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Stochastic Steiner Tree with Non-uniform Inflation

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2007, RANDOM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4627))

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: 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 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 S ←π [ 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).

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Authors and Affiliations

  1. Computer Science Department, Carnegie Mellon University, Pittsburgh PA 15213, USA

    Anupam Gupta & Amit Kumar

  2. Department of Computer Science and Engineering, Indian Institute of Technology, New Delhi, 110016, India

    MohammadTaghi Hajiaghayi

Authors
  1. Anupam Gupta
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  2. MohammadTaghi Hajiaghayi
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  3. Amit Kumar
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Editor information

Editors and Affiliations

  1. Computer Science Dept., Princeton University, 35 Olden Street, NJ 08540-5233, Princeton, USA

    Moses Charikar

  2. Institut für Informatik, Christian-Albrechts-Universität zu Kiel,, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Omer Reingold

  4. Centre Universitaire d’Informatique,, Battelle bâtiment A, route de Drize 7, 1227, Geneva, Carouge, Switzerland

    José D. P. Rolim

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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. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_10

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  • DOI: https://doi.org/10.1007/978-3-540-74208-1_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74207-4

  • Online ISBN: 978-3-540-74208-1

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