Skip to main content
SpringerLink
Log in

Approximations for node-weighted Steiner tree in unit disk graphs

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

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 → ∞.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borchers A., Du D.-Z.: The k-Steiner ratio in graphs. SIAM J. Comput. 26(3), 857–869 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berman P., Ramaiyer V.: Improved approximations for the Steiner tree problem. J. Algorithms 17(3), 381–408 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gröpl C., Hougardy S., Nierhoff T., Prömel H.J.: Approximation algorithms for the Steiner tree problem in graphs. In: Cheng, X., Du, D.-Z. (eds) Steiner trees in Industry, pp. 235–279. Kluwer, Dordrecht (2001)

    Google Scholar 

  4. Guha S., Khuller S.: Improved methods for approximating node weighted Steiner trees and connected dominating sets. Inf. Comput. 150(1), 57–74 (1999)

    Article  MathSciNet  Google Scholar 

  5. Klein P., Ravi R.: A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19(1), 104–115 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Papadimitriou C.H., Vazirani U.V.: On two geometric problems relating to the traveling salesman problem. J. Algorithms 5, 231–246 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  7. Robins G., Zelikovsky A.: Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math. 19(1), 122–134 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Wan, P.-J., Du, D.-Z., Pardalos, P., Wu, W.: Greedy approximations for minimum submodular cover with submodular cost. Comput. Optim. Appl. (2009) (to appear)

  9. Zelikovsky A.: The 11/6-approximation algorithm for the Steiner problem on networks. Algorithmica 9, 463–470 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Zelikovsky, A.: Better approximation bounds for the network and Euclidean Steiner tree problems. Technical Report CS-96-06, Department of Computer Science, University of Virginia (1996)

  11. Zou, F., Li, X., Kim, D., Wu, W.: Two Constant Approximation Algorithms for Node-Weighted Steiner tree in Unit Disk Graphs. Lecture Notes in Computer Science, vol. 5165, pp. 278–285. COCOA (2008)

Download references

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P.-J. Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. F. Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. X. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. W. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

Search

Navigation