Stochastic Steiner Tree with Non-uniform Inflation

  • Anupam Gupta
  • MohammadTaghi Hajiaghayi
  • Amit Kumar
Conference paper

DOI: 10.1007/978-3-540-74208-1_10

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4627)
Cite this paper as:
Gupta A., Hajiaghayi M., Kumar A. (2007) Stochastic Steiner Tree with Non-uniform Inflation. In: Charikar M., Jansen K., Reingold O., Rolim J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg

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 cM and cT (the costs for Monday and Tuesday, respectively). We are also given a probability distribution π: 2V →[0,1] over subsets of V, and will be given a client setS 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) + ESπ[ cT( ET(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 cT = σ×cM 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(log4n) approximation guarantee given by Chekuri et al. (FOCS 2006).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Anupam Gupta
    • 1
  • MohammadTaghi Hajiaghayi
    • 2
  • Amit Kumar
    • 1
  1. 1.Computer Science Department, Carnegie Mellon University, Pittsburgh PA 15213 
  2. 2.Department of Computer Science and Engineering, Indian Institute of Technology, New Delhi, 110016India

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