Towards minimizing k-submodular functions

Authors:
Anna Huber;Vladimir Kolmogorov
Affiliations:
School of Engineering and Computing Sciences, Durham University, Durham, UK;Institute of Science and Technology Austria, Klosterneuburg, Austria
Venue:
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
Year:
2012

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Abstract

In this paper we investigate k-submodular functions. This natural family of discrete functions includes submodular and bisubmodular functions as the special cases k=1 and k=2 respectively. In particular we generalize the known Min-Max-Theorem for submodular and bisubmodular functions. This theorem asserts that the minimum of the (bi)submodular function can be found by solving a maximization problem over a (bi)submodular polyhedron. We define a k-submodular polyhedron, prove a Min-Max-Theorem for k-submodular functions, and give a greedy algorithm to construct the vertices of the polyhedron.