Journal of Combinatorial Optimization

, Volume 36, Issue 3, pp 709–741 | Cite as

A compact representation for minimizers of k-submodular functions

  • Hiroshi Hirai
  • Taihei OkiEmail author


A k-submodular function is a generalization of submodular and bisubmodular functions. This paper establishes a compact representation for minimizers of a k-submodular function by a poset with inconsistent pairs (PIP). This is a generalization of Ando–Fujishige’s signed poset representation for minimizers of a bisubmodular function. We completely characterize the class of PIPs (elementary PIPs) arising from k-submodular functions. We give algorithms to construct the elementary PIP of 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 arises from a Potts energy function. Furthermore, we provide an efficient enumeration algorithm for all maximal minimizers of a Potts k-submodular function. Our results are applicable to obtain all maximal persistent labelings in actual computer vision problems. We present experimental results for real vision instances.


k-submodular function Birkhoff representation theorem Poset with inconsistent pairs (PIP) Potts energy function 



We thank Kazuo Murota, Satoru Fujishige, and the referees for helpful comments. This work was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 26280004, and by JST, ERATO, Kawarabayashi Large Graph Project.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyThe University of TokyoTokyoJapan

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