The facets of the polyhedral set determined by the GaleHoffman inequalities

The Gale-Hoffman lemma characterizes feasible external flow (supply, demand) in a directed, capacitated network in terms of a number of linear inequalities. These inequalities are generated by considering all node subsets and requiring that the external flow associated with each subset be less or equal to the total capacity of the arcs exiting :from it. For a particularly appealing statement of this lemma consult [11]. This paper is devoted to a refinement of this result. We are concerned with characterizing feasibility in terms of a minimal number of these inequalities. We are going to provide a simple criterion that identifies the linear inequalities that are redundant. One can also express this geometrically: the Gale-Hoffman inequalities determine a (convex) polyhedral set, and we are now interested in finding its facets. Thus, one can view this note as providing the description of the facets of one more polyhedral set encountered in combinatorial optimization, see e.g., [5], [10]; at a formal level, the criterion is similar in nature to the one obtained by Balas and Pulleyblank [1] for the perfectly matchable subgraph polytope of a bipartite graph. The framework of this paper is somewhat more general than that in the Gale [3], Hoffman [6] papers. We shall be interested in the polyhedral set (which turns out to be a polyhedral convex cone) of feasible external flows and capacities. This requires a minor reformulation of Gale-Hoffman result, but no new argumentation is necessary.