On unions and dominants of polytopes
A well-known result on unions of polyhedra in the same space gives an extended formulation in a higher-dimensional space whose projection is the convex hull of the union. Here in contrast we study the unions of polytopes in different spaces, giving a complete description of the convex hull without additional variables. When the underlying polytopes are monotone, the facets are described explicitly, generalizing results of Hong and Hooker on cardinality rules, and an efficient separation algorithm is proposed. These results are based on an explicit representation of the dominant of a polytope, and an algorithm for the separation problem for the dominant. For non-monotone polytopes, both the dominant and the union are characterized. We also give results on the unions of polymatroids both on disjoint ground sets and on the same ground set generalizing results of Conforti and Laurent.
Keywordsunions of polytopes cardinality constraints separation convex hulls dominant matroids polymatroids
Unable to display preview. Download preview PDF.
- 2.Balas, E.: Logical Constraints as Cardinality Rules: Tight Representations. #MSRR-628, Carnegie Mellon University, 2000Google Scholar
- 4.Boyd, S.C., Pulleyblank, W.R.: Facet generating techniques, Department of Combinatorics and Optimization. University of Waterloo, 1991Google Scholar
- 8.Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J., eds. Combinatorial Structures and their applications, Gordon and Breach, New York, 1970, pp. 69–87Google Scholar
- 12.Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer Verlag 1988Google Scholar
- 16.Rockafellar, T.: Convex analysis. Princeton University Press, 1970Google Scholar
- 17.Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combinatorial Theory, Ser. B 80, 346–355 (2000)Google Scholar