Advertisement

Distributed Local Approximation of the Minimum k-Tuple Dominating Set in Planar Graphs

  • Andrzej Czygrinow
  • Michal Hanćkowiak
  • Edyta Szymańska
  • Wojciech Wawrzyniak
  • Marcin Witkowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8878)

Abstract

In this paper we consider a generalization of the classical dominating set problem to the k-tuple dominating set problem (kMDS). For any positive integer k, we look for a smallest subset of vertices D ⊆ V with the property that every vertex in V ∖ D is adjacent to at least k vertices of D. We are interested in the distributed complexity of this problem in the model, where the nodes have no identifiers. The most challenging case is when k = 2, and for this case we propose a distributed local algorithm, which runs in a constant number of rounds, yielding a 7-approximation in the class of planar graphs. On the other hand, in the class of algorithms in which every vertex uses only its degree and the degree of its neighbors to make decisions, there is no algorithm providing a (5 − ε)-approximation of the 2MDS problem. In addition, we show a lower bound of (4 − ε) for the 2MDS problem even if unique identifiers are allowed.

For k ≥ 3, we show that for the problem kMDS in planar graphs, a trivial algorithm yields a k/(k − 2)-approximation. In the model with unique identifiers this, surprisingly, is optimal for k = 3,4,5, and 6, as we provide a matching lower bound.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Czygrinow, A., Hańćkowiak, M., Wawrzyniak, W.: Fast distributed approximations in planar graphs. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 78–92. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Diestel, R.: Graph Theory, 4th edn. Graduate texts in mathematics, vol. 173, pp. I–XVIII, 1–436. Springer (2012) ISBN 978-3-642-14278-9Google Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman (1979)Google Scholar
  4. 4.
    Göös, M., Hirvonen, J., Suomela, J.: Lower bounds for local approximation. J. ACM 60(5), 39 (2013)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Harary, F., Haynes, T.W.: Double domination in graphs. Ars Combinatoria 55, 201–213 (2000)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Hilke, M., Lenzen, C., Suomela, J.: Brief announcement: Local approximability of minimum dominating set on planar graphs. In: 33rd ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC 2014, Paris, France (July 2014)Google Scholar
  7. 7.
    Johnson, D.: Approximation Algorithms for Combinatorial Problems. Journal of Computer and System Sciences 9(3), 256–278 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Klasing, R., Laforest, C.: Hardness results and approximation algorithms of k-tuple domination in graphs. Information Processing Letters 89(2), 75–83 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Lenzen, C., Pignolet, Y.A., Wattenhofer, R.: Distributed minimum dominating set approximations in restricted families of graphs. Distrib. Comput. 26(2), 119–137 (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, Philadelphia (2000)CrossRefGoogle Scholar
  11. 11.
    Raz, R., Safra, S.: A Sub-Constant Error-Probability Low-Degree Test, and a Sub-Constant Error-Probability PCP Characterization of NP. In: Proc. 29th Symposium on Theory of Computing (STOC), pp. 475–484 (1997)Google Scholar
  12. 12.
    Wawrzyniak, W.: Brief announcement: A local approximation algorithm for MDS problem in anonymous planar networks. In: PODC 2013, pp. 406–408 (2013)Google Scholar
  13. 13.
    Wawrzyniak, W.: A strengthened analysis of a local algorithm for the minimum dominating set problem in planar graphs. Inf. Process. Lett. 114(3), 94–98 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andrzej Czygrinow
    • 1
  • Michal Hanćkowiak
    • 2
  • Edyta Szymańska
    • 2
  • Wojciech Wawrzyniak
    • 2
  • Marcin Witkowski
    • 2
  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland

Personalised recommendations