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We consider the single-source (or single-sink) buy-at-bulk problem with anunknown concave cost function. We want to route a set of demands along a graphto or from a designated root node, and the cost of routing x units of flowalong an edge is proportional to some concave, non-decreasing function f suchthat f(0) = 0. We present a polynomial time algorithm that finds a distributionover trees such that the expected cost of a tree for any f is within anO(1)-factor of the optimum cost for that f. The previous best simultaneousapproximation for this problem, even ignoring computation time, was O(log |D|),where D is the multi-set of demand nodes.  We design a simple algorithmic framework using the ellipsoid method thatfinds an O(1)-approximation if one exists, and then construct a separationoracle using a novel adaptation of the Guha, Meyerson, and Munagala algorithmfor the single-sink buy-at-bulk problem that proves an O(1) approximation ispossible for all f. The number of trees in the support of the distributionconstructed by our algorithm is at most 1+log |D|.

An Oblivious O(1)-Approximation for Single Source Buy-at-Bulk