Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G( V, E) with a set of terminals T subset of V including a particular vertex s called the root, and an integer k <= |T|. There are two cost functions on the edges of G, a buy cost b : E -> R+ and a distance cost r : E -> R+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost Sigma(e is an element of H) b(e) + Sigma(t is an element of T-s) dist(t, s) is minimized, where dist(t, s) is the distance from t to s in H with respect to the r cost. We present an O(log(4) n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b( e) and distance costs r( e), and an integer k. Our goal is to find a minimum cost ( under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an ( O( log n), O(log(3) n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least k/8 terminals. Using this we obtain an (O(log(2) n), O(log(4) n))-approximation algorithm for the shallow-light k-Steiner tree and an O(log(4) n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. Our results are recently used to give the first polylogarithmic approximation algorithm for the non-uniform multicommodity buy-at-bulk problem (Chekuri, C., et al. in Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS' 06), pp. 677-686, 2006).