Abstract
We consider two convex polyhedra related to the vertex packing problem for a finite, undirected, loopless graphG with no multiple edges. A characterization is given for the extreme points of the polyhedron\(\mathcal{L}_G = \{ x \in R^n :Ax \leqslant 1_m ,x \geqslant 0\} \), whereA is them × n edge-vertex incidence matrix ofG and 1 m is anm-vector of ones. A general class of facets of
= convex hull{x∈R n:Ax≤1 m ,x binary} is described which subsumes a class examined by Padberg [13]. Some of the results for
are extended to a more general class of integer polyhedra obtained from independence systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.L. Balinski, “Integer programming: Methods, uses, computation”,Management Science 12 (1965) 253–313.
M.L. Balinski, “Establishing the matching polytope”,Journal of Combinatorial Theory 13 (1972) 1–13.
M.L. Balinski and K. Spielberg, “Methods of integer programming: Algebraic, combinatorial and enumerative”, in:Progress in operations research, Vol. III, Ed. J. Aronofsky (Wiley, New York, 1969) pp. 195–292.
V. Chvátal, “On certain polytopes associated with graphs”, Centre de Recherches Mathématiques-238, Université de Montréal (October 1972).
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.
J. Edmonds, “Matroids and the greedy algorithm”,Mathematical Programming 1 (1971) 127–136.
D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra”,Mathematical Programming 1 (1971) 168–194.
D.R. Fulkerson, “Anti-blocking polyhedra”,Journal of Combinatorial Theory 12 (1972) 50–71.
R.S. Garfinkel and G.L. Nemhauser, “A survey of integer programming emphasizing computation and relations among models”, in:Mathematical Programming Eds. T.C. Hu and S.M. Robinson (Academic Press, New York, 1973) pp. 77–155.
R.E. Gomory, “Some polyhedra related to combinatorial problems”,Linear Algebra and Its Applications 2 (1969) 451–558.
L. Lovász, “Normal hypergraphs and the perfect graph conjecture”,Discrete Mathematics 2 (1972) 253–267.
M.W. Padberg, “On the facial structure of set packing polyhedra”,Mathematical Programming 5 (1973) 199–215.
L.E. Trotter, Jr., “Solution characteristics and algorithms for the vertex packing problem”, Technical Report No. 168. Department of Operations Research, Cornell University (January 1973).
L.E. Trotter, Jr., “A class of facet producing graphs for vertex packing polyhedra”, Technical Report No. 78, Department of Administrative Sciences, Yale University, (February 1974).
Author information
Authors and Affiliations
Additional information
This research was supported by the National Science Foundation under Grant GK-32282X to Cornell University and by the United States Army under Contract No. DA-31-124-ARO-D-462 to the Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, U.S.A.
Rights and permissions
About this article
Cite this article
Nemhauser, G.L., Trotter, L.E. Properties of vertex packing and independence system polyhedra. Mathematical Programming 6, 48–61 (1974). https://doi.org/10.1007/BF01580222
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580222