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Network Flow and Systems of Representatives

  • L. R. Ford and D. R. Fulkerson

Extract

The theory developed for the study of flows in networks (2; 3; 4; 5; 6; 7) sometimes provides a useful tool for dealing with certain kinds of combinatorial problems, as has been previously indicated in (3; 4; 6; 7). In particular, Hall-type theorems for the existence of systems of distinct representatives which contain a prescribed set of marginal elements (10; 11), or, more generally, whose intersection with each member of a given partition of the fundamental set has a cardinality between prescribed lower and upper bounds (9), can be obtained in this way (7).

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References

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1. 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), 183-195.
2. Dantzig, G. B. and Fulkerson, D. R., Computation of maximal flows in networks, Naval Research Logistics Quarterly, 2 (1955), 277-283.
3. Dantzig, G. B., On the min cut max flow theorem of networks, Annals of Mathematics Study No. 38, Linear Inequalities and Related Systems, ed. H. W. Kuhn and A. W. Tucker (Princeton, 1956), 215-221.
4. Ford, Jr. L. R. , and Fulkerson, D. R., A simple algorithm for finding maximal network flows and an application to the Hitchcock Problem, Can. J. Math., 9 (1957), 210-218.
5. Ford,Jr. L. R. , Maximal flow through a network, Can. J. Math., 8 (1956), 399-404.
6. Gale, D., A Theorem on flows in networks, RAND Corporation, Research Memorandum RM-1737, 1956 (to appear in Pacific J. Math.).
7. Gale, D. and Hoffman, A., Circulation in networks (unpublished notes).
8. Hall, P., On representatives of subsets, J. Lond. Math. Soc, 10 (1935), 26-30.
9. Hoffman, A. J. and Kuhn, H. W., On systems of distinct representatives, Annals of Mathematics Study, No. 38, Linear Inequalities and Related Systems, ed. H. W. Kuhn and A. W. Tucker (Princeton, 1956), 199-206.
10. Hoffman, A. J., Systems of distinct representatives and linear programming, Amer. Math. Monthly, 68 (1956), 455-460.
11. Mann, H. B. and Ryser, H. J., Systems of distinct representatives, Amer. Math. Monthly, 60 (1953), 397-401.
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