Smallest compact formulation for the permutahedron

In this note, we consider the permutahedron, the convex hull of all permutations of {1, 2 · · · , n}. We show how to obtain an extended formulation for this polytope from any sorting network. By using the optimal Ajtai-Komlós-Szemerédi (AKS) sorting network, this extended formulation has Θ(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 Ω(n log n) inequalities. The results easily extend to the generalized permutahedron.