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 $T\subseteq V$ including a particular vertex $s$ called the root, and an integer $k\leq |T|$. There are two cost functions on the edges of $G$, a buy cost $b:E\longrightarrow ... more >>>