Computing The Extension Complexities of All 4-Dimensional 0/1-Polytopes

TL;DR

This paper refines bounds on extension complexities of 0/1-polytopes and computes exact complexities for all 4-dimensional cases, revealing that minimal extensions can also be 0/1-polytopes with strong properties.

Contribution

It provides the first complete computation of extension complexities for all 0/1-polytopes up to dimension 4, including geometric constructions and properties of minimal extensions.

Findings

All 0/1-polytopes up to dimension 4 have minimum size extensions that are also 0/1-polytopes.

Refined bounds improve understanding of extension complexities.

Explicit geometric constructions for minimal extensions are provided.

Abstract

We present slight refinements of known general lower and upper bounds on sizes of extended formulations for polytopes. With these observations we are able to compute the extension complexities of all 0/1-polytopes up to dimension 4. We provide a complete list of our results including geometric constructions of minimum size extensions for all considered polytopes. Furthermore, we show that all of these extensions have strong properties. In particular, one of our computational results is that every 0/1-polytope up to dimension 4 has a minimum size extension that is also a 0/1-polytope.