Extension Complexity of the Correlation Polytope

TL;DR

This paper establishes a tight bound on the extension complexity of the correlation polytope based on the treewidth of the graph, linking geometric complexity to graph-theoretic properties.

Contribution

It provides a tight bound on the extension complexity of correlation polytopes in terms of graph treewidth, advancing understanding of polyhedral complexity in combinatorial optimization.

Findings

Extension complexity is bounded by 2^{O(tw(G) + log n)} for any n-vertex graph G.

The bound is tight for graphs in minor-closed classes.

Connects geometric complexity of polytopes with graph-theoretic parameters.

Abstract

We prove that for every $n$-vertex graph $G$, the extension complexity of the correlation polytope of $G$ is $2^{O(\mathrm{tw}(G) + \log n)}$, where $\mathrm{tw}(G)$ is the treewidth of $G$. Our main result is that this bound is tight for graphs contained in minor-closed classes.