Maximum gradient embeddings and monotone clustering
Manor Mendel, Assaf Naor
Abstract
Let (X,dX) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio dT(f(x),f(y)) / dX(x,y) is at most C (log n)2, 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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