Maximum gradient embeddings and monotone clustering

TL;DR

This paper introduces maximum gradient embeddings into trees for metric spaces, providing a probabilistic framework that enables improved approximation algorithms for monotone clustering problems like fault-tolerant k-median.

Contribution

It presents the concept of maximum gradient embeddings into trees with provable bounds, advancing the design of approximation algorithms for monotone clustering problems.

Findings

Existence of distributions over non-contractive tree embeddings with bounded expected maximum gradient.

Quadratic dependence on log n in the embedding bound is shown to be tight.

Framework enables approximation algorithms for various monotone clustering problems.

Abstract

Let (X,d_X) 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 d_T(f(x),f(y)) / d_X(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.