License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.MFCS.2019.60
URN: urn:nbn:de:0030-drops-110041
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Bulatov, Andrei A. ; ZivnĂ˝, Stanislav

Approximate Counting CSP Seen from the Other Side

LIPIcs-MFCS-2019-60.pdf (0.5 MB)


In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C,-), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C,-) is solvable in polynomial time if C has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT != #W[1], there are no other cases of #CSP(C,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04].
We show that, assuming FPT != W[1] (under randomised parametrised reductions) and for C satisfying certain general conditions, #CSP(C,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C,-). In particular, our condition generalises the case when C is closed under taking minors.

BibTeX - Entry

  author =	{Andrei A. Bulatov and Stanislav Zivn{\'y}},
  title =	{{Approximate Counting CSP Seen from the Other Side}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{60:1--60:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-110041},
  doi =		{10.4230/LIPIcs.MFCS.2019.60},
  annote =	{Keywords: constraint satisfaction, approximate counting, homomorphisms}

Keywords: constraint satisfaction, approximate counting, homomorphisms
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019

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