Abstract
There are many examples of optimization problems whose associated polyhedra can be described much nicer, and with way less inequalities, by projections of higher dimensional polyhedra than this would be possible in the original space. However, currently not many general tools to construct such extended formulations are available. In this paper, we develop the framework of polyhedral relations that generalizes inductive constructions of extended formulations via projections, and we particularly elaborate on the special case of reflection relations. The latter ones provide polynomial size extended formulations for several polytopes that can be constructed by iteratedly forming convex hulls of polytopes and (slightly modified) reflections of them at hyperplanes. We demonstrate the use of the framework by deriving small extended formulations for the G-permutahedra of all finite reflection groups G (generalizing both Goemans’ extended formulation of the permutahedron of size O(nlogn) and Ben-Tal and Nemirovski’s extended formulation with O(k) inequalities for the regular 2k-gon) and for Huffman-polytopes (the convex hulls of the weight-vectors of Huffman codes). This work is an extension of an extended abstract presented at IPCO XV (2011).
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Acknowledgements
Martin Grötschel is our academic grandfather (Volker Kaibel) and great-grandfather (Kanstantsin Pashkovich). Next to our deep appreciation of his scientific work and of his influence on the development of the field of Combinatorial Optimization, Volker Kaibel is in particular grateful for Martin Grötschel’s great support and for the excellent conditions (except for occasionally having to prepare PowerPoint-slides) he enjoyed while working at Zuse-Institute Berlin in 2005 and 2006. It is a true pleasure to contribute to this volume dedicated to Martin’s 65th birthday.
With respect to the work presented in this paper, we thank Samuel Fiorini, Michel Goemans, Günter Rote and Dirk Oliver Theis for valuable hints and discussions as well as the referee for all her or his efforts.
Volker Kaibel acknowledges funding by Deutsche Forschungsgemeinschaft (KA 1616/4-1 Extended Formulations in Combinatorial Optimization) and Kanstantsin Pashkovich is grateful for support by the International Max Planck Research School (IMPRS) for Analysis, Design and Optimization in Chemical and Biochemical Process Engineering Magdeburg and by the Progetto di Eccellenza 2008–2009 of the Fondazione Cassa Risparmio di Padova e Rovigo.
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Kaibel, V., Pashkovich, K. (2013). Constructing Extended Formulations from Reflection Relations. In: Jünger, M., Reinelt, G. (eds) Facets of Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38189-8_4
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