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DOI: 10.4230/LIPIcs.ISAAC.2016.36

A Gap Trichotomy for Boolean Constraint Problems: Extending Schaefer's Theorem

Authors: Lucy Ham

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract

In this paper, we investigate "gap problems", which are promise problems where YES instances are flexibly satisfiable in a certain sense, and NO instances are not satisfiable at all. These gap problems generalise a family of constraint-related decision problems, including the constraint satisfaction problem itself, the separation problem (can distinct variables be validly assigned distinct values?) and the 2-robust satisfiability problem (does any assignment on two variables extend to a full satisfying assignment?). We establish a Gap Trichotomy Theorem, which on Boolean domains, completely classifies the complexity of the gap problems considered. As a consequence, we obtain several well-known dichotomy results, as well as dichotomies for the separation problem and the 2-robust satisfiability problem: all are either polynomial-time tractable or NP-complete. Schaefer’s original dichotomy is a notable particular case.

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Lucy Ham. A Gap Trichotomy for Boolean Constraint Problems: Extending Schaefer's Theorem. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ham:LIPIcs.ISAAC.2016.36,
  author =	{Ham, Lucy},
  title =	{{A Gap Trichotomy for Boolean Constraint Problems: Extending Schaefer's Theorem}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{36:1--36:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.36},
  URN =		{urn:nbn:de:0030-drops-68060},
  doi =		{10.4230/LIPIcs.ISAAC.2016.36},
  annote =	{Keywords: Constraint Satisfaction Problem, Robust satisfiability, Clone theory, Dichotomy, Trichotomy, Boolean}
}

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A Gap Trichotomy for Boolean Constraint Problems: Extending Schaefer's Theorem

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