Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design

We consider approximation algorithms for non-uniform buy-at-bulk network design problems. The first non-trivial approximation algorithm for this problem is due to Charikar and Karagiozova (STOC 05); for an instance on h pairs their algorithm has an approximation guarantee of exp(O(radic(log h log log h)))for the uniform-demand case, and log D middot exp(O(radic(log h log log h))) for the general demand case, where D is the total demand. We improve upon this result, by presenting the first poly-logarithmic approximation for this problem. The ratio we obtain is O(log³ h middot min{log D, gamma(h²)}) where his the number of pairs and gamma(n) is the worst case distortion in embedding the metric induced by a n vertex graph into a distribution over its spanning trees. Using the best known upper bound on gamma(n) we obtain an O(min{log³ h middot log D, log⁵ h log log h}) ratio approximation. We also give poly-logarithmic approximations for some variants of the single-source problem that we need for the multicommodity problem