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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 costof 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 singletree T such that for all f the cost f(T) is a 47.45-approximation of theoptimal cost for that f. This is within a factor of 2.33 of the bestapproximation ratio currently achievable when the tree can be optimized for aspecific function. Trees achieving simultaneous O(1)-approximations for allconcave functions were previously not known to exist regardless of computationtime.

One Tree Suffices: A Simultaneous O(1)-Approximation for Single-Sink Buy-at-Bulk