Multi-Embedding of Metric Spaces

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of its size. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size. We make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric. We define "multi-embeddings" of metric spaces in which a point is mapped onto a set of points, while keeping the target metric of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In particular, for expander graphs, we are able to obtain constant distortion embeddings into trees in contrast with the Omega(log n) lower bound for all previous notions of embeddings. We demonstrate the algorithmic application of the new embeddings for two optimization problems: group Steiner tree and metrical task systems

Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of B\v{a}doiu, Indyk, and Sidiropoulos (2007) and B\v{a}doiu, Demaine, Hajiaghayi, Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of metric relaxed minor and show that if G contains an alpha-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is at meast alpha. Then, for H=K_{2,3}, we present an algorithm which either finds an alpha-relaxed minor, or produces an O(alpha)-embedding into an outerplanar metric.Comment: 27 pages, 4 figires, extended abstract to appear in the proceedings of APPROX-RANDOM 201

Maximum gradient embeddings and monotone clustering

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.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica