The complexity of approximating conservative counting CSPs

TL;DR

This paper classifies the complexity of approximately solving weighted counting CSPs with arbitrary finite domains, identifying conditions under which the problem is tractable, #BIS-hard, or as hard as #SAT.

Contribution

It extends the classification of conservative weighted counting CSPs from Boolean domains to arbitrary finite domains, introducing new notions of weak log-modularity and supermodularity.

Findings

#CSP(F) is in FP if F is weakly log-modular.

If not, #CSP(F) is at least as hard as #BIS.

Full trichotomy for arity-2 cases: FP, #BIS-equivalent, or #SAT-hard.

Abstract

We study the complexity of approximately solving the weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, there is a classification known for the case in which the domain of functions in F is Boolean. In this paper, we give a classification for the more general problem where functions in F have an arbitrary finite domain. We define the notions of weak log-modularity and weak log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem of counting independent sets in bipartite graphs. #BIS is complete with respect to approximation-preserving reductions for a logically-defined complexity class #RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in difficulty to #SAT, the problem of approximately counting the satisfying assignments of a Boolean formula in conjunctive normal form. We also discuss the algorithmic aspects of our classification.