Fooling Sets and the Spanning Tree Polytope

TL;DR

This paper demonstrates that the fooling set method cannot improve the known quadratic lower bound on the size of extended formulations for the Minimum Spanning Tree problem.

Contribution

It proves that all fooling sets for the Spanning Tree polytope are at most quadratic in size, limiting this method's effectiveness for lower bound improvements.

Findings

Fooling set size for the Spanning Tree polytope is O(n^2)

Limits the use of fooling sets to improve lower bounds

Supports existing quadratic lower bounds for extended formulations

Abstract

In the study of extensions of polytopes of combinatorial optimization problems, a notorious open question is that for the size of the smallest extended formulation of the Minimum Spanning Tree problem on a complete graph with $n$ nodes. The best known lower bound is $\Omega(n^2)$, the best known upper bound is $O(n^3)$. In this note we show that the venerable fooling set method cannot be used to improve the lower bound: every fooling set for the Spanning Tree polytope has size $O(n^2)$.