The Complexity of Approximating Bounded-Degree Boolean #CSP (Extended Abstract)

TL;DR

This paper classifies the complexity of approximately counting solutions for bounded-degree Boolean CSPs with specific constraints, revealing polynomial-time solvability or hardness depending on the constraint language and degree bound.

Contribution

It provides a complete complexity classification for Boolean CSPs with degree at least 25, identifying cases solvable in polynomial time and those with hardness results, extending understanding of approximate counting.

Findings

Polynomial-time solvability when all relations are affine.

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

Hardness results unless NP=RP for other cases.

Abstract

The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum degree is at least 25 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 in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs.