On the Combinatorial Lower Bound for the Extension Complexity of the Spanning Tree Polytope

TL;DR

This paper investigates the lower bounds on the extension complexity of the spanning tree polytope, showing that current combinatorial methods can only improve trivial bounds by a logarithmic factor.

Contribution

It introduces a nondeterministic communication protocol for the spanning tree slack matrix support, establishing a near-tight bound on combinatorial lower bounds.

Findings

Lower bound can only be improved by a factor of O(log n)

Supports the conjecture that trivial bounds are close to optimal

Provides a new communication protocol for the spanning tree problem

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 the trival (dimension) bound, $\Omega(n^2)$, the best known upper bound is the extended formulation by Wong (1980) of size $O(n^3)$ (also Martin, 1991). In this note we give a nondeterministic communication protocol with cost $\log_2(n^2\log n)+O(1)$ for the support of the spanning tree slack matrix. This means that the combinatorial lower bounds can improve the trivial lower bound only by a factor of (at most) $O(\log n)$.