Approximating partition functions of bounded-degree Boolean counting Constraint Satisfaction Problems

TL;DR

This paper classifies the complexity of approximately counting solutions to bounded-degree Boolean CSPs, showing tractability, equivalence to #BIS, or NP-hardness depending on the constraint language and degree bound.

Contribution

It extends the complexity classification of bounded-degree Boolean CSP counting problems beyond the conservative case, identifying conditions for tractability, approximation equivalence, or hardness.

Findings

Affine functions lead to polynomial-time counting.

IM2 class functions are AP-equivalent to #BIS for large degree bounds.

Otherwise, the problem is NP-hard to approximate.

Abstract

We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language $\Gamma$ and a degree bound $\Delta$, we study the complexity of #CSP$_\Delta(\Gamma)$, which is the problem of counting satisfying assignments to CSP instances with constraints from $\Gamma$ and whose variables can appear at most $\Delta$ times. Our main result shows that: (i) if every function in $\Gamma$ is affine, then #CSP$_\Delta(\Gamma)$ is in FP for all $\Delta$, (ii) otherwise, if every function in $\Gamma$ is in a class called IM$_2$, then for all sufficiently large $\Delta$, #CSP$_\Delta(\Gamma)$ is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large $\Delta$, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP$_\Delta(\Gamma)$, even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.