Vertex cuts, random walks, and dimension reduction in series-parallel graphs

Abstract

We consider questions about vertex cuts in graphs, random walks in metric spaces, and dimension reduction in L1 and L2; these topics are intimately connected because they can each be reduced to the existence of various families of realvalued Lipschitz maps on certain metric spaces. We view these issues through the lens of shortest-path metrics on series-parallel graphs, and we discuss the implications for a variety of well-known open problems. Our main results follow.-Every n-point series-parallel metric embeds into dom 1 with O( √ log n) distortion, matching a lower bound of Newman and Rabinovich. Our embeddings yield an O( √ log n) approximation algorithm for vertex sparsest cut in such graphs, as well as an O( √ log k) approximate max-flow/min-vertexcut theorem for series-parallel instances with k terminals, improving over the O(log n) and O(log k) bounds for general graphs.-Every n-point series-parallel metric embeds with distortion D into d 1 with d = n 1/Ω(D 2 ) , matching the dimension reduction lower bound of Brinkman and Charikar.-There exists a constant C > 0 such that if (X, d) is a series-parallel metric then for every stationary, reversible Markov chain {Zt} ∞ t=0 on X, we have for all t ≥ 0,More generally, we show that series-parallel metrics have Markov type 2. This generalizes a result of Naor, Peres, Schramm, and Sheffield for trees. * brinkmwj@muohio.edu †

Coarse Differentiation and Multi-flows in Planar Graphs

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Metric Dimension Reduction: A Snapshot of the Ribe Program

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Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

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