Min CSP on Four Elements: Moving beyond Submodularity

Abstract

We report new results on the complexity of the valued constraint satisfaction problem (VCSP). Under the unique games conjecture, the approximability of nite-valued VCSP is fairly well-understood. However, there is yet no characterisation of VCSPs that can be solved exactly in polynomial time. This is unsatisfactory, since such results are interesting from a combinatorial optimisation perspective; there are deep connections with, for instance, submodular and bisubmodular minimisation. We consider the Min and Max CSP problems (i.e. where the cost functions only attain values in {0, 1}) over four-element domains and identify all tractable fragments. Similar classi cations were previously known for twoand three-element domains. In the process, we introduce a new class of tractable VCSPs based on a generalisation of submodularity. We also extend and modify a graph-based technique by Kolmogorov and šivný (originally introduced by Takhanov) for e ciently obtaining hardness results in our setting. This allow us to prove the result without relying on computer-assisted case analyses (which is fairly common when studying VCSPs). The hardness results are further simpli ed by the introduction of powerful reduction techniques.

DOI: 10.1007/978-3-642-23786-7_34

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Cite this paper

@inproceedings{Jonsson2011MinCO, title={Min CSP on Four Elements: Moving beyond Submodularity}, author={Peter Jonsson and Fredrik Kuivinen and Johan Thapper}, booktitle={CP}, year={2011} }