Generalized skew bisubmodularity: A characterization and a min-max theorem

Huber, Krokhin, and Powell (Proc. SODA2013) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain. In this paper we consider a natural generalization of the concept of skew bisubmodularity and show a connection between the generalized skew bisubmodularity and a convex extension over rectangles. We also analyze the dual polyhedra, called skew bisubmodular polyhedra, associated with generalized skew bisubmodular functions and derive a min-max theorem that characterizes the minimum value of a generalized skew bisubmodular function in terms of a minimum-norm point in the associated skew bisubmodular polyhedron.