P versus NP problems 数学问题.docx
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P versus NP problems 数学问题.docx
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PversusNPproblems数学问题
PversusNOproblems
http:
//en.wikipedia.org/wiki/P_versus_NP_problem
DiagramofcomplexityclassesprovidedthatP≠NP.TheexistenceofproblemswithinNPbutoutsidebothPandNP-complete,underthatassumption,wasestablishedbyLadner'stheorem.[1]
Thegeneralclassofquestionsforwhichsomealgorithmcanprovideananswerinpolynomialtimeiscalled"classP"orjust"P".Forsomequestions,thereisnoknownwaytofindananswerquickly,butifoneisprovidedwithinformationshowingwhattheansweris,itmaybepossibletoverifytheanswerquickly.TheclassofquestionsforwhichananswercanbeverifiedinpolynomialtimeiscalledNP.
Considerthesubsetsumproblem,anexampleofaproblemthatiseasytoverify,butwhoseanswermaybedifficulttocompute.Givenasetofintegers,doessomenonemptysubsetofthemsumto0?
Forinstance,doesasubsetoftheset{−2,−3,15,14,7,−10}addupto0?
Theanswer"yes,becausethesubset{−2,−3,−10,15}addsuptozero"canbequicklyverifiedwiththreeadditions.However,thereisnoknownalgorithmtofindsuchasubsetinpolynomialtime(thereisone,however,inexponentialtime,whichconsistsof2n-1tries),butsuchanalgorithmexistsifP=NP;hencethisproblemisinNP(quicklycheckable)butnotnecessarilyinP(quicklysolvable).
AnanswertotheP = NPquestionwoulddeterminewhetherproblemsthatcanbeverifiedinpolynomialtime,likethesubset-sumproblem,canalsobesolvedinpolynomialtime.IfitturnedoutthatP≠NP,itwouldmeanthatthereareproblemsinNP(suchasNP-completeproblems)thatarehardertocomputethantoverify:
theycouldnotbesolvedinpolynomialtime,buttheanswercouldbeverifiedinpolynomialtime.
Asidefrombeinganimportantproblemincomputationaltheory,aproofeitherwaywouldhaveprofoundimplicationsformathematics,cryptography,algorithmresearch,artificialintelligence,gametheory,multimediaprocessing,philosophy,economicsandmanyotherfields.
Context
TherelationbetweenthecomplexityclassesPandNPisstudiedincomputationalcomplexitytheory,thepartofthetheoryofcomputationdealingwiththeresourcesrequiredduringcomputationtosolveagivenproblem.Themostcommonresourcesaretime(howmanystepsittakestosolveaproblem)andspace(howmuchmemoryittakestosolveaproblem).
Insuchanalysis,amodelofthecomputerforwhichtimemustbeanalyzedisrequired.Typicallysuchmodelsassumethatthecomputerisdeterministic(giventhecomputer'spresentstateandanyinputs,thereisonlyonepossibleactionthatthecomputermighttake)andsequential(itperformsactionsoneaftertheother).
Inthistheory,theclassPconsistsofallthosedecisionproblems(definedbelow)thatcanbesolvedonadeterministicsequentialmachineinanamountoftimethatispolynomialinthesizeoftheinput;theclassNPconsistsofallthosedecisionproblemswhosepositivesolutionscanbeverifiedinpolynomialtimegiventherightinformation,orequivalently,whosesolutioncanbefoundinpolynomialtimeonanon-deterministicmachine.[5]Clearly,P⊆NP.Arguablythebiggestopenquestionintheoreticalcomputerscienceconcernstherelationshipbetweenthosetwoclasses:
IsPequaltoNP?
Ina2002pollof100researchers,61believedtheanswertobeno,9believedtheanswerisyes,and22wereunsure;8believedthequestionmaybeindependentofthecurrentlyacceptedaxiomsandthereforeisimpossibletoproveordisprove.[6]
In2012,10yearslater,thesamepollwasrepeated.Thenumberofresearcherswhoansweredwas151:
126(83%)believedtheanswertobeno,12(9%)believedtheanswerisyes,5(3%)believedthequestionmaybeindependentofthecurrentlyacceptedaxiomsandthereforeisimpossibletoproveordisprove,8(5%)saideitherdon'tknowordon'tcareordon'twanttheanswertobeyesnortheproblemtoberesolved.[7]
NP-complete
EulerdiagramforP,NP,NP-complete,andNP-hardsetofproblems
Mainarticle:
NP-complete
ToattacktheP=NPquestiontheconceptofNP-completenessisveryuseful.NP-completeproblemsareasetofproblemstoeachofwhichanyotherNP-problemcanbereducedinpolynomialtime,andwhosesolutionmaystillbeverifiedinpolynomialtime.Thatis,anyNPproblemcanbetransformedintoanyoftheNP-completeproblems.Informally,anNP-completeproblemisanNPproblemthatisatleastas"tough"asanyotherprobleminNP.
NP-hardproblemsarethoseatleastashardasNPproblems,i.e.,allNPproblemscanbereduced(inpolynomialtime)tothem.NP-hardproblemsneednotbeinNP,i.e.,theyneednothavesolutionsverifiableinpolynomialtime.
Forinstance,thebooleansatisfiabilityproblemisNP-completebytheCook–Levintheorem,soanyinstanceofanyprobleminNPcanbetransformedmechanicallyintoaninstanceofthebooleansatisfiabilityprobleminpolynomialtime.ThebooleansatisfiabilityproblemisoneofmanysuchNP-completeproblems.IfanyNP-completeproblemisinP,thenitwouldfollowthatP=NP.Unfortunately,manyimportantproblemshavebeenshowntobeNP-complete,andnotasinglefastalgorithmforanyofthemisknown.
BasedonthedefinitionaloneitisnotobviousthatNP-completeproblemsexist.AtrivialandcontrivedNP-completeproblemcanbeformulatedas:
givenadescriptionofaTuringmachineMguaranteedtohaltinpolynomialtime,doesthereexistapolynomial-sizeinputthatMwillaccept?
[8]ItisinNPbecause(givenaninput)itissimpletocheckwhetherMacceptstheinputbysimulatingM;itisNP-completebecausetheverifierforanyparticularinstanceofaprobleminNPcanbeencodedasapolynomial-timemachineMthattakesthesolutiontobeverifiedasinput.Thenthequestionofwhethertheinstanceisayesornoinstanceisdeterminedbywhetheravalidinputexists.
ThefirstnaturalproblemproventobeNP-completewasthebooleansatisfiabilityproblem.Asnotedabove,thisistheCook–Levintheorem;itsproofthatsatisfiabilityisNP-completecontainstechnicaldetailsaboutTuringmachinesastheyrelatetothedefinitionofNP.However,afterthisproblemwasprovedtobeNP-complete,proofbyreductionprovidedasimplerwaytoshowthatmanyotherproblemsarealsoNP-complete,includingthesubset-sumproblemdiscussedearlier.Thus,avastclassofseeminglyunrelatedproblemsareallreducibletooneanother,andareinasense"thesameproblem".
Harderproblems
AlthoughitisunknownwhetherP=NP,problemsoutsideofPareknown.Anumberofsuccinctproblems(problemsthatoperatenotonnormalinput,butonacomputationaldescriptionoftheinput)areknowntobeEXPTIME-complete.BecauseitcanbeshownthatP⊊EXPTIME,theseproblemsareoutsideP,andsorequiremorethanpolynomialtime.Infact,bythetimehierarchytheorem,theycannotbesolvedinsignificantlylessthanexponentialtime.Examplesincludefindingaperfectstrategyforchess(onanN×Nboard)[9]andsomeotherboardgames.[10]
TheproblemofdecidingthetruthofastatementinPresburgerarithmeticrequiresevenmoretime.FischerandRabinprovedin1974thateveryalgorithmthatdecidesthetruthofPresburgerstatementshasaruntimeofatleast
forsomeconstantc.Here,nisthelengthofthePresburgerstatement.Hence,theproblemisknowntoneedmorethanexponentialruntime.Evenmoredifficultaretheundecidableproblems,suchasthehaltingproblem.Theycannotbecompletelysolvedbyanyalgorithm,inthesensethatforanyparticularalgorithmthereisatleastoneinputforwhichthatalgorithmwillnotproducetherightanswer;itwilleitherproducethewronganswer,finishwithoutgivingaconclusiveanswer,orotherwiserunforeverwithoutproducinganyansweratall.
ProblemsinNPnotknowntobeinPorNP-complete
ItwasshownbyLadnerthatifP≠NPthenthereexistproblemsinNPthatareneitherinPnorNP-complete.[1]SuchproblemsarecalledNP-intermediateproblems.Thegraphisomorphismproblem,thediscretelogarithmproblemandtheintegerfactorizationproblemareexamplesofproblemsbelievedtobeNP-intermediate.TheyaresomeoftheveryfewNPproblemsnotknowntobeinPortobeNP-complete.
Thegraphisomorphismproblemisthecomputationalproblemofdeterminingwhethertwofinitegraphsareisomorphic.AnimportantunsolvedproblemincomplexitytheoryiswhetherthegraphisomorphismproblemisinP,NP-complete,orNP-intermediate.Theanswerisnotknown,butitisbelievedthattheproblemisatleastnotNP-complete.[11]IfgraphisomorphismisNP-complete,thepolynomialtimehierarchycollapsestoitssecondlevel.[12]Sinceitiswidelybelievedthatthepolynomialhierarchydoesnotcollapsetoanyfinitelevel,itisbelievedthatgraphisomorphismisnotNP-complete.Thebestalgorithmforthisproblem,duetoLaszloBabaiandEugeneLukshasruntime2O(√nlog(n))forgraphswithnvertices.
Theintegerfactorizationproblemisthecomputationalproblemofdeterminingtheprimefactorizationofagiveninteger.Phrasedasadecisionproblem,itistheproblemofdecidingwhethertheinputhasafactorlessthank.Noefficientintegerfactorizationalgorithmisknown,andthisfactformsthebasisofseveralmoderncryptographicsystems,suchastheRSAalgorithm.TheintegerfactorizationproblemisinNPandinco-NP(andeveninUPandco-UP[13]).IftheproblemisNP-complete,thepolynomialtimehierarchywillcollapsetoitsfirstlevel(i.e.,NP=co-NP).Thebestknownalgorithmforintegerfactorizationisthegeneralnumberfieldsieve,whichtakesexpectedtime
tofactorann-bitinteger.However,thebestknownquantumalgorithmforthisproblem,Shor'salgorithm,doesruninpolynom
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