Mathematical Programming

, Volume 5, Issue 1, pp 199–215 | Cite as

On the facial structure of set packing polyhedra

  • Manfred W. Padberg
Article

Abstract

In this paper we address ourselves to identifying facets of the set packing polyhedron, i.e., of the convex hull of integer solutions to the set covering problem with equality constraints and/or constraints of the form “⩽”. This is done by using the equivalent node-packing problem derived from the intersection graph associated with the problem under consideration. First, we show that the cliques of the intersection graph provide a first set of facets for the polyhedron in question. Second, it is shown that the cycles without chords of odd length of the intersection graph give rise to a further set of facets. A rather strong geometric property of this set of facets is exhibited.

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References

  1. [1]
    J.P. Arabeyre, J. Fearnley, F. Steiger and W. Teather, “The airline crew scheduling problem: A survey”,Transportation Science 3 (1969) 140–163.Google Scholar
  2. [2]
    E. Balas and M.W. Padberg, “On the set covering problem”,Operations Research 20 (1972) 1152–1161.Google Scholar
  3. [3]
    E. Balas and M.W. Padberg, “On the set covering problem II: An algorithm”, Management Sciences Research Report No. 278, GSIA, Carnegie—Mellon University, Pittsburgh, Pa. (presented at the Joint National Meeting of ORSA, TIMS, AIEE, at Atlantic City, November 8–10, 1972).Google Scholar
  4. [4]
    M.L. Balinski, “On recent developments in integer programming”, in:Proceedings of the Princeton Symposium on Mathematical Programming, Ed. H.W. Kuhn (Princeton University Press, Princeton, N.J., 1970).Google Scholar
  5. [5]
    M.L. Balinski, “On maximum matching, minimum covering and their connections”, in:Proceedings of the Princeton Symposium on Mathematical Programming, Ed. H.W. Kuhn (Princeton University Press, Princeton, N.J., 1970).Google Scholar
  6. [6]
    L.W. Beineke, “A Survey of packings and coverings of graphs”, in:The many facets of graph theory, Eds. G. Chartrand and S.F. Kapoor (Springer, Berlin, 1969).Google Scholar
  7. [7]
    J.Edmonds, “Covers and packings in a family of sets”,Bulletin of the American Mathematical Society 68 (1962) 494–499.Google Scholar
  8. [8]
    J. Edmonds, “Maximum matching and a polyhedron with 0, 1 vertices”,Journal of Research of the National Bureau of Standards 69B (1965) 125–130.Google Scholar
  9. [9]
    D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra”,Mathematical Programming 1 (1971) 168–194.Google Scholar
  10. [10]
    R. Garfinkel and G.L. Nemhauser, “The set-partitioning problem: Set covering with equality constraints”,Operations Research 17 (1969) 848–856.Google Scholar
  11. [11]
    R.S. Garfinkel and G.L. Nemhauser, “A survey of integer programming emphasizing computation and relations among models”, Technical Report No. 156, Department of Operations Research, Cornell University, Ithaca, N.Y. (1972).Google Scholar
  12. [12]
    R.E. Gomory, “Some polyhedra connected with combinatorial problems”,Linear Algebra 2 (1969) 451–558.Google Scholar
  13. [13]
    D.K. Guha, “Set covering problem with equality constraints”, The Port of New York Authority, New York (1970).Google Scholar
  14. [14]
    F. Harary,Graph Theory (Addison—Wesley, Reading, Mass., 1969).Google Scholar
  15. [15]
    F. Harary and I.C. Ross, “A procedure for clique detection using the group matrix”,Sociometry 20 (1957) 205–215.Google Scholar
  16. [16]
    R.M. Karp, “Reducibility and combinatorial problems”, Technical Report 3, Department of Computer Science, University of California, Berkeley, Calif. (1972).Google Scholar
  17. [17]
    J. Messier and P. Robert, “Recherche des cliques d'un graphe irreflexif fini”, Publication No. 73, Departement d'Informatique, Université de Montréal (Novembre 1971).Google Scholar
  18. [18]
    O. Ore,Theory of graphs, American Mathematical Society Colloquium Publications 38 (American Mathematical Society, Providence, R.I., 1962).Google Scholar
  19. [19]
    M.W. Padberg, Essays in integer programming, Ph. D. Thesis, Carnegie—Mellon University, Pittsburgh, Pa. (April 1971), unpublished.Google Scholar
  20. [20]
    M.W. Padberg, “On the facial structure of set covering problems”, IIM Preprint No. I/72-13, International Institute of Management, Berlin (1972), (presented at the 41st National Meeting of ORSA, New Orleans, La., April 26–28, 1972).Google Scholar
  21. [21]
    J.F. Pierce, “Application of combinatorial programming to a class of all-zero–one integer programming problems”,Management Science 15 (1968) 191–212.Google Scholar
  22. [22]
    H. Thiriez, Airline crew scheduling, a group theoretic approach, Ph. D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., FTL-R69-1 (1969) (published in RIRO 5 (1971) 83–103).Google Scholar
  23. [23]
    V.A. Trubin, “On a method of solution of integer linear programming problems of a special kind”,Soviet Mathematics Doklady 10 (1969) 1544–1546.Google Scholar
  24. [24]
    L.A. Wolsey, “Extensions of the group theoretic approach in integer programming”,Management Science 18 (1971) 74–83.Google Scholar

Copyright information

© The Mathematical Programming Society 1973

Authors and Affiliations

  • Manfred W. Padberg
    • 1
  1. 1.International Institute of ManagementBerlinWest Germany

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