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 with a version of the standard distribution is truth-table reducible but not many-one reducible to 3SAT with 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 \subseteq AveZPP.

(3) C has a ≤ ^p _m -complete set.

(4) C has a ≤ ^p _tt -complete set that is not ≤ ^p _m -complete unless P = NP.

This shows that under the assumption that P ≠ NP, the two completeness notions differ on some nontrivial subclass of DistNP.