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

TL;DR

This paper establishes a fundamental lower bound on the size of symmetric extended formulations for permutahedra, showing they must be at least quadratic in size, which clarifies limitations of symmetric approaches.

Contribution

It proves a tight Omega(n^2) lower bound on the size of symmetric extended formulations for permutahedra, advancing understanding of their complexity.

Findings

Symmetric extended formulations of Pi_n require at least Omega(n^2) size.

Non-symmetric formulations can be significantly smaller, as shown by Goemans.

The result delineates the limitations of symmetric approaches for permutahedra.

Abstract

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.