The relative complexity of NP search problems

Abstract

Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, and a number of them are complete problems. We consider the question of the relative complexity of these search problem classes. We prove several separations which show that in a generic relativized world, the search classes are distinct and there is a standard search problem in each of them that is not computationally equivalent to any decision problem. (Naturally, absolute separations would imply that P 6 = NP.) Our separation proofs have interesting combinatorial content and go to the heart of the combinatorial principles on which the classes are based. We derive one result via new lower bounds on the degrees of polynomials asserted to exist by Hilbert's Nullstellensatz over nite elds.

Computing equilibria: a computational complexity perspective

“…An interesting and important research question is whether or not P P AD ⊆ P . There is reason to believe that P P AD ⊆ P , or at least that a proof of P P AD ⊆ P is nowhere in sight: as with P LS ⊆ P (Section 3.5), a proof that P P AD ⊆ P would have to be "non-relativizing" [8,43] and would yield a generic method of "shortcutting" the (exponentialtime) guided search procedure in every P P AD problem. A concrete negative result was given by Hirsch, Papadimitriou, and Vavasis [61] for computing approximate Brouwer fixed points, which applies even to the special case of the Cubical Brouwer problem (Section 4.4): every algorithm that computes an approximate fixed point and merely queries the supplied Brouwer function as a "black box", as opposed to examining the details of its description, requires exponential time in the worst case.…”

“…This motivates the class N P , which we define next. 8 Informally, a problem is in N P if a purported solution to an instance can be verified for correctness in polynomial time. For example, in an instance of the Maximum Clique problem with graph G and target k, a purported solution is a subset K of k vertices, and it is easy to check whether or not K is a k-clique of G in time polynomial in the input size.…”

“…Remarkably, there are also concrete and important problems unsolvable by any algorithm -undecidable is the technical term -such as the "Halting Problem", which asks whether or not a given program halts on a given input [130]. 8 The class N P is traditionally defined in terms of decision problems; we work with the search problem analog, which is sometimes called F N P , or "functional N P ". The letters N P stand for "nondeterministic polynomial", reflecting the class's origin story via nondeterministic Turing machines.…”

Settling the complexity of computing approximate two-player Nash equilibria

“…the size of the circuits S and P . 3 Given our current understanding of complexity, refuting ETH for PPAD seems unlikely: there are matching black-box lower bounds [45,19]. Recall that the NP-analogue ETH [47] is widely used (e.g.…”

A note on propositional proof complexity of some Ramsey-type statements

“…Finally, such a proof complexity separation can be usually turned into a non-reducibility result for corresponding N P search problems, cf. [4,7,8,14].…”