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A Simple Algorithm for Finding Maximal Network Flows and an Application to the Hitchcock Problem

  • L. R. Ford (a1) and D. R. Fulkerson (a1)

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The network-flow problem, originally posed by T. Harris of the Rand Corporation, has been discussed from various viewpoints in (1; 2; 7; 16). The problem arises naturally in the study of transportation networks; it may be stated in the following way. One is given a network of directed arcs and nodes with two distinguished nodes, called source and sink, respectively. All other nodes are called intermediate. Each directed arc in the network has associated with it a nonnegative integer, its flow capacity. Source arcs may be assumed to be directed away from the source, sink arcs into the sink. Subject to the conditions that the flow in an arc is in the direction of the arc and does not exceed its capacity, and that the total flow into any intermediate node is equal to the flow out of it, it is desired to find a maximal flow from source to sink in the network, i.e., a flow which maximizes the sum of the flows in source (or sink) arcs.

Thus, if we let P 1 be the source, P n the sink, we are required to find x ij (i,j =1, . . . , w) which maximize

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References

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1. Dantzig, G. B. and Fulkerson, D. R., On the max flow min cut theorem of networks, Linear Inequalities and Related Systems, 215-221. Annals of Math. Study 38 (Princeton, 1956).
2. Dantzig, G. B. and Fulkerson, D. R., Computation of maximal flows in networks, Naval Research Logistics Quarterly, 2 (1955), 277283.
3. Dantzig, G. B., Orden, A., and Wolfe, P., The generalized simplex method for minimizing a linear form under linear inequality constraints, Pacific J. Math., 5 (1955), 183195.
4. Dantzig, G. B., Application of the simplex method to a transportation problem, Activity Analysis of Production and Allocation, 359–373. Cowles Commission Monograph 13 (New York, 1951).
5. Dilworth, R. P., A decomposition theorem for partially ordered sets, Annals Math., 51 (1950), 161166.
6. Egerváry, E., Matrixok kombinatorius tulajdonságáirol, Mat. és Fiz. Lapok 38 (1931), 16–28 (translated as On combinatorial properties of matrices, by H. W. Kuhn, Office of Naval Research Logistics Project Report (Princeton, 1953)).
7. Ford, L. R. Jr. and Fulkerson, D. R., Maximal flow through a network, Can. J. Math., 8 (1956), 399404.
8. Fulkerson, D. R., Note on Dilworth's chain decomposition theorem for partially ordered sets, Proc. Amer. Math. Soc, 7 (1956), 701702.
9. Gale, D., A theorem onflows in networks, to appear in Pacific J. Math.
10. Gale, D., Kuhn, H. W., and Tucker, A. W., Linear programming and the theory of games, Activity Analysis of Production and Allocation, 317-329. Cowles Commission Monograph 13 (New York, 1951).
11. Hall, P., On representatives of subsets, Jour. London Math. Soc, 10 (1935), 2630.
12. Hitchcock, F. L., The distribution of a product from several sources to numerous localities, Jour. Math. Phys., 20 (1941), 224230.
13. König, D., Theorie der endlichen und unendlichen Graphen (Chelsea, New York, 1950).
14. Koopmans, T. C. and Reiter, S., A model of transportation, Activity Analysis of Production and Allocation, 222-259. Cowles Commission Monograph 13 (New York, 1951).
15. Kuhn, H. W., A combinatorial algorithm for the assignment problem. Issue 11 of Logistics Papers, George Washington University Logistics Research Project, 1954.
16. Robacker, J. T., On network theory. Rand Corp., Research Memorandum RM-1498, 1955.
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