Abstract
We discuss several methods of determining all facets of “small” polytopes and polyhedra and give several criteria for classifying the facets into different facet types, so as to bring order into the multitude of facets as, e.g., produced by the application of the double description algorithm (DDA). Among the forms that we consider are the normal, irreducible and minimum support representations of facets. We study symmetries of vertex and edge figures under permissible permutations that leave the underlying polyhedron unchanged with the aim of reducing the numerical effort to find all facets efficiently. Then we introduce a new notion of the rank of facets and integer polyhedra. In the last section, we present old and new results on the facets of the symmetric traveling salesman polytope \(\mathcal{Q}_{T}^{n}\) with up to n=10 cities based on our computer calculations and state a conjecture that, in the notion of rank ρ(P) introduced here, asserts \(\rho(\mathcal{Q}_{T}^{n}) = n - 5\) for all n≥5. This conjecture is supported by our calculations up to n=9 and, possibly, n=10.
Dedicated to Martin Grötschel, my first and best former doctoral student and a personal friend for almost forty years now. Ad multos annos, Martin!
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Acknowledgements
Supported in part by ONR grant N00014-96-0327 and by visits to Cologne U (1996) and IASI-CNR Rome (1997). This paper was presented in preliminary form in a plenary session at the XVIth ISMP in August 1997 at the EPFL in Lausanne, Switzerland, under the title “Facets, Rank of Integer Polyhedra and Other Topics”.
I wish to thank the referee for his constructive criticism.
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Padberg, M.W. (2013). Facets and Rank of Integer Polyhedra. In: Jünger, M., Reinelt, G. (eds) Facets of Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38189-8_2
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