License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2022.27
URN: urn:nbn:de:0030-drops-168250
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16825/
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Cai, Jin-Yi ; Szabo, Daniel P.

Bounded Degree Nonnegative Counting CSP

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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 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.

BibTeX - Entry

@InProceedings{cai_et_al:LIPIcs.MFCS.2022.27,
  author =	{Cai, Jin-Yi and Szabo, Daniel P.},
  title =	{{Bounded Degree Nonnegative Counting CSP}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{27:1--27:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16825},
  URN =		{urn:nbn:de:0030-drops-168250},
  doi =		{10.4230/LIPIcs.MFCS.2022.27},
  annote =	{Keywords: Computational Counting Complexity, Constraint Satisfaction Problems, Counting CSPs, Complexity Dichotomy, Nonnegative Counting CSP, Graph Homomorphisms}
}

Keywords: Computational Counting Complexity, Constraint Satisfaction Problems, Counting CSPs, Complexity Dichotomy, Nonnegative Counting CSP, Graph Homomorphisms
Collection: 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)
Issue Date: 2022
Date of publication: 22.08.2022


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