Advertisement

Local PTAS for Independent Set and Vertex Cover in Location Aware Unit Disk Graphs

(Extended Abstract)
  • Andreas Wiese
  • Evangelos Kranakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5067)

Abstract

We present the first local approximation schemes for maximum independent set and minimum vertex cover in unit disk graphs. In the graph model we assume that each node knows its geographic coordinates in the plane (location aware nodes). Our algorithms are local in the sense that the status of each node v (whether or not v is in the computed set) depends only on the vertices which are a constant number of hops away from v. This constant is independent of the size of the network. We give upper bounds for the constant depending on the desired approximation ratio. We show that the processing time which is necessary in order to compute the status of a single vertex is bounded by a polynomial in the number of vertices which are at most a constant number of vertices away from it. Our algorithms give the best possible approximation ratios for this setting.

The technique which we use to obtain the algorithm for vertex cover can also be employed for constructing the first global PTAS for this problem in unit disk graph which does not need the embedding of the graph as part of the input.

Keywords

Approximation Ratio Local Algorithm Class Number Vertex Cover Unit Disk Graph 
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.
    Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–45 (1985)MathSciNetGoogle Scholar
  2. 2.
    Basagni, S.: Finding a maximal weighted independent set in wireless networks. Telecommunication Systems 18(1-3), 155–168 (2001)MATHCrossRefGoogle Scholar
  3. 3.
    Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Computational Geometry. Theory and Applications 9(1-2), 3–24 (1998)MATHMathSciNetGoogle Scholar
  4. 4.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math 86(1-3), 165–177 (1990)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Czyzowicz, J., Dobrev, S., Fevens, T., González-Aguilar, H., Kranakis, E., Opatrny, J., Urrutia, J.: Local algorithms for dominating and connected dominating sets of unit disc graphs with location aware nodes. In: Proceedings of LATIN 2008. LNCS, vol. 4957 (2008)Google Scholar
  6. 6.
    Dinur, I., Safra, S.: The importance of being biased. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC 2002), May 19–21, 2001, pp. 33–42. ACM Press, New York (2002)CrossRefGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness (1979)Google Scholar
  8. 8.
    Håstad, J.: Clique is hard to approximate within n 1 − ε. Electronic Colloquium on Computational Complexity (ECCC) 4(38) (1997)Google Scholar
  9. 9.
    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. J. Algorithms 26(2), 238–274 (1998)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: Unit disk graph approximation. In: DIALM-POMC 2004: Proceedings of the 2004 joint workshop on Foundations of mobile computing, pp. 17–23. ACM Press, New York (2004)CrossRefGoogle Scholar
  11. 11.
    Kuhn, F., Nieberg, T., Moscibroda, T., Wattenhofer, R.: Local approximation schemes for ad hoc and sensor networks. In: DIALM-POMC 2005: Proceedings of the 2005 joint workshop on Foundations of mobile computing, pp. 97–103. ACM Press, New York (2005)CrossRefGoogle Scholar
  12. 12.
    Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25(1), 59–68 (1995)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nieberg, T., Hurink, J.L.: A PTAS for the minimum dominating set problem in unit disk graphs. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 296–306. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Nieberg, T., Hurink, J.L., Kern, W.: A robust PTAS for maximum weight independent sets in unit disk graphs. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 214–221. Springer, Heidelberg (2004)Google Scholar
  15. 15.
    Wiese, A., Kranakis, E.: Local PTAS for Dominating and Connected Dominating Set in Location Aware UDGs (to appear, 2007)Google Scholar
  16. 16.
    Wu, J., Li, H.: On calculating connected dominating set for efficient routing in ad hoc wireless networks. In: DIAL-M, pp. 7–14. ACM, New York (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andreas Wiese
    • 1
  • Evangelos Kranakis
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinGermany
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations