On the Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems
In this paper we separate many-one reducibility from truth- table reducibility for distributional problems in Dist NP under the hy- pothesis 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 Dist NP , this relation is proper unless P = NP. (2) C contains Dist P , but it is not contained in Ave P unless Dist NP ≠ Ave ZPP. (3) C has a ≤pm-complete set. (4) C has a ≤ptt-complete set that is not ≤pm-complete unless P = NP. This shows that under the assumption that P ≠ NP , the two complete- ness notions differ on some non-trivial subclass of Dist NP.
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