Genuinely Polynominal Simplex and Non-Simplex Algorithms for the Minimum Cost Flow Problem

We consider the minimum cost network flow problem min(cx : Ax=b , x > 0) on a graph G = (V,E). First we give a minor modification of Edmonds-Karp scaling technique, and we show that it solves the minimum cost flow problem in 0((|V|'^ log 1v|)(|e| + |V| log |v|)) steps. We also provide two dual simplex algorithms that solve the minimum cost flow problem in 0(|v| log |v|) pivots and 0(|v| log |v|) pivots respectively. Moreover, this latter dual simplex algorithm may be implemented so that the running time is proportional to that of Edmonds-Karp scaling technique.