Integrality gaps of semidefinite programs for Vertex Cover and relations to $\ell_1$ embeddability of Negative Type metrics

TL;DR

This paper investigates the limitations of semidefinite programming relaxations for Vertex Cover, revealing that certain strengthened formulations do not improve approximation bounds and establishing connections to metric embeddability and isoperimetric inequalities.

Contribution

It demonstrates that adding pentagonal inequalities does not improve approximation, shows that $ ext{L}_1$ embeddability can lead to exact relaxations, and constructs new examples of negative type metrics with poor $ ext{L}_1$ embeddings.

Findings

Pentagonal inequalities do not improve Vertex Cover approximation.

Strengthening SDP with $ ext{L}_1$ metric constraints yields exact relaxation.

Constructs negative type metrics with $ ext{L}_1$ embedding distortion at least 8/7.

Abstract

We study various SDP formulations for {\sc Vertex Cover} by adding different constraints to the standard formulation. We show that {\sc Vertex Cover} cannot be approximated better than $2-o(1)$ even when we add the so called pentagonal inequality constraints to the standard SDP formulation, en route answering an open question of Karakostas~\cite{Karakostas}. We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution is an $\ell_1$ metric, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily close to being $\ell_1$ embeddable, the integrality gap may be as big as $2-o(1)$. Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar \cite{Char02} to provide a family of simple examples for negative type metrics that cannot be embedded into $\ell_1$ with distortion better than $8/7-\eps$. To this end we prove a new isoperimetric inequality for the hypercube.