Nonnegative Weighted #CSP: An Effective Complexity Dichotomy

Abstract

We prove a complexity dichotomy theorem for counting constraint satisfaction problems (#CSPs) with nonnegative and algebraic weights. This caps a long series of important results on counting problems including counting unweighted and weighted graph homomorphisms and the celebrated dichotomy theorem for unweighted #CSPs. Our dichotomy theorem gives a succinct criterion for tractability. If a set $\mathcal{F}$ of constraint functions satisfies this criterion, then the problem #CSP$(\mathcal{F})$ defined by $\mathcal{F}$ is solvable in polynomial time; if $\mathcal{F}$ does not satisfy this criterion, then the problem is #P-hard. Furthermore, we show that the question of whether a given $\mathcal{F}$ satisfies the criterion or not is decidable in NP. Surprisingly, our tractability criterion is simpler than the previous criteria for the more restricted classes of counting problems, although when specialized to those classes, they are logically equivalent. Our proof mainly uses linear algebra and represents a departure from universal algebra, the dominant methodology in recent years for the study of #CSPs on large domains.