Bounded Degree Nonnegative Counting CSP

Abstract

Constraint satisfaction problems (CSP) encompass an enormous variety of computational problems. In particular, all partition functions from statistical physics, such as spin systems, are special cases of counting CSP (#CSP). We prove a complete complexity classification for every counting problem in #CSP with nonnegative valued constraint functions that is valid when every variable occurs a bounded number of times in all constraints. We show that, depending on the set of constraint functions \(\mathcal {F} \) , every problem in the complexity class # \(\text{CSP}(\mathcal {F}) \) defined by \(\mathcal {F} \) is either polynomial-time computable for all instances without the bounded occurrence restriction, or is #P-hard even when restricted to bounded degree input instances. The constant bound in the degree depends on \(\mathcal {F} \) . The dichotomy criterion on \(\mathcal {F} \) is decidable. As a second contribution, we prove a slightly modified but more streamlined decision procedure (from [14]) to test for the tractability of # \(\text{CSP}(\mathcal {F}) \) . This procedure on an input \(\mathcal {F} \) tells us which case holds in the dichotomy for # \(\text{CSP}(\mathcal {F}) \) . This more streamlined decision procedure enables us to fully classify a family of directed weighted graph homomorphism problems. This family contains both P-time tractable problems and #P-hard problems. To our best knowledge, this is the first family of such problems explicitly classified that are not acyclic , thereby the Lovász-goodness criterion of Dyer-Goldberg-Paterson [24] cannot be applied.