The complexity of counting locally maximal satisfying assignments of Boolean CSPs

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.Comment: V2 adds contextual material relating the results obtained here to earlier work in a different but related setting. The technical content is unchanged. V3 (this version) incorporates minor revisions. The title has been changed to better reflect what is novel in this work. This version has been accepted for publication in Theoretical Computer Science. 19 page