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Approximability of Clausal Constraints

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Abstract

We study a family of problems, called Maximum Solution (Max Sol), where the objective is to maximise a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. When the domain is Boolean (i.e. restricted to {0,1}), the maximum solution problem is identical to the well-studied Max Ones problem, and the complexity and approximability is completely understood for all restrictions on the underlying constraints. We continue this line of research by considering the Max Sol problem for relations defined by regular signed logic over finite subsets of the natural numbers; the complexity of the corresponding decision problem has recently been classified by Creignou et al. (Theory Comput. Syst. 42(2):239–255, 2008). We give sufficient conditions for when such problems are polynomial-time solvable and we prove that they are APX-hard otherwise. Similar dichotomies are also obtained for variants of the Max Sol problem.

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Authors and Affiliations

  1. Department of Computer and Information Science, Linköpings Universitet, 581 83, Linköping, Sweden

    Peter Jonsson & Gustav Nordh

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  1. Peter Jonsson
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  2. Gustav Nordh
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Correspondence to Peter Jonsson.

Additional information

A preliminary version of this paper appears in Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science (MFCS’06), Lecture Notes in Computer Science, Vol. 4162, Springer, Berlin, 2006, pp. 549–560.

P. Jonsson partially supported by the Center for Industrial Information Technology (CENIIT) under grant 04.01, and by the Swedish Research Council (VR) under grant 621-2003-3421.

G. Nordh partially supported by the Swedish-French Foundation and the National Graduate School in Computer Science (CUGS), Sweden. Part of the research was carried out while working at Lix, Ecole Polytechnique, Paris.

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Jonsson, P., Nordh, G. Approximability of Clausal Constraints. Theory Comput Syst 46, 370–395 (2010). https://doi.org/10.1007/s00224-008-9145-7

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