Local Global Tradeoffs in Metric Embeddings
Abstract
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.
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Copyright © 2010 Society for Industrial and Applied Mathematics.
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Submitted: 2 January 2008
Accepted: 13 April 2009
Published online: 30 April 2010
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