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 ≤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 complete- ness notions differ on some non-trivial subclass of Dist NP.
Unable to display preview. Download preview PDF.
- [AT00]S. Aida and T. Tsukiji, On the difference among polynomial-time reducibilities for distributional problems (Japanese), in Proc. of the LA Symposium, Winter, RIMS publication, 2000.Google Scholar
- [BDG88]J. Balcázar, J. Díaz, and J. Gabarró, Structural Complexity I, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1988.Google Scholar
- [Coo71]S.A. Cook, The complexity of theorem proving procedures, in the Proc. of the third ACM Sympos. on Theory of Comput., ACM, 151–158, 1971.Google Scholar
- [Hom97]S. Homer, Structural properties of complete problems for exponential time, in Complexity Theory Retrospective 2 (A.L. Selman Ed.), Springer-Verlag, 135–154, 1997.Google Scholar
- [Imp95]R. Impagliazzo, A personal view of average-case complexity, in Proc. 10th Conference Structure in Complexity Theory, IEEE, 134–147, 1995.Google Scholar
- [Lev73]L.A. Levin, Universal sequential search problem, Problems of Information Transmission, 9:265–266, 1973.Google Scholar
- [Wang97]J. Wang, Average-case computational complexity theory, in Complexity Theory Retrospective 2 (A.L. Selman Ed.), Springer-Verlag, 295–328, 1997.Google Scholar