The diameters of network-flow polytopes satisfy the Hirsch conjecture

We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a network-flow polytope for a network with n nodes and m arcs is never more than m+n−1. A key step to prove this is to show the same result for classical transportation polytopes.