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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Mendel, M., Naor, A. Maximum gradient embeddings and monotone clustering. Combinatorica 30, 581–615 (2010). https://doi.org/10.1007/s00493-010-2302-z
Mathematics Subject Classification (2000)