Maximal Flow through a Network

Introduction. The problem discussed in this paper was formulated by T. Harris as follows: "Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other." While this can be set up as a linear programming problem with as many equations as there are cities in the network, and hence can be solved by the simplex method (1), it turns out that in the cases of most practical interest, where the network is planar in a certain restricted sense, a much simpler and more efficient hand computing procedure can be described. In §1 we prove the minimal cut theorem, which establishes that an obvious upper bound for flows over an arbitrary network can always be achieved. The proof is non-constructive. However, by specializing the network (§2), we obtain as a consequence of the minimal cut theorem an effective computational scheme. Finally, we observe in §3 the duality between the capacity problem and that of finding the shortest path, via a network, between two given points.