Abstract
We introduce a concept of dual consistency of systems of linear inequalities with full generality. We show that a cardinality constrained polytope is represented by a certain system of linear inequalities if and only if the systems of linear inequalities associated with the cardinalities are dual consistent. Typical dual consistent systems of inequalities are those which describe polymatroids, generalized polymatroids, and dual greedy polyhedra with certain choice functions. We show that the systems of inequalities for cardinality-constrained ordinary bipartite matching polytopes are not dual consistent in general, and give additional inequalities to make them dual consistent. Moreover, we show that ordinary systems of inequalities for the cardinality-constrained (poly)matroid intersection are not dual consistent, which disproves a conjecture of Maurras, Spiegelberg, and Stephan about a linear representation of the cardinality-constrained polymatroid intersection.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Camion, P., Maurras, J.: Polytopes à sommets dans l’ensemble {0,1}n. Cahiers du Centre d’Études de Recherche Opérationnelle 24, 107–120 (1982)
Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, New York (1970)
Fujishige, S.: Dual greedy polyhedra, choice functions, and abstract convex geometries. Discrete Optimization 1, 41–49 (2004)
Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Annals of Discrete Mathematics, vol. 58. Elsevier (2005)
Grötschel, M.: Cardinality homogeneous set systems, cycles in matroids, and associated polytopes. In: Grötschel, M. (ed.) The Sharpest Cut. The Impact of Manfred Padberg and his Work. MPS-SIAM Series on Optimization, vol. 4, pp. 199–216 (2004)
Kaibel, V., Stephan, R.: On cardinality constrained cycle and path polytopes. Mathematical Programming, Ser. A 123, 371–394 (2010)
Maurras, J.F.: An example of dual polytopes in the unit hypercube. Annals of Discrete Mathematics 1, 391–392 (1977)
Maurras, J.F., Spiegelberg, I., Stephan, R.: On cardinality constrained polymatroids. Discrete Applied Mathematics (2011), doi:10.1016/j.dam.2011.10.007
Maurras, J.F., Stephan, R.: On the cardinality constrained matroid polytope. Networks 57, 240–246 (2011)
Santos, F.: A counterexample to the Hirsch conjecture. ArXiv:1006.2814v3 (math.CO) (November 8, 2011)
Stephan, R.: Cardinality constrained combinatorial optimization: Complexity and polyhedra. Discrete Optimization 7, 99–113 (2010)
Stephan, R., Spiegelberg, I.: On cardinality constrained polymatroids. Electronic Notes in Discrete Mathematics 36, 1017–1024 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fujishige, S., Maßberg, J. (2012). Dual Consistent Systems of Linear Inequalities and Cardinality Constrained Polytopes. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds) Combinatorial Optimization. ISCO 2012. Lecture Notes in Computer Science, vol 7422. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32147-4_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-32147-4_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32146-7
Online ISBN: 978-3-642-32147-4
eBook Packages: Computer ScienceComputer Science (R0)