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

Understand

It is well known that the permutahedron Pi_n has 2^n-2 facets.

• The Birkhoff polytope provides a symmetric extended formulation of Pi_n of size Theta(n^2).
• Recently, Goemans described a non-symmetric extended formulation of Pi_n of size Theta(n log(n)).
• In this paper, we prove that Omega(n^2) is a lower bound for the size of symmetric extended formulations of Pi_n.