NDLI: Local Global Tradeoffs in Metric Embeddings

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Makarychev, Konstantin Makarychev, Yury Charikar, Moses
Copyright Year	2010
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.
Starting Page	2487
Ending Page	2512
Page Count	26
File Format	PDF
ISSN	00975397
DOI	10.1137/070712080
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6 (Special Section On Foundations of Computer Science)
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-04-30
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Distance in graphs Approximation algorithms Linear programming lift and project Special maps on metric spaces global local properties metric embeddings
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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