The densest subgraph problem on clique graphs M - PDF document

Thedensestk-subgraphproblemoncliquegraphsM.Liazi1?,I.Milis2,F.Pascual3andV.Zissimopoulos11DepartmentofInformaticsandTelecommunications,UniversityofAthens,15784Athens,Greecefmliazi,vassilisg@di.uoa.gr2DepartmentofInformatics,AthensUniversityEconomicsandBusiness,10434Athens,Greecemilis@aueb.gr3IBISC,UniversityofEvry,523,PlacedesTerrassesdel'agora,91000Evry,Francefpascual@lami.univ-evry.frAbstract.TheDensestk-Subgraph(DkS)problemasksforak-vertexsubgraphofagivengraphwiththemaximumnumberofedges.TheproblemisstronglyNP-hard,asageneralizationofthewellknownCliqueproblemandwealsoknowthatitdoesnotadmitaPolynomialTimeApproximationScheme(PTAS).Inthispaperwefocusonspecialcasesoftheproblem,withrespecttotheclassoftheinputgraph.Especially,towardstheelucidationoftheopenquestionsconcerningthecomplexityoftheproblemforintervalgraphsaswellasitsapproximabilityforchordalgraphs,weconsidergraphshavingspecialcliquegraphs.WepresentaPTASforstarsofcliquesandadynamicprogrammingalgorithmfortreesofcliques.KeyWords:Densestk-subgraph,Cliquegraph,PolynomialTimeApproximationScheme,Dynamicprogramming1IntroductionIntheDensestk-subgraph(DkS)problemwearegivenagraphG=(V;E),jVj=n,andanintegerkn,andweaskforasubgraphofGinducedbyexactlykofitsverticessuch ?Theprojectisco-nancedwithinOp.EducationbytheESF(EuropeanSocialFund)andNationalResources. thatthenumberofedgesofthissubgraphismaximized.TheproblemisdirectlyNP-hardasgeneralizationofthewellknownMaximumCliqueproblem.IntheweightedversionoftheDkSwealsogivennonnegativeweightsontheedgesofGandthegoalistondak-vertexinducedsubgraphofmaximumtotaledgeweight.Duringlastyearsalargebodyofwork[2,3,5{7,10,14,19,20]hasbeenconcentratedonthedesignofapproximationalgorithmsforboththeDkSproblemanditsweightedversion,basedonavarietyoftechniquesincludinggreedyalgorithms,LPrelaxationsandsemidef-initeprogramming.Forabriefpresentationofthisbodyofworkthereaderisreferredtothemostrecentofthesearticles[3].However,thebestknownapproximationratiofortheDkSproblem,whichperformswellforallvaluesofk,isO(n),forsome1 3[5].Ontheotherhand,ithasbeenshownthattheDkSproblemsdoesnotadmitaPolynomialTimeApproximationScheme(PTAS)[13].However,thereisnotanegativeresultthatachievinganapproximationratioofO(n),forsome�0,isNP-hard.ConcerningapproximationalgorithmsforspecialcasesoftheproblemitisknownthattheDkSproblemadmitsaPTASforgraphsofminimumdegree\n(n)aswellasfordensegraphs(of\n(n2)edges)whenkis\n(n)[1].Moreover,algorithmsachievingapproximationfactorsof4[18]and2[11]havebeenproposedfortheweightedDkSproblemoncompletegraphswheretheweightssatisfythetriangleinequality.TheDkSproblemistrivialontrees(anysubtreeofkverticescontainsexactlyk1edges).ItisalsoknownthatDkSispolynomialforgraphsofmaximaldegreetwo[7]aswellasforcographs,splitgraphsandk-trees[4].OntheotherhandtheDkSproblemremainsNP-hardforbipartitegraphs[4],evenofmaximumdegreethree[7],aswellasforcomparabilitygraphs,chordalgraphs[4]andplanargraphs[12].TheweightedversionoftheDkSproblemispolynomialontreeseitherifweaskforaconnectedsolution[9,15,16]ornot[17].Infact,theresultforthelatercaseisimpliedbyaresultforthesolutionofthequadratic0-1knapsackproblemonedgeseries-parallelgraphsin[17]. AnoutstandingopenquestionconcernsthecomplexityoftheDkSproblemonintervalgraphsaswellasitsapproximabilityforchordalgraphs.Towardsthisdirectionwefocus,inthispaper,onchordalorintervalgraphshavingspecialcliquegraphs.Acliqueofanundirectedgraph,G=(V;E),isasubsetofitsverticesinducingacompletesubgraphinG.Theintersectiongraphofafamily,F,ofsubsetsofasetisdenedasagraph,G,whoseverticescorrespondtothesubsetsinF,andthereisanedgebetweentwoverticesofGifthecorrespondingpairofsubsetsintersect.Giventhesedenitions,thecliquegraphofagraphGisdenedastheintersectiongraphofthemaximalcliquesofG.Itiswellknownthatallmaximalcliques,andhencethecliquegraph,ofachordalgraphcanbefoundinpolynomialtime[8].Itis,clearly,convenienttostudytheDkSproblemonthecliquegraphofachordalgraphGinsteadontheGitself.Inthispaperweconsiderchordalorintervalgraphshavingspecialcliquegraphs,inordertofurtheridentifythefrontierbetweenhardandpolynomialsolvable/approximablevariantsofthetheDkSproblem.InthenextsectionwepresentaPTASforgraphsthathavingascliquegraphastar(starofcliques)andinSection3wepresentanO(nkm+1)timedynamicprogrammingalgorithmforgraphshavingascliquegraphatree(treeofcliques)ofmaximumdegreem.ThisalgorithmgivesanO(nk3)timealgorithmforgraphshavingascliquegraphapath(pathofcliques).Notethat,ingeneral,starsofcliquesaswellastreesofcliquesareneithergraphsofminimumdegree\n(n)nordensegraphs(of\n(n2)edges)forwhichaPTASisalreadyknown[1].2TheDkSproblemonstarsofcliquesInthissectionwestudygraphshavingascliquegraphastarofcliques.LetC0;C1;:::;Cm1bethemaximalcliquesofsuchastarsuchthatC0intersectswitheachothercliqueandnootherintersectionexists(byconventionwedenotebyCiboththecliqueCiandtheset ofitsvertices).SincesuchastaristhecliquegraphofagraphG,thereisnoedgeofGbetweenverticesbelongingtodierentcliques.WeshallcalltheC0centralcliqueandallothercliques,Ci;1im1,exteriorcliques.ForeachexteriorcliqueCiwedenotebyaithenumberofverticesinitsintersectionwithC0i.e.,ai=jCi\C0jandbybithenumberofitsverticesoutsideC0i.e.,bi=jCijai�0.ByC00wedenotethecliqueconsistingoftheverticesofC0notbelongingtoanyothercliquei.e.,C00=C0nSm1i=1Ci.BySwedenoteasolutiontotheDkSproblemi.e.,asubsetofjSj=kvertices,andbyE(S)wedenotethenumberofedgesinthesubgraphinducedbyS.BySwedenoteanoptimalsolutiontotheDkSproblem.Byn�kisdenotedthetotalnumberofverticesinallcliques.WesaythatacliqueCi;0im1;iscompletelyinasolutionSifallitsverticesareinS.Ontheotherhand,wesaythatthecliquesC0andC00arepartiallyinasolutionSifanon-emptysubsetoftheirvertices,butnotall,areinS.However,wesaythatanexteriorcliqueCi;1im1;ispartiallyinSifanon-emptysubsetofitsCinC0vertices,butnotall,areinS.WedistinguishthedenitionofthepartialinclusioninasolutionSforanexteriorcliqueCibecauseifonlysomeofitsCi\C0verticesareinS,theycanbeconsideredasverticesofC0.IngeneralwesaythatacliqueisparticipatinginasolutionSifitiseithercompletelyorpartiallyinS.ConcerninganoptimalsolutionSweobservethatifanexteriorcliqueCiispartiallyinS,thenallitsjCi\C0j=aiverticesareinS.Otherwisereplacingavertexy2CinC0;y2Sbyavertexx2Ci\C0;x=2Syieldsabettersolution,acontradiction.Inthefollowingweassumethat:(i)k�jCij;i=0;1;:::;m1.OtherwiseSconsistsofanysubsetofkverticesofsomecliqueforwhichjCijk. (ii)m�2.Form=1thepoint(i)holds.Form=2,ifk�jC0jjC1j,thenSconsistsoftheverticesofC0plusanysubsetofkjC0jverticesofC1nC0.UsingthesedenitionsandassumptionswegiveinthenextpropositionssomestructuralpropertiesofanoptimalsolutionS.Proposition1.AtmostoneofthecliquesC00;C1;:::;Cm1ispartiallyinanoptimalsolution.proof:WeproverstthatatmostoneoftheexteriorcliquesispartiallyinS.SupposethattwoexteriorcliquesCi;Cj;1i=jm1arepartiallyinSandassumew.l.o.g.thatjS\CijjS\Cjj.Letx=2SbeavertexinCinC0andy2SbeavertexinCjnC0.ThenconsiderthesolutionSinwhichwereplaceybyx.Then,E(S)=E(S)(jS\Cjj1)+jS\CijE(S)+1,acontradictiontotheoptimalityofS.TocompletetheproofitsucestoprovethatisnotpossiblebothcliqueC00andanexteriorcliqueCjtobepartiallyinS.Thisfactfollowsbyusingthesameargumentsasbefore,butnowweconsiderC0insteadofCiandx=2StobeavertexinC00.Proposition2.(i)IfC0isthelargestcliquei.e.,jC0j�jCij;1im1,thenC0belongscompletelytoeveryoptimalsolution.(ii)IfC0ispartiallyinanoptimalsolutionS,thenjC0jjCijforeverycliqueCiparticipatinginS.proof:(i)SupposethatSdoesnotcontainsomeq�0verticesofC0andconsiderasolutionSobtainedfromSbyreplacingqverticesofexteriorcliquesnotinC0bytheqverticesofC0notinS.LetusdenotebyEandE+thenumberofedgeswhichareremovedandinserted,respectively,toE(S)bythisreplacement.Then,E(S)=E(S)E+E+. EequalstothenumberofedgesthatqverticesofexteriorcliquescontributetoE(S).Thisnumber,evenifalltheqverticesbelongtothesameexteriorclique,isstrictlylessthanq2
+(jC0jq)q.Ontheotherhand,E+equalstothenumberofedgesthattheqverticesofC0notinSwillcontributetoE(S).Thisnumberisequaltoq2
+(jC0jq)q.Therefore,E(S)�E(S),acontradictiontotheoptimalityofS.(ii)SupposethatthereisanexteriorcliqueCiinSsuchthatjC0j�jCijandthatSdoesnotcontainsomeq�0verticesofC0.NoticethatjC0jq�ai,sinceifjC0jqai,thennootherexteriorcliqueparticipatesinS,thatisSispartofasingleclique(eitherCiorC0).ConsiderasolutionSobtainedfromSbyreplacingverticesofS\CinotinC0byverticesofC0notinS.Letb0i=jS\(CinC0)j,0b0ibi.UsingagainEandE+asinpart(i)wehaveE(S)=E(S)E+E+.Nowwedistinguishbetweentwocasesw.r.t.thevaluesofqandb0i.Ifqb0ithenEequalstothenumberofedgesthatb0iverticesoftheexteriorcliqueCicontributestoE(S)whileE+equalstothenumberofedgesthattheb0iverticesofC0notinSwillcontributetoE(S).ThenE(S)=E(S)E+E+=E(S)(b0i2+b0iai)+(b0i2+b0i(jC0jq))=E(S)+b0i((jC0jq)ai)&#x-372;&#x.688;E(S),acontradictiontotheoptimalityofS.Ifqb0ithenEequalstothenumberofedgesthatqverticesoftheexteriorcliqueCicontributestoE(S)whileE+equalstothenumberofedgesthattheqverticesofC0notinSwillcontributetoE(S).ThenE(S)=E(S)E+E+=E(S)(q2
+q(ai+b0iq))+(q2
+q(jC0jq))=E(S)+q(jC0j(ai+b0i))E(S)+q(jC0jjCij)&#x-272;&#x.314;E(S),acontradictiontotheoptimalityofS. DespitethenicestructuralpropertiesofanoptimalsolutioninPropositions1and2,manynaturalgreedycriteriabasedonthesizesofthecliquesor/andthesizesofintersec-tionsfailtogivesuchanoptimalsolution.InthefollowingweareabletogiveapolynomialtimedynamicprogrammingalgorithmforthecasewherethecentralcliqueiscompletelyintheoptimalsolutionandaPTASforthegeneralcase.2.1CliqueC0iscompletelyintheoptimalsolution.Lemma1.IfcliqueC0iscompletelyintheoptimalsolution,thenthereisanO(nk2)dynamicprogrammingalgorithmfortheDkSproblemonastarofcliques.proof:SincecliqueC0iscompletelyintheoptimalsolutionwehavetochoosek0=kjC0jverticesfromexteriorcliques.IfwechooseqverticesfromanexteriorcliqueCj,thentheycontributeq
aj+q2
edgestothesolution.Letf(i;j)bethemaximumnumberofedgesinasolutionchoosingiverticesfromtherstjexteriorcliques(recallthattherearem1exteriorcliques).Thusfori=0;1;2;:::;k0andj=2;3;:::;m1f(i;j)=max0qminfi;bjgff(iq;j1)+q
aj+q2gForj=1thefollowingboundaryconditionsholdfor0iminfk0;b1gf(i;1)=8�&#x-4.4;内:i2
+i
a1;ifiminfk0;b1g1;otherwiseThecomplexityofthedynamicprogrammingalgorithmisO(nk2).Thecomputationofasinglef(i;j)valuetakesO(k)timeduetothepossiblevaluesofq(0qminfi;bjgk0k)andf(i;j)valuesarecomputedforeveryik0kandjm1n.The optimalsolution,fortheDkSproblemisf(k0;m1)+jC0j2:NoticethatifC0isthelargestclique,then,byProposition2(i),C0belongscompletelytoeveryoptimalsolutionandtheabovedynamicprogrammingalgorithmapplies.2.2APTASforthegeneralcase.Inthegeneralcase,C0ispartiallyintheoptimalsolutionand,byProposition2(i),thereareexteriorcliqueslargerthanC0.LetcbethenumberofthosecliquesofsizeatleastjC0j.Moreover,byProposition2(ii),thecliquesparticipatingintheoptimalsolutionaresomeoftheseccliques.Nextpropositiongivesaweakupperboundforthenumberc.Proposition3.IfC0ispartiallyinanoptimalsolution,thenthenumberofexteriorcliquesofsizeatleastjC0jisatmostp n.proof:Thenumber,c,ofexteriorcliquesissmallerthanorequaltojC0j,sinceCi\C0=;andCj\Ci=;;1i=jm1.Thus,ifjC0jp n,thencp n.OtherwisejC0j�p n.Then,thetotalnumberofverticesintheseccliquesisatleastcp nandatmostn.Hence,cp n:ToproceedtowardsaPTASwearguefurtheronthenumberoftheexteriorcliquesofsizeatleastjC0j.Wedener=bk jC0jc.ThenthenumberofexteriorcliquesofsizeatleastjC0jthatcanbeinvolvedinanoptimalsolutionisatmostr.Letalsobeaxednumberwhichwillbedenedlater.Comparingrwithwedistinguishbetweentwocases.Case1:rIfris"small",thenweproceedinanexhaustivemanner.WeexamineallthepossiblesetsofrcliquesoutofccliquesofsizeatleastjC0ji.e.,cr
setsofcliques.Atechnicaldetail hereisthatcliqueC00shouldbealsoconsideredasoneoftheccliques.ItcanbeeasilydonebyconsideringcliqueC00asanexternalcliquewithzeroverticesoutsidecliqueC0.ByProposition3itfollowsthatthenumberofallthecr
setsofcliquesisO(nr 2).Foreachoneofthesesetsofrcliqueswecomputethekverticesthatmaximizethenumberofedgesasfollows:LetRbeasetofrcliques.ByProposition1,atmostoneofthecliquesinRispartiallyinS.Weconsiderallthe2r1subsetsofR.LetRibeoneofthesesubsetsandletCjibethejth;1jjRij,cliqueofthesetRi.ClearlyifPjRijj=1jCjijk,wediscardthesetRi.Otherwise,letk(j)=PjRijt=1;t=jjCtij,foreachj=1;2;:::;jRij.Ifk(j)&#x-282;&#x.344;kthenwediscardthisj.Otherwise(ifk(j)k)weobtainak-vertexsolutionbytakingkk(j)verticesfromcliqueCji,startingfromverticeswhichbelongtoitsintersectionwithC0.Considernowallthesolutionsobtainedforeachj=1;2;:::;jRij,andforeachRiR.Bytheirconstruction,thesesolutionsareallthepossiblek-vertexsolutionsforthesetRofcliques,undertherestrictionthatatmostoneofthemispartiallytaken.Therefore,tondtheoptimalsolutionwesimplyhavetochoosetheonewiththemaximumnumberofedges.ForasetRofrcliques,thereare2r1subsetsRi,andforeachsubsetthereareatmostrpossiblesolutions.Therefore,thenumberofsolutionstocompareisO(r2r).RecallingthatwehavetoexamineO(nr 2)setsofrcliques,thenextlemmafollows.Lemma2.Forthecaser,beaxednumber,anoptimalsolutionfortheDkSprob-leminastarofcliquescanbefoundinO(r2rnr 2)time.Case2:rIfris"large",thenweproceedinagreedymanner.Weconsiderthesolution,S,obtainedbythefollowingsimplealgorithm: LetC1C2:::Cm1andtbethelargestintegernumbersuchthatkPti=1jCij=k0.ReturnalltheverticesofthecliquesC1C2:::Ctandkk0verticesofcliqueCt+1.Nextpropositionforthecaseofindependentcliqueswillbeusefulforboundingthedeviationofoursolutionfromtheoptimalone.Proposition4.LetR1andR2betwosetsofindependentcliqueswithallcliquesinR1ofsizeatleastLandallcliquesinR2ofsizeexactlyL.ForanypairofsetsofkverticesS1andS2inR1andR2,respectively,suchthatinbothsetsatmostonecliqueistakenpartially,itholdsthatE(S1)E(S2).proof:TransformS1toanequivalenttoS2setasfollowing.First,removefromeachcliqueinS1someverticessuchthateachcliqueinS1hasnowsizeexactlyL.Letk0thenumberoftheremovedvertices.Then,replacethek0verticeswithdk0 Lecliques,all,butone,ofsizeexactlyL.AlltheremovedverticesnowhavedegreeatmostL1whileinS1theyhaddegreeatleastL1.Thus,E(S1)E(S2).LetusnowconsiderthesolutionSobtainedbyouralgorithm.ByProposition2(ii),theoptimalsolutionSinvolvesexteriorcliquesofsizeatleastjC0j.SinceouralgorithmndsasolutionSbychoosingkverticesfromthelargerexteriorcliques,itfollowsthatallcliquesinSareofsizeatleastjC0j.Moreover,sincer=bk jC0jc,weneedatleastrcliquesofsizejC0jinordertollk.Hence,choosingkverticesfromasetofindependentcliquesofsizejC0j,yieldsatleastrE(C0)edges.Therefore,byProposition4,itfollowsthatE(S)rE(C0).Clearly,anoptimalsolution,S,couldcontaincliquesofsmallersizethanthosechosenbyouralgorithm.ThesesmallcliquesareselectedbySduetotheedgesbetweentheir overlapswithC0.SincetheseedgesbelongtoC0,theoptimalsolutioncannotbegreaterthanE(S)plustheedgesofC0i.e.,E(S)E(S)+E(C0)E(S)+E(S) rE(S)+E(S) =E(S)+1 :Thus,nextlemmafollows.Lemma3.Forthecaser,where=1 ,01,thereisan(1)-approximationalgorithmfortheDkSprobleminastarofcliques.ThecomplexityofthegreedyapproximationalgorithmofLemma3isO(nlogn).ThecomplexityoftheexhaustiveoptimalalgorithmofLemma2isexponentialinr=1 ,thatisexponentialin1 .Hence,weobtainTheorem1.ThereisapolynomialtimeapproximationschemefortheDkSprobleminstarsofcliques.3TheDkSproblemontreesofcliques.InthissectionwepresentadynamicprogrammingalgorithmwhichyieldsanoptimalsolutionfortheDkSproblemforgraphshavingascliquegraphatree.LetC1;C2;:::;Ctbethecliquesofsuchatreeandmitsmaximumdegree.WeconsiderjCijk;i=1;:::;t,otherwisetheproblemistrivial.Weconsiderthetreerootedataleafclique,saycliqueCt.ThiswaytherootcliqueCthasatleastonevertexoutsideitsintersectionswithitschildrencliques.LetCibeanon-leafcliquewithmi1children,Ci1;:::;Cimi.WedenotebyQhtheintersectionofCiwithitshthchildclique,Ch,forh=i1;:::;imii.e.,Qh=Ci\Ch.WedenotebyFitheintersectionofthecliqueCiwithitsfatherclique,Cf,inthetreei.e.,Fi=Ci\CfandbyitheverticesofacliqueCinotbelongingtoanyintersectioni.e.,i=CiFiSimih=i1Qh. ByconventionweconsiderFt,fortherootcliqueCt,tobeconsistedofasinglevertexi.e.,jFtj=1.Thealgorithmtraversesthetreeofcliquesstartingfromtheleavescliques.Ineachstepitcomputesanoptimalsolutionforallthej-vertexdensestsubgraph(DjS)problems,forj=1;:::;k,onthesubtreerootedatcliqueCi.Wedenotebyfi(j)thevalueoftheoptimalsolutionoftheDjSproblemonthesubtreerootedatcliqueCi.Byfi(j;a)wedenotethevalueofanoptimalsolutiontotheDjSproblemonthesubtreerootedatcliqueCiincludingexactlyaverticesfromthecliqueFi.Itisclearthatfi(j)=max0ajFijffi(j;a)g:Tocomputeanfi(j;a)valueforanonleavecliqueCiweconsideritschildrencliquesCi1;:::;Cimi;mi1.Letfh(jh;ah)bethevalueofanoptimalsolutionofthejh-vertexdensestsubgraph,forjh=1;2;:::;k,onthesubtreerootedatcliqueCh,h=i1;:::;imi,usingahverticesofFh.NotethatfortheintersectionofthecliqueCiwithitschildChitholdsthatQh=Ci\Ch=Fh.Wecomputethevalueoffi(j;a),asfollows:Ifa=0,thennovertexofFi[ibelongstotheoptimalsolutionofthecorrespondingDjSproblemonthesubtreerootedatCi.Therefore,thevalueofthissolutionisthesameasthevalueoftheoptimalsolutiontotheDjSproblemonthesubgraphwhichistheunionofthesubtreesrootedatthechildrenofCiplustheedgesbetweentheverticesintheirQh's,thatisfi(j;a)=maxPimih=i1jh=jfimiXh=i1fh(jh;ah)+imiXi;j=i1i=jai
aj 2g: Ifa�0,thenanoptimalsolutiontotheDjSproblemincludingaverticesofFicanalsoincludeb0verticesofi,andah1vertices4ofeachQh,h=i1;:::;imi.Then,fi(j;a)=8������������&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃&#x-4.4;傃:j2;ifjPimih=i1jQhj+jij+amaxa+b+Pimih=i1jh=jfimiXh=i1fh(jh;ah)+a+b2+(a+b)imiXh=i1ah+imiXi;j=i1i=jai
aj 2g;otherwise:ForallcliquesCithatareleavesinthetreethefollowingboundaryconditionsholdfor1jk:Ifa=0,thenfi(j;a)=0.If1ajFij,thenfi(j;a)=8�&#x-4.4;内:j2
;ifjjij+a1;otherwise:Thealgorithmterminatesbycomputingthevalueft(k)fortherootcliqueCt.RecallthatweconsiderjFtj=1andthustheoptimalsolutionforthek-vertexdensestsubgraphproblemisft(k)=maxa=0;1fft(k;a)g.Thecomputationofasinglefi(j)valueforacliqueCiwithmichildrentakesO(kmi+1)timeduetothecombinationsofa;bandPimih=i1jh,suchthata+b+Pimih=i1jh=j.Thealgorithmcomputesfi(j),foreveryj=1;2;:::;kandforeveryi=1;2;:::;t.SinceintheworstcasetisO(n)andmaxifmig=m1thenexttheoremfollows:Theorem2.ThereisanO(nkm+1)algorithmfortheDkSproblemonatreeofcliquesofmaximumdegreem. 4orah0foradisconnectedsolution. NextcorollaryfollowsdirectlyfromTheorem2.Corollary1.ThereisanO(nk3)optimalalgorithmfortheDkSproblemonapathofcliques.4ConclusionsWehavepresentedaPTASforthedensestk-subgraphproblemonastarofcliquesandanO(nkm+1)timeoptimalalgorithmforthesameproblemontreesofcliques,wherenisthetotalnumberofverticesinallthecliquesandmthemaximumdegreeofthetree.ThislastalgorithmgivesanO(nk3)optimalalgorithmforpathsofcliques.SinceintervalandchordalgraphscanbeseenascliquegraphsourresultcouldbeexploitedinthedirectionofexploringthecomplexityandtheapproximabilityoftheDkSproblemintheseclassesofgraphs.Acknowledgment:Theauthorswouldliketothankthetwoanonymousrefereesfortheirvaluablecommentsthatsignicantlyimprovedthepaper.References1.S.Arora,D.Karger,andM.Kaprinski.PolynomialtimeapproximationschemesfordenseinstancesofNP-hardproblems.InProceedingsofthe27thannualACMsymposiumonTheoryofComputing,pages284{293,1995.2.Y.Asahiro,K.Iwama,H.Tamaki,andT.Tokuyama.Greedilyndingadensesubgraph.JournalofAlgorithms,34(2):203{221,2000.3.A.BillionnetandF.Roupin.Adeterministicalgorithmforthedensestk-subgraphproblemusinglinearprogramming.TechnicalReportNo486,CEDRIC,CNAM-IIE,Paris,2004.4.D.G.CorneilandY.Perl.Clusteringanddominationinperfectgraphs.DiscreteAppliedMathematics,9:27{39,1984.5.U.Feige,G.Kortsarz,andD.Peleg.Thedensek-subgraphproblem.Algorithmica,29(3):410{421,2001.6.U.FeigeandM.Langberg.Approximationalgorithmsformaximizationproblemsarisingingraphpartitioning.JournalofAlgorithms,41(2):174{211,2001. 7.U.FeigeandM.Seltser.Onthedensestk-subgraphproblem.TechnicalReportCS97-16,WeizmannInstitute,1997.8.F.Gavril.Algorithmsforminimumcoloring,maximumclique,minimumcoveringbycliquesandmaximumindependentsetofchordalgraph.SIAMJournalonComputing,1(2):180{187,1972.9.O.GoldschmidtandD.Hochbaum.k-edgesubgraphproblems.DiscreteAppliedMathematicsandCombina-torialOperationsResearchandComputerScience,74(2):159{169,1997.10.Q.Han,Y.Ye,andJ.Zhang.Animprovedroundingmethodandsemideniteprogrammingrelaxationforgraphpartition.MathematicalProgramming,92(3):509{535,2002.11.R.Hassin,S.Rubinstein,andA.Tamir.Approximationalgorithmsformaximumdispersion.OperationsResearchLetters,21(3):133{137,1997.12.J.M.KeilandT.B.Brecht.Thecomplexityofclusteringinplanargraphs.JournalofCombinatorialMathematicsandCombinatorialComputing,9:155{159,1991.13.S.Khot.RulingoutPTASforgraphmin-bisection,densestsubgraphandbipartiteclique.InProceedingsofthe45thAnnualIEEESymposiumonFoundationsofComputerScience,pages136{145,2004.14.G.KortsarzandD.Peleg.Onchoosingadensesubgraph.InProceedingsofthe34thAnnualIEEESymposiumonFoundationsofComputerScience,pages692{701,1993.15.F.Maoli.Findingabestsubtreeofatree.TechnicalReport91.041,PolitechnicodiMilano,DipartimentodiElektronica,1991.16.Y.PerlandY.Shiloach.Ecientoptimizationofmonotonicfunctionsontrees.SIAMJournalofAlgebricandDiscreteMethods,4(4):512{516,1983.17.D.J.Jr.RaderandG.J.Woeginger.Thequadratic0-1knapsackproblemwithseries-parallelsupport.OperationResearchLetters,30(3):159{166,2002.18.S.S.Ravi,D.J.Rosenkrantz,andG.K.Tayi.Heuristicandspecialcasealgorithmsfordispersionproblems.OperationsResearch,42(2):299{310,1994.19.A.SrivastavandK.Wolf.Findingdensesubgraphswithsemideniteprogramming.InProceedingsoftheInternationalWorkshoponApproximationAlgorithmsforCombinatorialOptimization,pages181{191,1998.20.Y.YeandJ.Zhang..519approximationofdense-n/2-subgraph.WorkingPaper,DepartmentofManagementSciences,HenryB.TippieCollegeofBusiness,TheUniversityofIowa,1999.