In the buy-at-bulk k-Steiner tree (or rent-or-buy 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 rent 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 ∑
