Maximum gradient embeddings and monotone clustering

Authors:
Manor Mendel;Assaf Naor
Affiliations:
Computer Science Division The Open University of Israel, P.O. Box 808, 1 University Rd., 43107, Raanana, Israel;Courant Institute New York University, 251 Mercer Street, 10012, New York, NY, USA
Venue:
Combinatorica
Year:
2010

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Maximum Gradient Embeddings and Monotone Clustering

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

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Abstract

Let (X,d X ) be an n-point metric space. We show that there exists a distribution **image** over non-contractive embeddings into trees f: X → T such that for every x ∈ X, **image** where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot 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.