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 TV 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 ∑ _{e H} b(e)+ ∑ _{t 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 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).