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A Compact Representation for Minimizers of k-Submodular Functions (Extended Abstract)

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Combinatorial Optimization (ISCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9849))

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Abstract

k-submodular functions were introduced by Huber and Kolmogorov as a generalization of bisubmodular functions. This paper establishes a compact representation of minimizers of k-submodular functions by posets with inconsistent pairs (PIPs), and completely characterizes the class of PIPs (elementary PIPs) corresponding to minimizers of k-submodular functions. Our representation coincides with Birkhoff’s representation theorem if \(k=1\) and with signed-poset representation by Ando and Fujishige if \(k=2\). We also give algorithms to construct the elementary PIP representing the minimizers of a k-submodular function f for three cases: (i) a minimizing oracle of f is available, (ii) f is network-representable, and (iii) f is the objective function of the relaxation of multiway cut problem. Furthermore, we provide an efficient algorithm to enumerate all maximal minimizers from the PIP representation. Our results are applied to obtain all maximal persistent assignments in labeling problems arising from computer vision.

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Acknowledgement

This work was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 26280004, and by the JST, ERATO, Kawarabayashi Large Graph Project.

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Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, 113-8656, Japan

    Hiroshi Hirai & Taihei Oki

Authors
  1. Hiroshi Hirai
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  2. Taihei Oki
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Correspondence to Taihei Oki .

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Editors and Affiliations

  1. University of Salerno , Fisciano, Italy

    Raffaele Cerulli

  2. Kyoto University , Kyoto, Japan

    Satoru Fujishige

  3. LAMSADE, Université Paris-Dauphine , Paris , France

    A. Ridha Mahjoub

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Hirai, H., Oki, T. (2016). A Compact Representation for Minimizers of k-Submodular Functions (Extended Abstract). In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_33

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  • DOI: https://doi.org/10.1007/978-3-319-45587-7_33

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45586-0

  • Online ISBN: 978-3-319-45587-7

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