Theory of Computing ------------------- Title : One Tree Suffices: A Simultaneous $O(1)$-Approximation for Single-Sink Buy-at-Bulk Authors : Ashish Goel and Ian Post Volume : 8 Number : 15 Pages : 351-368 URL : https://theoryofcomputing.org/articles/v008a015 Abstract -------- We study the single-sink buy-at-bulk problem with an unknown cost function. We wish to route flow from a set of demand nodes to a root node, where the cost of routing x total flow along an edge is proportional to f(x) for some concave, non-decreasing function f satisfying f(0)=0. We present a simple, fast, combinatorial algorithm that takes a set of demands and constructs a single tree T such that for all f the cost f(T) is a 47.45-approximation of the optimal cost for that particular f. This is within a factor of 2.33 of the best approximation ratio currently achievable when the tree can be optimized for a specific function. Trees achieving simultaneous O(1)-approximations for all concave functions were previously not known to exist regardless of computation time.