Subtractive Reductions and Complete Problems for Counting Complexity Classes
We introduce and investigate a new type of reductions between counting problems, which we call subtractive reductions. We show that the main counting complexity classes #P, #NP, as well as all higher counting complexity classes #·ΠP k, k≥2, are closed under subtractive reductions. We then pursue problems that are complete for these classes via subtractive reductions. We focus on the class #NP (which is the same as the class #·coNP) and show that it contains natural complete problems via subtractive reductions, such as the problem of counting the minimal models of a Boolean formula in conjunctive normal form and the problem of counting the cardinality of the set of minimal solutions of a homogeneous system of linear Diophantine inequalities.
KeywordsConjunctive Normal Form Truth Assignment Boolean Formula Complete Problem Counting Problem
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- [HJK99]M. Hermann, L. Juban, and P. G. Kolaitis. On the complexity of counting the Hilbert basis of a linear Diophantine system. In Proc. 6th LPAR, volume 1705 of LNCS (in AI), pages 13–32, September 1999. Springer.Google Scholar
- [HO92]L. A. Hemachandra and M. Ogiwara. Is #P closed under subtraction? Bulletin of the EATCS, 46:107–122, February 1992.Google Scholar
- [KST89]J. Köbler, U. Schöning, and J. Torán. On counting and approximation. Acta Informatica, 26(4):363–379, 1989.Google Scholar
- [Sch86]A. Schrijver. Theory of linear and integer programming. John Wiley & Sons, 1986.Google Scholar
- [Tod91]S. Toda. Computational complexity of counting complexity classes. PhD thesis, Tokyo Institute of Technology, Dept. of Computer Science, Tokyo, 1991.Google Scholar