171MulticutFirstconsiderthemulticutproblemGivenagraphGVEKpairsofterminalverticesfsitigandacostfunctionontheedgescERthemulticutproblemasksforaminimumcostcutofGthatseparatessiandtiforalli ID: 837398
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1 CS880:ApproximationsAlgorithmsScribe:Mat
CS880:ApproximationsAlgorithmsScribe:MattElderLecturer:ShuchiChawlaTopic:SparsestCutandBalancedCutDate:3/20/07 17.1MulticutFirst,considerthemulticutproblem.GivenagraphG=(V;E),Kpairsofterminalverticesfsi;tig,andacostfunctionontheedgesc:E!R,themulticutproblemasksforaminimum-costcutofGthatseparatessiandtiforalli.Lasttime,wegaveaO(logk)approximationforthisproblem.The(relaxed)linearprogramforthisproblemisasfollows;callit\Primal1".minimizeXe2Ecedewhered(si;ti)18idisametricWecanrewriteasfollows:minimizeXe2EcedewherePi=fAllpathsfromsitotigXe2Pde18i8P2Pide08eThedualofthisLP,whichwe'llcall\Dual1",isasfollows:maximizeXiXP2Pifi;PwherePi=fAllpathsfromsitotigXiXP2PiP3efi;Pce8efi;P0Dual1solvesthemax-summulti-commodity\rowproblem:cerepresentsthecapacityofanedge,andfi;Pistheamountof\rowdirectedfromsitotialongthepathP.TheLPtriestomaximizethetotalamountofcommodity\row.Lemma17.1.1Multicutisalwayslargerthanthecorrespondingmax-summulti-commodity\row.Lemma17.1.2MulticutisatmostO(logK)timesthecorrespondingmax-summulti-commodity\row.Theorem17.1.3Whenk=2,multicutequalsmax-summulti-commodity\row.1 17.2MaximumConcurrentMulticommodityFlowAsolutiontoDual1maystarvesomecommoditieswhileroutingothers.Incontrast,max-concurrentmulticommodity\rowroutesequalfractionsofallcommoditieswhilerespectingca-pacities.Thus,wedevisethefollowingLPformax-concurrentmulticommodity\row,whichwecallDual2:maximizetwhereXiXP2PiP3efi;Pce8eXP2Pifi;Prit8ifi;P08i;8P2PIntuitively,thedi erencebetweenDual1andDual2isthatDual1seekstomaxim
2 izethetotal\rowacrossindependentcommodit
izethetotal\rowacrossindependentcommodities,whileDual2seekstomaximizetheminimumofasetofweighted\rows.Thisisthemaximumconcurrentmulticommodity\rowproblem.Primal2,thedualofthemaximumconcurrentmulticommodity\rowproblem,isasfollows:minimizeXedecewhereXe2Pdeyi8i;8P2PiXiriyi1de08eyi08iThecostsceareconstantsoftheprobleminstance,soPrimal2willseektominimizethevaluesforde.Thus,theywillbenolargerthantheyareconstrainedtobe,soyi=d(si;ti),theshortestdistancefromsitotiwhereeachedgeehaslengthde.So,wecandevisethefollowinglinearprogram,equivalenttoPrimal2:minimizeXecedewhereXirid(si;ti)1disametricUptoscalingd,thisisthesameasthefollowingprogram:minimizePecede Pirid(si;ti)wheredisametric,2 whichwilltiedirectlytothesparsestcutproblem.17.3SparsestCutLetG=(E;V)besomegraph.LetT,thesetofterminals,beasetofpairsofvertices.ForanycutS,asubsetofV,let (S)denotethesparsityofS,with (S)= E(S;S) (SS)\T :Thus, (S)isthesizeofthecutSdividedbythenumberofterminalsthatSseparates.ThesparsestcutproblemtakesGandT,and ndsthecutSthatminimizes (S).Wecangeneralizethisfurther,byintroducingacostcontheedgesandaweightriforeachterminali.Then,ourmoregeneral (S)lookslikethis: (S)=c(E(S;S)) Pfijsi2S,ti=2Sgri:Considertheuniformversionofsparsestcut,whereweletthesetofterminalsbeallpossiblepairs.Thatis,weletT=VVandru;v=1forallu=v.Then,thesparsityis (S)=c(E(S;S)) jSjjSj;whichisquitesimilartotheexpansionofaset,fromthecontextofexpandergraphs.Here,wegiveaO(logKlogD)-approximationforthisproblem,whereD=PiriandK=jTj.Nexttime,we'
3 ll ndaO(logK)-approximation.Thebestknown
ll ndaO(logK)-approximation.ThebestknownecientapproximationisaO(p logKloglogK)-approximation.SupposewesolvePrimal2,yieldingthemetricd.Letyidef=d(si;ti).WeknowthatPiriyi1andthat8e;de1.Weneedtorounddintoacutmetric|ametricwithonlyonesandzeroes|withoutmuchincreasingthesparsity.Considerthespecialcasewhere8i;yi1 2ymax.Letymaxdef=maxiyi,d0def=d=ymin,andy0idef=d0(si;ti)foralli.Then,8i;y01,andd0isafeasiblesolutiontothemulticutLP.So,wecanfeedd0tothemulticutapproximationalgorithmwesawlastlecture.ThatalgorithmwillyieldacutofvalueO(logK)Peced0e.Wededuce:O(logK)Xeced0eO(logK)1 yminXecedeO(logK)1 ymaxXecede:ThedemandthisseparatesisthusPiri(1=ymax)Piriyi.So,thesparsityofthisalgorithm,inthisspecialcase,isatmostO(logK)Pcede Priyi:3 So,nowconsiderthegeneralcase,withanarbitrarilylargeratiobetweenymaxandymin.De neIx,thesetofallyiinaconveniently-de nedinterval,as:Ix=nijyi2ymax 2x+1;ymax 2xio:Whenxisconstrainedtotheintegers,it'sclearthateveryyiiscontainedinexactlyoneIx.ForeachIx,ouralgorithmwillconstructamulticutinstanceasinthespecialcase,butitwillscaledby2x+1=ymaxinsteadof1=ymax.Thesparsityforeachoftheseinstancesisnottoolarge: (Ix)O(logK)Pecede Pi2Ixriyi:Ifthereexistsaconstant andanxsuchthatPi2Ixriyi 1Piriyi,thenthesparsityofthisinstanceisatleastO(logK) Pcede=(Priyi).Weclaimthatwecanignoreallisuchthatyiymax=D2.Again,DisthetotaldemandPri.De nethesetWcontainingweevaluesofyi,W=ijyiymax=D2 .IgnoringWcanresultinthelossofatmostPi2WriyiPi2Wriymax=D2ymax=D1=Dfromthedenominatorofouralgorithm'sspa
4 rsity.AssumingD2,ignoringWhasonlyasmall
rsity.AssumingD2,ignoringWhasonlyasmallconstantapproximationcost.(IfD2,thisisaneasyboundarycase,whichwecane ecientlyhandleinanad-hocway.)Thus,weneedtoconsideronlythoseIxwhere2xD2.Thereareatmost2logDsuchsets,so 1=(2logD).ThisyieldsaO(logKlogD)-approximation.17.4BalancedCutGivenagraphG=(V;E),thebalancedcutproblemdemandsthemin-costcutsuchthateachsidehasatleast nnodes,forsomeconstantvalueof 1 2.Itisknownthatthisproblemisinapproximableton2 =OPTifP=NP.Thisisanabsurdlypoorapproximation.So,weconsiderinsteadapseudo-approximationalgorithm,inwhichweapproximateboththeobjectivefunctionoftheoptimalsolutionandtheparametersofitsinstance.So,inthiscase,whenaskedforacutwithabalanceof ,weinsteadoutputacutwithabalance 0,suchthat 0 and 01=3.Iftheoptimalcutofbalance hascostC ,ourcutwillhavecostnogreaterthanO(logn)C =( 0).Noticethat,thoughwehaveareasonableboundontheratiobetweenthesizeofourcutandC ,theratiobetweenthesizeofourcutandC0 maybeunbounded.Thealgorithmemploysadirectreductiontothesparsestcutproblem,lettingT=VVandri=1.Then,thesparsityofacutSis,asbefore,c(E(S;S))=(jSjjSj).Wediscussfurtherdetailsnexttime.Eventhoughthisalgorithmisfarfromoptimal,itisactuallyuseful.Thispseudo-approximationhasapplicationsindivide-and-conqueralgorithmsongraphs.Itensuresthatwecanalwaysdivideagraphintotwopieces,eachwithsizeroughlylinearinthesizeoftheoriginalgraph,suchthatthecutbetweenthepiecesisn'ttoolarge.Thisyieldslog-depthrecursion,whichdivide-and-conqueralgorithmsdemand,whileboundingthecostofrecombiningpieces.