Extended formulations for polygons

TL;DR

This paper investigates the extension complexity of polygons, providing new proofs for known bounds on regular polygons, establishing lower bounds for generic polygons, and constructing polygons with high extension complexity on integer grids.

Contribution

It offers a simplified proof for the $O(\log n)$ extension complexity of regular polygons, extends lower bounds to generic polygons, and constructs polygons with high extension complexity on integer grids.

Findings

Regular $n$-gons have extension complexity $O(\log n)$

Generic $n$-gons have a lower bound of $\sqrt{2n}$ on extension complexity

Existence of $n$-gons on integer grids with extension complexity $\Omega(\sqrt{n}/\sqrt{\log n})$

Abstract

The extension complexity of a polytope $P$ is the smallest integer $k$ such that $P$ is the projection of a polytope $Q$ with $k$ facets. We study the extension complexity of $n$-gons in the plane. First, we give a new proof that the extension complexity of regular $n$-gons is $O(\log n)$, a result originating from work by Ben-Tal and Nemirovski (2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), and Kaibel and Pashkovich (2011). Second, we prove a lower bound of $\sqrt{2n}$ on the extension complexity of generic $n$-gons. Finally, we prove that there exist $n$-gons whose vertices lie on a $O(n) \times O(n^2)$ integer grid with extension complexity $\Omega(\sqrt{n}/\sqrt{\log n})$.