NP vs coNP NP hard problems not in NP The Chromatic ......NP and coNP • These are the way that we...

Outline

•  NPvscoNP•  NPhardproblemsnotinNP•  TheChromaticnumberproblem

11/6/19 TheoryofComputation-Fall'19

LorenzoDeStefani 1

FromSipserChapter7

Recall•  NPisclassoflanguagesthathavepolytimeverifiers–  NPcomesfromNondeterministicPolynomialtime–  Alternateformulation:nondeterministicTMacceptsdecidesthelanguageinpolynomialtime

•  ANon-DeterministicTuringMachine(NDTM)isapolynomial-timedeciderifithaltsonanyinputoneveryexecutionbranchinpolynomialtime

11/6/19 TheoryofComputation-Fall'19LorenzoDeStefani 2

Recall•  ConsideralanguageLinNP•  Thenthereexistsapoly-timeNDTMdeciderDforL

•  Wait...canweusethistoconstructadeciderfor?

•  Waitwait...butthenand!!•  Waitwaitwait...ifthisworkswecandoitforanyLinNP–  SoNP=coNP?????


D

Accept

Rejectw

D

Accept

Reject

ww Accept

Reject

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L̄ 2 coNP<latexit sha1_base64="JdSG5GMd1a1nTG+r86g5d/3nR7M=">AAAB+XicbVDLSgMxFM3UV62vUZdugkVwVWas+NgV3bgQqWAf0BlKJs20oZlkSDKFMvRP3LhQxK1/4s6/MTMdRK0HAodzzuXenCBmVGnH+bRKS8srq2vl9crG5tb2jr2711YikZi0sGBCdgOkCKOctDTVjHRjSVAUMNIJxteZ35kQqajgD3oaEz9CQ05DipE2Ut+2vQDJ9HYGPcohFnfNvl11ak4OuEjcglRBAZP/8AYCJxHhGjOkVM91Yu2nSGqKGZlVvESRGOExGpKeoRxFRPlpfvkMHhllAEMhzeMa5urPiRRFSk2jwCQjpEfqr5eJ/3m9RIcXfkp5nGjC8XxRmDCoBcxqgAMqCdZsagjCkppbIR4hibA2ZVXyEi4znH1/eZG0T2puvVa/P602roo6yuAAHIJj4IJz0AA3oAlaAIMJeATP4MVKrSfr1XqbR0tWMbMPfsF6/wLDSJM6</latexit>

L̄ 2 NP<latexit sha1_base64="p4KuvY/tXp73bqvwav6dVTZHK/Q=">AAAB9XicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKblyIVLAPaKJMppN26GQSZiZKCf0PNy4Uceu/uPNvnKRF1Hpg4HDOudw7x485U9q2P63C3PzC4lJxubSyura+Ud7caqkokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH55nfvueSsUicaNHMfVC3BcsYARrI926Ppbp5Ri5TKCrxl25YlftHGiWOFNSgSlM/sPtRSQJqdCEY6W6jh1rL8VSM8LpuOQmisaYDHGfdg0VOKTKS/Orx2jPKD0URNI8oVGu/pxIcajUKPRNMsR6oP56mfif1010cOKlTMSJpoJMFgUJRzpCWQWoxyQlmo8MwUQycysiAywx0aaoUl7CaYaj7y/PktZB1alVa9eHlfrZtI4i7MAu7IMDx1CHC2hAEwhIeIRneLEerCfr1XqbRAvWdGYbfsF6/wK4eZIj</latexit>

D’

Soyay!!


Notquite!


Whatgoeswrong


Canweactuallyconstructthisnon-deterministicpolynomialtimeTMdeciderD’for?•  Twopossibleapproaches:

1.  TakeDandinverttheuniqueacceptandrejectstates.•  Doesthisactuallywork?•  No!IngeneraltheconstructedmachineD1willnotcorrectlyrecognize

–  Theremaybeexception(someproblemsarebothNPandcoNP)

•  Example:givenLinNPand–  RunDonw:ingeneralsome(atleastone)executionbrancheswillendin

acceptandsomeinreject–  NowrunD1onw,–  ThebranchesthatwereacceptingforDwillrejectforD1andviceversa–  IftherewasonerejectbranchwhenrunningDthentherewillbean

acceptingbranchonD1–  ButthenD1willacceptw!Thisisanabsurdaseitheror

L̄<latexit sha1_base64="/CNvENZPOPLT00e+zFMRermS+34=">AAAB7nicbVDLSsNAFL2pr1pfUZduBovgqiRafOyKbly4qGAf0IYymU7aoZNJmJkIJfQj3LhQxK3f486/cZIGUeuBC4dz7uXee/yYM6Ud59MqLS2vrK6V1ysbm1vbO/buXltFiSS0RSIeya6PFeVM0JZmmtNuLCkOfU47/uQ68zsPVCoWiXs9jakX4pFgASNYG6nT97FMb2cDu+rUnBxokbgFqUKB5sD+6A8jkoRUaMKxUj3XibWXYqkZ4XRW6SeKxphM8Ij2DBU4pMpL83Nn6MgoQxRE0pTQKFd/TqQ4VGoa+qYzxHqs/nqZ+J/XS3Rw4aVMxImmgswXBQlHOkLZ72jIJCWaTw3BRDJzKyJjLDHRJqFKHsJlhrPvlxdJ+6Tmntbqd/Vq46qIowwHcAjH4MI5NOAGmtACAhN4hGd4sWLryXq13uatJauY2YdfsN6/AHo2j80=</latexit>

w 2 L̄<latexit sha1_base64="a6mRnm+AzWHlG0TuhBoLbLncwpM=">AAAB8nicbVDLSsNAFJ3UV62vqks3g0VwVRItPnZFNy5cVLAPSEKZTCft0MkkzNwoJfQz3LhQxK1f486/cZoG8XXgwuGce7n3niARXINtf1ilhcWl5ZXyamVtfWNzq7q909Fxqihr01jEqhcQzQSXrA0cBOslipEoEKwbjC9nfveOKc1jeQuThPkRGUoeckrASO69x6UXEJVdT/vVml23c+C/xClIDRVo9avv3iCmacQkUEG0dh07AT8jCjgVbFrxUs0SQsdkyFxDJYmY9rP85Ck+MMoAh7EyJQHn6veJjERaT6LAdEYERvq3NxP/89wUwjM/4zJJgUk6XxSmAkOMZ//jAVeMgpgYQqji5lZMR0QRCialSh7C+QwnXy//JZ2junNcb9w0as2LIo4y2kP76BA56BQ10RVqoTaiKEYP6Ak9W2A9Wi/W67y1ZBUzu+gHrLdPogORnw==</latexit>

w 2 L<latexit sha1_base64="KdpedO6KeMtcKlvAgsC42cgrV+A=">AAAB7XicbVDLSsNAFL2pr1pfVZduBovgqiRafOyKbly4qGAf0IYymU7asZOZMDNRSug/uHGhiFv/x51/Y5IGUeuBC4dz7uXee7yQM21s+9MqLCwuLa8UV0tr6xubW+XtnZaWkSK0SSSXquNhTTkTtGmY4bQTKooDj9O2N75M/fY9VZpJcWsmIXUDPBTMZwSbRGo99JhA1/1yxa7aGdA8cXJSgRyNfvmjN5AkCqgwhGOtu44dGjfGyjDC6bTUizQNMRnjIe0mVOCAajfOrp2ig0QZIF+qpIRBmfpzIsaB1pPASzoDbEb6r5eK/3ndyPhnbsxEGBkqyGyRH3FkJEpfRwOmKDF8khBMFEtuRWSEFSYmCaiUhXCe4uT75XnSOqo6x9XaTa1Sv8jjKMIe7MMhOHAKdbiCBjSBwB08wjO8WNJ6sl6tt1lrwcpnduEXrPcvMYmPBA==</latexit>

w 2 L<latexit sha1_base64="KdpedO6KeMtcKlvAgsC42cgrV+A=">AAAB7XicbVDLSsNAFL2pr1pfVZduBovgqiRafOyKbly4qGAf0IYymU7asZOZMDNRSug/uHGhiFv/x51/Y5IGUeuBC4dz7uXee7yQM21s+9MqLCwuLa8UV0tr6xubW+XtnZaWkSK0SSSXquNhTTkTtGmY4bQTKooDj9O2N75M/fY9VZpJcWsmIXUDPBTMZwSbRGo99JhA1/1yxa7aGdA8cXJSgRyNfvmjN5AkCqgwhGOtu44dGjfGyjDC6bTUizQNMRnjIe0mVOCAajfOrp2ig0QZIF+qpIRBmfpzIsaB1pPASzoDbEb6r5eK/3ndyPhnbsxEGBkqyGyRH3FkJEpfRwOmKDF8khBMFEtuRWSEFSYmCaiUhXCe4uT75XnSOqo6x9XaTa1Sv8jjKMIe7MMhOHAKdbiCBjSBwB08wjO8WNJ6sl6tt1lrwcpnduEXrPcvMYmPBA==</latexit>

Whatgoeswrong


Canweactuallyconstructthisnon-deterministicpolynomialtimeTMdeciderD’for?•  Twopossibleapproaches:

2. JustinvertthefinaldecisionofD.Thatis,acceptifatleastonebranchacceptorrejectotherwise•  Doesthisactuallywork?•  No!SuchamachineisNOTaNDTMbydefinition•  Inordertodosowewouldneedtoevaluateallpossiblenon-deterministic

branches!

L̄<latexit sha1_base64="/CNvENZPOPLT00e+zFMRermS+34=">AAAB7nicbVDLSsNAFL2pr1pfUZduBovgqiRafOyKbly4qGAf0IYymU7aoZNJmJkIJfQj3LhQxK3f486/cZIGUeuBC4dz7uXee/yYM6Ud59MqLS2vrK6V1ysbm1vbO/buXltFiSS0RSIeya6PFeVM0JZmmtNuLCkOfU47/uQ68zsPVCoWiXs9jakX4pFgASNYG6nT97FMb2cDu+rUnBxokbgFqUKB5sD+6A8jkoRUaMKxUj3XibWXYqkZ4XRW6SeKxphM8Ij2DBU4pMpL83Nn6MgoQxRE0pTQKFd/TqQ4VGoa+qYzxHqs/nqZ+J/XS3Rw4aVMxImmgswXBQlHOkLZ72jIJCWaTw3BRDJzKyJjLDHRJqFKHsJlhrPvlxdJ+6Tmntbqd/Vq46qIowwHcAjH4MI5NOAGmtACAhN4hGd4sWLryXq13uatJauY2YdfsN6/AHo2j80=</latexit>

…

•  ThisisnotaNDTM!•  EquivalentDTMwouldrunin

exponentialtime

NPandcoNP•  ThesearethewaythatwecanapproachconstructingD1and

theybothfail.•  OurcurrentknowledgeisthatisnotclearwhetherNP=coNP•  Thisdiscussionshouldhighlightwhyitisingeneralbetterto

reasononNPandcoNPintermsofcertificates!


P<latexit sha1_base64="5tsa+7MfOU5B0wZ6R6OOKoTg+TU=">AAAB6HicbVDLSsNAFL2pr1pfVZdugkVwVRIrPnZFNy5bsA9og0ymN+3YySTMTIQS+gVuXCji1k9y5984aYOo9cDA4ZxzmXuPH3OmtON8WoWl5ZXVteJ6aWNza3unvLvXVlEiKbZoxCPZ9YlCzgS2NNMcu7FEEvocO/74OvM7DygVi8StnsTohWQoWMAo0UZqNu7KFafqzGAvEjcnFchh8h/9QUSTEIWmnCjVc51YeymRmlGO01I/URgTOiZD7BkqSIjKS2eLTu0jowzsIJLmCW3P1J8TKQmVmoS+SYZEj9RfLxP/83qJDi68lIk40Sjo/KMg4baO7Oxqe8AkUs0nhhAqmdnVpiMiCdWmm9KshMsMZ98nL5L2SdWtVWvN00r9Kq+jCAdwCMfgwjnU4QYa0AIKCI/wDC/WvfVkvVpv82jBymf24Res9y/CYY0L</latexit>

NP<latexit sha1_base64="YKcq6WgzuJxbLng1udU8T2FMURg=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKblxJFfuANshkOmmHTiZhZiKE0D9w40IRt/6RO//GSRpErQcGDuecy9x7vIgzpW370yotLC4tr5RXK2vrG5tb1e2djgpjSWibhDyUPQ8rypmgbc00p71IUhx4nHa9yWXmdx+oVCwUdzqJqBvgkWA+I1gb6fa6dV+t2XU7B5onTkFqUMDkPwbDkMQBFZpwrFTfsSPtplhqRjidVgaxohEmEzyifUMFDqhy03zTKTowyhD5oTRPaJSrPydSHCiVBJ5JBliP1V8vE//z+rH2z9yUiSjWVJDZR37MkQ5RdjYaMkmJ5okhmEhmdkVkjCUm2pRTyUs4z3DyffI86RzVnUa9cXNca14UdZRhD/bhEBw4hSZcQQvaQMCHR3iGF2tiPVmv1tssWrKKmV34Bev9C13ajWM=</latexit>

NPCOMPLETE

coNPCOMPLETE

coNP<latexit sha1_base64="GndNh/StfCDjf0WbsAGCXWmT4Vs=">AAAB63icbVDLSsNAFL2pr1pfVZdugkVwVRIVH7uiG1dSwT6gDTKZTtqh8wgzE6GE/oIbF4q49Yfc+TdO0iBqPTBwOOdc5t4Txoxq43mfTmlhcWl5pbxaWVvf2Nyqbu+0tUwUJi0smVTdEGnCqCAtQw0j3VgRxENGOuH4KvM7D0RpKsWdmcQk4GgoaEQxMpmE5U3zvlrz6l4Od574BalBAZv/6A8kTjgRBjOkdc/3YhOkSBmKGZlW+okmMcJjNCQ9SwXiRAdpvuvUPbDKwI2ksk8YN1d/TqSIaz3hoU1yZEb6r5eJ/3m9xETnQUpFnBgi8OyjKGGukW52uDugimDDJpYgrKjd1cUjpBA2tp5KXsJFhtPvk+dJ+6juH9ePb09qjcuijjLswT4cgg9n0IBraEILMIzgEZ7hxeHOk/PqvM2iJaeY2YVfcN6/AOgGjkk=</latexit>

NP-hardproblemsnotinNP•  BydefinitionsNP-HardproblemsareNP-Completebutarenot

necessarilyNP•  WewilldiscussaNP-hardlanguageforwhichwecannotsay

whetheritisNP•  ChromaticnumberofaGraphGistheminimumvalueksuch

thatGisk-colorable•  CN={<G,k>|kisG’schromaticnumber}

–  Careful!Weareinterestedintheminimumvaluekforwhichthegraphiscolorable

•  WeshallclaimthatCNisNOTinNPunlessNP=coNP


•  Lasttimewesawthat3SAT≤p3COLORING•  Thereductionproceedsgivenaninputbooleanexpression

ΦtocreateagraphGΦ


CNisNP-hard

Observations–  Gφismustrequireatleast3colors

•  Asbyconstructionwehavesometriangles–  φissatisfiableimpliesGφis3-colorable->Itschromaticnumberis3

•  foreachclauseCj=(a∨b∨c)atleastoneofa,b,ciscoloredTrue.OR-gadgetforCjcanbe3-coloredsuchthatoutputisTrue.

–  Gφis3-colorableimpliesφissatisfiable•  consideranyclauseCj=(a∨b∨c).itcannotbethatalla,b,careFalse.Ifso,outputofOR-gadgetforCjhastobecoloredFalsebutoutputisconnectedtoBaseandFalseandhencethegraphwouldnotbe3-colorable

–  ThisimpliesthatifφisNOTsatisfiablethenGφisnot3-colorable•  Itrequiresatleastfourcolors!

–  TheconstructionissuchthatφisNOTsatisfiablethenGφis4-colorable->Itschromaticnumberis4


•  Weusethesamereductionfunctionusedwhenshowing

that3SAT≤p3COLinordertoshowthat3SAT≤pCN•  Thereductionproceedsgivenaninputbooleanexpression

ΦtocreateagraphGΦ

•  WethencreateaninstanceoftheCNproblemas<GΦ,3>•  Byconstruction:–  ΦissatisfiableimpliesGΦ’schromaticnumberis3–  ΦisNOTsatisfiableimpliesGΦ’schromaticnumberis4

•  Asshownbeforethereductionrequirespolynomialtime•  Hencewehave3SAT≤pCNàCNisNP-Hard!


CNisNP-hard

� 2 3SAT () < G�, 3 >2 CN<latexit sha1_base64="WQgRco85n/IxZ/FudPV4d0dGuV4=">AAACDnicbZDLSgMxGIUz9VbrbdSlm2ApuJAytcULiFS70JVU7A3aUjJppg3NZIYkI5ShT+DGV3HjQhG3rt35Nmamg6j1QODjnD8k/7F9RqWyrE8jNTe/sLiUXs6srK6tb5ibWw3pBQKTOvaYJ1o2koRRTuqKKkZaviDItRlp2qNKlDfviJDU4zU19knXRQNOHYqR0lbPzHX8IYUdymHx9rymwXHg6WUvjOzJfvEsSirXPTNr5a1YcBYKCWRBomrP/Oj0PRy4hCvMkJTtguWrboiEopiRSaYTSOIjPEID0tbIkUtkN4zXmcCcdvrQ8YQ+XMHY/XkjRK6UY9fWky5SQ/k3i8z/snagnONuSLkfKMLx9CEnYFB5MOoG9qkgWLGxBoQF1X+FeIgEwko3mIlLOIl0+L3yLDQO8oVivnRTypYvkjrSYAfsgj1QAEegDK5AFdQBBvfgETyDF+PBeDJejbfpaMpI7myDXzLevwDxTppJ</latexit>

•  RecallthatweshowedthatalanguageLisNP-Completeiffits

complementLiscoNP-complete–  3SATNP-CompleteiscoNP-complete

•  Weusethesamereductionfunctionusedwhenshowingthat3SAT≤p3COLinordertoshowthat3SAT≤pCN

•  ThereductionproceedsgivenaninputBooleanexpressionΦtocreateagraphGΦ

•  WethencreateaninstanceoftheCNproblemas<GΦ,4>–  ΦisNOTsatisfiableimpliesGΦ’schromaticnumberis4–  ΦissatisfiableimpliesGΦ’schromaticnumberis3

•  Asshownbeforethereductionrequirespolynomialtime•  Hencewehave3SAT≤pCNàCNiscoNP-Hard!


CNiscoNP-hard

� 2 3SAT () < G�, 4 >2 CN<latexit sha1_base64="l3aPqaDEUsrcZBCDJsuTANB7QaY=">AAACGXicbVBLSwMxGMz6rPVV9eglWAQPUra2+ACRag96kop9QXcp2TTbhmazS5IVytK/4cW/4sWDIh715L8xu11ErQOBYWa+JN84AaNSmeanMTM7N7+wmFnKLq+srq3nNjab0g8FJg3sM1+0HSQJo5w0FFWMtANBkOcw0nKG1dhv3REhqc/rahQQ20N9Tl2KkdJSN2dawYBCi3Jo+ToXXxOVbs/rY625Ljy97EZxYrxfPotD1etuLm8WzARwmhRTkgcpat3cu9XzcegRrjBDUnaKZqDsCAlFMSPjrBVKEiA8RH3S0ZQjj0g7SjYbw12t9KDrC324gon6cyJCnpQjz9FJD6mB/OvF4n9eJ1TusR1RHoSKcDx5yA0ZVD6Ma4I9KghWbKQJwoLqv0I8QAJhpcvMJiWcxDj8XnmaNA8KxVKhfFPOVy7SOjJgG+yAPVAER6ACrkANNAAG9+ARPIMX48F4Ml6Nt0l0xkhntsAvGB9fcNqfcA==</latexit>

() 3SAT<latexit sha1_base64="VsKcV0Aepmh1UfJQwMQ88XGW/kQ=">AAAB/XicbVDJTsMwFHTKVsoWlhsXiwqJU5XQiuVW4MKxiG5SE1WO67RWnTiyHaQSVfwKFw4gxJX/4Mbf4KQRAspIlkYz8+zn8SJGpbKsT6OwsLi0vFJcLa2tb2xumds7bcljgUkLc8ZF10OSMBqSlqKKkW4kCAo8Rjre+Cr1O3dESMrDpppExA3QMKQ+xUhpqW/uOdT3ocN1Jr0iqd5eNKd9s2xVrAxwntg5KYMcjb754Qw4jgMSKsyQlD3bipSbIKEoZmRacmJJIoTHaEh6moYoINJNsu2n8FArA+hzoU+oYKb+nEhQIOUk8HQyQGok/3qp+J/Xi5V/5iY0jGJFQjx7yI8ZVBymVcABFQQrNtEEYUH1rhCPkEBY6cJKWQnnKU6+vzxP2scVu1qp3dTK9cu8jiLYBwfgCNjgFNTBNWiAFsDgHjyCZ/BiPBhPxqvxNosWjHxmF/yC8f4FLKaVLw==</latexit>

Recap•  SofarweshowedthatCNisbothNP-hardandcoNP-hard

•  WeneverarguedwhetherCNisinNPorcoNP!

•  Thisisactuallyaveryhardquestion•  InthefollowingwewillshowitistiedtoPvsNPandtoNPvscoNP!


GraphK-coloring•  Recall:AgraphGisk-colorableifitsverticescanbepartitionedinkpartsV1,V2,...Vksuchthatanypairofnodeconnectedbyanedgeareinadifferentpart–  Ispossibletocolortheverticesofthegraphsothatallneighborverticesareassigneddifferentcolors

•  Givenagraph,thegoalofthek-coloringproblemistodecidewhetherthegraphiskcolorable

•  Equivalentlygiven<G>,thegoalistodecideifitbelongstothelanguage


kCOL = {< G, k > |G is k-colorable}<latexit sha1_base64="ZyhuZPkQPPbWXkSI1Y9kfWU3Bzc=">AAACEXicbVDJSgNBEO2JW4xb1KOXxiDkoGGiwQVUgjnEg2AEs0AmhJ5OJ2lmpbtGDGN+wYu/4sWDIl69efNv7EwGUeODhserqlfVz/RtLkHXP7XE1PTM7FxyPrWwuLS8kl5dq0kvEJRVqWd7omESyWzusipwsFnDF4w4ps3qplUa1es3TEjuudcw8FnLIT2XdzkloKR2OmuVLi/wCTbC4/K2dXpXNoDdQoi5xNZOZE+U1dAYttMZPadHwJMkH5MMilFppz+MjkcDh7lAbSJlM6/70AqJAE6VY8oIJPMJtUiPNRV1icNkK4xWDvGWUjq46wn1XMCR+nMiJI6UA8dUnQ6BvvxbG4n/1ZoBdA9bIXf9AJhLx4u6gY3Bw6N4cIcLRsEeKEKo4OpWTPtEEAoqxFQUwtEI+99fniS13Vx+L1e4KmSKZ3EcSbSBNlEW5dEBKqJzVEFVRNE9ekTP6EV70J60V+1t3JrQ4pl19Ava+xe475yE</latexit>

kCOLORisNP

– Certificate:foreachnodeanassignmentofoneofthekcolors

– Verifier:foreachedge(u,v)checkthatthereareassigneddifferentcolors

– Runsinpolynomial(linear)timewithrespecttothenumberofedges!

11/7/19 16TheoryofComputation-Fall'19LorenzoDeStefani

IfP=NPàCNP•  P=NPandà•  Thatis,thereexistsapolytimealgorithmAtoverifyk-colorabilityforagivenk

•  ConsiderthefollowingalgorithmforCN–  GivenG=(V,E)–  Fori=1,2,…|V|

•  RunAonininput<G,i>•  HaltforthefirstisuchthatGisi-colorable

•  Polynomialruntime–  atmost|V|loops–  EachlooprequirestimeP(|V|)since

•  Thenwewouldhave


2<latexit sha1_base64="OGutpjBGqXjN2B7YqOyu423H7bU=">AAAB6nicbVDLSsNAFL2pr1pfUZduBovgqiRafOyKblxWtA9oQ5lMJ+3QySTMTIQS+gluXCji1i9y5984SYOo9cCFwzn3cu89fsyZ0o7zaZWWlldW18rrlY3Nre0de3evraJEEtoiEY9k18eKciZoSzPNaTeWFIc+px1/cp35nQcqFYvEvZ7G1AvxSLCAEayNdNdnYmBXnZqTAy0StyBVKNAc2B/9YUSSkApNOFaq5zqx9lIsNSOczir9RNEYkwke0Z6hAodUeWl+6gwdGWWIgkiaEhrl6s+JFIdKTUPfdIZYj9VfLxP/83qJDi68lIk40VSQ+aIg4UhHKPsbDZmkRPOpIZhIZm5FZIwlJtqkU8lDuMxw9v3yImmf1NzTWv22Xm1cFXGU4QAO4RhcOIcG3EATWkBgBI/wDC8Wt56sV+tt3lqyipl9+AXr/QtlyI4D</latexit>

kCOL 2 NP<latexit sha1_base64="w2k0BaqPbKtC9uCVBFJoFxa6di8=">AAAB8XicbVDLSsNAFL2pr1pfVZduBovgqiRafOyK3bgQrWAf2AaZTCft0MkkzEyEEvoXblwo4ta/ceffOEmD+DowcDjnXObe40WcKW3bH1Zhbn5hcam4XFpZXVvfKG9utVUYS0JbJOSh7HpYUc4EbWmmOe1GkuLA47TjjRup37mnUrFQ3OhJRN0ADwXzGcHaSLfjxtVFnwl02bwrV+yqnQH9JU5OKpDD5N/7g5DEARWacKxUz7Ej7SZYakY4nZb6saIRJmM8pD1DBQ6ocpNs4ynaM8oA+aE0T2iUqd8nEhwoNQk8kwywHqnfXir+5/Vi7Z+4CRNRrKkgs4/8mCMdovR8NGCSEs0nhmAimdkVkRGWmGhTUikr4TTF0dfJf0n7oOocVmvXtUr9LK+jCDuwC/vgwDHU4Rya0AICAh7gCZ4tZT1aL9brLFqw8plt+AHr7ROJFZBQ</latexit>

kCOL 2 P<latexit sha1_base64="fV/gJqsxAlP0PdbFdR2q3PdVkto=">AAAB8HicbVDLSsNAFL2pr1pfVZduBovgqiRafOyK3bgQrGAf0gaZTCft0MkkzEyEEvoVblwo4tbPceffOEmDqPXAwOGcc5l7jxdxprRtf1qFhcWl5ZXiamltfWNzq7y901ZhLAltkZCHsuthRTkTtKWZ5rQbSYoDj9OON26kfueBSsVCcasnEXUDPBTMZwRrI92NG9dXfSZQ875csat2BjRPnJxUIIfJf/QHIYkDKjThWKmeY0faTbDUjHA6LfVjRSNMxnhIe4YKHFDlJtnCU3RglAHyQ2me0ChTf04kOFBqEngmGWA9Un+9VPzP68XaP3MTJqJYU0FmH/kxRzpE6fVowCQlmk8MwUQysysiIywx0aajUlbCeYqT75PnSfuo6hxXaze1Sv0ir6MIe7APh+DAKdThEprQAgIBPMIzvFjSerJerbdZtGDlM7vwC9b7F+rwj/g=</latexit>

kCOL 2 P<latexit sha1_base64="fV/gJqsxAlP0PdbFdR2q3PdVkto=">AAAB8HicbVDLSsNAFL2pr1pfVZduBovgqiRafOyK3bgQrGAf0gaZTCft0MkkzEyEEvoVblwo4tbPceffOEmDqPXAwOGcc5l7jxdxprRtf1qFhcWl5ZXiamltfWNzq7y901ZhLAltkZCHsuthRTkTtKWZ5rQbSYoDj9OON26kfueBSsVCcasnEXUDPBTMZwRrI92NG9dXfSZQ875csat2BjRPnJxUIIfJf/QHIYkDKjThWKmeY0faTbDUjHA6LfVjRSNMxnhIe4YKHFDlJtnCU3RglAHyQ2me0ChTf04kOFBqEngmGWA9Un+9VPzP68XaP3MTJqJYU0FmH/kxRzpE6fVowCQlmk8MwUQysysiIywx0aajUlbCeYqT75PnSfuo6hxXaze1Sv0ir6MIe7APh+DAKdThEprQAgIBPMIzvFjSerJerbdZtGDlM7vwC9b7F+rwj/g=</latexit>

CN 2 P<latexit sha1_base64="KsCAhyxZfwYFoVWQeAGyQbf73ZU=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiRafOyK3biSCqYttEEm00k7dDIJMxOhhH6EGxeKuPV73Pk3TtIgaj0wcDjnXObe48ecKW3bn1ZpaXllda28XtnY3Nreqe7udVSUSEJdEvFI9nysKGeCupppTnuxpDj0Oe36k1bmdx+oVCwSd3oaUy/EI8ECRrA2Urd1M2ACte+rNbtu50CLxClIDQqY/MdgGJEkpEITjpXqO3asvRRLzQins8ogUTTGZIJHtG+owCFVXpqvO0NHRhmiIJLmCY1y9edEikOlpqFvkiHWY/XXy8T/vH6igwsvZSJONBVk/lGQcKQjlN2OhkxSovnUEEwkM7siMsYSE20aquQlXGY4+z55kXRO6s5pvXHbqDWvijrKcACHcAwOnEMTrqENLhCYwCM8w4sVW0/Wq/U2j5asYmYffsF6/wKEtI8s</latexit>

IfCNNPàNP=coNPToprovetheclaimweshowthatifCNNPthen:

1.  2. 

•  Weprovidetheprooffor2.:ForanyLcoNP,ifCNNPthenLNP–  ToproveitweconstructaNDTMNthatdecidesLasfollows:–  SinceCNNPthereexistaNDTMMdecidingCNinpolynomialtime–  OninputwforL:

1.  ReducetoancoNP-Hardlanguage,suchasCN–  OnesuchreductionexisitsforanyLgiventhedefinitionofcoNP-Hardness–  ForinputwobtaininstanceofCN<G,k>

2.  UseMdecidingCNinpolynomialtimeon<G,k>3.  Naccepts/rejectsifMdoes

–  NdecidesLinpolynomialtimeàLNP



NP ✓ coNP<latexit sha1_base64="5eKB7EP2QY4u8wpCS3DePHOUu80=">AAAB+nicbVDLSgMxFM34rPXV6tJNsAiuyowWH7uiG1elgn1AO5RMeqcNzWTGJKOU2k9x40IRt36JO//GzHQQtR4IHM45l3tzvIgzpW3701pYXFpeWc2t5dc3Nre2C8WdpgpjSaFBQx7KtkcUcCagoZnm0I4kkMDj0PJGl4nfugOpWChu9DgCNyADwXxGiTZSr1Cs1XFXxZ4CDbeYhrV6r1Cyy3YKPE+cjJRQBpP/6PZDGgcgNOVEqY5jR9qdEKkZ5TDNd2MFEaEjMoCOoYIEoNxJevoUHxilj/1Qmic0TtWfExMSKDUOPJMMiB6qv14i/ud1Yu2fuRMmoliDoLNFfsyxDnHSA+4zCVTzsSGESmZuxXRIJKHatJVPSzhPcPL95XnSPCo7x+XKdaVUvcjqyKE9tI8OkYNOURVdoTpqIIru0SN6Ri/Wg/VkvVpvs+iClc3sol+w3r8AeqSTow==</latexit>

coNP ✓ NP<latexit sha1_base64="9ffZrXJOX4akvpY5R1UYoXS273k=">AAAB+nicbVDLSgMxFM34rPXV6tJNsAiuyowWH7uiG1elgn1AO5RMeqcNzWTGJKOU2k9x40IRt36JO//GzHQQtR4IHM45l3tzvIgzpW3701pYXFpeWc2t5dc3Nre2C8WdpgpjSaFBQx7KtkcUcCagoZnm0I4kkMDj0PJGl4nfugOpWChu9DgCNyADwXxGiTZSr1CkYa2Ouyr2FGi4xbV6r1Cyy3YKPE+cjJRQBpP/6PZDGgcgNOVEqY5jR9qdEKkZ5TDNd2MFEaEjMoCOoYIEoNxJevoUHxilj/1Qmic0TtWfExMSKDUOPJMMiB6qv14i/ud1Yu2fuRMmoliDoLNFfsyxDnHSA+4zCVTzsSGESmZuxXRIJKHatJVPSzhPcPL95XnSPCo7x+XKdaVUvcjqyKE9tI8OkYNOURVdoTpqIIru0SN6Ri/Wg/VkvVpvs+iClc3sol+w3r8AfAKTow==</latexit>






WhatifNP=coNP!


P<latexit sha1_base64="5tsa+7MfOU5B0wZ6R6OOKoTg+TU=">AAAB6HicbVDLSsNAFL2pr1pfVZdugkVwVRIrPnZFNy5bsA9og0ymN+3YySTMTIQS+gVuXCji1k9y5984aYOo9cDA4ZxzmXuPH3OmtON8WoWl5ZXVteJ6aWNza3unvLvXVlEiKbZoxCPZ9YlCzgS2NNMcu7FEEvocO/74OvM7DygVi8StnsTohWQoWMAo0UZqNu7KFafqzGAvEjcnFchh8h/9QUSTEIWmnCjVc51YeymRmlGO01I/URgTOiZD7BkqSIjKS2eLTu0jowzsIJLmCW3P1J8TKQmVmoS+SYZEj9RfLxP/83qJDi68lIk40Sjo/KMg4baO7Oxqe8AkUs0nhhAqmdnVpiMiCdWmm9KshMsMZ98nL5L2SdWtVWvN00r9Kq+jCAdwCMfgwjnU4QYa0AIKCI/wDC/WvfVkvVpv82jBymf24Res9y/CYY0L</latexit>

NP = coNP<latexit sha1_base64="1i6wTgNpq/NmYFkfoVtjqBHHVmU=">AAAB7nicbVDLSgMxFL3js9ZX1aWbYBFclRktPhZC0Y2rUsE+oB0kk2ba0EwyJBmhDP0INy4Ucev3uPNvTKeDqPVA4HDOueTeE8ScaeO6n87C4tLyymphrbi+sbm1XdrZbWmZKEKbRHKpOgHWlDNBm4YZTjuxojgKOG0Ho+up336gSjMp7sw4pn6EB4KFjGBjpXa9cUlkvXFfKrsVNwOaJ15OypDD5j96fUmSiApDONa667mx8VOsDCOcToq9RNMYkxEe0K6lAkdU+2m27gQdWqWPQqnsEwZl6s+JFEdaj6PAJiNshvqvNxX/87qJCc/9lIk4MVSQ2UdhwpGRaHo76jNFieFjSzBRzO6KyBArTIxtqJiVcDHF6ffJ86R1XPFOKtXbarl2lddRgH04gCPw4AxqcAMNaAKBETzCM7w4sfPkvDpvs+iCk8/swS8471+nfo9D</latexit>

NP-HardandcoNP-Hard

•  Careful!WedidnotprovethatNP=coNP!•  WejustarguedthatifCNwasNPthatwouldimplythatNP=coNP

•  Challengingopenresearchquestion!

NP vs coNP NP hard problems not in NP The Chromatic ......NP and coNP • These are the way that we...

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