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 ⊆ 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 ∑_e∈Hb(e) + ∑_t∈T-s dist(t, s) is minimize, 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 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) over the edges, 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(log³ 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(log² n), O(log⁴ n))-approximation for the shallow-light k-Steiner tree and an O(log⁴ n)-approximation for the buy-at-bulk k-Steiner tree problem.