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Smallest compact formulation for the permutahedron. (English) Zbl 1322.90048
Summary: In this note, we consider the permutahedron, the convex hull of all permutations of \(\{1,2\dots ,n\}\). We show how to obtain an extended formulation for this polytope from any sorting network. By using the optimal Ajtai-Komlós-Szemerédi sorting network [M. Ajtai et al., Combinatorica 3, 1–19 (1983; Zbl 0523.68048)], this extended formulation has \(\varTheta (n\log n)\) variables and inequalities. Furthermore, from basic polyhedral arguments, we show that this is best possible (up to a multiplicative constant) since any extended formulation has at least \(\varOmega (n \log n)\) inequalities. The results easily extend to the generalized permutahedron.

MSC:
90C10 Integer programming
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