## 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.

## Preview

Unable to display preview. Download preview PDF.

### References

- [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]E. Balas and M.W. Padberg, “On the set covering problem”,
*Operations Research*20 (1972) 1152–1161.Google Scholar - [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]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]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]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]J.Edmonds, “Covers and packings in a family of sets”,
*Bulletin of the American Mathematical Society*68 (1962) 494–499.Google Scholar - [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]D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra”,
*Mathematical Programming*1 (1971) 168–194.Google Scholar - [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]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]R.E. Gomory, “Some polyhedra connected with combinatorial problems”,
*Linear Algebra*2 (1969) 451–558.Google Scholar - [13]D.K. Guha, “Set covering problem with equality constraints”, The Port of New York Authority, New York (1970).Google Scholar
- [14]F. Harary,
*Graph Theory*(Addison—Wesley, Reading, Mass., 1969).Google Scholar - [15]F. Harary and I.C. Ross, “A procedure for clique detection using the group matrix”,
*Sociometry*20 (1957) 205–215.Google Scholar - [16]R.M. Karp, “Reducibility and combinatorial problems”, Technical Report 3, Department of Computer Science, University of California, Berkeley, Calif. (1972).Google Scholar
- [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]O. Ore,
*Theory of graphs,*American Mathematical Society Colloquium Publications 38 (American Mathematical Society, Providence, R.I., 1962).Google Scholar - [19]M.W. Padberg, Essays in integer programming, Ph. D. Thesis, Carnegie—Mellon University, Pittsburgh, Pa. (April 1971), unpublished.Google Scholar
- [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 41
^{st}National Meeting of ORSA, New Orleans, La., April 26–28, 1972).Google Scholar - [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]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]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]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