1. 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] 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… 1 TFNP and LeafCovering

2. Outline 1. Generally on Search Problems 2. The Class TFNP 3. Subclasses of TFNP part I: PPA, PPAD Problems in PPA, PPAD Completeness in PPAD 4. Subclasses of TFNP part II: PPP, PLS 5. PPAD-completeness of NASH & the complexity of computing equilibria in congestion games 2TFNP and LeafCovering

3. Outline 1. Generally on Search Problems 2. The Class TFNP 3. Subclasses of TFNP part I: PPA, PPAD Problems in PPA, PPAD Completeness in PPAD 4. Subclasses of TFNP part II: PPP, PLS 5. PPAD-completeness of NASH & the complexity of computing equilibria in congestion games 3TFNP 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 4TFNP 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 5TFNP and LeafCovering

6. The Classes FP and FNP L € NP iff there exists poly-time computable R L (x,y) s.t. X € L  y { |y| ≤ p(|x|) & R L (x,y) } Note how R L defines the problem-language L The corresponding search problem Π R(L) € FNP is: given an x find any y s.t. R L (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 6TFNP and LeafCovering

7. Reductions and completeness A function problem Π R reduces to a function problem Π S 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 Π R to Π S and g transforms a solution of Π S to one of Π R Standard notion of completeness works fine… 7TFNP 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?? 8TFNP and LeafCovering

9. Outline 1. Generally on Search Problems 2. The Class TFNP 3. Subclasses of TFNP part I: PPA, PPAD Problems in PPA, PPAD Completeness in PPAD 4. Subclasses of TFNP part II: PPP, PLS 5. PPAD-completeness of NASH & the complexity of computing equilibria in congestion games 9TFNP 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 10TFNP and LeafCovering

11. Basic stuff about TFNP 1. FP TFNP FNP 2. TFNP = F(NP coNP) NP = problems with “yes” certificate y s.t. R 1 (x,y) coNP = problems with “no” certificate z s.t. R 2 (x,y) for TFNP F(NP coNP) take R = R 1 U R 2 for F(NP coNP) TFNP take R 1 = R and R 2 = ø 3. There is an FNP-complete problem in TFNP iff NP = coNP  : If NP = coNP then trivially FNP = TFNP  : If the FNP-complete problem Π R is in TFNP then: FSAT reduces to Π R via f and g, hence any unsatisfiable formula φ has a “no” certificate y, s.t. R(f(φ),y) (y exists since Π R is in TFNP) and g(y)=“no” 4. TFNP is a semantic complexity class  no complete problems! note how telling whether a relation is total is undecidable (and not even RE!!) 11TFNP and LeafCovering