# Subtractive Reductions and Complete Problems for Counting Complexity Classes

## Abstract

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 #·Π^{P} _{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.

## Keywords

Conjunctive Normal Form Truth Assignment Boolean Formula Complete Problem Counting Problem## Preview

Unable to display preview. Download preview PDF.

## References

- [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 - [HV95]L. A. Hemaspaandra and H. Vollmer. The satanic notations: Counting classes beyond #P and other definitional adventures.
*SIGACT News*, 26(1):2–13, 1995.CrossRefGoogle Scholar - [KST89]J. Köbler, U. Schöning, and J. Torán. On counting and approximation.
*Acta Informatica*, 26(4):363–379, 1989.Google Scholar - [McC80]J. McCarthy. Circumscription-A form of non-monotonic reasoning.
*Artficial Intelligence*, 13(1-2):27–39, 1980.zbMATHCrossRefMathSciNetGoogle Scholar - [OH93]M. Ogiwara and L.A. Hemachandra. A complexity theory for feasible closure properties.
*Journal of Computer and System Science*, 46(3):295–325, 1993.zbMATHCrossRefMathSciNetGoogle 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 - [TW92]S. Toda and O. Watanabe. Polynomial-time 1-Turing reductions from #PH to #P.
*Theoretical Computer Science*, 100(1):205–221, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - [Val79a]L. G. Valiant. The complexity of computing the permanent.
*Theoretical Computer Science*, 8(2):189–201, 1979.zbMATHCrossRefMathSciNetGoogle Scholar - [Val79b]L. G. Valiant. The complexity of enumeration and reliability problems.
*SIAM Journal on Computing*, 8(3):410–421, 1979.zbMATHCrossRefMathSciNetGoogle Scholar - [Wra76]C. Wrathall. Complete sets and the polynomial-time hierarchy.
*Theoretical Computer Science*, 3(1):23–33, 1976.CrossRefMathSciNetGoogle Scholar