On the Complexity of Matrix Rank and Rigidity Meena Mahajan and Jayalal Sarma M - PDF document

OntheComplexityofMatrixRankandRigidityMeenaMahajanandJayalalSarmaM.N.TheInstituteofMathematicalSciences,Chennai600113,India.fmeena,jayalalg@imsc.res.inAbstractWerevisitawellstudiedlinearalgebraicproblem,computingtherankanddetermi-nantofmatrices,inordertoobtaincompletenessresultsforsmallcomplexityclasses.Inparticular,weprovethatcomputingtherankofaclassofdiagonallydominantmatricesiscompleteforL.WeshowthatcomputingthepermanentanddeterminantoftridiagonalmatricesoverZisinGapNC1andishardforNC1.Wealsoinitiatethestudyofcomputingtherigidityofamatrix:thenumberofentriesthatneedstobechangedinordertobringtherankofamatrixbelowagivenvalue.Weshowthatsomerestrictedversionsoftheproblemcharacterizesmallcomplexityclasses.Wealsolookatavariantofrigiditywherethereisaboundontheamountofchangeallowed.Usingideasfromthelinearintervalequationsliterature,weshowthatthisproblemisNP-hardoverQandthatacertainrestrictedversionisNP-complete.Restrictingtheproblemfurther,weobtainvariationswhichcanbecomputedinPLandarehardforC=L.1IntroductionAseriesofseminalpapersbyavarietyofpeopleincludingValiant,Mulmuley,Toda,Vinay,Grigoriev,Cook,andMcKenzie,setthestageforstudyingthecomplexityofcomputingmatrixproperties(inparticular,determinantandrank)intermsoflogspacecomputationandpoly-sizepolylog-depthcircuits.Thisareahasbeenactiveformanyyears,andanNCupperboundisknownformanyrelatedproblemsinlinearalgebra;seeforinstance[All04].SomeofthemajorresultsinthisareaarethatcomputingthedeterminantofintegermatricesisGapL-completeandthattestingsingularityofintegermatricesisC=L-complete.Inparticular,thecomplexityofcomputingtherankofagivenmatrixoverQhasbeenwellstudied.Forgeneralmatrices,checkingiftherankisatmostrisC=L-complete[ABO99].Completeproblemsforcomplexityclassesarealwayspromising,sincetheyprovideasetofpossibletechniquesthatareassociatedwiththeproblemtoattackvariousquestionsregardingthecomplexityclass.Suchresultscanbeexpectedto\rourishwhenthecompleteproblemhaswell-developedtoolsassociatedwithit.Withthismotivation,welookatspecial1 Matrixtype(overQ) rankbound singular determinant general C=L-complete C=L-complete GapL-complete (even0-1) [ABO99] [ABO99] [Dam91,Vin91] [Tod91,Val92] symmetricnon-neg. C=L-complete C=L-complete GapL-hardunder logTreductions [ABO99] [ABO99] [Kul07] symmetricnon-neg. L-complete L-complete diag.dominant(d.d.) (Theorem5) (Theorem5) ? symmetricd.d. L-hardevenwhendet2f0;1g(Thm.10) ? diagonal TC0-complete AC0 TC0-complete (Prop1) (Prop1) (Prop1) tridiagonal ? C=NC1(Theorem11) GapNC1(Thm.11) tridiagonalnon-neg. non-negativepermequivalenttoplanar#BWBP(Thm.11) Table1:rankbound,singular,anddeterminantforspecialmatricescasesofthematrixrankproblemandtrytocharacterizesmallcomplexityclasses.Wecon-siderrestrictionswhicharecombinationsofnon-negativity,0-1entries,symmetry,diagonaldominance,andtridiagonalsupport,andweconsiderthecomplexitiesofthreeproblems:computingtherank,computingthedeterminantandtestingsingularity.These,thoughintimatelyrelated,canhavedieringcomplexities,asTable1shows.However,thecorrespondingoptimizationsearchproblemscanbeconsiderablyharder.Considerthefollowingexistentialsearchquestion:GivenamatrixMoveraeldK,atargetrankrandaboundk,decidewhethertherankofMcanbebroughtdowntobelowrbychangingatmostkentriesofM.Intuitively,onewouldexpectsuchaquestiontobein9
NC:guessklocationswhereMistobechanged,guessthenewentriestobeinsertedthere,andcomputetherankinNC([Mul87]).However,thisintuition,whilecorrectforniteelds(thiscasewasrecentlyshowntobeNP-complete[Des07]),doesnotdirectlytranslatetoaproofforQandZ1sincetherequirednewentriesmaynothaverepresentationspolynomially-boundedintheinputsize.Usingresultsaboutmatrixcompletionproblems,wecanobtainupperboundsofPSPACEoverrealsandcomplexnumbers,anddecidabilityoverp-adicnumbers,[BFS99].However,inthecaseofarbitraryinniteelds,thebestupperboundwecanseeinthegeneralcaseisrecursiveenumerability,andinparticular,thisisthesituationoverQ.WealsodonotknowanylowerboundsforthisquestionoverQ.Inthispaper,weexplorethecomputationalcomplexityofseveralvariantsofthisproblem.Theabovequestionisacomputationalversionofrigidityofamatrix,whichisthesmallestvalueofkforwhichtheanswertotheabovequestionisyes.ThenotionofrigiditywasintroducedbyValiant[Val77]andindependentlyproposedbyGrigoriev[Gri76].The 1Technically,rankoverZisnotdened,sinceZisnotaeld.InSection2,wedeneanaturalnotionofrankoverrings.Underthis,sinceZisanintegraldomain,therankisthesameasoverthecorrespondingdivisionringQ.2 mainmotivationforstudyingrigidityisthatgoodlowerboundsonrigiditygiveimportantcomplexity-theoreticresultsinothercomputationalmodels,likelinearalgebraiccircuitsandcommunicationcomplexity.Thoughthequestionweaddressisinfactacomputationalversionofrigidity,ithasnodirectimplicationsfortheselowerbounds.However,itprovidesnaturalcompleteproblemsbasedonlinearalgebraforimportantcomplexityclasses.Animportantaspectofcomputingrigidityisitspossibleconnectiontothetheoryofnat-uralproofsdevelopedbyRazborovandRudich[RR97].Valiant'sreduction[Val77]identies\highrigidity"asacombinatorialpropertyoffunctions,basedonwhichheproveslinear-sizelowerboundsforlog-depthcircuits.However,themodelofarithmeticcircuitshasnotbeenstudiedinsucientdetailsuchthatinthesettingofnaturalproofsthiscandirectlyprovidesomeevidenceaboutthepoweroftheprooftechnique.Nevertheless,thiscouldbethoughtofasmotivationforthecomputationalquestionofrigidity.Ourquestionbearscloseresemblancetothebodyofproblemsconsideredundermatrixcompletion,seeforinstance[BFS99,Lau01].Givenamatrixwithindeterminatesinsomelocations,canweinstantiatetheminsuchawaythatsomedesiredproperty(e.g.non-singularity)isachieved?InSection4,wediscusshowresultsfrommatrixcompletioncanyieldupperboundsforourquestion.Inthispaper,werestrictourattentiontoZandQ(someextensionstoniteeldsarediscussedattheend).SinceevenanupperboundofNPisnotobvious,werestrictthechoiceavailableinchangingmatrixentries.Weconsidertwovariants:(1).Intheinput,anitesubsetSKisgiven.MhasentriesoverS,andthechangedentriesmustalsobefromS;rankcomputationcontinuestobeoverK.(Forinstance,wemayconsiderBooleanmatrices,soS=f0;1g,whilerankcomputationisoverK.)ItiseasytoseethatthisvariantisindeedinNP.(2).Intheinput,aboundisgiven.Werequirethatthechangesbeboundedby;wemayapplytheboundtoeachchange,ortothetotalchange,ortothetotalchangeperrow/column.(Seeforinstance[Lok95].)Thisversionhascloseconnectionswithanotherwell-studiedareacalledlinearintervalequationswhicharisesnaturallyinthecontextofcontrolsystemstheory(see[Roh89]).Weobtaintighterlowerandupperboundsforsomeofthesequestions.Weshowcom-pletenessforC=Lwhenk2O(1)intherstvariant,forNPwhenthetargetrankrequalsninthesecondvariant,andforC=Lwhenr=ninthegeneralcase.Table2summarizestheresults.2PreliminariesOveranyeldF,therankofamatrixM2Fnn(weconsideronlysquarematricesinthispaper)hasthefollowingequivalentdenitions:(1)Themaximumnumberoflinearlyinde-pendentrowsorcolumnsinM.(2)Themaximumsizeofanon-singularsquaresubmatrixofM(3)TheminimumrsuchthatM=ABforsomeA2FnrandB2Frn.(4)TheminimumrsuchthatMisthesumofrrank-1matrices,wherearank-1matrixisoneforwhichthereexistsavectorv(notnecessarilyinthematrix)suchthateveryrowinthematrixcanbeexpressedasamultipleofv.Thesedenitionsneednotbeequivalentwhen3 K,SK restriction bound (if,thenS=K) ZorQ,f0,1g inNP ZorQ,f0,1g k2O(1) C=L-complete(Thm13) ZorQ, k2O(1) C=L-hard[ABO99] Q, r=n C=L-complete[ABO99] witness-searchinLGapL(Thm15) Z, r=nandk=1 inLGapL(Thm16) ZorQ, boundedrigidity,r=n NP-complete(Thm18) ZorQ, boundedrigidity,r=n;k=1 InPL,andC=L-hard(Thm22) Table2:Ourboundsonrigidwhenk2O(1)orr=ntheunderlyingalgebraicstructureisnotaeld.Hence,thenotionofrankisnotwell-denedoverarbitraryrings.However,iftheringunderconsiderationisaninniteintegraldomain(likeZ)(noticethataniteintegraldomainhastobeaeld),thentheabovedenitionsareindeedequivalent,andcanbetakenasadenitionofrank.Infact,therankinthatcasecanbeeasilyseentobesameastherankoverthecorrespondingquotienteld;thusrankoverZasdenedaboveisthesameasrankoverQ.Nowweintroducethebasicnotionsincomplexitytheorythatweneed.LandNLdenotelanguagesacceptedbydeterministicandnondeterministiclogspaceclassesrespectively,andFListheclassoflogspace-computablefunctions.#ListheclassoffunctionsthatcountthenumberofacceptingpathsofanNLmachine,andGapLisitsclosureundersubtrac-tion.ComputingthedeterminantoverZiscompleteforGapL.Incontrast,computingthepermanentiscompletefor#P,theclassoffunctionscountingacceptingpathsofanNPma-chine.NC1istheclassoflanguageswithpolynomial-sizelogarithmic-depthBooleancircuits.#NC1istheclassoffunctionscomputedbyarithmeticcircuits(gatescompute+and)withthesamesizeanddepthboundsasNC1,andGapNC1isitsclosureundersubtraction.AC0(TC0)istheclassoflanguageswithpolynomial-sizeconstant-depthunboundedfaninBooleancircuits,wheregatescomputeand,or,not(andmajority).Formoredetails,see[Vol99].AlanguageLisintheexactcountinglogspaceclassC=L(orprobabilisticlogspacePL)ifandonlyifitconsistsofexactlythosestringswhereacertainGapLfunctioniszero(positive,respectively).Thelanguagessingular(K)=fMjOverK,Misnotfullrankgrankbound(K)=f(M;r)jOverK,rank(M)rgforK=ZorQarecompleteforC=L[ABO99].Notethatforanytypeofmatrices,andanycomplexityclassC,C-hardnessofsingularimpliesC-hardnessofrankbound.Howevertheconverseisnottrue:Proposition1.(folklore)Restrictedtodiagonalmatrices,singular(Z)isinAC0whilerankbound(Z)anddeterminantareTC0-complete.4 Therigidityfunction,anditsdecisionversion,areasdenedbelow2.(Heresupport(N)=#f(i;j)jN(i;j)=0g.)RM(r)def=minNfsupport(N):rank(M+N)rgrigidK=f(M;r;k)jRM(r)kgLemma2.(Valiant,folklore)OveranyeldF,rank(M)rRM(r+1)(nr)2.Theinequalityontheleftalsofollowsfromthefollowinglemmawhichwewilluselater:Lemma3.(folklore)OveranyeldF,foranytwomatricesMandNofthesameorder,support(MN)=1=)jrank(M)rank(N)j1Toseethis,usethefactsthatrankissub-additiveandthatrank(A)=rank(A).HenceforanytwomatricesAandB,rank(A)rank(B)rank(A+B)rank(A)+rank(B).Further,rank(A)support(A),yieldingtheclaim.3ComputingtherankforspecialmatricesComputationofrankisintimatelyrelatedtocomputationofthedeterminant.Mulmuley[Mul87]showedthatoverarbitraryelds,rankcanbecomputedinNC(withthetheeldoperationsasprimitives).OverZandQ,rankboundisC=L-complete([ABO99]),andwewishtocharacterizeitssubclassesbyrestrictingthetypesofmatrices.Anaturalapproachistousecharacterizationsofmatrixrankintermsofassociatedcombinatorialobjects,likegraphs.However,noknownparameterofthegraphofamatrixcharacterizesthematrixrankingeneral.Thefollowingiseasytosee:Proposition4.Thelanguagesrankbound(Z)andsingular(Z)remainC=L-hardeveniftheinstancesarerestrictedtobesymmetric0-1matrices.Proof.LetA0bethesymmetricmatrix0AAT0.Sincerank(A0)=2(rank(A)),rankbound(Z)remainsC=L-hardwhenrestrictedtosymmetricmatrices.Further,determinantremainsGapL-hardevenwhenthematricesarerestrictedtobe0-1(seeforinstance[Tod91]).ThussingularremainsC=L-hardevenwhenrestrictedto0-1matrices.SinceMisinsingularifandonlyif(M;n)isinrankboundifandonlyif(M0;2n)isinrankbound,itfollowsthatrankbound(Z)remainsC=L-hardforsymmetric0-1matricesaswell. 2Inmuchoftherigidityliterature,rank(M+N)risrequired.Weusestrictinequalitytobeconsistentwiththedenitionofrankboundfrom[ABO99].5 determinantremainsGapL-hardfor0-1matrices,butitisnotclearthatsymmetricinstancesareGapL-hardundermany-onereductions.Theabovetrickdoesnotworkforcomputingdeterminants,becausedet(A0)willequaldet(A)2andGapLisnotknowntobeclosedundertakingsquare-roots.Wedonotknow(anyotherwayofshowing)many-onehardnessforsymmetricdeterminant.Recently,Kulkarni[Kul07]hasobservedthatsymmetricinstancesareGapL-hardunderTuringreductions.TheideaistorstuseChineseRemaindering:anydeterminantcanbecomputedinLifitsresiduesmodulopolynomiallymanyprimesareavailable.Smallprimes(logarithmicallymanybits)suceandcanbeobtainedexplicitly.Nowtondthedeterminantmoduloasmallprimep,rangeoveralla2f0;1;:::;p1gandtestifitequalsamodulop.Butthiscanberecast,usingtheGapL-completenessproofsofthedeterminant,asaskingifarelateddeterminantis0modulop.Finally,usingtheideaintheproofofProposition4,wecanasktheoracleforthedeterminantofarelatedsymmetricmatrixandtest(inL)ifitis0modulop.Wenowconsideranadditionalrestriction.AmatrixMissaidtobediagonallydominantifforeveryi,jmi;ijPj=ijmi;jj.(Ifalltheinequalitiesarestrict,thenMissaidtobestrictlydiagonallydominant.)Weshow:Theorem5.singular(Z)restrictedtonon-negativediagonallydominantsymmetricma-tricesisL-complete.ThehardnessisviauniformAC0many-onereductions.Proof.Thisresultexploitsaverynicecombinatorialconnectionbetweensuchmatricesandgraphs.Foranon-negativesymmetricdiagonally-dominantmatrixM,itssupportgraphGM=(V;EM)hasV=fv1;:::vng,andEM=f(vi;vj)ji=jmi;j�0g[f(vi;vi)jmi;i�Pi=jmi;jg.Thefollowingisshownin[Dah99]forR,anditcanbeveriedthatthesameholdsoverQ.Lemma6([Dah99]).LetMbeanon-negativesymmetricdiagonallydominantmatrixofordernoverQorR.Thenrank(M)=nc,wherecisthenumberofbipartitecomponentsinthesupportgraphGM.Hardness:ThereductionisfromundirectedforestaccessibilityUFA,whichisL-completeandremainsL-hardevenwhenthegraphhasexactly2components[CM87].Withoutlossofgenerality,wecanassumethattheinputinstanceshaveaniceform,asstatedinthefollowinglemma.Lemma7.([CM87])GivenanundirectedforestG,ofboundeddegreewithexactlytwocomponents,andthreespecialverticess;tandq,withtheguaranteethattandqareindierentcomponents,decidingwhichcomponentsbelongstoisL-hard.Proof.ThereductionisfromthemachinemodelforL,andisessentiallyreproducedfrom[CM87].Werephrasetheproofheretohighlightthefactthatthenormalformweneedisindeedachievable.Tobeginwith,modifythemachinedescriptionsuchthatwheneverthecomputationisonaninniteloop,themachineclearsothework-tapeandgoestoanerrorstatee.Thusthereareonlytwopossiblenalstatesforthemachine,oneistheerrorcongurationse,andtheotheristheacceptingcongurationt.6 ThesetofcongurationsofaTuringmachinewithaxedinputwformstheverticesofsuchagraphG,andthe(unique)acceptingcongurationisaccessiblefromtheinitialcongurationifandonlyiftheTuringmachineacceptstheinputw.Gcanbemadeacyclicbyassociatingatimestampwiththecongurations,andinsistingthatanedgealwaysjoinsacongurationattimeitoacongurationattimei+1.Ifp(n)isanupperboundonthecomputationtimeoftheTuringmachinewithinputw,thenweletthenodetinthegraphbetheacceptingcongurationwithtimestampp(n),andswillbetheinitialcongurationwithtimestamp0.Bydenition,thenumberofpossible(in/out)-neighboursofanynodeisboundedbyaconstant.Inadditionthereareexactlytwonodesofoutdegree0,andtheycorrespondtothecongurationseandt.Viewingeachedgeintheresultingdigraphasundirectedyieldsanundirectedforestsuchthatsandtbelongtothesametreeifandonlyifadirectedpathexistedfromstotintheoriginaldigraph.Notethattheresultingundirectedforesthaspreciselytwocomponents,andthethreeverticessatisfytherequiredpropertiesofthereduction. WenowconstructG0asfollows:MaketwocopiesG1andG2ofG.Addanewvertexu.Addedges(s1;s2);(t1;u);(t2;u).Addself-loopsatq1andq2.G0hasatmostthreecomponents(copiesofthecomponentscontainingtjoinupviau).Thecomponent(s)containingcopiesofqarenecessarilynon-bipartite.Ifthereisans;tpathinG,theninG0thetwocopiesofthepath,alongwiththeedges(s1;s2);(t1;u);(u;t2)createanoddcycle,sothenewjoinedupcomponentisalsonotbipartite.HenceG0hasnobipartitecomponents.Ifthereisnos;tpathinG,thecomponentcontainingt1andt2willremainbipartite.Thusthereisexactlyonebipartitecomponentnow.Tocompletetheproof,weneedtoproduceamatrixMsuchthatG0isitssupportgraph.WeconstructMasfollows:Foreachi=jmi;j=1if(i;j)2E00otherwiseForeachimi;i=1+Pj=imi;jif(i;i)2E0Pj=imi;jotherwiseFromLemma6,Missingularifandonlyifthereisnos;tpathinG.ItisclearthatMcanbeconstructedfromG0,andhencefromG,byauniformTC0circuit.NowweshowthatinfactitcanbeconstructedinAC0.First,observethattheforestthatwestartwith(astheL-hardinstance)hasboundeddegree.SowewouldliketorewritethesummationPj=imi;jasPj=i;mi;j=0mi;j.Buthowdoweknowaprioriwhichentriesarenon-zero?Foranodei,deneLitobethelistofnodesforwhichmi;jcanpossiblybenon-zero.SincethelogspaceTuringmachinealtersonlyasmallpartofthecongurationinonestep,thislistisofboundedlength,withtheboundldependingonlyonthemachine'sdescriptionandnotontheinputlength.Letlist(i;t)denotethetthelementinalexicographicalenumerationofLi;oninputi;t,list(i;t)canbedeterminedinAC0.Now7 therequiredsummationisexactlyPj2Limi;j=Plt=1mi;list(i;t),andthusitcanbecomputedbyanAC0circuit.MembershipinL:GivenamatrixMsatisfyingthestatedconditions,itisstraightforwardtoconstructthesupportgraphGM.By[AG00,NTS95,Rei05],checkingwhethertwoverticesbelongtothesamecomponentinanundirectedgraph,countingthenumberofcomponents,andcheckingbipartitenessofanamedcomponentareallinL.Hence,byLemma6,rank(M)canbecomputedinL. Corollary8.Thelanguagerankbound(Z),restrictedtosymmetricnon-negativediago-nallydominantinstances,isL-complete.However,thehardnessofrankbound(Z)isnotderivedjustfromthehardnessofsingular.Anobviouswaytoobtainhardnessatothervaluesofrank(ratherthanr=ninthecaseofsingular)istopadoutthematrixwithzerorowsand/orcolumns.WepresenthereaslightlydierentproofofTheorem5,establishinghardnessofdecidingwhethertherankisn1orn2.Proof.ThereductionisagainfromundirectedforestaccessibilityUFA,usingniceinstancesasguaranteedbyLemma7.LetG;s;tbeaninstanceofUFA,whereGhastwotrees.WeconstructanewgraphG0=(V0;E0)asfollows:taketwodisjointcopiesofG.Addanewvertexuandconnectittobothcopiesoft.Connectthetwocopiesofs.Also,addself-loopsatbothcopiesoft.Ifthereisans;tpathinG,thenG0hasthreecomponents:thecopiesofthecompo-nentcontainingsandtjoinup,whilethecopiesoftheothercomponentremaindisconnected(andhencebipartite).Thetwocopiesofthepath,alongwiththeedges(s1;s2);(t1;u);(u;t2)createanoddcycle,sothenewjoinedupcomponentisnotbipartite.HenceG0hasexactlytwobipartitecomponents.Ifthereisnos;tpathinG,thecomponentcontainings1ands2willremainbipartite.Theothercomponentisnotbipartiteduetotheselfloopsatt;t2.Thusthereisexactlyonebipartitecomponentnow.Tocompletetheproof,weneedtoproduceamatrixMsuchthatG0isitssupportgraph.ThiscanbedoneinAC0asdescribedinthepreviousproof. NotethatthoughrankforthesematricescanbecomputedinL,wedonotknowhowtocomputetheexactvalueofthedeterminantitself.(NotethatbyTheorem5,thisishardforFL.)Inabriefdigression,wenotethefollowing:ifamatrixistohavenotrivial(all-zero)rows,andyetbediagonallydominant,thenitcannothaveanyzeroesonthediagonal.Onemayaskifitiseasiertocomputedeterminantswhentherecanbenozerosonthemaindiagonal.Wedonotknowifcomputingdeterminantsofsuchmatricesiscompleteundermany-onereductions,althoughthefollowinglemmashowsthatitiscompleteunderarestrictivetypeofTuringreduction.Lemma9.ForeveryGapLfunctionfandeveryinputx,f(x)canbeexpressedasdet(M)1,whereMhasnozeroesonthediagonal.Mcanbeobtainedfromxviaprojections(eachoutputbitisdependentonatmostonebitofx).8 Proof.ConsiderToda'sproof[Tod91]forshowingthatdeterminantisGapLhard(seealso[ABO99,MV97]).GivenanyGapLfunctionfandinputx,itconstructsadirectedgraphGwithself-loopsateveryvertexexceptaspecialvertexs.Galsohasthepropertythateverynon-trivialcycle(notaself-loop)inGpassesthroughs.IfAistheadjacencymatrixofG,thentheconstructionsatisesf(x)=det(A).NowconsiderthematrixBobtainedbyaddingaself-loopats.Whatadditionaltermsdoesdet(B)havethatwereabsentindet(A)?Suchtermsmustcorrespondtocyclecoversusingtheself-loopats;i.e.cyclecoversinGnfsg.ButGnfsghasnonon-trivialcycles,sotheonlyadditionalcyclecoverisallself-loops,contributinga+1.Thusdet(B)=1+det(A),andBistherequiredmatrix. WealsoshowviaasomewhatdierentreductionthattheL-hardnessofsingularinTheorem5holdsevenifweallownegativevalues,butdisallowmatriceswithdeterminantotherthan0or1.Theorem10.singular(Z)forsymmetricdiagonallydominantmatricesisL-hard,evenwhenrestrictedtoinstanceswith0-or-1determinant.Proof.AsintheproofofTheorem5,webeginwithaninstance(G;s;t)ofUFAwhereGhasexactlytwocomponents.Addedge(s;t)toobtaingraphH.Bythematrix-treetheorem,(seeforexampleTheoremII-12in[Bol84]),ifAistheLaplacianmatrixofH(denedbelow),andBisobtainedbydeletingthetopmostrowandleftmostcolumnofA,thendet(B)equalsthenumberofspanningtreesofH.TheLaplacianmatrixAisdenedasfollows:ai;i=thedegreeofvertexiinHai;j=1ifi=jand(i;j)isanedgeinHai;j=0ifi=jand(i;j)isnotanedgeinHClearly,Aisdiagonallydominant(infact,foreachi,theconstraintisanequality);also,sinceHisanundirectedgraph,Aissymmetric.NowthenumberofspanningtreesinHis1ifs;Gt(Hitselfisatree)andis0ifs;Gt(Hstillhastwocomponents,sincetheedgeinHnGjoinsverticesinthesamecomponentofG). Thenextrestrictionweconsideristridiagonalmatrices:mi;j=0=)jijj1.WeshowthatdeterminantandpermanentareinGapNC1,byusingbounded-widthbranchingprogramsBWBP.IntheBooleancontext,BWBPequalsNC1.However,inthearithmeticcontext,theyarenotthatwellunderstood.Itisstillopen([All04,CMTV98])whetherthecontainment#BWBP#NC1isinfactanequality(thoughitisknownthatGapBWBP=GapNC1).LayeredplanarBWBParetheG-graphsreferredtoin[AAB+99].CountingpathsinG-graphsmaywellbesimplerthanGapNC1duetoplanarity.However[AAB+99](seealso[All04])showsthatevenoverwidth-2G-graphs,itishardforNC1(underAC0[5]reductions).Weshowthatthepermanentanddeterminantoftridiagonalmatricesareessentiallyequivalenttocountinginwidth-2G-graphs.Inwhatfollowswehaveaweighted9 BWBP,wheretheweightofapathistheproductoftheweightsoftheedgesonthepath.ThevalueofaweightedBWBPisthesum,overalls-tpaths,oftheweightsofthepaths.Theorem11.ComputingthepermanentanddeterminantofatridiagonalmatrixoverZisequivalenttoevaluatingalayeredplanarweightedBWBPofwidth2.Proof.GivenatridiagonalmatrixA,letAibethetop-leftsubmatrixofAoforderi,andletXnandYndenoteitspermanentanddeterminantrespectively.Wehavethefollowingrecurrences:X0=Y0=1X1=Y1=a1;1Xi=ai;iXi1+ai1;iai;i1Xi2Yi=ai;iYi1ai1;iai;i1Yi2Figure1showsaweightedbranchingprogramforXnthathaswidth2andcanbedrawninalayeredplanarfashion.TheconstructionforthedeterminantYnissimilar,usingsomenegativeweights.Thiscompletestheproofofonedirection.X0a12 a11??????????????a21 X2a33??????????????a34 a42 X4an;n1 annXn X1a22a23 a32 X3a45 a44Xn1Figure1:Width-2branchingprogramfortridiagonalpermanentWeremarkthatintheconstructionforthepermanent(Xn),whenallentriesarenon-negative,thisproblemreducestocountingpathsinunweightedplanarbranchingprogramsofwidth5.Toseethis,replaceeachweightededgeinFigure1withawidth-threegadgethavingtheappropriatenumberofpathsinastandardway.Toseetheotherdirection,noticethatanylayerofaplanarwidth-2BWBPshouldlooklikeoneofthefollowingstructures.a b@@@@@@@@d c f e~~~~~~~DUFigure2:Componentsofwidth-2layeredplanargraphsAnywidth-2graphGcorrespondingtotheBPcanbeencodedasasequenceofandUcomponentsasindicatedingure2.FirstconsiderthecasewherethesequenceconsistsofalternatingDandU;thatisconsidersequencesin(DU).Eachsuchsequencelooksexactlylikethegraphingure1.Byjustreadingotheweightsonthecorrespondingedgesin10 thegraph,wecanproducetwomatricesM1andM2suchthatpermanentofM1andthedeterminantofM2(byputtinginappropriatenegations)equalthevalueoftheweightedBWBP.NowitissucienttoarguethatthegraphcorrespondingtoanyBWBPcanbetrans-formedtothisform.Ifthestringdoesnotstartwithacomponent,wewilljustputinaprexwithabc=101.Similarly,addasuxUcomponentwithdef=101ifnecessary.Weneedtohandlethecasewhentherearetwoconsecutivecomponentsofthesametype;UUorDD.Simplyputinacomponentwithabc=101betweentwoUs,andaUcompo-nentwithdef=101betweentwos.Noticethatthenewwidth-2graphwhenencodedwillbeanelementof(DU),andtheweightsofthepathsarepreservedinthetransformation.TheabovereductionnowgivesthetwomatricesM1andM2.Inaddition,observethatiftheBWBPisunweighted,thenthematrixM1thatweproducehasonly0,1entries,andM2willhaveentriesfromf1;0;1g. FromTheorem11andthediscussionprecedingit,wehavethefollowingcorollary.Corollary12.ComputingthepermanentanddeterminantofatridiagonalmatrixoverZisinGapNC1,andishardforNC1underAC0[5]reductions.4ComplexityresultsonrigidityInthissectionwestudytheproblemofcomputingmatrixrigidity,rigidK,andalsoitsrestrictionrigidK;Sdenedbelow,wherethematricescanhaveentriesonlyfromSK.rigidK;S=(M;r;k)MoverS;9M0overS:rank(M0)r^support(MM0)kWewillmostlyconsiderStobeeitherallofK,oronlyB=f0;1g.Wealsoconsiderthecomplexityofrigidwhenkisxed,viathefollowinglanguage:rigidK;S(k)=f(M;r)j(M;r;k)2rigidK;SgAsmentionedintheintroduction,matrixrigidityandmatrixcompletionarerelated.TheMinRankproblemtakesasinputamatrixwithvariables,andasksfortheminimumrankachievableunderallinstantiationsofthevariablesintheunderlyingeld,seeforinstance[BFS99].1-MinRankisitsrestrictionwhereeveryvariableoccursatmostonce,andisalsocalledminimumrankcompletion.MaxRankand1-MaxRankaresimilarlydened.Thenaivealgorithmforrigidity,mentionedintheintroduction,easilytranslatestoanupperboundofNP(1-MinRank).Moreprecisely,rigidisin9
1-MinRank.WhileMinRankoverZisundecidable[BFS99],thishardnessproofdoesnotcarryoverfor1-MinRank.Nonetheless,thebestknownupperboundfor1-MinRankisrecursiveenumerability.Thusthenaivealgorithmdoesnotgiveanyreasonableupperboundforrigid.Theorem13.Foreachxedk,rigidZ;B(k)iscompleteforC=L.11 Proof.Membership:Weshowthatforeachk,rigidZ;B(k)isinC=L.Aninstance(M;r)isinrigidZ;B(k)ifthereisasetof0skentriesofM,which,when\ripped,yieldamatrixofranklessthanr.Thenumberofsuchsetsisboundedbyks=0ns
=t2nO(1).LetthecorrespondingmatricesbedenotedM1;M2:::Mt;thesecanbegeneratedfromMinlogspace.Now(M;r)2rigidZ;B(k)()9i:(Mi;r)2rankbound(Z).HencerigidZ;B(k)logdttrankbound(Z).Sincerankbound(Z)isinC=L,andsinceC=Lisclosedunderlogspacedisjunctivetruth-tablereductions(see[AO96]),itfollowsthatrigidZ;B(k)isinC=L.Hardness:Toshowacorrespondinghardnessresult,weuseLemma3below.ThehardnessforrigidZ;B(0)holdsbecausesingularremainsC=L-hardevenwhenrestrictedto0-1matrices(Proposition4).Hardnessforallthelanguagesmentionedinthelemmaalsofollowsfromthisfact,andfromthefollowingclaim:(1)M2singular(Z)=)(M\nIk+1;n(k+1)k)2rigidZ;B(0)rigidZ;B(k)(2)M62singular(Z)=)(M\nIk+1;n(k+1)k)62rigidZ(k)Here\ndenotestensorproductandIk+1denotesthe(k+1)(k+1)identitymatrix.Notethatrank(M\nIk+1)=(k+1)rank(M).Toseetheclaim,observethatifM2singular(Z),thenrank(M)n1andsorank(M\nIk+1)(k+1)(n1)n(k+1)k.IfM62singular(Z),thenrank(M\nIk+1)=n(k+1).Thuswewanttoreducetherankbyatleastk+1.ByLemma3,weneedtochangeatleastk+1entries. Remark14.ThemembershipboundofTheorem13cruciallyusesthefactthatC=Lisclosedunderlogdttreductions.WealsoobservethatthisresultholdsforanyniteS,evenifSisnotxedaprioributsuppliedexplicitlyaspartoftheinput.ThehardnessofTheorem13essentiallyexploitsthehardnessoftestingsingularity.Thereforewenowconsiderthecomplexityofrigidatthesingular-vs-non-singularthreshold,i.e.whenr=n.FromLemma2weknowthatoveranyeldF,(M;n;k)isinrigidwhen-everk1.And(M;n;0)isinrigidifandonlyifM2singular(F).SothecomplexityofdecidingthispredicateoverQisalreadywellunderstood.Wethenaddressthequestionofhowdicultitistocomeupwithawitnessingmatrix.Theorem15.Givenanon-singularmatrixMoverQ,asingularmatrixNsatisfyingsupport(MN)=1canbeconstructedinLGapL.Proof.Foreach(i;j),letM(i;j)bethematrixobtainedfromMbyreplacingmi;jwithanindeterminatex.Thendet(M(i;j))isoftheformax+b,andaandbcanbedeterminedinGapL(seeforinstance[AAM03]).SinceRM(n)=1(Lemma2),thereisatleastoneposition(i;j)wherethedeterminantissensitivetotheentry,andhencea=0.Settingmi;jtobeb=agivesthedesiredN. Anotherquestionthatarisesnaturallyisthecomplexityofrigidatthesingularitythresh-oldoverrings.NotethatLemma2doesnotnecessarilyholdforrings.Forinstance,changing12 oneentryofanon-singularrationalmatrixMsucestomakeitsingular.ButevenifMisintegral,thechangedmatrixmaynotbeintegral,andoverZ,RM(n)maywellexceed1.(Itdoes,forthematrix2357.)Thus,thequestionofdecidingRM(n)overZisnon-trivial.Weshow:Theorem16.GivenM2Znn,decidingif(M;n;k)isinrigid(Z)is(1)trivialforkn,(2)C=Lcompletefork=0,and(3)inLGapLfork=1.Proof.(1)holdsbecausezeroingoutanentirerowalwaysgetssingularity.(2)merelysaysthatsingular(Z)isC=L-complete.(3)followsfromtheproofofTheorem15andaddition-allycheckingtheintegralityofb=a. Inparticular,(3)aboveimpliesthatifoverZ,RM(n)=1,thenthenon-zeroentryofawitnessingmatrixispolynomiallyboundedinthesizeofM.However,ifRM(n)�1wedonotknowsuchasizebound.Todemonstratethisdiculty,considerthecaseinwhichk=2.FollowingthegeneralideainTheorem15,foreachchoiceoftwoentriesinthematrix,replacethembyvariablesxandy.Thisdenesafamilyofn2
=O(n2)matricesandafamilyPofbilinearbivariatepolynomialsrepresentingthecorrespondingdeterminants.Thecoecientsofeachp2PcanbecomputedinGapL.Now,totestifRM(n)2,itsucestocheckifatleastoneoftheDiophantineequationsdenedbyp2P(orequivalently,thesinglemultilinearDiophantineequationq(x1;x2:::;y1;y2:::)=Qp2Pp(xp;yp)=0)hasanintegralsolution.However,wedonotknowhowtodothis.5ComputingBoundedRigidityWenowconsidertheboundednormvariantofrigiditydescribedinSection1:changedmatrixentriescandierfromtheoriginalentriesbyatmostapre-speciedamount.Formally,thefunctionsofinterestarethenormrigidity
M(r)andtheboundedrigidityRM(r;),asdenedin[Lok95],andtheirdecisionversion,asgivenbelow.
M(r)def=infN(Xi;jjni;jj2:rank(M+N)r)RM(r;)def=minNfsupport(N):rank(M+N)r;8i;j:jni;jjgb-rigidK=f(M;r;k;)jRM(r;)kgOverZ,thenaivealgorithmforb-rigidZisnowinNP.HoweveroverQ,theboundstilldoesnotimplyanaprioripoly-sizeboundonthechangedentries.Thus,unlikeinSection4,herecomputationoverQappearsharderthanoverZ.Thefollowinglemmashowsthattheboundedrigidityfunctionscanbehaveverydier-entlyfromthestandardrigidityfunction.13 Lemma17.Forany,andforanysucientlylargensuchthatn logn�+1,thereisannnmatrixMoverQsuchthatRM(n)=1,
M(n)=(4n),andtheboundedrigidityRM(n;n)isundened.Proof.LetMbeannndiagonalmatrixwithmi;i=2nandmi;j=0fori=j.Clearly,RM(n)=1;justzerooutanydiagonalentry.Thisinvolvesanormchangeof4n.CanMbemadesingularbyasmallernorm-change,evenallowingmoreentriestobechanged?RecallthedenitionofstrictdiagonaldominancefromSection3.WeinvoketheLevy-Desplanquestheorem(seeforinstanceTheorem2.1in[MM64])thatsaysthatthedeterminantofastrictlydiagonallydominantmatrixisnon-zero.Now,atotalnorm-changelessthan4nwillnotdestroystrictdiagonallydominance,andthematrixwillremainnon-singular.Hence
M(n)=4n,andRM(n;n)isundened. SinceforagivenmatrixM,arankrandabound,RM(r;)canbeundened,weexaminehowdicultisittocheckthis.Weshowthefollowing:Theorem18.1.GivenamatrixM2Qnn,andarationalnumber�0,testingifRM(n;)isdenedisNP-complete.2.GivenMandasabove,andfurthergivenanintegerk,testingifRM(n;)isatmostkisNP-complete.Proof.Tobeginwith,noticethatRM(r;)isdenedifandonlyifRM(r;)n2.Membership:WerstshowthemembershipinNPfor(2).Membershipin(1)followsbyusingthiswithk=n2.Weusearesultfromthelinearintervalequationsliterature.FortwomatricesAandB,wesaythatABifforeachi,j,AijBij.ForAB,theintervalofmatrices[A;B]isthesetofallmatricesCsuchthatACB.Anintervalissaidtobesingularifitcontainsatleastonesingularmatrix;otherwiseitissaidtoberegular.ByTheorem2.8of[PR93](ordirectlyfromLemma21),checkingsingularityofagivenintervalmatrixisinNP.GivenM,andk,wewanttotestwhetherRM(n;)isatmostk.InNP,weguesskpositions(i1;j1);(i2;j2);:::(ik;jk)andconstructthematrixVimjm=forall1mkand0elsewhere.NowletA =MVand A=M+V.ThenRM(n;)kifandonlyifforsomesuchguessedV,theinterval[A ; A]issingular,andthiscanbetestedinNP.Hardness:Itsucestoprovehardnessfor(1),sincehardinstancesof(1)alongwithk=n2giveshardinstancesof(2).Westartwiththemaximumbipartitesubgraphproblem:GivenanundirectedgraphG=(V;E),withnverticesandmedgesandanumberk,checkwhetherthereisbipartitesubgraphwithatleastkedges.ThisproblemisknowntobeNP-complete(see[GJ79]).In[PR93],thereisareductionfromthisproblemtocomputingtheradiusofnon-singularity,denedasfollows:GivenamatrixA,itsradiusofnon-singularityd(A)istheminimum�0suchthattheinterval[AJ;A+J]issingular,whereJistheall-1smatrix.Wesketchthereductionof[PR93]belowandobservethatityieldsNP-hardnessforourproblemaswell.14 GivenaninstanceG;kofthemaximumbipartitesubgraphproblem,wedenethematrixNas,Nij=8:1ifi=jandiandjareadjacentinG2m+1ifi=j0otherwiseNoticethatsinceNisdiagonallydominant,byLevy-Desplanquestheorem(seeforinstanceTheorem2.1in[MM64]),Nisinvertible.LetM=N1.ByTheorems2.6and2.2of[PR93],(G;k)isaYesinstance()1=d(M)(2m+1)n+4k2m()d(M)=1 (2m+1)n+4k2m()theinterval[MJ;M+J]issingular()RM(n;)isdened. Remark19.1.Itiseasytoseethat,byclearingdenominators,wehavehardinstanceswhereM;takeintegralvalues.Thus,thehardnessresultholdsforZaswell.2.Thematricesthatareproducedintheabovereductionareallsymmetricaswell.Rohn[Roh94]consideredthecasewhentheintervalofmatricesunderconsiderationissym-metric;thatisboththeboundarymatricesaresymmetric.Noticethattheintervalcanstillcontainnon-symmetricmatrices.Heprovedthatinsuchaninterval,ifthereisasingularmatrix,thentheremustbeasymmetricsingularmatrixtoo.UnravellingtheNPalgorithmdescribedinthemembershippartofTheorem18,anditsproofofcorrectness,isilluminating.Essentially,whatisestablishedin[Roh89]andusedin[PR93]isthefollowing:Lemma20([Roh89]).Ifaninterval[A;B]issingular,i.e.thedeterminantvanishesforsomematrixCwithintheboundsACB,thenthedeterminantvanishesforamatrix2[A;B]which,atallbutatmostoneposition,takesanextremevalue(dijiseitheraijorbij).Inparticular,thisimpliesthatthereisamatrixintheintervalwhoseentrieshaverepre-sentationspolynomiallylonginthatofAandB.Toseethis,letbethematrixclaimedtoexistasabove,andletk;lbethe(only)positionwhereakldklbkl.TheotherentriesofmatchthoseofAorBandhencearepolynomiallyboundedanyway.Nowreplacedklbyavariablextogetmatrixx.Itsdeterminantisaunivariatelinearpolynomialx+whichvanishesatx=dkl.NowandcanbecomputedfromxinGapL,andhencearepolynomiallybounded.If=0,then=0andthepolynomialisidenticallyzero.Other-wise,thezeroofthepolynomialis=.Eitherway,thereisazerowithapolynomiallylongrepresentation.In[Roh89],theabovelemmaisestablishedaspartofalongchainofequivalencescon-cerningdeterminantpolynomials.However,itisinfactageneralpropertyofarbitrarymultilinearpolynomials,asweshowbelow.15 Lemma21(Zero-on-an-EdgeLemma).Letp(x1:::xt)beamultilinearpolynomialoverQ.IfithasazerointhehypercubeHdenedby[`1;u1];:::[`t;ut],thenithasazeroonanedgeofH,i.e.azero(a1;:::;at)suchthatforsomek,8(i=k),ai2f`i;uig.Proof.Theproofisbyinductiononthedimensionofthehypercube.Thecasewhent=1isvacuouslytrue,sinceHisitselfanedge.Considerthecaset=2.Letp(x1;x2)bethemultilinearpolynomialwhichhasazero(z1;z2)inthehypercubeH;`iziuifori=1;2.Assume,tothecontrary,thatphasnozeroonanyedgeofH.Denetheunivariatepolynomialq(x1)=p(x1;z2).Sinceq(x1)islinearandvanishesatz1,p(`1;z2)andp(u1;z2)mustbeofoppositesign.Buttheunivariatelinearpolynomialsp(`1;x2)andp(u1;x2)donotchangesignsontheedgeseither,andsop(`1;u2)andp(u1;u2)alsohaveoppositesign.Bylinearityofp(x1;u2),theremustbeazeroontheedgex2=u2,contradictingourassumption.Letusassumethestatementforhypercubesofdimensionlessthant.Considerthehypercubeofdimensiontandthepolynomialp(x1;:::xt).Let(z1:::zt)bethezeroinsidethehypercube.Themultilinearpolynomialrcorrespondingtop(x1;:::xn1;zt)hasazeroinsidethe(t1)-dimensionalhypercubeH0denedbyintervals[`1;u1];:::[`t1;ut1].Byinduction,rhasazeroonanedgeofH0.Withoutlossofgenerality,assumethatthiszerois(z01;2:::t1)wherei2f`i;uig.Thusthepolynomialq(x1;xt)=p(x1;2:::t1;xt)hasazerointhehypercubedenedbyintervals[`1;u1];[`t;ut].Hencethebasecaseappliesagain,completingtheinduction. AnalogoustoTheorems13,15and16,weconsiderb-rigidKwhenk2O(1).Theorem22.b-rigidQandb-rigidZareC=L-hardforeachxedchoiceofk,andremainhardwhenr=n.Whenk=1andr=n,b-rigidQisinPL,whileb-rigidZisinLGapL.Proof.Foranyk,(M;n;k;0)2b-rigidK()Missingular;henceC=L-hardness.ToseethePLupperboundoverQ,let=p q.Foreachelement(i;j),denethe(i;j)thelementasvariablexandthenwritethedeterminantasax+b.Thus,ifjxj=jb ajp qforatleastonesuch(i;j)pair,wearedone.Thisisequivalenttocheckingif(bq)2(ap)2.Thevaluesofaandbcanbewrittenasdeterminants,hence(ap)2and(bq)2areGapLfunctions,andcomparisonoftwoGapLfunctionscanbedoneinPL.SincePLisclosedunderdisjunction(see[AO96]),theentirecomputationcanbedoneinPL.OverZ,q=1and=p,butweneedanintegralvalueforxaswell.Thatis,wewantan(i;j)pairwherejb ajandadividesb.ThiscanbecheckedinLGapL. 6DiscussionWhilethematrixrigidityproblemoverniteeldsisNP-complete([Des07]),wecanconsiderrestrictedversionstheretoo.Itisknown[BDHM92]thatsingular(Fp)iscompleteforModpL(computingtheexactvalueofthedeterminantoverFpisinModpL),andthat(seee.g.[All04]),foranyprimep,rankbound(Fp)isinModpL.Usingthis,andclosurepropertiesofModpL,wecanobtainanaloguesofTheorem13and15overFp:(1)foreachk,andeach16 primep,rigidFp(k)iscompleteforModpL,and(2)givenanon-singularmatrix,asingularmatrixcanbeobtainedbychangingjustoneentry,andthechangecanbecomputedinModpL.Wecanalsoconsiderthecomplementaryquestiontomatrixrigidity,namely,computingthenumberofentriesthatneedtobechangedtoincreasetherankaboveagivenvalue.Usingargumentssimilartothecaseofdecreasingrank,wecanobtainsimilarcomplexityresultsinthiscasealso.However,notablyinthiscase,wenotonlyhavedecidability,wealsohaveanupperboundofNP.Thisfollowsfromtheframeworkofmaximumrankmatrixcompletion,whichisknowntobeinP[Gee99,Mur93].Forthemostgeneralquestionoftestingrigidityoverarbitraryinniteelds,asanopti-mizationproblem,anaturaldirectiontoexploreistheexistenceofxedparametertractablealgorithms.Morespecically,givenannnmatrix,andrankrandavaluek,isitpossibletotestRM(r)kintimenc:f(k)foraconstantcandanarbitraryfunctionf.However,inthisproblemwedonotseehowsuchadditionaltimecanused.7AcknowledgementsWethankV.Arvind,N.S.Narayanaswamy,R.Balasubramanian,KapilParanjapeandRaghavKulkarniformanyinsightfuldiscussions.Wethanktheanonymousrefereesforusefulcommentswhichhelpedimprovethereadabilityofthepaper.Further,oneoftherefereespointedouthowtoimprovetheboundonthereductioninTheorem5fromTC0toAC0.References[AAB+99]EricAllender,AndrisAmbainis,DavidA.MixBarrington,SamirDatta,andHuongLeThanh.Bounded-deptharithmeticcircuits:countingandclosure.InProc.26thICALP,LNCS1644,pages149{158,1999.[AAM03]EricAllender,VikramanArvind,andMeenaMahajan.Arithmeticcomplexity,Kleeneclosure,andformalpowerseries.TheoryComput.Syst.,36(4):303{328,2003.[ABO99]EricAllender,RobertBeals,andMitsunoriOgihara.Thecomplexityofma-trixrankandfeasiblesystemsoflinearequations.ComputationalComplexity,8(2):99{126,1999.[AG00]CAlvarezandRGreenlaw.Acompendiumofproblemscompleteforsymmetriclogarithmicspace.ComputationalComplexity,9:73{95,2000.[All04]EricAllender.Arithmeticcircuitsandcountingcomplexityclasses.InJanKra-jicek,editor,ComplexityofComputationsandProofs,QuadernidiMatematicaVol.13,pages33{72.SecondaUniversitadiNapoli,2004.17 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