On strongly polynomial variants of the network simplex algorithm for the maximum flow problem

We give a short proof that the network simplex algorithm with either the smaller label or the smallest label pivot rules proposed by D. Goldfarb and J. Hao, solves a maximum flow problem on an n-node, m-arc network in at most nm pivots and O(n^2m) time. We also show that a straightforward adaptation of a shortest augmenting path algorithm is not polynomial.