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Constructing extended formulations from reflection relations. (English) Zbl 1341.90148
Günlük, Oktay (ed.) et al., Integer programming and combinatoral optimization. 15th international conference, IPCO 2011, New York, NY, USA, June 15–17, 2011. Proceedings. Berlin: Springer (ISBN 978-3-642-20806-5/pbk). Lecture Notes in Computer Science 6655, 287-300 (2011).
Summary: 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 a 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 as convex hulls of the unions of (exponentially) many copies of an input polytope obtained via sequences of reflections 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 M. X. Goeman’s [Math. Program. 153, No. 1 (B), 5–11 (2015; Zbl 1322.90048)] extended formulation of the permutahedron of size \(O(n\log n)\) and A. Ben-Tal and A. Nemirovski’s [Math. Oper. Res. 26, No. 2, 193–205 (2001; Zbl 1082.90133)] extended formulation with \(O(k)\) inequalities for the regular \(2^{k }\)-gon) and for Huffman-polytopes (the convex hulls of the weight-vectors of Huffman codes).
For the entire collection see [Zbl 1216.90002].

MSC:
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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