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

It is well known that the permutahedron Πn has 2n − 2 facets. The Birkhoff polytope provides a symmetric extended formulation of Πn of size Θ(n2). Recently, Goemans described a non-symmetric extended formulation of Πn of size Θ(n log n). In this paper, we prove that Ω(n2) is a lower bound for the size of symmetric extended formulations of Πn. Moreover, we prove that the cardinality indicating polytope has the same tight lower bounds for the sizes of symmetric and nonsymmetric extended formulations as the permutahedron.