The Complexity of Approximating Bounded-Degree Boolean \sharp CSP

TL;DR

This paper classifies the complexity of approximately counting solutions for bounded-degree Boolean CSPs with specific relations, revealing polynomial-time solvability, equivalence to counting independent sets, or intractability depending on the constraint language and degree.

Contribution

It provides a complete complexity classification for Boolean CSPs with bounded degree and specific relations, extending understanding of approximate counting complexity.

Findings

Polynomial-time solvable when all relations are affine.

Equivalent to counting independent sets in bipartite graphs for certain relations.

No FPRAS exists unless NP=RP for other cases.

Abstract

The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.