Submodular Flow Problem with a Nonseparable Cost Function

The submodular flow problem is extended by considering a nonseparable cost function, which is assumed to enjoy a variant of the exchange property of the base polyhedron of a submodular system. Two optimality criteria are established, one in terms of potentials associated with vertices and the other in terms of negative cycles in an auxiliary graph. These are natural extensions of the well-known result for the conventional min-cost flow problem as well as the recent result of Fujishige for the submodular flow problem with a separable convex cost function. (