Extension complexity of low-dimensional polytopes

TL;DR

This paper investigates the extension complexity of low-dimensional polytopes, providing bounds for random polytopes, cyclic polygons, and constructing high extension complexity polytopes with few vertices.

Contribution

It offers new bounds on extension complexity for specific classes of low-dimensional polytopes and introduces techniques for analyzing this parameter.

Findings

Random d-dimensional polytopes have extension complexity on the order of the square root of their vertices.

Cyclic n-vertex polygons have extension complexity at most 24√n, tight up to a constant.

Existence of low-dimensional polytopes with n vertices and extension complexity n^{1-o(1)}.

Abstract

Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension $d$, the extension complexity of a random $d$-dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic $n$-vertex polygon (whose vertices lie on a circle) has extension complexity at most $24\sqrt n$. This bound is tight up to the constant factor $24$. Finally, we show that there exists an $n^{o(1)}$-dimensional polytope with at most $n$ vertices and extension complexity $n^{1-o(1)}$. Our theorems are proved with a range of different techniques, which we hope will be of further interest.