Towards Minimizing k-Submodular Functions

TL;DR

This paper extends the Min-Max-Theorem to k-submodular functions, a broad class including submodular and bisubmodular functions, by defining a new polyhedron and establishing a minimization-maximization relationship.

Contribution

It introduces a generalized Min-Max-Theorem for k-submodular functions and defines a corresponding k-submodular polyhedron, broadening the theoretical framework for discrete optimization.

Findings

Proves a Min-Max-Theorem for k-submodular functions

Defines a k-submodular polyhedron

Generalizes known results for submodular and bisubmodular functions

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 and investigate a k-submodular polyhedron and prove a Min-Max-Theorem for k-submodular functions.