A compact representation for minimizers of k-submodular functions

Hiroshi Hirai () and Taihei Oki ()
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Hiroshi Hirai: The University of Tokyo
Taihei Oki: The University of Tokyo

Journal of Combinatorial Optimization, 2018, vol. 36, issue 3, No 3, 709-741

Abstract: Abstract 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.

Keywords: k-submodular function; Birkhoff representation theorem; Poset with inconsistent pairs (PIP); Potts energy function (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-017-0142-0

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