We consider approximation algorithms for nonuniform 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 ( log h * log log h) for the uniform-demand case, and log D * 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 O(log 3 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 γ(n) we obtain an O(min{log 3 h * log D, log 5 h}) ratio approximation. Thus, for polynomial in n demands our ratio is O(log 4 h). This is achieved by a combinatorial (namely greedy) algorithm. For superpolynomial in n demands the ratio can be as bad as log 5 h and uses LP methods. Joint work with C. Chekuri, M. Hajiaghayi and M. Salavatipour.
The first polylogarithmic approximation algorithm for non-uniform (multi-commodity) buy-at-bulk network design problem was given in REF with an approximation ratio of O(log 4 n).