An out-of-kilter method for submodular flows

In the present paper we extend the out-of-kilter method for the ordinary minimum-cost flow problem to the submodular flow problem considered by Edmonds and Giles. From the algorithmic point of view, there is no essential difference between a submodular flow problem described by a submodular function on a crossing family and one described by a submodular function on a distributive lattice, if we are given any feasible flow. We clarify the aspects of the relevant polyhedron and of its representation as a system of linear inequalities in the submodular flow problem. The problem of finding a feasible flow is also discussed.