Maximum Flows and Minimum Cuts

In the mid-1950s, U. S. Air Force researcher Theodore E. Harris and retired U. S. Army general Frank S. Ross published a classified report studying the rail network that linked the Soviet Union to its satellite countries in Eastern Europe. The network was modeled as a graph with 44 vertices, representing geographic regions, and 105 edges, representing links between those regions in the rail network. Each edge was given a weight, representing the rate at which material could be shipped from one region to the next. Essentially by trial and error, they determined both the maximum amount of stuff that could be moved from Russia into Europe, as well as the cheapest way to disrupt the network by removing links (or in less abstract terms, blowing up train tracks), which they called “the bottleneck”. Their results, including the drawing of the network in Figure 10.1, were only declassified in 1999.1 This one of the first recorded applications of the maximum flow and minimum cut problems. For both problems, the input is a directed graph G = (V, E), along with special vertices s and t called the source and target. As in previous chapters, I will write u v to denote the directed edge from vertex u to vertex v. Intuitively, the maximum flow problem asks for the largest amount of material that can be transported from s to t; the minimum cut problem asks for the minimum damage needed to separate s from t.