Abstract
In this paper we describe several forms of thek-partition problem and give integer programming formulations of each case. The dimension of the associated polytopes and some basic facets are identified. We also give several valid and facet defining inequalities for each of the polytopes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Barahona, M. Grötschel and A.R. Mahjoub, “Facets of the bipartite subgraph polytope,”Mathematics of Operations Research 10 (1985) 340–358.
F. Barahona and A.R. Mahjoub, “On the cut polytope,”Mathematical Programming 36 (1986) 157–173.
F. Barahona and A. Casari, “On the Magnetisation of the ground states in two dimensional Ising spin glasses,” Research Report, University of Waterloo (1987).
J.A. Bondy and U.S.R. Murty,Graph Theory with Applications (North-Holland, Amsterdam, 1976).
R.C. Carlson and G.L. Nemhauser, “Scheduling to minimize interaction costs,”Operations Research 14 (1966) 52–58.
S. Chopra and M.R. Rao, “The Partition Problem I: Formulations, dimensions and basic facets,” Working Paper No. 89-27, Stern School of Business, New York University (New York, 1989).
M. Conforti, M.R. Rao and A. Sassano, “The Equipartition Polytope I,”Mathematical Programming 49 (1990/91) 49–70.
M. Conforti, M.R. Rao and A. Sassano, “The Equipartition Polytope II,”Mathematical Programming 49 (1990/91) 71–90.
M. Deza, M. Grötschel and M. Laurent, “Clique web facets for multicut polytopes,” Report No. 186, Institut für Mathematik, Universität Augsburg (1989).
M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979).
O. Goldschmidt and D.S. Hochbaum, “Polynomial algorithm for thek-cut problem,” preprint, University of California (Berkeley, CA, 1987).
M. Grötschel and Y. Wakabayashi, “Facets of the clique partitioning polytope,”Mathematical Programming 47 (1990) 367–387.
M. Grötschel and Y. Wakabayashi “A cutting plane algorithm for a clustering problem,”Mathematical Programming 45 (1989) 59–96.
M. Grötschel and Y. Wakayabashi “Composition of facets of the clique partitioning polytope,” Report No. 18, Institut für Mathematik, Universität Augsburg (1987).
D.S. Johnson, C.H. Papadimitriou, P. Seymour and M. Yannakakis, “The complexity of multi way cuts,” extended abstract (1983).
B.W. Kernighan and S. Lin, “An efficient heuristic procedure for partitioning graphs,”Bell Systems Technical Journal 49 (1970) 291–308.
Author information
Authors and Affiliations
Additional information
Partial Support from NSF Grants DMS 8606188 and ECS 8800281 is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Chopra, S., Rao, M.R. The partition problem. Mathematical Programming 59, 87–115 (1993). https://doi.org/10.1007/BF01581239
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01581239