Maximum Gradient Embeddings and Monotone Clustering

abstract Manor Mendel1 and Assaf Naor2 1 The Open University of Israel 2 Courant Institute 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 ∈ X, ED  max y∈X\{x} dT (f(x), f(y)) dX (x, y)  ≤ 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.

• Type: Conference
• Year: 2007
• Where: APPROX
• Authors: Manor Mendel, Assaf Naor
• Keywords: Algorithms, APPROX 2007, Maximum Gradient Embeddings, N-point Metric Space, Non-contractive Embeddings