A Min-Max Theorem for k-submodular Functions and Extreme Points of the Associated Polyhedra

A. Huber and V. Kolmogorov (ISCO 2012) introduced a concept of k-submodular function as a generalization of ordinary submodular (set) functions and bisubmodular functions. They presented a min-max relation for the k-submodular function minimization by considering l1 norm, which requires a non-convex set of feasible solutions associated with the k-submodular function. Our approach overcomes the trouble incurred by the non-convexity by means of a new norm composed of l1 and l∞ norms. We show another min-max relation that characterizes the minimum of a k-submodular function in terms of the maximum of the negative of the norm values over the associated convex set of feasible solutions. The min-max relation given in the present paper is simpler than that of Huber and Kolmogorov. We also show a counterexample to a characterization, given by Huber and Kolmogorov, of extreme points of the k-submodular polyhedron in their sense and make it a correct one by fixing a flaw therein.