Local Global Tradeoffs in Metric Embeddings

Suppose that every k points in a metric space X are D-distortion embeddable into _{lscr 1}. We give upper and lower bounds on the distortion required to embed the entire space X into _{lscr 1}. 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 _{lscr 1} with distortion O(D times log(|X|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from I. For D = 1 + delta we give a lower bound of Omega(log(|X|/k/ log( 1/delta)); and for D = 1, we give a lower bound of Omega( log |X|/(log k +log log | X|)). Our bounds significantly improve on the results of Arora, Jjovdsz, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.