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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.P. Arabeyre, J. Fearnley, F. Steiger and W. Teather, “The airline crew scheduling problem: A survey”,Transportation Science 3 (1969) 140–163.
E. Balas and M.W. Padberg, “On the set covering problem”,Operations Research 20 (1972) 1152–1161.
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).
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).
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).
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).
J.Edmonds, “Covers and packings in a family of sets”,Bulletin of the American Mathematical Society 68 (1962) 494–499.
J. Edmonds, “Maximum matching and a polyhedron with 0, 1 vertices”,Journal of Research of the National Bureau of Standards 69B (1965) 125–130.
D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra”,Mathematical Programming 1 (1971) 168–194.
R. Garfinkel and G.L. Nemhauser, “The set-partitioning problem: Set covering with equality constraints”,Operations Research 17 (1969) 848–856.
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).
R.E. Gomory, “Some polyhedra connected with combinatorial problems”,Linear Algebra 2 (1969) 451–558.
D.K. Guha, “Set covering problem with equality constraints”, The Port of New York Authority, New York (1970).
F. Harary,Graph Theory (Addison—Wesley, Reading, Mass., 1969).
F. Harary and I.C. Ross, “A procedure for clique detection using the group matrix”,Sociometry 20 (1957) 205–215.
R.M. Karp, “Reducibility and combinatorial problems”, Technical Report 3, Department of Computer Science, University of California, Berkeley, Calif. (1972).
J. Messier and P. Robert, “Recherche des cliques d'un graphe irreflexif fini”, Publication No. 73, Departement d'Informatique, Université de Montréal (Novembre 1971).
O. Ore,Theory of graphs, American Mathematical Society Colloquium Publications 38 (American Mathematical Society, Providence, R.I., 1962).
M.W. Padberg, Essays in integer programming, Ph. D. Thesis, Carnegie—Mellon University, Pittsburgh, Pa. (April 1971), unpublished.
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).
J.F. Pierce, “Application of combinatorial programming to a class of all-zero–one integer programming problems”,Management Science 15 (1968) 191–212.
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).
V.A. Trubin, “On a method of solution of integer linear programming problems of a special kind”,Soviet Mathematics Doklady 10 (1969) 1544–1546.
L.A. Wolsey, “Extensions of the group theoretic approach in integer programming”,Management Science 18 (1971) 74–83.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Padberg, M.W. On the facial structure of set packing polyhedra. Mathematical Programming 5, 199–215 (1973). https://doi.org/10.1007/BF01580121
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580121