# Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low-Dimensional Spaces

## TL;DR

This paper extends known embedding results to approximately low-dimensional points, showing that such embeddings into  can be achieved with distortion depending on the approximate dimension, which improves bounds on the Sparsest Cut problem.

## Contribution

It proves a robust analogue of low-dimensional embeddings into  with distortion depending on the approximate dimension, improving previous bounds on the integrality gap for Sparsest Cut.

## Key findings

- Embedding into  with (r) distortion under approximate low-dimensionality.
- Improved bound on the integrality gap of Goemans-Linial SDP for graphs with spectral properties.
- Enhanced understanding of embeddings and SDP relaxations in graph partitioning problems.

## Abstract

We consider the problem of embedding a finite set of points $\{x_1, \ldots, x_n\} \in \mathbb{R}^d$ that satisfy $\ell_2^2$ triangle inequalities into $\ell_1$, when the points are approximately low-dimensional. Goemans (unpublished, appears in a work of [Magen and Moharammi, 2008]) showed that such points residing in \emph{exactly} $d$ dimensions can be embedded into $\ell_1$ with distortion at most $\sqrt{d}$. We prove the following robust analogue of this statement: if there exists a $r$-dimensional subspace $\Pi$ such that the projections onto this subspace satisfy $\sum_{i,j \in [n]}\Vert \Pi x_i - \Pi x_j \Vert _2^2 \geq \Omega(1) \sum_{i,j \in [n]}\Vert x_i - x_j \Vert _2^2$, then there is an embedding of the points into $\ell_1$ with $O(\sqrt{r})$ average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is $O(\sqrt{r})$ on graphs $G$ whose $r$-th smallest normalized eigenvalue of the Laplacian satisfies $\lambda_r(G)/n \geq \Omega(1)\Phi_{SDP} (G)$. Our result improves upon the previously known bound of $O(r)$ on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in the previous works of [Deshpande and Venkat, 2014] and [Deshpande, Harsha and Venkat, 2016].