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Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees

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Abstract

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G(V,E) with a set of terminals T⊆V including a particular vertex s called the root, and an integer k≤|T|. There are two cost functions on the edges of G, a buy cost b:E→ℝ+ and a distance cost r:E→ℝ+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑e∈H b(e)+∑t∈T−sdist(t,s) is minimized, where dist(t,s) is the distance from t to s in H with respect to the r cost. We present an O(log 4 n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an (O(log n),O(log 3 n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least \(\frac{k}{8}\) terminals. Using this we obtain an (O(log 2 n),O(log 4 n))-approximation algorithm for the shallow-light k-Steiner tree and an O(log 4 n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. Our results are recently used to give the first polylogarithmic approximation algorithm for the non-uniform multicommodity buy-at-bulk problem (Chekuri, C., et al. in Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pp. 677–686, 2006).

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Authors and Affiliations

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

    Mohammad Taghi Hajiaghayi

  2. Department of Computer Science, Rutgers University-Camden, Camden, NJ, USA

    Guy Kortsarz

  3. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Mohammad R. Salavatipour

Authors
  1. Mohammad Taghi Hajiaghayi
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  2. Guy Kortsarz
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  3. Mohammad R. Salavatipour
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Correspondence to Mohammad Taghi Hajiaghayi.

Additional information

A preliminary version of this paper appeared in the Proceedings of 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) 2006, LNCS 4110, pp. 153–163, 2006.

M.T. Hajiaghayi supported in part by IPM under grant number CS1383-2-02.

M.R. Salavatipour supported by NSERC grant No. G121210990, and a faculty start-up grant from University of Alberta.

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Hajiaghayi, M.T., Kortsarz, G. & Salavatipour, M.R. Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees. Algorithmica 53, 89–103 (2009). https://doi.org/10.1007/s00453-007-9013-x

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  • DOI: https://doi.org/10.1007/s00453-007-9013-x

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