1 / 11

Complexity of Search Problems: TFNP, PPA, PPP, Complexity of Computations

Explore the complexity of search problems, TFNP class, PPA and PPP subclasses, PPAD completeness, and Nash equilibrium computations. Understand the relations between different classes and the challenges in solving these problems efficiently. Dive into the intricacies of function problems and their impact on computational complexity.

shirleyt
Download Presentation

Complexity of Search Problems: TFNP, PPA, PPP, Complexity of Computations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. On the Complexity of Search ProblemsGeorge Pierrakos Mostly based on: On the Complexity of the Parity Argument and Other Insufficient Proofs of Existence [Pap94] On total functions, existence theorems and computational complexity [MP91] How easy is local search? [JPY88] Computational Complexity [Pap92] The complexity of computing a Nash equilibrium [DGP06] The complexity of pure Nash equilibria [FPT04] slides and scribe notes from many people… TFNP and LeafCovering

  2. Outline • Generally on Search Problems • The Class TFNP • Subclasses of TFNP part I: PPA, PPAD • Problems in PPA, PPAD • Completeness in PPAD • Subclasses of TFNP part II: PPP, PLS • PPAD-completeness of NASH & the complexity of computing equilibria in congestion games TFNP and LeafCovering

  3. Outline • Generally on Search Problems • The Class TFNP • Subclasses of TFNP part I: PPA, PPAD • Problems in PPA, PPAD • Completeness in PPAD • Subclasses of TFNP part II: PPP, PLS • PPAD-completeness of NASH & the complexity of computing equilibria in congestion games TFNP and LeafCovering

  4. Decision Problems vs Search (or “function”) Problems • SAT • Input: boolean CNF-formula φ • Output: “yes” or “no” • FSAT • Input: boolean CNF-formula φ • Output: satisfying assignment or “no” if none exist TFNP and LeafCovering

  5. Are search problems harder? They are definitely not easier: a poly-time algorithm for FSAT can be easily tweaked to give a poly-time algorithm for SAT …and vice versa, FSAT “reduces” to SAT: we can figure out a satisfying assignment by running poly-time algorithm for SAT n-times TFNP and LeafCovering

  6. The Classes FP and FNP • L €NP iff there exists poly-time computable RL(x,y) s.t. X € L  y { |y| ≤ p(|x|) & RL(x,y) } • Note how RL defines the problem-language L • The corresponding search problem ΠR(L) €FNP is: given an x find any y s.t. RL(x,y) and reply “no” if none exist • FSAT € FNP… what about FTSP? • Are all FNP problems self-reducible like FSAT? [open?] • FP is the subclass of FNP where we only consider problems for which a poly-time algorithm is known TFNP and LeafCovering

  7. Reductions and completeness • A function problem ΠR reduces to a function problem ΠS if there exist log-space computable string functions f and g, s.t. R(x,g(y))  S(f(x),y) • intuitively f reduces problem ΠR to ΠS • and g transforms a solution of ΠS to one of ΠR • Standard notion of completeness works fine… TFNP and LeafCovering

  8. FP <?> FNP • A proof a-la-Cook shows that FSAT is FNP-complete • Hence, if FSAT € FP then FNP = FP • But we showed self-reducibility for SAT, so the theorem follows: • Theorem: FP = FNP iff P=NP • So, why care for function problems anyway?? TFNP and LeafCovering

  9. Outline • Generally on Search Problems • The Class TFNP • Subclasses of TFNP part I: PPA, PPAD • Problems in PPA, PPAD • Completeness in PPAD • Subclasses of TFNP part II: PPP, PLS • PPAD-completeness of NASH & the complexity of computing equilibria in congestion games TFNP and LeafCovering

  10. On total “functions”: the class TFNP • What happens if the relation R is total? i.e., for each x there is at least one y s.t. R(x,y) • Define TFNP to be the subclass of FNP where the relation R is total • TFNP contains problems that always have a solution, e.g. factoring, fix-point theorems, graph-theoretic problems, … • How do we know a solution exists? By an “inefficient proof of existence”, i.e. non-(efficiently)-constructive proof • The idea is to categorize the problems in TFNP based on the type of inefficient argument that guarantees their solution TFNP and LeafCovering

  11. Basic stuff about TFNP • FP TFNP FNP • TFNP = F(NP coNP) • NP = problems with “yes” certificate y s.t. R1(x,y) • coNP = problems with “no” certificate z s.t. R2(x,y) • for TFNP F(NP coNP) take R = R1 U R2 • for F(NP coNP) TFNP take R1 = R and R2 = ø • There is an FNP-complete problem in TFNP iff NP = coNP • : If NP = coNP then trivially FNP = TFNP • : If the FNP-complete problem ΠR is in TFNP then:FSAT reduces to ΠR via f and g, hence any unsatisfiable formula φ has a “no” certificate y, s.t. R(f(φ),y) (y exists since ΠR is in TFNP) and g(y)=“no” • TFNP is a semantic complexity class  no complete problems! • note how telling whether a relation is total is undecidable (and not even RE!!) TFNP and LeafCovering

More Related