A generalization of max flow—min cut

Abstract

We present a theorem which generalizes the max flow--min cut theorem in various ways. In the first place, all versions of m.f.--m.c. (emphasizing nodes or arcs, with graphs directed or undirected, etc.) will be proved simultaneously. Secondly, instead of merely requiring that cuts block all paths, we shall require that our general cuts be weight assignments which meet paths in amounts satisfying certain lower bounds. As a consequence, our general theorem will not only include as a special case m.f.--m.c., but also includes the existence of integral solutions to a transportation problem (with upper bounds on row and column sums) and its dual.