On the extension complexity of combinatorial polytopes

TL;DR

This paper investigates the extension complexity of combinatorial polytopes, demonstrating exponential complexity for various NP-complete problems and relating it to graph minors, with implications for quantum information and planar graphs.

Contribution

It introduces a lifting argument to prove exponential extension complexity for several NP-complete problems and relates this complexity to graph minors, extending previous results.

Findings

Exponential extension complexity for subset-sum and 3D matching.

Relationship between extension complexity of a graph's cut polytope and its minors.

Exponential complexity for cut polytopes of graphs used in quantum information and cubic planar graphs.

Abstract

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.