Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain

Let G = (V, E) be an undirected graph with maximum degree. 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 less than or similar to gamma*(P) less than or similar to 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 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 ').