatedthetabusearchheuristicintotheselectionofexchangepairs.Thetabumethodwouldpreventexchangeswithverticesthathadbeenexchangedrecently.Preliminaryexperimentswereperformedwithtabutenuresofvarioussizes.Afterthesetests,resultsshowedasignicantincreaseinprocesstimefortheexchangealgorithmwithtabu,witheithernoincreaseoradecreaseinsolutionsize.Therefore,nofurtherexperimentswereperformedonthiscombinedalgorithm. time=1000seconds. Thealgorithmwasappliedtothesamesetofbenchmarkcliqueinstancesasthetabalgorithm,fromtheSecondDIMACSchallenge[ 9 ].Again,experimentswereperformedwithvaluesk=1andk=2.Eachexperimentwastestedfor20independentruns. ResultsoftheexperimentscanbefoundinTable 3-3 ,showingaveragetimetosolution,maximumbestsolution,minimumbestsolution,andaveragebestsolution.Theseresultshavebeencomparedwiththetabusearchresults,alongwiththebenchmarkinstancesof[ 3 ]inTable 3-4 andTable 3-5 .Comparisonvaluesindicateaveragetimetosolutionandmaximumbestvalues.Valuesinsquarebracketsaretheboundsfornon-optimalterminationsinthebranch-and-cutinstancesfrom[ 3 ].Valuesinparenthesisarenewbounds.Experimentsthatdidnotndoptimalvalues(orcurrentlowerbounds)aremarkedwithanasterisk(*). Theexchangelocalsearchalgorithmperformedwellonmostproblems.Althoughmostsearchtimeswerecomparablewithtabusearch,someinstancesprovedtorequire 18

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3 12 Greedyalgorithmfork-plex 1 3 timedo Selectvi2Susingthegreedyalgorithm; 7 SelectarandomvifromS; 15 Tabualgorithmfork-plex 20

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3 timedo Selectvi2Susingthegreedyalgorithm; 7 SelectarandomvnewfromS; 10 SelectarandomexchangepartnervoldfromS; 12 Selectvoldasexchangepartner; 16 Svold Selectvoldasexchangepartner; 23 Svold Exchangealgorithmfork-plex 21

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ThetestedDIMACSinstances GraphjVjjEj a9.clq45918MANN a27.clq37870551MANN a45.clq1035533115san200 0.7 2.clq20013930keller4.clq1719435 22

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Taburesultsfork=1andk=2 c-fat200-1.clq0.00121212.000.00121212.00c-fat200-2.clq0.00242424.000.00242424.00c-fat200-5.clq0.00585858.000.00585858.00c-fat500-1.clq0.00141414.000.00141414.00c-fat500-2.clq0.00262626.000.00262626.00c-fat500-5.clq0.00646464.000.00646464.00c-fat500-10.clq1.00126126126.000.95126126126.00hamming6-2.clq0.00322029.400.00322228.90hamming6-4.clq0.00444.000.00666.00hamming8-2.clq1.9012880107.600.3512894112.80hamming8-4.clq0.05161214.800.00161215.60hamming10-2.clq331.10512330427.50101.40512369446.45hamming10-4.clq201.75403537.50244.80474445.25johnson8-2-4.clq0.00444.000.00555.00johnson8-4-4.clq0.00141012.600.00141414.00MANN a9.clq0.00161616.000.00262425.70MANN a27.clq248.80126125125.1558.10235234234.35MANN a45.clq307.10334330330.7040.40660660660.00san200 0.7 2.clq0.00121212.000.00242424.00keller4.clq0.00988.250.00151414.40 23

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Exchangeresultsfork=1andk=2 c-fat200-1.clq0.00121212.000.00121212.00c-fat200-2.clq0.00242424.000.00242424.00c-fat200-5.clq0.00585858.000.00585858.00c-fat500-1.clq0.00141414.000.05141414.00c-fat500-2.clq0.00262626.000.00262626.00c-fat500-5.clq0.85646464.000.20646464.00c-fat500-10.clq0.00126126126.000.50126126126.00hamming6-2.clq0.00323232.000.00323232.00hamming6-4.clq0.00444.000.00666.00hamming8-2.clq1.40128128128.002.25128128128.00hamming8-4.clq0.20161616.000.00161616.00hamming10-2.clq169.50512476510.20199.42512407489.57hamming10-4.clq63.05404040.00294.47484848.00johnson8-2-4.clq0.00444.000.00555.00johnson8-4-4.clq0.00141414.000.00141414.00MANN a9.clq0.00161616.000.00262626.00MANN a27.clq303.80126125125.70179.11235234234.55MANN a45.clq573.45335331332.3539.22660660660.00san200 0.7 2.clq60.05181417.400.00242424.00keller4.clq0.00111111.000.00151515.00 24

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Resultcomparisonfork=1 GraphTabuTime(s)ExgTime(s)BCTime(s) c-fat200-1.clq120.00120.001216.84c-fat200-2.clq240.00240.002419.68c-fat200-5.clq580.00580.005811.15c-fat500-1.clq140.00140.0014221.09c-fat500-2.clq260.00260.0026328.70c-fat500-5.clq640.00640.8564555.26c-fat500-10.clq1261.001260.00126279.70hamming6-2.clq320.00320.00320.01hamming6-4.clq40.0040.0040.29hamming8-2.clq1281.901281.401280.01hamming8-4.clq160.05160.201610788.10hamming10-2.clq512331.10512169.505120.18hamming10-4.clq(40)201.75(40)63.05[38,379]10800.20johnson8-2-4.clq40.0040.0040.17johnson8-4-4.clq140.00140.00142.99MANN a9.clq160.00160.00160.09MANN a27.clq(126)248.80(126)303.80[125,148]10800.30MANN a45.clq334*307.10335*573.45[342,422]10802.00san200 0.7 2.clq12*0.00(18)60.05[17,27]10800.90keller4.clq9*0.00110.00117510.53 25

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Resultcomparisonfork=2 GraphTabuTime(s)ExgTime(s)BCTime(s) c-fat200-1.clq120.00120.001230.92c-fat200-2.clq240.00240.002431.67c-fat200-5.clq580.00580.005833.62c-fat500-1.clq140.00140.0514557.10c-fat500-2.clq260.00260.0026885.21c-fat500-5.clq640.00640.20641087.75c-fat500-10.clq1260.951260.501261014.87hamming6-2.clq320.00320.00320.23hamming6-4.clq60.0060.0066.57hamming8-2.clq1280.351282.25[128,130]10800.30hamming8-4.clq160.00160.00[16,80]10800.40hamming10-2.clq512101.40512199.42[512,534]10800.20hamming10-4.clq(47)244.80(48)294.47[45,458]10802.20johnson8-2-4.clq50.0050.0051.63johnson8-4-4.clq140.00140.00[14,16]10800.30MANN a9.clq260.00260.00260.14MANN a27.clq235*58.10235*179.11[236,260]10800.50MANN a45.clq660*40.40660*39.22[662,739]10800.80san200 0.7 2.clq24*0.0024*0.00[62,86]10800.40keller4.clq15*0.0015*0.00[40,45]10800.50 26

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Thisstudydevelopednewmethodsforapplyingheuristicstothemaximumk-plexproblem.Therstproposedalgorithmusedtabusearch,alongwithagreedytechnique,toimplementalocalsearchprocedure. ThetabualgorithmwastestedonasetofproblemsprovidedbyDIMACS[ 9 ].Itshowedgoodresultsformostproblems,ndingoptimalsolutionsfor15of16knownsolutionsfork=1and11of11knownsolutionsfork=2.Forbenchmarkswithunknownoptimalsolutions,thealgorithmfoundnewboundsfortwooffoursolutionswithk=1andoneofninesolutionswithk=2.Asmallnumberofinstancesdidnotbreaklowerbounds. TheexchangealgorithmwasexperimentedonthesamesetofDIMACSinstancesastabusearch[ 9 ].Incomparisonwithtabusearch,theexchangealgorithmperformedmorereliably;terminatingsolutionswereconsistentlyhigher.Forvaluesofbothk=1andk=2,allknownoptimalsolutionswerefound.Forinstanceswithboundedsolutions,theexchangealgorithmmetorsurpassedlowerboundsforthreeoffoursolutionswithk=1andveofninesolutionswithk=2.Again,asmallnumberofinstancesdidnotmeetlowerbounds,andtheseexamplesofpoorperformanceareprimarilyshowninhighdensitygraphs. Itisoftennecessarytondsolutionstooptimizationproblemsinalimitedperiodoftime.Developmentofapproximationmethodstoestimatesolutionsquicklyisnecessaryforthequickevaluationoflargeandcomplexproblems.Theseheuristicsprovidenewmethodthatcanbeappliedtondingk-plexesinagraph. Extendingtheresearchoflocalsearchmethodsforthemaximumk-plexproblemisverypromising.Withadditionaltuning,thetabusearchalgorithmcouldimprovedramaticallyfromitscurrentperformance.Incorporatingtechniquesthatcanbeadjustedduringthesearchprocedureforboththetabuandexchangealgorithmsisexpectedto 27

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Furtherresearchregardingthemaximumk-plexproblemwillhavepositiveeectsontheeldofoptimizationandsocialsciences.Applicationsofndinglargek-plexvaluesingraphsareexpectedtohaveanimpactondeterminingcohesivesubgroupsinmanyreallifesituations.Otherthansocialnetworkanalysis,themaximumk-plexproblemhastheopportunityandpossibilityofbeingimplementedinmanyothereldsofstudy. 28

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Theauthor,ErikaShort,isamaster'sstudentattheUniversityofFlorida.ShestudiesinthedepartmentofIndustrialandSystemsEngineering,withherprimaryfocusinoperationsresearch. 31