Abstract 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 aparticular vertex s called the root, and an integer k < = |T |. There are two cost functions on theedges 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 Pe2H b(e)+Pt2T-s dist(t, s) is min-imized, where dist(t, s) is the distance from t to s in H with respect to the r cost. We present an O(log4 n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. The second andclosely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In theshallow-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 thatthe 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 solutionhas at least k 8 terminals. Using this we obtain an (O(log 2 n), O(log4 n))-approximation algorith
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