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DOI: 10.4230/LIPIcs.CSL.2015.60
A Definability Dichotomy for Finite Valued CSPs

Authors: Anuj Dawar and Pengming Wang

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Abstract
Finite valued constraint satisfaction problems are a formalism for describing many natural optimisation problems, where constraints on the values that variables can take come with rational weights and the aim is to find an assignment of minimal cost. Thapper and Zivny have recently established a complexity dichotomy for valued constraint languages. They show that each such languages either gives rise to a polynomial-time solvable optimisation problem, or to an NP-hard one, and establish a criterion to distinguish the two cases. We refine the dichotomy by showing that all optimisation problems in the first class are definable in fixed-point language with counting, while all languages in the second class are not definable, even in infinitary logic with counting. Our definability dichotomy is not conditional on any complexity-theoretic assumption.
Cite as

Anuj Dawar and Pengming Wang. A Definability Dichotomy for Finite Valued CSPs. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 60-77, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dawar_et_al:LIPIcs.CSL.2015.60,
  author =	{Dawar, Anuj and Wang, Pengming},
  title =	{{A Definability Dichotomy for Finite Valued CSPs}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{60--77},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.60},
  URN =		{urn:nbn:de:0030-drops-54078},
  doi =		{10.4230/LIPIcs.CSL.2015.60},
  annote =	{Keywords: descriptive complexity, constraint satisfaction, definability, fixed-point logic, optimization}
}
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