Asymptotic theory of finite dimensional normed spaces
Asymptotic theory of finite dimensional normed spaces
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Improved bandwidth approximation for trees and chordal graphs
Journal of Algorithms
Lectures on Discrete Geometry
Dimension Reduction in the \ell _1 Norm
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On the impossibility of dimension reduction in l1
Journal of the ACM (JACM)
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
How to Play Unique Games Using Embeddings
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Probability: Theory and Examples
Probability: Theory and Examples
Coarse Differentiation and Multi-flows in Planar Graphs
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
A node-capacitated okamura-seymour theorem
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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.