Limitations of the Hyperplane Separation Technique for Bounding the Extension Complexity of Polytopes

TL;DR

This paper critically examines the hyperplane separation technique for bounding polytope extension complexity, revealing its limitations and sensitivity to slack matrix choices, which affects the strength of derived bounds.

Contribution

It demonstrates the technique's limitations on specific polytopes and shows how normalization can improve the bounds, highlighting the method's sensitivity.

Findings

Hyperplane separation bounds are trivial for certain polytopes.

Normalization of slack matrices can strengthen bounds.

The technique's effectiveness depends on slack matrix choice.

Abstract

We illustrate the limitations of the hyperplane separation bound, a non-combinatorial lower bound on the extension complexity of a polytope. Most notably, this bounding technique is used by Rothvo{\ss} (J ACM 64.6:41, 2017) to establish an exponential lower bound for the perfect matching polytope. We point out that the technique is sensitive to the particular choice of slack matrix. For the canonical slack matrices of the spanning tree polytope and the completion time polytope, we show that the lower bounds produced by the hyperplane separation method are trivial. These bounds may, however, be strengthened by normalizing rows and columns of the slack matrices.