Extension Complexity of Independent Set Polytopes

TL;DR

This paper demonstrates that certain graphs have independent set polytopes requiring exponentially large extended formulations, revealing new insights into the complexity of these polytopes and connecting it to circuit depth.

Contribution

It provides explicit examples of high extension complexity in independent set polytopes, linking polyhedral complexity to circuit depth.

Findings

Existence of graphs with exponential extension complexity in independent set polytopes

First explicit examples surpassing previous exponential bounds

Connection established between extended formulations and monotone circuit depth

Abstract

We exhibit an $n$-node graph whose independent set polytope requires extended formulations of size exponential in $\Omega(n/\log n)$. Previously, no explicit examples of $n$-dimensional $0/1$-polytopes were known with extension complexity larger than exponential in $\Theta(\sqrt{n})$. Our construction is inspired by a relatively little-known connection between extended formulations and (monotone) circuit depth.