Realization of set functions as cut functions of graphs and hypergraphs

Abstract

We consider the problems of realizing set functions as cut functions on graphs and hypergraphs. We give necessary and sufficient conditions for set functions to be realized as cut functions on nonnegative networks, symmetric networks and nonnegative hypernetworks. The symmetry significantly simplifies the characterization of set functions of nonnegative network type. Set functions of nonnegative hypernetwork type generalize those of nonnegative network type (cut functions of ordinary networks) and are submodular functions. For any constant integer k⩾2, we can discern in polynomial time by a max-flow algorithm whether a given set function of order k is of nonnegative hypernetwork type.