Stochastic Steiner Tree with Non-Uniform Inflation

We study the Steiner Tree problem in the model of two-stage stochastic optimization with non-uniform inflation factors, and give a poly-logarithmic approximation factor for this problem. In this problem, we are given a graph G = (V,E), with each edge having two costs cM and cT (the costs for Monday and Tuesday, respectively). We are also given a probability distribution pi: 2V → [0, 1] over subsets of V, and will be given a client set S drawn from this distribution on Tuesday. The algorithm has to buy a set of edges EM on Monday, and after the client set S is revealed on Tuesday, it has to buy a (possibly empty) set of edges ET (S) so that the edges in EM ∪ ET (S) connect all the nodes in S. The goal is to minimize the cM (EM) +ES←pi [ cT (ET (S) )]. We give the first poly-logarithmic approximation algorithm for this prob-lem. Our algorithm builds on the recent techniques developed by Chekuri et al. (FOCS 2006) for multi-commodity Cost-Distance. Previously, the problem had been studied for the cases when cT = σ × cM for som