Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-06-01T17:05:26.742Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Extreme Rays of the Metric Cone

Published online by Cambridge University Press:  20 November 2018

David Avis
David Avis*
Affiliation:
McGill University, Montreal, Quebec
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 classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equations

Birkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 126 - 144
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Avis, D., On the Hamming cone, Technical Report 77-5 (1977), Department of Operations Research, Stanford University.Google Scholar
2. Birkhoff, G., Très observationes sobre et linear algebra, Rev. Univ. Nac. Tucuman A5 (1946), 147–158.Google Scholar
3. Chvätal, V., Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Mathematic. 4 (1973), 305–337.Google Scholar
4. Edmonds, J., Maximum matching and a polyhedron with integral vertices, J. Res. Nat. Bur. Standards B69 (1965), 125–130.Google Scholar
5. Erdos, P. and Spencer, J., Probabilistic methods in combinatorics (Academic Press, New York, 1974).Google Scholar
6. Harary, F., Graph theory (Addison-Wesley, Reading, Mass., 1969).Google Scholar
7. Stoer, J. and Witzgall, C., Convexity and optimization infinite dimensions (Springer-Verlag, Berlin, 1970).Google Scholar
8. Veinott, A. F. Jr., Extreme points of Leonlief substitution systems, Linear Algebra and Application. 1 (1968), 181–194.Google Scholar
9. Yao, A. and Rivest, R., On the polyhedral decision problem, manuscript, Computer Science Department, Stanford University (1977).Google Scholar
10. Graham, R., Yao, A. and Yao, F., Information bounds are weak in the shortest distance problem, to appear in J. ACM.Google Scholar