On the difference between polynomial-time many-one and truth-table reducibilities on distributional problems

Abstract

In this paper we separate many-one reducibility from truth-table reducibility for distributional problems in DistNP under the hypothesis that P ≠ NP. As a first example we consider the 3-Satisfiability problem (3SAT) with two different distributions on 3CNF formulas. We show that 3SAT using a version of the standard distribution is truth-table reducible but not many-one reducible to 3SAT using a less redundant distribution unless P = NP. We extend this separation result and define a distributional complexity class C with the following properties: (1) C is a subclass of DistNP, this relation is proper unless P = NP. (2) C contains DistP, but it is not contained in AveP unless DistNP Í AveZPP. (3) C has a £mp-complete set. (4) C has a £ttp-complete set that is not £mp-complete unless P = NP. This shows that under the assumption that P 6 ≠ NP, the two completeness notions differ on some non-trivial subclass of DistNP.