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Approximations for node-weighted Steiner tree in unit disk graphs

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Abstract

Given a node-weighted connected graph and a subset of terminals, the problem node-weighted Steiner tree (NWST) seeks a lightest tree connecting a given set of terminals in a node-weighted graph. While NWST in general graphs are as hard as Set Cover, NWST restricted to unit-disk graphs (UDGs) admits constant-approximations. Recently, Zou et al. (Lecture notes in computer science, vol 5165, COCOA, 2008, pp 278–285) showed that any μ-approximation algorithm for the classical edge-weighted Steiner tree problem can be used to produce 2.5 μ-approximation algorithm for NWST restricted to UDGs. With the best known approximation bound 1.55 for the classical Steiner tree problem, they obtained an approximation bound 3.875 for NWST restricted to UDGs. In this paper, we present three approximation algorithms for NWST restricted to UDGs, the k-Restricted Relative Greedy Algorithm whose approximation bound converges to 1 + ln 5 ≈ 2.61 as k → ∞, the 3-Restricted Greedy Algorithm with approximation bound \({4\frac{1}{3}}\) , and the k-Restricted Variable Metric Algorithm whose approximation bound converges to 3.9334 as k → ∞.

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Authors and Affiliations

  1. Department of Computer Science, Illinois Institute of Technology, Chicago, USA

    X. Xu, H. Du & P.-J. Wan

  2. Institute of Theoretical Computer Science, Tsinghua University, Beijing, People’s Republic of China

    Y. Wang

  3. Department of Computer Science, University of Texas at Dallas, Richardson, USA

    F. Zou & W. Wu

  4. School of Mathematics and Statistics, Lanzhou University, Lanzhou, People’s Republic of China

    X. Li

Authors
  1. X. Xu
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  2. Y. Wang
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  3. H. Du
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  4. P.-J. Wan
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  5. F. Zou
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  6. X. Li
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  7. W. Wu
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Corresponding author

Correspondence to P.-J. Wan.

Additional information

X. Xu, H. Du, P.-J. Wan were supported in part by NSF under grant CNS-0831831.

Y. Wang was supported in part by the National Basic Research Program of China Grants 2007CB807900 and 2007CB807901, NSFC under Grant 60604033, and the Hi-Tech Research Development Program of China Grant 2006AA10Z216.

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Xu, X., Wang, Y., Du, H. et al. Approximations for node-weighted Steiner tree in unit disk graphs. Optim Lett 4, 405–416 (2010). https://doi.org/10.1007/s11590-010-0194-x

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  • DOI: https://doi.org/10.1007/s11590-010-0194-x

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