ApproximatingBoundedDegree InstancesofNP-HardProblems MarekKarpinskiy. Abstract

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1 ApproximatingBoundedDegree InstancesofNP-HardProblems MarekKarpinskiy optimizationproblems. theintermediatestagesofprovingapproximationhardnessofsomeothergeneric wepresentthebestuptonowknownexplicitnonapproximabilityboundsonthe verysmalldegreeoptimizationproblemswhichareofparticularimportanceon proximatingboundeddegreecombinatorialoptimizationproblems.inparticular, Wepresentsomeoftherecentresultsoncomputationalcomplexityofap- Abstract 22-24,2001. grants,dimacs,andistgrant14036(rand-apx),andbythemax-planckresearchprize. ResearchpartiallydonewhilevisitingDepartmentofComputerScience,YaleUniversity. ToappearinProc.13thSymp.onFundamentalsofComputationTheory,FCT'01,August ydept.ofcomputerscience,universityofbonn,53117bonn.supportedinpartbydfg 1

2 Aninterestingapproximationhardnessphenomenonofcombinatorialoptimizationwasdiscoveredin[PY91]and[ALMSS92],totheeectthatthebounded asboundeddegreeinstancesofmanyoptimizationproblemswereknowntohave withinanarbitraryconstant.thisfactseemedtobeabitpuzzlingatthetime 1Introduction trivialapproximationalgorithmsdramaticallyimprovingperformancesofthebest knownapproximationalgorithmsongeneralinstances.aninterestingartifacton degreeinstancesofseveraloptimizationproblemsarehardtoapproximateto provinghardnessofapproximationofsomeotherimportantoptimizationproblems,likesetcover,somerestrictedversionsoftravelingsalesmanproblem,and turnedouttobeparticularlyimportantattheintermediatereductionstagesfor bestknownapproximationalgorithmsonthatinstances.theseinstanceshave forasurvey.wediscusshereexplicitapproximationlowerboundsforbounded degreeinstanceswithaverysmallboundsondegreeslike3or4,andalsothe mialtimeapproximationschemes(ptass)forthem[akk95],[kz97],see[k01] theircomplementary,i.e.dense,instanceswasalsotheexistenceofpolyno- interestingnewresultsonasymptoticrelationsbetweenhardnessofapproximationandboundsonadegreeofinstances[h00],[t01].theseresultsdonotyield thoughexplicitlowerapproximationboundsforsmalldegreeinstancesneededin [F98],[PY91],[PY93],[BK99],[FK99],[E99],[EK00].Wementionheresome theproblemofsortingbyreversalsmotivatedrecentlybymolecularbiology,cf. applicationsmentionedbefore. plicitlowerapproximationboundsfortheproblemofsortingbyreversals[bk99], BISECTION,MAX-2SAT,MAXIMUMINDEPENDENTSET,andMINIMUM NODECOVER[BK99],[BK01b].Wemoveon,andapplytheseresultstogetexisabilityofsystemsoflinearequationsmod2,MAX-CUT,MAX-andMINtionproblems,liketheproblemsofmaximizationorminimizationofthesat- lowerboundsforthesmalldegree(numberofvariableoccurrences)optimiza- Wesurveyinthispaperthebestknownuptonowexplicitapproximation wementionrecentimprovementonapproximationratiosofalgorithmsforsmall andthetravelingsalesmanproblemwithdistancesoneandtwo[ek00].finally, degreemax-cutandmax-bisectionproblemsbasedonlocalenhancing methodsforsemideniteprogramming[fkl00a],[fkl00b],[kkl00]. 2BoundedDegreeMaximizationProblems Wearegoingtodenebasicoptimizationproblemsofthissection. MAX-Ek-LIN2:Givenasetofequationsmod2withexactlykvariables perequation,constructanassignmentmaximizingthenumberofequations satised. 2

3 b-occ-max-hybrid-lin2:givenasetofequationsmod2withexactly b-occ-max-ek-lin2:givenasetofequationsmod2withexactlykvariablesperequationandthenumberofoccurrencesofeachvariablebounded twoorthreevariablesperequation,andthenumberofoccurrencesofeach equationssatised. variableboundedbyb,constructanassignmentmaximizingthenumberof byb,constructanassignmentmaximizingthenumberofequationssatised. b-occ-max-2sat:givenaconjunctivenormalformformulawithtwo d-max-cut:givenanundirectedgraphofdegreeboundedbyd,partition d-mis:givenanundirectedgraphofdegreeboundedbyd,constructa variablesperclause,constructanassignmentmaximizingthenumberof clausessatised. itsverticesintotwogroupssoastomaximizethenumberofedgeswith exactlyoneendpointineachgroup. ski[bk99]designedspeciallyforboundeddegreeproblems.thiskindofamplier reductionsdependsonanewwheel-amplierconstructionofbermanandkarpinducefromthemax-e2-lin2andthemax-e3-lin2problems.themethodof Wearegoingtodisplaynowapproximationpreservingreductionswhichre- adjacent. maximumcardinalitysubsetofverticessuchthatnotwoverticesofitare Theorem1.([H97])Forany0<<12,itisNP-hardtodecidewhether mizationproblems. e.g.,aroraandlund[al97])forsmalldegree,andnumberofoccurrences,opti- hasturnedouttobemoreecientthanthestandardexpanderampliers(cf. Theorem2.([H97])Forany0<<12,itisNP-hardtodecidewhetheran aninstanceofmax-e2-lin2with16nequationshasitsoptimumvalueabove (12 )norbelow(11+). WestartwiththefollowingknowninapproximabilityresultsofHastad[H97]. approximationpreservingreductionswereconstructed: orbelow(1+)n. instanceofmax-e3-lin2with2nequationshasitsoptimumvalueabove(2 )n f2:max-e2-lin2!3-max-cut, f1:max-e2-lin2!3-occ-max-e2-lin2, InBermanandKarpinski[BK99]thefollowingpolynomialtimerandomized 3

4 whereasaconstructionforf3usescertain3-hypergraphextensionofit.the followingoptimizingpropertiesoff1;f2,andf3wereprovenin[bk99]. Theorem3.([BK99])Forany0<<12,itisNP-hardtodecidewhetheran Theconstructionsforf1,andf2usevariantsofwheel-ampliermethods, f3:max-e3-lin2!3-occ-max-hybrid-lin2, instanceoff1(max-e2-lin2)23-occ-max-e2-lin2with336edgeshasits valueabove(332 )norbelow(331+)n. instanceoff2(max-e2-lin2)23-max-cutwith336edgeshasitsoptimum optimumvalueabove(332 )norbelow(331+)n. Theorem4.([BK99])Forany0<<12,itisNP-hardtodecidewhetheran Forf3andMAX-HYBRID-LIN2wehave Asimilarresultcanbeprovenforf2,andthe3-MAX-CUT-problem. 2SAT. optimumvalueabove(62 )norbelow(61+)n. Theorem5.([BK99])Forany0<<12,itisNP-hardtodecidewhethraninstanceoff3(MAX-E3-LIN2)23-OCC-MAX-HYBRID-LIN2with60nequations withexactlytwovariablesand2nequationswithexactlythreevariableshasits Theorem6.([BK99])Forany0<<12,itisNP-hardtodecidewhether (2012 )norbelow(2011+)n. aninstanceof3-occ-max-2sat,with2016nclauseshasitsoptimumabove Theorem4canbealsousedtoderivethefollowingboundfor3-OCC-MAXderivelowerboundsfor4-MISproblem,andusingsomemoresubtleconstruction, evenfor3-misproblem. Theorem7.([BK99])Forany0<<12,itisNP-hardtodecidewhetheran instanceof4-miswith152nnodeshasitsoptimumvalueabove(74 )norbelow (73+)n,andwhetheraninstanceof3-MISwith284nnodeshasitsoptimum The3-OCC-MAX-HYBRID-LIN2problemandTheorem5canbeusedto Corollary1.Forevery>0,itisNP-hardtoapproximate: valueabove(140 )norbelow(139+)n. (1)3-OCC-MAX-E2-LIN2and3-MAX-CUTtowithinafactor332=331, Theresultsaboveimplythefollowingexplicitnonapproximabilityresults. 4

5 (3)3-OCC-MAX-2SATtowithinafactor2012=2011, (2)3-OCC-MAX-HYBRID-LIN2towithinafactor62=61, sectionwillbeusedalsolateroninourpaper. arefrom[gw94],[bf94],[bf95],[fg95],[fkl00a].thetechnicalresultsofthis mationboundsaresummarizedintable1.theupperapproximationbounds (4)4-MIStowithinafactor74=73,and3-MIStowithinafactor140=139. Thebesttoourcurrentknowledgegapsbetweenupperandlowerapproxi- BoundedDegreeMaximizationProblems TABLE1: 3-OCC-MAX-HYBRID-LIN2 3-OCC-MAX-E2-LIN2 3-OCC-MAX-2SAT 3-MAX-CUT Approx.UpperApprox.Lower 3-MIS MIS Wearegoingtointroducenowthefollowingminimizationproblems. 3BoundedDegreeMinimizationProblems d-nodecover:givenanundirectedgraphofdegreeboundedbyd,constructaminimumcardinalitysubsetofverticessuchthateachedgeofa b-occ-min-ek-lin2:givenasetofequationsmod2withexactlykvariablesperequationandthenumberofoccurrencesofeachvariableexactly MIN-Ek-LIN2:Givenasetofequationsmod2withexactlykvariables perequation,constructanassignmentminimizingthenumberofequations graphhashatleastoneofitsendpointsinit. satised. equaltob,constructanassignmentminimizingthenumberofequations 5

6 MIN-BISECTION:Givenanundirectedgraph,partitiontheverticesinto boundsonboundeddegreeminimizationproblems. WewillspecializenowtechniquesofSection2toobtainlowerapproximation d-min-bisection:givenad-regulargraph,partitiontheverticesinto twoequalhalvessoastominimizethenumberofedgeswithexactlyone endpointineachhalf. 4-MISwith152nnodes.ItisNP-hard,forany0<<12,todecidewhether 4-NodeCoverhasitsoptimumvalueabove( )n=(79 )norbelow setiofg,vniisaminimumnodecoverofg.wetakenowaninstanceof lem.foragivenundirectedgraphg=(v;e),andamaximumindependent ( )n=(78+)n.Similarlyfor3-NodeCover.Thuswehave WestartwithadirectapplicationofTheorem7towardsd-NodeCoverprob- Theorem8.Forany0<<12,itisNP-hardtodecidewhetheraninstanceof valueabove(145 )norbelow(144+)n. and3-nodecoverproblems. Corollary2.Forevery>0,itisNP-hardtoapproximate (78+)n,andwhetheraninstanceof3-NodeCoverwith284nhasitsoptimum 4-NodeCoverwith152nnodeshasitsoptimumvalueabove(79 )norbelow 1.3-NodeCovertowithinafactor145=144, 2.4-NodeCovertowithinafactor79=78. Theorem8givesthefollowingapproximationlowerboundsfor4-NodeCover Theorem9.([DKRS00])MIN-LIN2isNP-hardtoapproximatetowithinafactor (seealso[dks98],[kst97]). nc=loglognforsomeconstantc. tions. Weturnnowtotheboundedoccurrenceminimumsatisabilityoflinearequa- MIN-LIN2isequivalenttothewellknownNearestCodewordproblem(cf. WeneedthefollowingrecentresultofDinur,Kindler,RazandSafra[DKRS00] functionsrandt,wecallanapproximationalgorithmaforanoptimization algorithmwasdesignedbybermanandkarpinski[bk01b]. problemp,an(r(n);t(n))-approximationalgorithmifaapproximatespwithin [ABSS93]).OnlyveryrecentlytherstsublinearapproximationratioO(n=logn) Weintroducenowanotionofan(r;t)-approximationalgorithm.Fortwo 6

7 instance. anapproximationratior(n)andaworksino(t(n))timefornasizeofan an(r(cn);t(cn))-approximationalgorithmformin-lin2. (r(n);t(n))-approximationalgorithmfor3-occ-min-e3-lin2,thenthereexists approximationsofthe3-occ-min-e3-lin2problem. Theorem10.([BK01b])Thereexistsaconstantcsuchthatifthereexistsan Theorem9entailsnow BermanandKarpinski[BK01b]provedthefollowingresultonthe(r;t)- Theorem11.Theproblem3-OCC-E3-LIN2isNP-hardtoapproximatetowithin afactornc=loglognforsomeconstantc. have Corollary3.The3-aryNearestCodewordproblemwiththenumberofoccurrencesofeachvariableexactlyequalto3isNP-hardtoapproximatetowithina occurrence3-arynearestcodewordproblem(c.f.[kst97]),andthereforewe The3-OCC-MIN-E3-LIN2problemisequivalenttotheexactly-3bounded factornc=loglognforsomeconstantc. resultwillbeonlyrelativetotheapproximationhardnessofmin-bisection, whichexcludesexistenceofaptasforthatproblem. sionofmin-bisection. thestatusofwhichiswideopen,andweknowcurrentlyofnoprooftechnique provedthefollowingresultonapproximationhardnessofboundeddegreever- Somewhatsurprisinglyinthatcontext,BermanandKarpinski[BK01b] WeapplyasimilartechniquefortheproblemofMIN-BISECTION.Hereour for3-min-bisection,thenthereexistsan(r(n3);t(n3))-approximationalgorithmformin-bisection. Theorem12.([BK01b])Ifthereexistsan(r(n);t(n))-approximationalgorithm forthegeneralplanarmin-bisectionoftheplanarmin-bisectionproblem improvementonapproximationratiorfor3-regulargraphs,sayr=o(log2n), isofratioo(log2n)duetofeigeandkrauthgamer[fk00].anyasymptotic willentail,bytheorem12,animprovementonanapproximationratioforthe generalmin-bisection. Asimilartechniquecanbealsousedtoproveapproximationhardnessresult ThebestcurrentlyknownapproximationalgorithmfortheMIN-BISECTION on3-regulargraphs. 7

8 Wearegoingtoapplyourpreviousresultsforsomeothergenericoptimization 4SomeApplication problems.therstproblemisoneofthemostimportantproblemsinanalysisofgenomerearrangements,anditisbeingrecentlyalsomotivatedbyother algorithmicproblemsofcomputationalmolecularbiology. MIN-SBR(SortingbyReversals):Givenapermutation,constructamini- boundoftheorem7for4-miscanbeadaptedtoproveforthersttimethe in[c99],calledmsbrandtreesbr(seethedenitionsthere). Theprooftechniqueusedin[BK99]toproveexplicitapproximationlower WereferalsotosomeothervariantsofSortingbyReversalsproblemsstudied formsittotheidentitypermutation. mumlengthsequenceofreversals(seefordenitions[bp96])whichtrans- inapproximabilityofmin-sbr,and,infact,alsogivinganexplicitapproximation withinafactor1237=1236. Theorem13.([BK99])Forevery>0,itisNP-hardtoapproximateMIN-SBR lowerboundforthatproblem. explicitapproximationlowerboundsforthoseproblems. bothbeforementionedproblems,msbr,andtreesbr,andtocomputetherst asymmetricversionby(1,2)-atsp(cf.[py93],[v92]). (1,2)-TSPtheTravelingSalesmanProblemwithdistancesoneandtwo,andits WeturnnowtoanotherapplicationoftheresultsofSection2.Wedenoteby EngebretsenandKarpinski[EK00]hasusedrecentlytheresulton3-OCC- Caprara[C99]hasusedtheaboveresulttoproveinapproximabilityofthe MAX-HYBRID-LIN2ofTheorem5toprovethefollowingexplicitinapproximabilityresultfor(1,2)-ATSPproblem. withinafactor743=742. ATSPwithinafactor321=320. adaptedtoyieldanexplicitresultfor(1,2)-tsp. Theorem14.([EK00])Forevery>0,itisNP-hardtoapproximate(1,2)- Theorem15.([EK00])Forevery>0,itisNP-hardtoapproximate(1,2)-TSP TheconstructionusedbyEngebretsenandKarpinski[EK00]couldbealso 8

9 5NewUpperApproximationBounds lowerapproximationboundshasstimulatedresearchonimprovingapproximation Theintricacyofprovingtherstexplicitapproximationlowerboundsforsmall ratiosforthoseproblemsaswellasforsomeothergenericproblems. degreeoptimizationproblems,andtheresultinghugegapsbetweenupperand recentlybyfeige,karpinskiandlangberg[fkl00a],seetable1.thetechnique of[fkl00a]isbasedonanewlocalenhancingmethodforsemideniteprograms. Theorem16.([FKL00a])Thereexistsapolynomialtimealgorithmapproximating3-MAX-CUTwithinafactor Therstgapfor3-MAX-CUT(and3-OCC-MAX-E2-LIN2)wasimproved problemongeneralgraphsis1.1383([gw94]),andthebestknownapproximation lowerboundis1.0624[h97].wenotealsothatforthesemideniterelaxation ofmax-cutusedin[gw94],theintegralitygapisatleast1.1312,evenfor 2-regulargraphs.ThustheboundofTheorem16beatstheintegralitybound evenfor2-regulargraphs. WenotethatthebestapproximationratiocurrentlyknownforMAX-CUT (partitioningofagraphintotwohalvessoastomaximizeanumberoftheedges investigateapproximationalgorithmsforthemax-cutandmax-bisection betweenthem). Weturnnowtothespecialcaseofregularboundeddegreegraphs,andwill formax-bisectionongeneralgraphsis1.4266[hz00]. MAX-CUT,andRd-MAX-BISECTION.Thebestknownapproximationratio BISECTIONproblems,respectively,restrictedtod-regulargraphs. thebestknownapproximationratiosforbothboundeddegreeproblems,rd- Rd-MAX-CUTandRd-MAX-BISECTIONaretheMAX-CUTandMAX- Theorem17.([FKL00a],[FKL00b])Therearepolynomialtimealgorithmsthat approximater3-max-cutandr3-max-bisectionproblemswithinfactor Feige,KarpinskiandLangberg[FKL00a],[FKL00b]wereabletoimprove MAX-BISECTIONproblems. Theorem18.([KKL00])ThereexistsapolynomialtimealgorithmapproximatingR3-MAX-BISECTIONwithinafactor and1.1991,respectively. gas[kkl00],havefurtherimprovedapproximationratiosofthelowdegreerd- Usinganadditionallocaladhancementmethod,Karpinski,KowalukandLin- BISECTIONonlowdegreeplanargraphsundertakenin[KKL00]hasleadto designoftherstptassforthegeneralplanarmax-bisectionaswellasfor othergeometricallydenedclassesofgraphs(see[jkls01]). Interestingly,therstimprovementsonapproximationratiosofMAX- 9

10 thebestuptonowexplicitapproximationlowerboundsforr3-max-cut,and R3-MAX-BISECTIONproblemsequaltothelowerapproximationboundfor3- MAX-CUTproblemofSection2. 6SummaryofApproximationResultson Onthelowerboundsside,wenotethatthetechniquesof[BK99]yieldalso Wepresenthere(Table2)theresultsofSection3and4onboundeddegreeminimizationproblemsandthebesttoourknowledgegapsbetweenupperandlower approximationboundsonthoseproblems.theupperapproximationboundsare from[bf94],[bf95],[bk01b],[fk00],[bhk01],[v92],[py93]. BoundedDegreeMinimizationProblems BoundedDegreeandWeightMinimizationProblems Approx.Upper TABLE2: 3-OCC-MIN-E3-LIN2 3-MIN-BISECTION 3-NodeCover 4-NodeCover Approx.Lower MIN-SBR O(n=logn) O(log2n) EqualtoMIN-BISECTION n(1)=loglogn (1,2)-ATSP (1,2)-TSP Anobviousopenproblemistoimproveonboththelowerandupperapproximationboundsofboundeddegreeoptimizationproblems,especiallyonthose withtheverysmalldegreebounds.theessentialimprovementsontheexplicit lowerboundsfortheseproblemsmightbeofparamountdicultythough,but mediateeectsontheexplicitlowerboundsforotheroptimizationproblems,as sametimetheyarealsoofgreatinterest.anysuchimprovementwouldhaveim- 7OpenProblemsandFurtherResearch

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