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URN: urn:nbn:de:0030-drops-106531
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Testing the Complexity of a Valued CSP Language

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Abstract

A Valued Constraint Satisfaction Problem (VCSP) provides a common framework that can express a wide range of discrete optimization problems. A VCSP instance is given by a finite set of variables, a finite domain of labels, and an objective function to be minimized. This function is represented as a sum of terms where each term depends on a subset of the variables. To obtain different classes of optimization problems, one can restrict all terms to come from a fixed set Gamma of cost functions, called a language. Recent breakthrough results have established a complete complexity classification of such classes with respect to language Gamma: if all cost functions in Gamma satisfy a certain algebraic condition then all Gamma-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language Gamma is known to be NP-hard. We thus study exponential algorithms for this meta-problem. We show that the tractability condition of a finite-valued language Gamma can be tested in O(sqrt[3]{3}^{|D|}* poly(size(Gamma))) time, where D is the domain of Gamma and poly(*) is some fixed polynomial. We also obtain a matching lower bound under the Strong Exponential Time Hypothesis (SETH). More precisely, we prove that for any constant delta<1 there is no O(sqrt[3]{3}^{delta|D|}) algorithm, assuming that SETH holds.

BibTeX - Entry

@InProceedings{kolmogorov:LIPIcs:2019:10653,
  author =	{Vladimir Kolmogorov},
  title =	{{Testing the Complexity of a Valued CSP Language}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{77:1--77:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10653},
  URN =		{urn:nbn:de:0030-drops-106531},
  doi =		{10.4230/LIPIcs.ICALP.2019.77},
  annote =	{Keywords: Valued Constraint Satisfaction Problems, Exponential time algorithms, Exponential Time Hypothesis}
}

Keywords: Valued Constraint Satisfaction Problems, Exponential time algorithms, Exponential Time Hypothesis
Seminar: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue date: 2019
Date of publication: 2019


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