2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 5156
author:  Vladimir Deineko, Peter Jonsson, Mikael Klasson and Andrei Krokhin 

title:  Supermodularity on chains and complexity of maximum constraint satisfaction 
keywords: 
maximum constraint satisfaction, complexity,
supermodularity, Monge properties, digraph
H
colouring

abstract: 
In the maximum constraint satisfaction problem (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 NPhard
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 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
NPcomplete. Moreover, we present a simple description of
all polynomialtime solvable cases of our problem. This
description uses the wellknown combinatorial property of
supermodularity.

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reference:  Vladimir Deineko and Peter Jonsson and Mikael Klasson and Andrei Krokhin (2005), Supermodularity on chains and complexity of maximum constraint satisfaction, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 5156 
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