Min CSP on four elements: moving beyond submodularity

  • Authors:

  • Peter Jonsson;Fredrik Kuivinen;Johan Thapper

  • Affiliations:

  • Department of Computer and Information Science, Linköpings universitet, Linköping, Sweden;Department of Computer and Information Science, Linköpings universitet, Linköping, Sweden;école polytechnique, Laboratoire d'informatique, Palaiseau Cedex, France

  • Venue:
  • CP'11 Proceedings of the 17th international conference on Principles and practice of constraint programming
  • Year:
  • 2011

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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 Živny (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.