Advertisement

Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Disk Graphs

  • Christoph Ambühl
  • Thomas Erlebach
  • Matúš Mihalák
  • Marc Nunkesser
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4110)

Abstract

For a given graph with weighted vertices, the goal of the minimum-weight dominating set problem is to compute a vertex subset of smallest weight such that each vertex of the graph is contained in the subset or has a neighbor in the subset. A unit disk graph is a graph in which each vertex corresponds to a unit disk in the plane and two vertices are adjacent if and only if their disks have a non-empty intersection. We present the first constant-factor approximation algorithm for the minimum-weight dominating set problem in unit disk graphs, a problem motivated by applications in wireless ad-hoc networks. The algorithm is obtained in two steps: First, the problem is reduced to the problem of covering a set of points located in a small square using a minimum-weight set of unit disks. Then, a constant-factor approximation algorithm for the latter problem is obtained using enumeration and dynamic programming techniques exploiting the geometry of unit disks. Furthermore, we also show how to obtain a constant-factor approximation algorithm for the minimum-weight connected dominating set problem in unit disk graphs.

Keywords

Unit Disk Steiner Tree Internal Vertex Unit Disk Graph Disk Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alzoubi, K., Wan, P.-J., Frieder, O.: Message-optimal connected dominating sets in mobile ad hoc networks. In: Proceedings of the 3rd ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc 2002), pp. 157–164 (2002)Google Scholar
  2. 2.
    Ambühl, C., Erlebach, T., Mihal’ák, M., Nunkesser, M.: Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. Research Report CS-06-008, Department of Computer Science, University of Leicester (June 2006)Google Scholar
  3. 3.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41(1), 153–180 (1994); Extended abstract published in the proceedings of FOCS 1983, pp. 265–273 (1983)MATHCrossRefGoogle Scholar
  4. 4.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete & Computational Geometry 14(4), 463–479 (1995)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Calinescu, G., Mandoiu, I., Wan, P.-J., Zelikovsky, A.: Selecting forwarding neighbors in wireless ad hoc networks. Mobile Networks and Applications 9(2), 101–111 (2004)CrossRefGoogle Scholar
  6. 6.
    Cheng, X., Huang, X., Li, D., Wu, W., Du, D.-Z.: A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks. Networks 42(4), 202–208 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Mathematics 86, 165–177 (1990)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feige, U.: A threshold of ln n for approximating set cover. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC 1996), pp. 314–318 (1996)Google Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)MATHGoogle Scholar
  10. 10.
    Guha, S., Khuller, S.: Improved methods for approximating node weighted Steiner trees and connected dominating sets. Information and Computation 150(1), 57–74 (1999)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. Journal of the ACM 32(1), 130–136 (1985)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hunt III, H.B., 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. Journal of Algorithms 26(2), 238–274 (1998)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lichtenstein, D.: Planar formulae and their uses. SIAM Journal on Computing 11(2), 329–343 (1982)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    van Leeuwen, E.J.: Approximation algorithms for unit disk graphs. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 351–361. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar
  17. 17.
    Wang, Y., Li, X.-Y.: Distributed low-cost backbone formation for wireless ad hoc networks. In: Proceedings of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc 2005), pp. 2–13 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christoph Ambühl
    • 1
  • Thomas Erlebach
    • 1
  • Matúš Mihalák
    • 1
  • Marc Nunkesser
    • 2
  1. 1.Department of Computer ScienceUniversity of Liverpool 
  2. 2.Institute of Theoretical Computer ScienceETH Zürich 

Personalised recommendations