Citation
Local Search Algorithms for the Maximum k-Plex Problem

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

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Pardalos, Panagote M.
Committee Members:
Smith, Jonathan
Graduation Date:
5/1/2008

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Greedy algorithms ( jstor )
Heuristics ( jstor )
Information search ( jstor )
Operations research ( jstor )
Optimal solutions ( jstor )
Social network analysis ( jstor )
Social networking ( jstor )
Taboos ( jstor )
Vertices ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
kplex, local, search, tabu
Genre:
Electronic Thesis or Dissertation
bibliography ( marcgt )
theses ( marcgt )
Industrial and Systems Engineering thesis, M.S.

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. ( en )
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
Statement of Responsibility:
by Erika J. Short

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University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright by Erika J. Short. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
5/31/2010
Classification:
LD1780 2008 ( lcc )

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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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[1] AbelloJ,PardalosPM,ResendeMGC.Onmaximumcliqueproblemsinverylargegraphs,Externalmemoryalgorithmsandvisualization.DIMACSSeriesonDiscreteMathematicsandTheoreticalComputerScience.AmericanMathematicalSociety1999;50:119-130. [2] AlbaRD.Agraph-theoreticdenitionofasociometricclique.JournalofMathematicalSociology1973;3:113-126. [3] BalasundaramB,ButenkoS,HicksIV,SachdevaS.Cliquerelaxationsinsocialnetworkanalysis:Themaximumk-plexproblem.Submitted;2006. [4] BattitiR,TecchiolliG.TheReactiveTabuSearch.ORSAJournalonComputing1994;6:126-140. [5] BellareM,GoldreichO,SudanM.FreeBits,PCPs,andNonapproximabilityTowardsTightResults.SIAMJournalonComputing1998;27(3):804915. [6] BomzeIM,BudinichM,PardalosPM,PelilloM.Themaximumcliqueproblem.HandbookofCombinatorialOptimization,KluwerAcademicPublishers,Dordrecht,TheNetherlands1999;1:1-74. [7] DellAmicoM,TrubianM.Applyingtabusearchtothejob-shopschedulingproblem.AnnalsofOperationsResearch1993;41:23152. [8] DeoN.Graphtheorywithapplicationstoengineeringandcomputerscience.EnglewoodClis,NJ:Prentice-Hall;1974. [9] DIMACS.Cliques,Coloring,andSatisability:SecondDIMACSImplementationChallenge.http://dimacs.rutgers.edu/Challenges/;1995.AccessedFebrurary2008. [10] FreemanLC.Thesociologicalconceptofa"group":Anempiricaltestoftwomodels.AmericanJournalofSociology1992;98:152-166. [11] GendreauM,LaporteG,MusaraganyiC,TaillardED.Atabusearchheuristicfortheheterogeneouseetvehicleroutingproblem.ComputersandOperationsResearch1999;26:115373. [12] GloverF,LagunaM.TabuSearch.HandbookofCombinatorialOptimization,KluwerAcademicPublishers,Dordrecht,TheNetherlands1999;3:621-757. [13] GloverF.TabuSearchFundamentalsandUses.UniversityofColorado,Boulder.http://leeds-faculty.colorado.edu/glover/TS%20-%20Fundamentals&Uses.pdf;June1995.AccessedMarch2008. [14] GloverF.Tabusearch:PartI.ORSAJournalonComputing1989;1(3):190-206. [15] GloverF.Tabusearch:PartII.ORSAJournalonComputing1990;2(1):4-32. 29

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