Minimum Cost Flows on Hypergraphs

Directed hypergraphs have been introduced quite recently in connection with different application areas, such as propositional satisfiability, deductive data bases and Leontief substitution systems. In [Jeroslow, Martin, Rardin&Wang, 1989], the uncapacitated min-cost flow problem on hypergraphs is considered, and its properties are studied. Here we address the capacitated case, and show how the simplex operations can be specialized in order to exploit the structure of the problem. In particular, we define spanning hypertrees of a hypergraph so generalizing the spanning tree of a standard graph, and show that, like in the standard min-cost flow problem, a correspondence exists between bases and spanning hypertrees. We show also that, as happens in the network simplex algorithms for the standard min-cost flow problem, most of the computations performed at each pivot operation have a direct hypergraph interpretation.