Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain

Abstract

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 $d^{\prime }$ 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 $(V,E,d^{\prime })$.