Cuts, trees and l1-embeddings of graphs

Motivated by many recent algorithmic applications, the paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred where the graph is embedded into l₁ space. The main results are: 1. Explicit constant-distortion embeddings of all series parallel graphs, and all graphs with bounded Euler number. These are thus the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, we obtain algorithms to approximate the sparsest cut in such graphs to within a constant factor. 2) A constant-distortion embedding of outerplanar graphs into the restricted class of l₁-metrics known as “dominating tree metrics”. We also show a lower bound of Ω(log n) on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics. This shows, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low tree width, and excludes the possibility of using them to explore the finer structure of l₁-embeddability