Abstract
In the maximum constraint satisfaction problem ($\mathrm{Max \; CSP}$), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximise the number (or the total weight) of satisfied constraints. This problem is $\mathrm{NP}$-hard in general so it is natural to study how restricting the allowed types of constraints affects the complexity of the problem. In this paper, we show that any $\mathrm{Max \; CSP}$ problem with a finite set of allowed constraint types, which includes all constants (i.e. constraints of the form $x=a$), is either solvable in polynomial time or is $\mathrm{NP}$-complete. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known combinatorial property of supermodularity.
Highlights
Introduction and Related WorkThe constraint satisfaction problem (CSP) is a general framework in which a variety of combinatorial problems, including propositional satisfiability and graph colourability, can be expressed in a natural way [5, 8]
We show that any maximum constraint satisfaction problem (Max CSP) problem with a finite set of allowed constraint types, which includes all constants, is either solvable in polynomial time or is NP-complete
We study the maximum constraint satisfaction problem (MAX CSP) which is a natural optimization version of CSP
Summary
The constraint satisfaction problem (CSP) is a general framework in which a variety of combinatorial problems, including propositional satisfiability and graph colourability, can be expressed in a natural way [5, 8]. This form subsumes the previous two forms, but it is not so widely used in combinatorics This form of sub- and supermodularity is popular in mathematical economics where it is used to model games in which an optimal strategy can be found efficiently (e.g., supermodular games) [11]. This general form of supermodularity has been recently shown to be highly relevant in the study of MAX CSP where it captures all currently known tractable cases [4, 9]. The only form of supermodularity applicable in this case is supermodularity on chains, and our main result states that it precisely characterizes tractable MAX CSP problems (under the above assumption). We conclude the paper with a discussion of the list H-colouring optimization problem for digraphs
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