Supermodular functions and the complexity of M AX CSP

Abstract

In this paper we study the complexity of the maximum constraint satisfaction problem (M AX CSP) over an arbitrary finite domain. An instance of M AX 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 M AX CSP for arbitrary finite domains. In addition, we provide the first dichotomy result for a special class of non-Boolean M AX 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 M AX CSP which are hard even to approximate.