Abstract
We report new results on the complexity of the valued constraint satisfaction problem (VCSP). Under the unique games conjecture, the approximability of finite-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 classifications were previously known for two- and 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 Živný (originally introduced by Takhanov) for efficiently 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 simplified by the introduction of powerful reduction techniques.
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Jonsson, P., Kuivinen, F., Thapper, J. (2011). Min CSP on Four Elements: Moving beyond Submodularity. In: Lee, J. (eds) Principles and Practice of Constraint Programming – CP 2011. CP 2011. Lecture Notes in Computer Science, vol 6876. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23786-7_34
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DOI: https://doi.org/10.1007/978-3-642-23786-7_34
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