Numbers of Primal and Dual Bases of Network Flow and Unimodular Integer Programs

To integer programming, a ideals have been problems, which Hiroki Nakayama The University of Tokyo (Received October 27, 2003) Hiroshi Imai jases and standard pairs via toric ilgebraic approaches using Grobner k studied in recent years. In this paper, we consider a unimodular case, e.g., network flow enables us to analyze primal and dual problems in an equal setting. By combining existing results in an algebraic approach, we prove a theorem that the maximum number of dual feasible bases is obtained by computing the normalized volume of the convex hull generated from column vectors of a coefficient matrix in the primal standard form. We apply the theorem, partly with Grobner bases theory, to transportation problems and minimum cost flow problems on acyclic tournament graphs. In consequence, we show new algebraic proofs to the Balinski and Russakoff's result on the dual transportation polytope and Klee and Witzgall's result on the primal transportation polytope. Similarly results for the primal case of acyclic tournament graphs are obtained by using Gelfand, Graev and Postnikov's result for nilpotent hypergeometric functions. We also give a bound of the number of feasible bases for its dual case.