AdvancedApproximationAlgorithms(CMU18-854B,Spring2008)Lecture27:Algori - PDF document

AdvancedApproximationAlgorithms(CMU18-854B,Spring2008)Lecture27:AlgorithmsforSparsestCutApr24,2008Lecturer:AnupamGuptaScribe:VarunGupta Inlecture19,wesawanLPrelaxationbasedalgorithmtosolvethesparsestcutproblemwithanapproximationguaranteeofO(logn).Inthislecture,wewillshowthattheintegralitygapoftheLPrelaxationisO(logn)andhencethisisthebestapproximationfactoronecangetviatheLPrelaxation.WewillalsostartdevelopinganSDPrelaxationbasedalgorithmwhichprovidesanO(p logn)approximationfortheuniformsparsestcutproblem(wheredemandsbetweenallpairsofverticesisij=1),andanO(p lognloglogn)algorithmforthesparsestcutproblemwithgeneraldemands.1ProblemDenitionandLPrelaxationreviewRecallthatthethesparsestcutproblemisdenedasfollows.WearegivenanundirectedgraphG=(V;E)withnon-negativeedgecosts(orcapacities)ce=cijforalle=fi;jg2V2
,non-negativedemandsijbetweeneverypairofverticesfi;jg2V2
.Withtheedgecapacitiesandthedemands,wecanassociatevectorsc;2(n2)(n=jVj).Thesparsestcutproblemseekstond=minSVc
S 
S=cap(S;S) dem(S;S)(1)wherecap(S;S)=Pi2S;j2Scijdem(S;S)=Pi2S;j2SijandS2(n2)isthecutmetricassociatedwithS:Sij=(0ifi;j2Sori;j2S1otherwiseToformtheLPrelaxation,werelaxtherequirementofminimizingoverthecutmetricstomini-mizingoverallmetrics.Thatis,=minmetricsdc
d 
d1 Theaboverelaxationissolvedbythefollowinglinearprogram:minPi;jcijdijsubjecttoPi;jijdij=1dij+djkdik8i;j;kdij08i;j(2)Clearly,.Inlecture19,weproved
O(logn)byembeddingthemetricreturnedbytheLPinto`1withdistortionO(logn).2IntegralitygapforsparsestcutLPrelaxationAnaturalquestiontoaskis,canwegetabetterapproximationratiothanO(logn)usingtheLPrelaxation?Inthissectionwewillseethattheanswerisno,sincetheLPrelaxationhasanintegralitygapofO(logn).Claim2.1.Theintegralitygapbetweenandis\n(logn).Toprovetheaboveclaim,wewillrsttakeasmalldigressionandintroducethemaximumconcurrentowproblemwhichtakesthesameinputasthesparsestcutproblem.WewillshowthattheoptimalvalueofthemaximumconcurrentowproblemisequaltotheoptimalvalueofthesparsestcutLPrelaxationonthesameinputgraph.Finallywewillprovethattheintegralitygapofand=is\n(logn).2.1ThemaximumconcurrentowproblemDenition2.2.GivenanundirectedgraphG=(V;E)withnon-negativeedgecapacitiesce=cijforalle=fi;jg2V2
,non-negativedemandsijbetweeneverypairofverticesfi;jg2V2
.themaximumconcurrentproblemseekstomaximize,suchthatwecansend
ijowbetweenverticesiandjsimultaneouslyforallfi;jg2V2
whilesatisfyingtheedgecapacityconstraints.Letdenotetheoptimalvalueofthemaximumconcurrentowproblem.Considerapartition(S;S)ofV.Thetotalowcrossingthispartitionis
dem(S;S)whereasthecapacityofthepartitioniscap(S;S).Sincewecan'thavemoreowthanthecapacity,
dem(S;S)cap(S;S)8SVandhence,minSVcap(S;S) dem(S;S)=Infact,themaximumconcurrentowvalueisexactlythesameastheoptimumvaluefortheLPrelaxation.2 Capacity edges Figure1:Aninstanceofthesparsestcut/maximumconcurrentowproblem.Fact2.3.Givenaninstanceofthesparsestcutproblem,andmaximumconcurrentowproblemonthesameinputgraph,=.Proof.TheprooffollowsbyobservingthattheLPrelaxationofsparsestcutproblemandtheLPforthemaximumconcurrentowproblemaredualsofeachother.TheLPforthemaximumconcurrentowhasavariablef(P)foreachsimplepathintheinputgraph.minsubjecttoPpathsPbetweeni;jf(P)
ij8i;jPpathsPcontaininge=fu;vgf(P)Cuv8u;vf(P)08PThedualhasvariablesijanduvcorrespondingtothetwosetsofcontraintsabove:maxPijCuvuvsubjecttoPfu;vg2Puvij8pathsPbetweeni;jPijijij1;0Nowifweconsiderijtobethedistancebetweeni;janduvtobetheedgelengthoftheedgefu;vg,thenthisLPcanbecheckedtobethesameas. 2.2IntegralityGapsForexample,considerthegraphinFigure1.Thesolidedgesrepresentedgeswithcapacity1.Thedottededgesrepresentpairsfi;jgwithij=1.Remainingcapacitiesanddemandsare0.NotethatthevalueofthesparsestcutinthegraphinFigure1is=1(chooseanysolidvertexasthesetS).Furthermore,3 4.Ifsendunitsofowbetweeneverydemandpair,thetotalvolumeoftheowis8,sinceeachpathbetweenademandpairhastwoedgesandtherearefourdemandpairs.Thetotalcapacityis6,andhence3 4.Infact=3 4,bysending1 4unitsofowoneachofthethreepathsbetweenthewhiteverticesand3 8unitsoneachofthetwopathsbetweeneverypairofsolidvertices.3 Theorem2.4.[7]Theintegralitygapbetweenand(=)is\n(logn).Proof.LetG=(V;E)beaconstantdegreeexpanderwithunitcapacityoneachedgeci;j=18fi;jg2E,andunitdemandbetweeneverypairofverticesij=18i=j.WewillshowthatGhasalargebutsmall.(A)=minScap(S;S) dem(S;S)=minjSjn 2cap(S;S) jSjjSj=minjSjn 2cap(S;S) jSjq21 n;2 n=\n(1)\n1 nwherethelaststepfollowssinceminjSjn 2cap(S;S) jSjistheedgeexpansionwhichis\n(1)foracon-stantdegreeexpander.(B)Recallthefollowingclaim,whichfollowsfromproblem7inhomework5:Claim2.5.Inaconstantdegreeexpander(saydegree=10),\n(n2)pairsofverticesareatadis-tancegreaterthan1 10logn.Sinceallijare1,atleast
\nn2lognvolumeofowisneededtosendunitsofowbetweenthese\n(n2)pairs.However,sincethegraphisaconstantdegreeexpander,totaledgecapacityisO(n).Therefore,O1 nlogn:Thiscompletestheproofof\n(logn)gapbetweenand,andhenceofClaim2.1. 3SDPrelaxationforsparsestcutToobtaintheLPrelaxation,wehadrelaxedtherequirementofminimizingoverallcutmetricstominimizingoverallmetrics.ToobtaintheSDPrelaxationweconsiderthefollowingtighterrelaxation:=mind2metric\`22c
d 
dRecallthatann-pointmetricdisin`22(itisa“squared-Euclidean”metric)ifthereexistpointsv1;v2;


;vn2ksuchthatthedistancesaredij=kvivjk22:Notethattheconditionthatthesquareddistancesformametric(i.e.,satisfythetrianglein-equality)isequivalenttosayingthatinthespacek,noneofthetrianglesbetweenthesenpointshaveobtuseangles.Notethat`22\metricformsaconvexcone.Somemorepropertiesof`22\metric:4 1.Ifd2`1,thend2`22.(Why?)Thisiswhatwerequiresinceweneedtooptimizeover`1andhencethefeasiblesetoftheSDPrelaxationshouldbeasupersetof`1.2.Ink,wecanhaveatmost2kpointswith`22metric(infact,anynegativetypemetric).Thisisachievedbythehypercube.3.Givennpointsontherealline1withthe`1metric,the`22embeddingofthesepointsrequiresndimensions—anewdimensionforeachpoint.(Usethistoinferthatd2`1)d2`22.)TheSDPtocomputeisgivenby:minPi;jcijkxixjk2subjecttoPi;jijkxixjk2=1kxixjk2+kxjxkk2kxixkk28i;j;kxi2t8i(3)TheapproximationratiooftheSDPrelaxationnaturallydependsonhowwell(lowdistortion)onecanembedan`22metricinto`1.ThefollowingtheoremsgiveupperboundsontheintegralitygapfortheSDPrelaxation(3).Theorem3.1(Goemans,unpublished).IftheSDPreturnsasolutionink,thentheintegralitygapisO(p k).Theorem3.2([2]).Fortheuniformsparsestcutproblem(ij=18i=j),theSDPintegralitygapisO(p logn).Theorem3.3.[1]Forgeneralsparsestcut,theSDPintegralitygapisO(p lognloglogn).ThetechniquesusedinprovingabovetheoremsareusefulastoolstoroundSDPrelaxationsinminimizationsproblems(earlierwehaveseenroundingtechniquesformaximizationproblems).Goemans,andindependently,LinialmadethefollowingconjectureontheintegralitygapoftheSDPrelaxation:Conjecture3.4.[4,8]Theintegralitygapbetweenandis(1).TheGoemans-LinialconjecturewasrstdisprovedbyKhotandVishnoi[5]whoprovedan\nloglogn)1=6
integralitygapforthenon-uniformcase.Thiswasthenimprovedto\n(loglogn)byKrauthgamerandRabani[6].Foruniformsparsestcut,\n(loglogn)integralitygapwasshownbyDevanur,Khot,SaketandVishnoi[3].3.1FromSDPrelaxationtosparsecutsInthissectionwewillseeanimportantstructuretheoremandsomeintuitionofhowthisstructuretheoremcanleadtoaO(p logn)approximationforthesparsestcutproblem;theproofwillbegiveninthenextlecture.5 Lemma3.5.StructureLemma[2]:Letv1;v2;


;vnbepointsintheunitballinksatisfyingdij=kvivjk2isametric.Supposethepointssatisfythefollowing“well-spread-outproperty”:1 n2Pi;jdij=\n(1)ThenthereexistdisjointsetsSandTsuchthatjSj;jTj\n(n)andmini2S;j2Tdij\n1 p logn IntuitionfortheO(p logn)approximationintheuniformcase.Sincealldemandsarethesame,wecanscalethedemandsandsetthemtoij=1 n28i=j.NowtheSDPensuresthatXijdij=Xijdij=1 n2Xdij=1:Nowsupposewegotlucky,andtheSDPembeddingliesontheunitball,sothatwecanusetheStructurelemma.(Thiswillnothappeningeneral:we'llgivearigorousproofnexttime.)IfnowwepicktheSandTsatisfyingtheStructureLemmaandcutatarandomdistancefromS,theprobabilityanedgeeiscutis:Pr[edgeeiscut]=de 1 p logn=dep lognThustheexpectedtotalcapacitycrossingthecutis
O(p logn).Furthermore,sinceSandTareboth\n(n)andthedemandsareequal,weloseaconstantinthedemandcrossingthecut. S log n1cut at a random T 3.2TheStructureLemmaisTightAnaturalquestionis,canwetightentheStructureLemmatoobtainabetterapproximation?Theanswertothisisno:anexamplewheretheStructureLemmaistightisthehypercubef1 p K;1 p KgK6 whereK=log2n.The`22distancebetweentwoverticesviandvjisgivenby:dij=kvivjk2=4Hammingdist(i;j) KToprovethis,considertwosetsS;TwithjSj;jTj=sforsomeparameterstobespeciedsoon.Thevertex-isoperimetricinequalityforthehypercubesaysthatforallsetswithsizes,thesetXthathasthefewestneighborsoutsideX(i.e.,thesmallestjN(X)nXj)isaballaroundsomevertex.Therefore,onesuchsetXwiththesmallestvertex-expansionisthesetjXjcontainingexactlythespointsclosestto1 p Kf1;1;


;1g.AndhencejS[N(S)jjX[N(X)jforalljSj=s=jXj.Nowsupposewechoose:s=PiK 2q Klog(1 )Ki
thensnviatailboundsonthebinomialdistribution.LetSbeanysetofthissize,then:jS[N(S)jjX[N(X)j=PiK 2q Klog(1 )+1Ki
Iteratingthis,ifwedeneSttobeallelementsatHammingdistancetfromS,wewouldhavejStjPiK 2q Klog(1 )+tKi
Fort=p Klog(1=)thiswouldbeatleastn=2.Similarly,jTtjwouldbeatleastn=2forthesamevalue.Sinceboththesesetscontainatleasthalftheelements,StintersectsTt,andhenceSandTare2t-closeinHammingdistance.Butt=O(p K),whichmeansthatthe`22distancebetweenSandTis4
O(p K) K=O(1 p logn),whichprovesthefactthattheStructureLemmaistightuptoconstants.3.3ProvingaSmallIntegralityGapintheUniformCaseWeendwiththefollowinglemmaduetoRabinovich[9].Lemma3.6.[9]Fortheuniformsparsestcutproblemij=18i=j,supposethemetricdgivenbytheSDPembedsinto2`1suchthat,1.d2.Pi;jijPijdij ,thentheintegralitygapfortheuniformsparsestcutSDPisatmost.7 Proof.TheproofisverysimilartotheproofwesawfortheSparsestCutprobleminLecture19thatanembeddinginto`1withdistortionimpliesanintegralitygapof.Sinceherewearedealingwiththeuniformcase,weshowthattheaverageconditionabovesufces.Indeed,c
 
c
d Pijc
d Pdij=c
d 
d
= Inthenextlecture,wewillseeatechniquetoembedtheSDPmetricinto1(andhence`1).References[1]S.Arora,J.Lee,andA.Naor.Euclideandistortionandthesparsestcut.InProc.37thSymp.onTheoryofComputation(STOC),pages553–562,2005.[2]S.Arora,S.Rao,andU.Vazirani.Expanderows,geometricembeddingsandgraphparti-tioning.InProc.36thSymp.onTheoryofComputation(STOC),pages222–231,2004.[3]N.Devanur,S.Khot,R.Saket,andN.Vishnoi.Integralitygapsforsparsestcutandminimumlineararrangementproblems.InProc.38thSymp.onTheoryofComputation(STOC),pages537–546,2006.[4]M.Goemans.Semideniteprogrammingincombinatorialoptimization.MathematicalPro-gramming,79:143–161,1997.[5]S.KhotandN.Vishnoi.Theuniquegamesconjecture,integralitygapforcutproblemsandembeddabilityofnegativetypemetricsinto`1.InProc.46thSymp.onFoundationsofCom-puterScience(FOCS),pages53–62,2005.[6]R.KrauthgamerandY.Rabani.Improvedlowerboundsforembeddingsintol1.InProceedingsofthe17thannualACM-SIAMsymposiumonDiscretealgorithm,pages1010–1017.ACMPress,2006.[7]F.T.LeightonandS.Rao.Anapproximatemax-owmin-cuttheoremforuniformmulticom-modityowproblemswithapplicationstoapproximationalgorithms.InProc.29thSymp.onFoundationsofComputerScience(FOCS),pages422–431,1988.[8]N.Linial.Finitemetricspaces-combinatorics,geometryandalgorithms.InProc.Interna-tionalCongressofMathematiciansIII,pages573–586,2002.[9]Y.Rabinovich.Onaveragedistortionofembeddingmetricsintothelineandinto`1.InProc.35thSymp.onTheoryofComputation(STOC),pages456–462,2003.8