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Unformatted text preview: Algorithms Lecture 15: Maximum Flows and Minimum Cuts Col. Hogan: One of these wires disconnects the fuse, the other one fires the bomb. Which one would you cut, Shultz? Sgt. Schultz: Don’t ask me, this is a decision for an officer. Col. Hogan: All right. Which wire, Colonel Klink? Col. Klink: This one. [points to the white wire] Col. Hogan: You’re sure? Col. Klink: Yes. [Hogan cuts the black wire; the bomb stops ticking] Col. Klink: If you knew which wire it was, why did you ask me? Col. Hogan: I wasn’t sure which was the right one, but I was certain you’d pick the wrong one. — “A Klink, a Bomb, and a Short Fuse", Hogan’s Heroes (1966) 15 Maximum Flows and Minimum Cuts In the mid-1950s, Air Force researchers T. E. Harris and F. 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 figure at the top of the page) were only declassified in 1999. 1 Figure 2 From Harris and Ross : Schematic diagram of the railway network of the Western So- viet Union and Eastern European countries, with a maximum flow of value 163,000 tons from Russia to Eastern Europe, and a cut of capacity 163,000 tons indicated as ‘The bottleneck’. The max-flow min-cut theorem In the RAND Report of 19 November 1954, Ford and Fulkerson  gave (next to defining the maximum flow problem and suggesting the simplex method for it) the max-flow min- cut theorem for undirected graphs, saying that the maximum flow value is equal to the minimum capacity of a cut separating source and terminal. Their proof is not constructive, but for planar graphs, with source and sink on the outer boundary, they give a polynomial- time, constructive method. In a report of 26 May 1955, Robacker [1955a] showed that the max-flow min-cut theorem can be derived also from the vertex-disjoint version of Menger’s theorem. As for the directed case, Ford and Fulkerson  observed that the max-flow min-cut theorem holds also for directed graphs. Dantzig and Fulkerson  showed, by extending the results of Dantzig [1951a] on integer solutions for the transportation problem to the 25 Harris and Ross’s map of the Warsaw Pact rail network 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...
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- Spring '09
- Graph Theory, Glossary of graph theory, Flow network, Maximum flow problem, Max-flow min-cut theorem, maximum flow