Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain

Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$ and vertex conductance $\Psi^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap $\gamma^*(P)$ satisfies $$\Psi^*(G)^{2}/\log\Delta \lesssim \gamma^*(P) \lesssim \Psi^*(G).$$ Our bound is optimal under the Small Set Expansion Hypothesis, and answers a question of Olesker-Taylor and Zanetti, who obtained such a result with $\log\Delta$ replaced by $\log|V|$. In order to obtain our result, we show how to embed a negative-type semi-metric $d$ defined on $V$ into a negative-type semi-metric $d'$ supported in $\mathbb{R}^{O(\log\Delta)}$, such that the (fractional) matching number of the weighted graph $(V,E,d)$ is approximately equal to that of $(V,E,d')$.