Random zero sets with local growth guarantees

TL;DR

This paper constructs random subsets in metric spaces that embed into Hilbert spaces, providing new guarantees on local growth and applications to Euclidean distortion and semidefinite programming gaps.

Contribution

It introduces a novel probabilistic method for analyzing metric spaces with quasisymmetric embeddings, refining existing rounding techniques and deriving bounds on Euclidean distortion and SDP gaps.

Findings

Largest Euclidean distortion of n-point subsets of ℓ₁ is Θ(√log n)

Integrality gap of Goemans–Linial SDP for Sparsest Cut is Θ(√log n)

Provides probabilistic guarantees for local growth in metric spaces

Abstract

We prove that if $(\mathcal{M},d)$ is an $n$-point metric space that embeds quasisymmetrically into a Hilbert space, then for every $\tau>0$ there is a random subset $\mathcal{Z}$ of $\mathcal{M}$ such that for any pair of points $x,y\in \mathcal{M}$ with $d(x,y)\ge \tau$, the probability that both $x\in \mathcal{Z}$ and $d(y,\mathcal{Z})\ge \beta\tau/\sqrt{1+\log (|B(y,\kappa \beta \tau)|/|B(y,\beta \tau)|)}$ is $\Omega(1)$, where $\kappa>1$ is a universal constant and $\beta>0$ depends only on the modulus of the quasisymmetric embedding. The proof relies on a refinement of the Arora--Rao--Vazirani rounding technique. Among the applications of this result is that the largest possible Euclidean distortion of an $n$-point subset of $\ell_1$ is $\Theta(\sqrt{\log n})$, and the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut problem on inputs of size $n$ is $\Theta(\sqrt{\log n})$. Multiple further applications are given.