Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain
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Let G = (V,E) be an undirected graph with maximum degree Δ and vertex conductance Ψ^*(G). We show that there exists a symmetric, stochastic matrix P, with off-diagonal entries supported on E, whose spectral gap γ^*(P) satisfies Ψ^*(G)^2/logΔ≲γ^*(P) ≲Ψ^*(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 logΔ 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(logΔ), such that the (fractional) matching number of the weighted graph (V,E,d) is approximately equal to that of (V,E,d').
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