Skip to main content
SpringerLink
Log in

On connected domination in unit ball graphs

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

Abstract

Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio approximation algorithm for the minimum connected dominating set problem in unit ball graphs and then introduces and studies the edge-weighted bottleneck connected dominating set problem, which seeks a minimum edge weight in the graph such that the corresponding bottleneck subgraph has a connected dominating set of size k. In wireless network applications this problem can be used to determine an optimal transmission range for a network with a predefined size of the virtual backbone. We show that the problem is hard to approximate within a factor better than 2 in graphs whose edge weights satisfy the triangle inequality and provide a 3-approximation algorithm for such graphs. We also show that for fixed k the problem is polynomially solvable in unit disk and unit ball graphs.

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.

Similar content being viewed by others

References

  1. Breu H., Kirkpatrick D.G.: Unit disk graph recognition is NP-hard. Comput. Geom. Theory Appl. 9, 9–31 (1998)

    MathSciNet  Google Scholar 

  2. Cheng X., Huang X., Li D., Wu W., Du D.-Z.: Polynomial-time approximation scheme for minimum connected dominating set in ad hoc wireless networks. Networks 42, 202–208 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Clark B.N., Colbourn C.J., Johnson D.S.: Unit disk graphs. Discrete Math. 86, 165–177 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Conway J.H., Sloane N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, New York (1999)

    MATH  Google Scholar 

  5. Deng J., Han Y.S., Chen P., Varshney P.K.: Optimal transmission range for wireless ad hoc networks based on energy efficiency. IEEE Trans. Commun. 55(9), 1772–1781 (2007)

    Article  Google Scholar 

  6. Guha S., Khuller S.: Approximation algorithms for connected dominating sets. Algorithmica 20(4), 374–387 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hliněný, P.: Touching graphs of unit balls. In: Proceedings of the 5th International Symposium on Graph Drawing. Lecture Notes in Computer Science, vol. 1353, pp. 350–358 (1997)

  8. Hochbaum D.S., Shmoys D.B.: A unified approach to approximation algorithms for bottleneck problems. J. ACM 33(3), 533–550 (1986)

    Article  MathSciNet  Google Scholar 

  9. Hunt H.B. III, Marathe M.V., Radhakrishnan V., Ravi S.S., Rosenkrantz D.J., Stearns R.E.: NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms 26, 238–274 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Li L., Halpern J.Y.: Minimum-energy mobile wireless networks revisited. IEEE Int. Conf. Commun. 1, 278–283 (2001)

    Google Scholar 

  11. Rodoplu V., Meng T.H.: Minimum energy mobile wireless networks. IEEE J. Selected Areas Commun. 17(8), 1333–1344 (1999)

    Article  Google Scholar 

  12. Sanchez, M., Manzoni, P., Haas, Z.J.: Determination of critical transmission range in ad hoc networks. In: Multiaccess Mobility and Teletraffic for Wireless Communications 1999 Workshop (1999)

  13. Lee C.-M., Kloks T., Kratsch D., Liu J.: Improved bottleneck domination algorithms. Discrete Appl. Math. 154, 1578–1592 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wu, J., Li, H.: On calculating connected dominating set for efficient routing in ad hoc wireless networks. In: Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (1999)

  15. Wu J., Wu B.: A transmission range reduction scheme for power-aware broadcasting in ad hoc networks using connected dominating sets. IEEE Vehicular Technol. Conf. 5, 2906–2909 (2003)

    Google Scholar 

  16. Yen W.C.K.: Bottleneck domination and bottleneck independent domination on graphs. J. Inf. Sci. Eng. 18, 311–331 (2002)

    MathSciNet  Google Scholar 

  17. Zhang Z., Gao X., Wu W., Du D.: PTAS for minimum connected dominating set in unit ball graph. Lecture Notes Comput. Sci. (LNCS) 5258, 154–161 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX, 77843-3131, USA

    Sergiy Butenko, Sera Kahruman-Anderoglu & Oleksii Ursulenko

Authors
  1. Sergiy Butenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sera Kahruman-Anderoglu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oleksii Ursulenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergiy Butenko.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Butenko, S., Kahruman-Anderoglu, S. & Ursulenko, O. On connected domination in unit ball graphs. Optim Lett 5, 195–205 (2011). https://doi.org/10.1007/s11590-010-0211-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-010-0211-0

Keywords

Search

Navigation