An Improved Approximation to Sparsest Cut

In this paper, we study metrics of negative type, which are metrics $(V,d)$ such that $\sqrt{d}$ is an Euclidean metric; these metrics are thus also known as "${\ell }_2$-squared" metrics. We show how to embed $n$-point negative-type metrics into Euclidean space ${\ell }_2$ with distortion $D=O\left({log}^{3/4}n\right)$. This embedding result, in turn, implies an $O\left({log}^{3/4}k\right)$-approximation algorithm for the Sparsest Cut problem with non-uniform demands. Another corollary we obtain is that $n$-point subsets of ${\ell }_1$ embed into ${\ell }_2$ with distortion $O\left({log}^{3/4}n\right)$.