Bo Brinkman
Adriana Karagiozova
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 ofvarious families of real-valued Lipschitz maps on certain metric spaces. We view these issues through the lens of shortest-path metricson series-parallel graphs, and we discussthe implications for a variety of well-known open problems. Our main results follow.
Every n-point series-parallel metric embeds into l1dom with O(√ log n) distortion, matchinga lower bound of Newman and Rabinovich. Our embeddings yield an O(√log n) approximation algorithm for vertex sparsestcut in such graphs, as well as an O(√log k) approximate max-flow/min-vertex-cut theorem for series-parallel instances withk terminals, improving over the O(log n) and O(log k) boundsfor general graphs.
Every n-point series-parallel metric embeds withdistortion D into l1d with d = n1/Ω(D2),matching the dimension reduction lower bound of Brinkman andCharikar.
There exists a constant C > 0 such that if (X,d) is aseries-parallel metric then for every stationary, reversible Markovchain Ztt=0∞ on X, we have for all t ≥ 0, E[d(Zt,Z0)2] ≤ Ct ·, E[d(Z0,Z1)2]. More generally, we show thatseries-parallel metrics have Markov type 2. This generalizesa result of Naor, Peres, Schramm, and Sheffield for trees.
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- Vertex cuts, random walks, and dimension reduction in series-parallel graphs
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