Towards Minimizing k-Submodular Functions

Anna Huber and Vladimir Kolmogorov.

In International Symposium on Combinatorial Optimization (ISCO), April 2012.


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.



Note: the second part of the conference paper contains a mistake (thanks to Satoru Fujishige and Shin-ichi Tanigawa for pointing it out). See arxiv version for details.