Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T00:44:53.832Z Has data issue: false hasContentIssue false
  • English
  • Français

Widths and Heights of (0,1) -Matrices

Published online by Cambridge University Press:  20 November 2018

D. R. Fulkerson  and
H. J. Ryser
D. R. Fulkerson
Affiliation:
The RAND Corporation
H. J. Ryser
Affiliation:
Ohio State University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A number of combinatorial problems may be regarded as particular instances of the following rather general situation. Given a set X composed of n elements x1, x2, ..., xn, and m subsets X1 X2, … , Xm of X, find a minimal system of representatives for X1, X2, … , Xm. That is, single out a subset X* of X such that Xi ∩ X* is non-empty for i = 1,2, … ,m, and no subset of X containing fewer elements than X* has this property. To illustrate, each of the following can be thought of in these terms.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 239 - 255
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Berge, C., Two theorems in graph theory, Proc. Nat. Acad. Sci., 43 (1957), 842–844.Google Scholar
2. Ford, L. R., Jr. 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.Google Scholar
3. Fulkerson, D. R., A network flow feasibility theorem and combinatorial applications, Can. J. Math., 11 (1959), 440–451.Google Scholar
4. Fulkerson, D. R., Zero-one matrices with zero trace, Pac. J. Math., 10 (1963), 831–836.Google Scholar
5. Gale, D., A theorem onflows in networks, Pac. J. Math., 7 (1957), 1073–1082.Google Scholar
6. Gomory, R., All-integer integer programming algorithm, I.B.M. Report RC-189 (1960), to appear in a special issue of the I.B.M. Journal.Google Scholar
7. Haber, R. M., Term rank of 0,1 matrices, Rend. Sem. Mat. Padova, 30 (1960), 24–51.Google Scholar
8. Hoffman, A. J., Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, Proceedings of Symposia in Applied Mathematics, Amer. Math. Soc, 10 (1969) 113–128.Google Scholar
9. Roth, J. P., Combinatorial topological methods in the synthesis of switching circuits, I.B.M. Report RC-11 (1957).Google Scholar
10. Ryser, H. J., Combinatorial properties of matrices of zeros and ones, Can. J. Math., 9 (1957), 371–377.Google Scholar
11. Ryser, H. J., The term rank of a matrix, Can. J. Math., 10 (1958), 57–65.Google Scholar
12. Ryser, H. J., Traces of matrices of zeros and ones, Can. J. Math., 12 (1960), 463–476.Google Scholar
13. Ryser, H. J., Matrices of zeros and ones, Bull. Amer. Math. Soc, 66 (1960), 442–464.Google Scholar
14. Tucker, A. W., On directed graphs and integer programs, to appear in a special issue of the I.B.M. Journal.Google Scholar