Stochastic Steiner Tree with Non-uniform Inflation

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Abstract

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 c M and c T (the costs for Monday and Tuesday, respectively). We are also given a probability distribution π: 2 V →[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 E M on Monday, and after the client set S is revealed on Tuesday, it has to buy a (possibly empty) set of edges E T (S) so that the edges in E M  ∪ E T (S) connect all the nodes in S. The goal is to minimize the c M (E M ) + E Sπ [ c T ( E T (S) ) ].

We give the first poly-logarithmic approximation algorithm for this problem. 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 c T  = σ×c M for some constant σ ≥ 1 (i.e., the uniform case), or for the case when the goal was to find a tree spanning all the vertices but Tuesday’s costs were drawn from a given distribution \(\widehat{\pi}\) (the so-called “stochastic MST case”).

We complement our results by showing that our problem is at least as hard as the single-sink Cost-Distance problem (which is known to be Ω(loglogn) hard). Moreover, the requirement that Tuesday’s costs are fixed seems essential: if we allow Tuesday’s costs to dependent on the scenario as in stochastic MST, the problem becomes as hard as Label Cover (which is \(\Omega(2^{\log^{1-\varepsilon} n})\) -hard). As an aside, we also give an LP-rounding algorithm for the multi-commodity Cost-Distance problem, matching the O(log4 n) approximation guarantee given by Chekuri et al. (FOCS 2006).