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 nontrivial 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(√log h log log h))for the uniform-demand case, and log · exp(O(√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 0(log³ h · min{log D, γ(h ²)}) where h is the number of pairs and γ(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 7(71) we obtain an O(min{log³ h· log D, log⁵ h log log h}) ratio approximation. We also give poly-logarithmic approximations for some variants of the singe-source problem that we need for the multicommodity problem.