Extension complexity of polytopes with few vertices or facets

TL;DR

This paper classifies the extension complexity of polytopes with few vertices or facets, revealing precise bounds and classifications for certain classes of polytopes based on their size and structure.

Contribution

It provides a complete classification of $d$-polytopes with at most $d+4$ vertices regarding their extension complexity and establishes lower bounds for generic realizations of certain polytopes.

Findings

Most $d$-polytopes with $d+4$ vertices have extension complexity $d+4$

Certain families of $d$-polytopes with $d+4$ vertices have lower extension complexity of $ heta(d^2)$

Generic realizations of specific polytopes have extension complexity at least $2 \sqrt{d(d+ ext{alpha})} - d + 1$

Abstract

We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most $d+4$ vertices according to their extension complexity: Out of the super-exponentially many $d$-polytopes with $d+4$ vertices, all have extension complexity $d+4$ except for some families of size $\theta(d^2)$. On the other hand, we show that generic realizations of simplicial/simple $d$-polytopes with $d+1+\alpha$ vertices/facets have extension complexity at least $2 \sqrt{d(d+\alpha)} -d + 1$, which shows that for all $d>(\frac{\alpha-1}{2})^2$ there are $d$-polytopes with $d+1+\alpha$ vertices or facets and extension complexity $d+1+\alpha$.