Polynomial-time primal simplex algorithms for the minimum cost network flow problem

Abstract

We present two variants of the primal network simplex algorithm which solve the minimum cost network flow problem in at mostO(n 2 logn) pivots. Here we define the network simplex method as a method which proceeds from basis tree to adjacent basis tree regardless of the change in objective function value; i.e., the objective function is allowed to increase on some iterations. The first method is an extension of theminimum mean augmenting cycle-canceling method of Goldberg and Tarjan. The second method is a combination of a cost-scaling technique and a primal network simplex method for the maximum flow problem. We also show that the diameter of the primal network flow polytope is at mostn 2 m.