A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Approximation algorithms for the 0-extension problem
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Lectures on Discrete Geometry
A tight bound on approximating arbitrary metrics by tree metrics
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
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
Local versus global properties of metric spaces
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Metric structures in L1: dimension, snowflakes, and average distortion
European Journal of Combinatorics
Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Local embeddings of metric spaces
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Reconstructing approximate tree metrics
Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computing
Linear programming relaxations of maxcut
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Integrality gaps for Sherali-Adams relaxations
Proceedings of the forty-first annual ACM symposium on Theory of computing
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Suppose that every $k$ points in a $n$ point metric space $X$ are $D$-distortion embeddable into $\ell_1$. We give upper and lower bounds on the distortion required to embed the entire space $X$ into $\ell_1$. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that $X$ can be embedded into $\ell_1$ with distortion $O(D\times\log(n/k))$. Moreover, we give a lower bound showing that this result is tight if $D$ is bounded away from 1. For $D=1+\delta$ we give a lower bound of $\Omega(\log(n/k)/\log(1/\delta))$; and for $D=1$, we give a lower bound of $\Omega(\log n/(\log k+\log\log n))$. Our bounds significantly improve on the results of Arora et al. who initiated a study of these questions.