An approximation trichotomy for Boolean #CSP

TL;DR

This paper establishes a three-way classification of the complexity for approximately counting solutions in Boolean CSPs, identifying cases with polynomial-time exact counting, cases with approximation complexity equivalent to known hard problems, and cases that are computationally intractable.

Contribution

It provides a comprehensive trichotomy theorem that characterizes the approximate counting complexity of Boolean CSPs based on the constraint language used.

Findings

Exact counting is polynomial-time for affine relations.

Approximate counting is as hard as #RH for relations in co-clone IM_2.

Other relations lead to #P-complete approximate counting problems.

Abstract

We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the co-clone IM_2 from Post's lattice, then the problem of counting satisfying assignments is complete with respect to approximation-preserving reductions in the complexity class #RH\Pi_1. This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately counting the number of independent sets in a bipartite graph. For every other fixed constraint language, the problem is complete for #P with respect to approximation-preserving reductions, meaning that there is no fully polynomial randomised approximation scheme for counting satisfying assignments unless NP=RP.