Abstract
Padberg (Math Program 137:593–599, 2013) introduced a geometric notion of ranks for (mixed) integer rational polyhedrons and conjectured that the geometric rank of the matching polytope is one. In this work, we prove that this conjecture is true.
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Notes
The matching polytope of a cycle with four nodes has a rank 0, every facet is required in the minimal formulation.
References
Lovász, L., Plummer, M.: Matching Theory. Akadémiai Kiadó, Budapest (1986)
Padberg, M.: The rank of (mixed-) integer polyhedra. Math. Program. 137, 593–599 (2013)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)
Acknowledgments
The author would like to thank Daniel Karch for the valuable discussions and comments on the proof. The author would also like to thank the two anonymous reviewers for their useful comments that helped in considerably improving the presentation of the ideas. The Proof of Lemma 3.2 was sketched by the anonymous reviewer. The author was supported by the DFG research center Matheon “Mathematics for key technologies” in Berlin.
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Arulselvan, A. On the geometric rank of matching polytope. Math. Program. 152, 189–200 (2015). https://doi.org/10.1007/s10107-014-0782-0
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DOI: https://doi.org/10.1007/s10107-014-0782-0