The complexity of counting locally maximal satisfying assignments of Boolean CSPs

TL;DR

This paper studies the computational complexity of counting maximal satisfying assignments in Boolean CSPs, providing a complete classification for exact and approximate counting, and contrasting it with related problems in graph theory.

Contribution

It establishes a complexity dichotomy and trichotomy for counting maximal solutions in Boolean CSPs, revealing cases where the problem is easier or as hard as counting all solutions.

Findings

Exact counting has a clear complexity classification.

Approximate counting exhibits a three-way complexity classification.

Maximal counting can be easier than counting all solutions, but never harder.

Abstract

We investigate the computational complexity of the problem of counting the maximal satisfying assignments of a Constraint Satisfaction Problem (CSP) over the Boolean domain {0,1}. A satisfying assignment is maximal if any new assignment which is obtained from it by changing a 0 to a 1 is unsatisfying. For each constraint language Gamma, #MaximalCSP(Gamma) denotes the problem of counting the maximal satisfying assignments, given an input CSP with constraints in Gamma. We give a complexity dichotomy for the problem of exactly counting the maximal satisfying assignments and a complexity trichotomy for the problem of approximately counting them. Relative to the problem #CSP(Gamma), which is the problem of counting all satisfying assignments, the maximal version can sometimes be easier but never harder. This finding contrasts with the recent discovery that approximately counting maximal independent sets in a bipartite graph is harder (under the usual complexity-theoretic assumptions) than counting all independent sets.