Minimizing a submodular function arising from a concave function Original Research Article

We consider a class of submodular functions on distributive lattices that are defined in terms of concave functions and modular functions. The minimization of such a submodular function is made in time required for a max-flow computation on an associated network by means of the parametric max-flow algorithm of Gallo, Grigoriadis and Tarjan. The problem is closely related to the classic majorization and the majorization on posets. As an application of the parametric approach, we improve the time complexity of a capacity scaling algorithm for the submodular flow problem. We also discuss a generalization and its relation to the principal partition or the lexicographically optimal base of a submodular system.