Computability and computational complexity

Transcription

1 Computability ad computatioal complexity Lecture 12: O P vs NP Io Petre Computer Sciece, Åbo Akademi Uiversity Fall December 9,

2 Cotet The map of NP Oracle machies Settlig P vs NP through oracles The P vs NP uestio some historical poiters sigificace possible aswers possible strategies possible coseueces December 9,

3 The map of NP The class of NP-complete problems is very useful i classifyig problems i NP Crucial i settlig the uestio of whether or ot P¹NP There are problems that are ot kow to be i either P or NP-complete GRAPH ISOMORPHISM is i NP but ot kow to be either NP-complete or i P Laszlo Babai (2015): GI ca be solved with a uasipolyomial algorithm ruig i 2 O(logc ())!! SUBGRAPH ISOMORPHISM is NP-complete! (for graphs G ad H, does G have a subgraph that is isomorphic to H?) Questio: are there problems that are either i P or NP-complete? Three cases possible i priciple: 1. NP cotais three distict subclasses: P, NP-complete, problems that are either 2. NP cotais oly two classes: P ad NP-complete 3. NP cotais oly oe class, i.e. P=NP I fact, the secod case is ot possible! December 9,

4 The map of NP NP NP Theorem. If P¹NP, the there is a laguage i NP which is either P, or is it NP-complete. NP-complete Proof skip it. ¹Æ P=NP= NP-complete Note the uary case: Theorem (Berma, 1978). If there exists a NP-complete uary laguage UÍ{0}*, the P=NP. P NP NP-complete P December 9,

5 Oracles Aalogy is a favorite method of reasoig ad of provig results Give a difficult problem, we look for a similar problem that has occurred elsewhere, i a alterative uiverse Questio: How would the problem P¹NP look i a alterative uiverse? Additioal uestios: What is a alterative uiverse for computatioal complexity? What is our uiverse for computatioal complexity? Our uiverse: o computatio is for free! Alterative uiverse: some (ot all) computatios are perhaps for free! For example, at some poit i a computatio, the algorithm asks a uestio, such as whether a certai Boolea formula it curretly cosiders is satisfiable, ad it gets a istataeous aswer to it This is the world of oracles, where a oracle for SAT ca aswer all our SAT uestios for free (istataeously) Is P¹NP i the alterative uiverse? Surprisig aswer! December 9,

6 Oracles cotiued Defiitio. A Turig machie M? with oracle is a multi-tape, determiistic TM that has a special tape called the uery tape, ad three special states:? (uery state) YES, NO (aswer states) Let AÍS* be a arbitrary laguage that will fuctio as a oracle for M?. The computatio of M? with oracle A proceeds like a ordiary TM, except for trasitios from the uery state: from?, M? moves to either YES or NO depedig o whether the curret strig o the uery tape is i A or ot the aswer thus allows the machie to chage its computatio The computatio of M? with oracle A o iput x is deoted M A (x). Time complexity defied exactly as for ordiary TM eve the uery steps oly cout as oe, o matter how difficult the uery is to decide Nodetermiistic TM with oracles defied exactly as before If C is ay (determiistic or odetermiistic) time complexity class, we ca defie C A to be the class of laguages decided (or accepted) by machies of the same type ad time boud as i C, oly that they have access to oracle A December 9,

7 Decidig P vs. NP with oracles Theorem. There is a oracle A for which P A =NP A. Proof. The idea is to cosider a oracle that either makes determiistic computatios much more powerful, or reders odetermiism powerless There is oe complexity result that we discussed i Lecture 6 where odetermiism makes o differece: PSPACE=NPSPACE Cosider as a oracle A ay PSPACE-complete laguage The PSPACE Í P A Í NP A Í NPSPACE = PSPACE First iclusio: sice A is PSPACE-complete, ay laguage i PSPACE ca be decided by a polyomial-time determiistic machie that reduces it first to A ad the uses the oracle oce Secod iclusio: trivial Third iclusio: ay odetermiistic machie with oracle A ca be simulated by a odetermiistic polyomial space-bouded machie where the uery to A is replaced with the computatio A(x) (which is polyomially space-bouded because A is i PSPACE) Fourth euality: by Savitch s theorem (see Lecture 6) December 9,

8 Decidig P vs NP with oracles cotiued Theorem. There is a oracle B for which P B ¹NP B. Proof. We skip it here. Oly poit out that we eed a oracle B that greatly ehaces the power of odetermiism The laguage used i the proof with LÎ NP B -P B is L={0 : there is xîb with x =} December 9,

9 Coclusios. Reasoig by aalogy does ot work for the P ¹ NP problem The uestio P¹NP will ot be settled by a proof that ca be carried over oracle machies, i.e., by proofs that trasced worlds ote that may of the techiues discussed i this course ca be carried over verbatim from oe world to aother Usig oracles ca be useful as a exploratory proof techiue. Provig that C A ¹D A shows that it is uite possible that C¹D ad that probably there is o trivial proof that C=D December 9,

10 The remaiig of this lecture is based o R.J.Lipto The P=NP uestio ad Gödel s lost letter, Spriger 2010 THE P=NP QUESTION December 9,

11 History 1956: Gödel writes a letter to vo Neuma where he formulates a problem very closely related to P vs NP Gödel s lost letter 1971: Stephe Cook formalizes the otio of reductio, defies NP-completeess, proves that SAT is NP-complete, formulates the uestio of whether P=NP (albeit ot i this form) Very few pay attetio Results formulated from the perspective of logic (similarly as Gödel) Same results/otios idepedetly by Leoid Levi i USSR 1972: Richard Karp writes a paper where he formulates the uestio of whether P=NP i the form that we ow kow, proves that 21 combiatorial problems are NP-complete Places Cook s results i a cetral positio i Computer Sciece Approaches the results from the poit of view of practical computer sciece Shows that it has wider coseueces tha Cook ad Gödel thought Helps galvaize the computer sciece commuity uickly 1972: Michael Rabi uickly orgaizes a coferece where Karp presets his work, publishes a book icludig Karp s paper, givig it more visibility December 9,

12 Solvig the problem Provig P=NP Direct method: Choose your favourite NP-complete problem ad give it a polyomial algorithm Idirect method: cotradict a result that holds if P=NP Provig P¹NP Direct method: Choose your favourite NP-complete problem ad prove that it caot have a polyomial-time determiistic algorithm Idirect method: cotradict a result that holds if P¹NP December 9,

13 Is P=NP well posed? Compare with the Riema Hypothesis Prime umber theorem: p(x), the umber of primes less tha x, is well approximated by the logarithmic itegral: p(x)=ò 0 x dt/log(t)+e(x) ad E(x)=o(x/log x) is the error term RH implies that the error term is order x 1/2, igorig logarithmic terms RH is biary : havig eve oe sigle zero ot o the critical lie destroys the boud December 9,

14 Is P vs NP well posed? cotiued P vs NP: very differet type of behavior There might be a algorithm for SAT ruig i a reasoable time boud This would be a great result, with extraordiary practical coseueces There might be a algorithm for SAT that is polyomial but with a very high expoet, say O( 10 ) It would solve the problem without much coseuece for practical computer sciece (e.g. for cryptography) There might be a algorithm for SAT that is polyomial i a very weak sese, e.g. with a ruig time No coseueces for practical computer sciece There might be a algorithm for SAT ruig i time c, with a ukow costat c December 9,

15 So what if P¹NP? Possible proofs that P¹NP. What are their coseueces? SAT might reuire expoetial time. Perhaps somethig like 1.2. Or how about ? SAT might have wild complexity For example, polyomial for a ifiite umber of, expoetial for a ifiite umber of SAT might have a barely super-polyomial complexity, say log log log All of these would solve the problem, but the aswer would ot uite settle whether SAT is truly itractable December 9,

16 P vs NP: a problem with three possible aswers? P=NP: there is a practical polyomial algorithm for SAT Questio: would there be practical algorithms for all NP-complete problems? P»NP: there is a o-practical polyomial algorithm for SAT The algorithm would be polyomial but with a huge expoet This would prove that P=NP but it would have little coseueces i practice P¹NP: there is o polyomial time algorithm for SAT (or ay other NPcomplete problem) December 9,

17 Why do people believe that P¹NP 1. P=NP would be too good to be true. Some argue that this would ed huma creativity 2. P=NP would imply that thousads of NP-complete problems are i P. Sice amog them are problems of very differet ature, it would be surprisig to fid a uiversal method for all 3. P=NP might put a ed to moder cryptography 4. P=NP would go agaist kow results i automata theory where odetermiism is kow to be very powerful expoetial blow-up i state complexity from DFA to NFA; it adds power to pushdow automata 5. December 9,

18 SAT i practice O the practical side of the CS commuity, may claim that i practice, all istaces of SAT that arise aturally are easily hadled by existig SAT solvers Questio: Why? Is this a sig that i fact P=NP? O the theoretical side of the CS commuity, may believe that P¹NP This clearly cotradicts the belief of the practical side of the commuity Uclear what makes real SAT istaces easy (whe they are ideed easy) Possibly a problem i the very defiitio of our complexity classes Worst-case complexity ot fully relevat i practice December 9,

19 The worst case model for complexity Advatage It gives a guaratee of the behavior of the algorithm i all possible cases Disadvatage Because it has to say somethig about ALL possible cases, it caot say that the vast majority of the cases (or perhaps the real-world cases) ca i fact have a much better behavior Alteratives Average case complexity: complexity for iputs that come from radom distributios How ca we capture the real-world cases? This is perhaps oe of the great challeges i complexity theory today. December 9,

20 What happes whe P vs NP is resolved? Compare to the saga of Adrew Wiles s proof of Fermat s theorem Periodically there are excited aoucemets of proofs Some are easily dismissed, especially whe comig from hobbyists Some are actually serious attempts by professioal mathematicias How to hadle the excitemet, the possible embarrassmet How may ca check the proof? Computer-based proofs It is all i the details of the proof Recall the earlier argumet o the possible solutios Havig a simple, oracle-like aswer to the problem is ot uite satisfactory December 9,

21 Learig objectives Oracle machies The status of P vs NP for oracle machies Perspectives o the P=NP problem types of possible aswers possible coseueces December 9,