A Dichotomy Theorem for Constraints on a Three-Element Set

  • Authors:

  • Andrei A. Bulatov

  • Affiliations:

  • -

  • Venue:
  • FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
  • Year:
  • 2002

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Abstract

The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certainrestrictions on the 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 which are tractable, from those which remain NP-complete.In 1978 Schaefer gave an exhaustive solution of this problem for the CSP on a 2-element domain. In this paper we generalise this result to a classification of the complexity of CSPs on a 3-element domain. The main result states that every subclass of the CSP defined by a set of allowed constraints is either tractable or NP-complete, and the criterion separating them is that conjectured in [6, 8]. We also exhibit a polynomial time algorithmwhich, for a given set of allowed constraints, determines whether 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 [17] and [6, 8].