|
PAGE 1
1
PAGE 2
2
PAGE 3
3
PAGE 4
Iwouldliketothankmyadvisor,Dr.PanosM.Pardalos,forallofhisencouragementandsupport.Also,IwouldliketoacknowledgeDr.J.ColeSmithforservingonmycommitteeandforallofhishonestadvice.Lastly,Iwouldliketothankmyfamilyandfriendsfortheirnever-endingmoralsupport. 4
PAGE 5
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
PAGE 6
Table page 3-1ThetestedDIMACSinstances ............................ 22 3-2Taburesultsfork=1andk=2 .......................... 23 3-3Exchangeresultsfork=1andk=2 ........................ 24 3-4Resultcomparisonfork=1 ............................. 25 3-5Resultcomparisonfork=2 ............................. 26 6
PAGE 7
Figure page 3-1Greedyalgorithmfork-plex ............................. 20 3-2Tabualgorithmfork-plex .............................. 20 3-3Exchangealgorithmfork-plex ............................ 21 7
PAGE 8
Themaximumk-plexproblemisoneofmanywaystoclassifyacohesivesubgroupinsocialnetworkanalysis.Consideredatypeofrelaxedclique,themaximumk-plexproblemisadegreebasedapproachtoidentifyingcloselyrelatedverticesinagraph.Therecentdevelopmentofapplyingoptimizationtothemaximumk-plexproblemhasthecapabilityforgrowth.Localsearchalgorithmsweredevelopedandappliedtobenchmarkproblemsforthemaximumk-plexproblem.Computationalexperimentswereperformed,andnewlowerboundshavebeensuccessfullyattained. 8
PAGE 9
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
PAGE 10
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
PAGE 11
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
PAGE 12
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
PAGE 13
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
PAGE 14
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
PAGE 15
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
PAGE 16
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
PAGE 17
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
PAGE 18
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
PAGE 19
19
PAGE 20
3 12 Greedyalgorithmfork-plex 1 3 timedo Selectvi2Susingthegreedyalgorithm; 7 SelectarandomvifromS; 15 Tabualgorithmfork-plex 20
PAGE 21
3 timedo Selectvi2Susingthegreedyalgorithm; 7 SelectarandomvnewfromS; 10 SelectarandomexchangepartnervoldfromS; 12 Selectvoldasexchangepartner; 16 Svold Selectvoldasexchangepartner; 23 Svold Exchangealgorithmfork-plex 21
PAGE 22
ThetestedDIMACSinstances GraphjVjjEj a9.clq45918MANN a27.clq37870551MANN a45.clq1035533115san200 0.7 2.clq20013930keller4.clq1719435 22
PAGE 23
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
PAGE 24
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
PAGE 25
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
PAGE 26
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
PAGE 27
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
PAGE 28
Furtherresearchregardingthemaximumk-plexproblemwillhavepositiveeectsontheeldofoptimizationandsocialsciences.Applicationsofndinglargek-plexvaluesingraphsareexpectedtohaveanimpactondeterminingcohesivesubgroupsinmanyreallifesituations.Otherthansocialnetworkanalysis,themaximumk-plexproblemhastheopportunityandpossibilityofbeingimplementedinmanyothereldsofstudy. 28
PAGE 29
[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
PAGE 30
HertzA,deWerraD.Usingtabusearchtechniquesforgraphcoloring.Computing1987;39:34551. [17] HubscherR,GloverF.Applyingtabusearchwithinuentialdiversicationtomultiprocessorscheduling.ComputersandOperationsResearch1994;21(8):87784. [18] LuceRD.Connectivityandgeneralizedcliquesinsociometricgroupstructure.Psychometrika1950;15:169-190. [19] MokkenRJ.Cliques,clubsandclans.QualityandQuantity1979;13:161-173. [20] OsmanIH,LaporteG.Metaheuristics:abibliography.AnnalsofOperationsResearch1996;63:513623. [21] PardalosPM,XueJ.Themaximumcliqueproblem.JournalofGlobalOptimization1994;4(3):301328. [22] ScottJ.SocialNetworkAnalysis:AHandbook.SagePublications,London,2ndedition;2000. [23] SeidmanSB,FosterBL.Agraphtheoreticgeneralizationofthecliqueconcept.JournalofMathematicalSociology1978;6:139-154. [24] TangH,Miller-HooksE.Atabusearchheuristicfortheteamorienteeringproblem.ComputersandOperationsResearch2005;32:13791407. [25] WassermanS,FaustK.SocialNetworkAnalysis.CambridgeUniversityPress;1994. 30
PAGE 31
Theauthor,ErikaShort,isamaster'sstudentattheUniversityofFlorida.ShestudiesinthedepartmentofIndustrialandSystemsEngineering,withherprimaryfocusinoperationsresearch. 31
|
