Supermodular functions and the complexity of MAX CSP

  • Authors:

  • David Cohen;Martin Cooper;Peter Jeavons;Andrei Krokhin

  • Affiliations:

  • Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK;IRIT, University of Toulouse III, Toulouse, France;Computing Laboratory, University of Oxford, Oxford, UK;Department of Computer Science, University of Durham, Durham, UK

  • Venue:
  • Discrete Applied Mathematics - Special issue: Boolean and pseudo-boolean funtions
  • Year:
  • 2005

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Abstract

In this paper we study the complexity of the maximum constraint satisfaction problem (MAX CSP) over an arbitrary finite domain. An instance of MAX CSP consists of a set of variables and a collection of constraints which are applied to certain specified subsets of these variables; the goal is to find values for the variables which maximize the number of simultaneously satisfied constraints. Using the theory of sub-and supermodular functions on finite lattice-ordered sets, we obtain the first examples of general families of efficiently solvable cases of MAX CSP for arbitrary finite domains. In addition, we provide the first dichotomy result for a special class of non-Boolean MAX CSP, by considering binary constraints given by supermodular functions on a totally ordered set. Finally, we show that the equality constraint over a non-Boolean domain is non-supermodular, and, when combined with some simple unary constraints, gives rise to cases of MAX CSP which are hard even to approximate.