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It is well known that the permutahedron Pi_n has 2^n-2 facets. The Birkhoffpolytope provides a symmetric extended formulation of Pi_n of size Theta(n^2).Recently, Goemans described a non-symmetric extended formulation of Pi_n ofsize Theta(n log(n)). In this paper, we prove that Omega(n^2) is a lower boundfor the size of symmetric extended formulations of Pi_n.

Tight Lower Bounds on the Sizes of Symmetric Extensions of Permutahedra and Similar Results