Local global tradeoffs in metric embeddings

Suppose that every k points in a n point metric space X are D-distortion embeddable into ℓ1. We give upper and lower bounds on the distortion required to embed the entire space X into ℓ1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded into ℓ1 with distortion O(D X log(n/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from 1. For D = 1 + δ we give a lower bound of ω(log(n/k)/log(1/δ)); and for D =1, we give a lower bound of ω(logn/(log k + loglogn)). Our bounds significantly improve on the results of Arora et al. who initiated a study of these questions.