article

A dichotomy theorem for constraint satisfaction problems on a 3-element set

Author:

Andrei A. BulatovAuthors Info & Claims

Journal of the ACM (JACM), Volume 53, Issue 1

Pages 66 - 120

https://doi.org/10.1145/1120582.1120584

Published: 01 January 2006 Publication History

Abstract

The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming to separate those subclasses of the CSP that are tractable and those which remain NP-complete.Schaefer gave an exhaustive solution of this problem for the CSP on a 2-element domain. In this article, we generalise this result to a classification of the complexity of the CSP on a 3-element domain. The main result states that every subproblem of the CSP is either tractable or NP-complete, and the criterion separating them is that conjectured in Bulatov et al. [2005] and Bulatov and Jeavons [2001b]. We also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure. Finally, we exhibit a polynomial time algorithm which, for a given set of allowed constraints, outputs if this set gives rise to a tractable problem class. To obtain the main result and the algorithm, we extensively use the algebraic technique for the CSP developed in Jeavons [1998b], Bulatov et al.[2005], and Bulatov and Jeavons [2001b].

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Index Terms

A dichotomy theorem for constraint satisfaction problems on a 3-element set
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
2. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness
  2. Logic
    1. Constraint and logic programming

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cover image Journal of the ACM

Journal of the ACM Volume 53, Issue 1

January 2006

206 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/1120582

Issue’s Table of Contents

Copyright © 2006 ACM.

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Published: 01 January 2006

Published in JACM Volume 53, Issue 1

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