Sets Computable in Polynomial Time on Average

In this paper, we discuss the complexity and properties of the sets which are computable in polynomial-time on average. This study is motivated by Levin's question of whether all sets in NP are solvable in polynomial-time on average for every reasonable (i.e., polynomial-time computable) distribution on the instances. Let PP-comp denote the class of all those sets which are computable in polynomial-time on average for every polynomial-time computable distribution on the instances. It is known that P (PP-comp (E. In this paper, we show that PP-comp is not contained in DTIME(2 cn) for any constant c and that it lacks some basic structural properties: for example, it is not closed under many-one reducibility or for the existential operator. From these results, it follows that PP-comp contains P-immune sets but no P-bi-immune sets; it is not included in P=cn for any constant c; and it is diierent from most of the well-known complexity classes, such as UP, NP, BPP, and PP. Finally, we show that, relative to a random oracle, NP is not included in PP-comp and PP-comp is not in PSPACE with probability 1. x1 Introduction. The theory of average-case complexity which was introduced by Levin 11] allows us to study the polynomial-time computability of NP problems in a more general setting. Several famous NP-complete problems have turned out to be easy on average with respect to some particular but natural distribution on the inputs 7]. Our main interest here is raised by the open question of whether some, or even all, NP-complete problems can be solved in polynomial-time on average with respect to every reasonably realistic probability distribution. Levin suggested that polynomial-time computable (P-computable, for short) distributions are reasonable (see 7]); he shows that a randomized version of the bounded tiling problem is complete for NP decision problems with P-computable distributions 11]. Later, a more general notion of polynomially samplable distributions has been proposed and studied in 4, 8]. In this paper, however, we restrict our attention to P-computable distributions. A rst partial answer to the above question { whether NP can be solved de-terministically in polynomial-time on average { was given by Ben-David, et al. 4]. They showed that this situation would be unlikely since it would imply E = NE (its negation implies P 6 = NP). In this paper, we take a closer look at P P-comp , the class of problems solvable in polynomial-time on …