eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-08-22
27:1
27:16
10.4230/LIPIcs.MFCS.2022.27
article
Bounded Degree Nonnegative Counting CSP
Cai, Jin-Yi
1
Szabo, Daniel P.
1
Department of Computer Sciences, University of Wisconsin-Madison, WI, USA
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 ℱ, every problem in the complexity class #CSP(ℱ) defined by 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 ℱ. The dichotomy criterion on ℱ is decidable. As a second contribution, we prove a slightly modified but more streamlined decision procedure (from [Jin-Yi Cai et al., 2011]) for tractability. This 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 [Martin E. Dyer et al., 2006] cannot be applied.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol241-mfcs2022/LIPIcs.MFCS.2022.27/LIPIcs.MFCS.2022.27.pdf
Computational Counting Complexity
Constraint Satisfaction Problems
Counting CSPs
Complexity Dichotomy
Nonnegative Counting CSP
Graph Homomorphisms