Completeness, approximability and exponential time results for counting problems with easy decision version

Abstract

Hard counting problems with easy decision version, like computing the permanent and counting independent sets, are of particular importance in counting complexity theory. TotP is the class of all self-reducible problems in #P with decision version in P. It has been a long standing open question to determine TotP-complete problems under strong reductions that preserve exactly the values of the functions (Karp), since Cook reductions blur structural differences between counting classes, specifically under Cook reductions TotP-complete problems are also #P-complete. In this paper we present the first such problems and we show some implications of these results to counting complexity.The problems that we prove to be TotP-complete under Karp reductions are:(a) #Tree-Monotone-Circuit-SAT, which further implies that it is hard to compute, both exactly and approximately, the number of satisfying assignments of monotone circuits, where the monotonicity is defined by a partial order which is given as part of the input.(b) Max-Lower-Set-Size, which implies that given a poset it is hard to compute and approximate the size of a principal upper set, and the size of the maximum lower set all elements of which share a given property P.(c) Size-of-Subtree, which directly implies that every problem in TotP admits a polynomial-time approximation algorithm the error of which depends on the amount of imbalance of the respective self-reducibility tree of the problem. We settle the worst case complexity of this problem, we provide a family of instances that are hard to approximate, and we show exponential time hardness results under variants of the exponential time hypothesis (ETH).(d) #Clustered-Monotone-SAT, which implies that approximating #SAT remains hard even on instances for which some kind of navigation among solutions is possible. Thus we narrow down any further research regarding the polynomial-time approximability of #SAT to instances with the aforementioned property.Among other implications, our results show that all these problems do not admit an FPRAS, unless NP=RP, something that we would not be able to conclude by showing them to be TotP-complete under Cook reductions.1