# Supermodularity on chains and complexity of maximum constraint satisfaction

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.
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https://hal.inria.fr/hal-01184377
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Vladimir Deineko, Peter Jonsson, Mikael Klasson, Andrei Krokhin. Supermodularity on chains and complexity of maximum constraint satisfaction. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.51-56. ⟨hal-01184377⟩

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