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On the Complexity of Search Problems George Pierrakos

On the Complexity of Search Problems George 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]

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On the Complexity of Search Problems George Pierrakos

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  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

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