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.

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