Maximum Gradient Embeddings and Monotone Clustering

Authors:
Manor Mendel;Assaf Naor
Affiliations:
The Open University of, Israel;Courant Institute,
Venue:
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
Year:
2007

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Abstract

Let (X,dX) be an n-point metric space. We show that there exists a distribution over non-contractive embeddings into trees f:X茂戮驴Tsuch that for every x茂戮驴 X, where Cis a universal constant. Conversely we show that the above quadratic dependence on logncannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.