Advantageous Properties of Dual Transhipment Polyhedra

The dual linear programs for the transhipment problem over a directed graph, G N, E }, are shown to have polyhedra with properties that make them well suited to vertex visiting solution techniques, like the simplex method. In particular, nondegenerate cases are shown to have feasible regions with considerably fewer extreme points than the feasible sets for primal problems. The adjacency structure of feasible bases is also shown to be quite favorable. In fact, the Hirsch Conjecture is valid when the network is complete. A dual-based simplex method for cases of the transhipment problem, which include the shortest paths problem, is presented that finds an optimal solution in no more than Min (I El INI + 1, INI IN11/2) pivots. Key words, linear programming, network flows, hirsch conjecture AMS(MOS) subject classification. 09C08