D-width, metric embedding, and their connections

• Authors: Mohammad Ali Safari
• Venue: D-width, metric embedding, and their connections
• Year: 2007

Embedding between metric spaces is a very powerful algorithmic tool and has been used for finding good approximation algorithms for several problems. In particular, embedding to an ℓ1 norm has been used as the key step in an approximation algorithm for the sparsest cut problem. The sparsest cut problem, in turn, is the main ingredient of many algorithms that have a divide and conquer nature and are used in various fields. While every metric is embeddable into ℓ1 with distortion O(log n) [13], and the bound is tight [39], for special classes of metrics better bounds exist. Shortest path metrics for trees and outerplanar graphs are isometrically embeddable into ℓ 1 [41], Series-parallel graphs [28] and k-outerplanar graphs [19] (for constant k) are embeddable intoℓ 1 with constant distortion planar graphs and bounded tree-width graphs are conjectured to have constant distortion in embedding to ℓ1. Bounded tree-width graphs are one of most general graph classes on which several hard problems are tractable. We study the embedding of series-parallel graphs (or, more generally, graphs with tree-width two) into ℓ1 and also the embedding between two line metrics. We then move our attention to the generalization of tree-width to digraphs and hypergraphs and study several relevant problems.