A Primal Simplex Algorithm that Solves the Maximum Flow Problem in at most nm Pivots and O(nm) Time

Donald Goldfarb, Jianxiu Hao

Abstract: We propose a primal network simplex algorithm for solving the maximum flow problem which chooses as the arc to enter the basis one that isclosest to the source node from amongst all possible candidates. We prove that this algorithm requires at mostnm pivots to solve a problem withn nodes andm arcs, and give implementations of it which run in O(n 2 m) time. Our algorithm is, as far as we know, the first strongly polynomial primal simplex algorithm for solving the maximum flow problem.