Abstract
Given a graph with weights on vertices, the vertex packing problem consists of finding a vertex packing (i.e. a set of vertices, no two of them being adjacent) of maximum weight. A linear relaxation of one binary programming formulation of this problem has these two well-known properties: (i) every basic solution is (0, 1/2, 1)-valued, (ii) in an optimum linear solution, an integer-valued variable keeps the same value in an optimum binary solution.
As an answer to an open problem from Nemhauser and Trotter, it is shown that there is a unique maximal set of variables which are integral in optimal (VLP) solutions.
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, “On maximum matching, minimum covering and their connections”, in: H. Kuhn, ed.,Proceedings of the Princeton symposium on mathematical programming (Princeton University Press, Princeton NJ, 1970) 303–312.
J. Edmonds and E.L. Johnson. “Matching: a well-solved class of integer linear programs”, in:Proceedings of the Calgary international conference on combinatorial structures and their applications (Gordon and Breach, New York, 1970).
L.R. Ford and D.R. Fulkerson,Flows in Networks (Princeton University Press, Princeton, NJ, 1962).
E.L. Johnson, “Programming in networks and graphs”, ORC 65.1, University of California, Berkeley (1965).
L.C. Lorentzen, “Notes on covering of arcs by nodes in an undirected graph”, ORC 66.16, University of California, Berkeley (1966).
G.L. Nemhauser and L.E. Trotter, “Vertex packings: structural properties and algorithms”,Mathematical Programming 8 (2) (1975) 232–248.
J.C. Picard and M. Queyranne, “On the integer-valued variables in the linear vertex packing problem”,Tech. Rept. EP75-R-35. Ecole Polytechnique, Montréal (July 1975).
J.C. Picard and M. Queyranne, “Vertex packings: (VLP)-reductions through alternate labeling”,Tech. Rept. EP75-R-47, Ecole Polytechnique, Montréal (Sept. 1975).
Author information
Authors and Affiliations
Additional information
This research was supported by National Research Council of Canada GRANT A8528 and RD 804.
Rights and permissions
About this article
Cite this article
Picard, JC., Queyranne, M. On the integer-valued variables in the linear vertex packing problem. Mathematical Programming 12, 97–101 (1977). https://doi.org/10.1007/BF01593772
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01593772