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Classes of submodular constraints expressible by graph cuts

Constraints volume 15, pages 430–452 (2010)Cite this article

Abstract

Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial time. However, the general algorithm is of order O(n 6) and hence rather impractical. In this paper, by using results from the theory of pseudo-Boolean optimisation, we identify several broad classes of submodular constraints over a Boolean domain which are expressible using binary submodular constraints, and hence can be minimised in cubic time. Furthermore, we describe how our results translate to certain optimisation problems arising in computer vision.

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Authors and Affiliations

  1. Computing Laboratory, University of Oxford, Oxford, UK

    Stanislav Živný & Peter G. Jeavons

Authors
  1. Stanislav Živný
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  2. Peter G. Jeavons
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Correspondence to Stanislav Živný.

Additional information

A preliminary version of this paper appeared in Proceedings of the 14th International Conference on Principles and Practice of Constraint Programming (CP), 2008, pp. 112–127.

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Živný, S., Jeavons, P.G. Classes of submodular constraints expressible by graph cuts. Constraints 15, 430–452 (2010). https://doi.org/10.1007/s10601-009-9078-z

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  • DOI: https://doi.org/10.1007/s10601-009-9078-z

Keywords

  • Valued constraint satisfaction problems
  • Submodular constraints
  • Minimisation of submodular functions
  • Min-cut
  • Pseudo-Boolean optimisation