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Maximum gradient embeddings and monotone clustering

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Abstract

Let (X,d X ) be an n-point metric space. We show that there exists a distribution over non-contractive embeddings into trees f: X → T such that for every x ∈ X, 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.

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Authors and Affiliations

  1. Computer Science Division The Open University of Israel, 1 University Rd., P.O. Box 808, Raanana, 43107, Israel

    Manor Mendel

  2. Courant Institute New York University, 251 Mercer Street, New York, NY, 10012, USA

    Assaf Naor

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  1. Manor Mendel
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  2. Assaf Naor
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Correspondence to Manor Mendel.

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Mendel, M., Naor, A. Maximum gradient embeddings and monotone clustering. Combinatorica 30, 581–615 (2010). https://doi.org/10.1007/s00493-010-2302-z

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  • DOI: https://doi.org/10.1007/s00493-010-2302-z

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