Linear-Time Computation of a Linear Problem Kernel for Dominating Set on Planar Graphs

  • René van Bevern
  • Sepp Hartung
  • Frank Kammer
  • Rolf Niedermeier
  • Mathias Weller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7112)

Abstract

We present a linear-time kernelization algorithm that transforms a given planar graph G with domination number γ(G) into a planar graph G′ of size O(γ(G)) with γ(G) = γ(G′). In addition, a minimum dominating set for G can be inferred from a minimum dominating set for G′. In terms of parameterized algorithmics, this implies a linear-size problem kernel for the NP-hard Dominating Set problem on planar graphs, where the kernelization takes linear time. This improves on previous kernelization algorithms that provide linear-size kernels in cubic time.

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References

  1. 1.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for Dominating Set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for dominating set. J. ACM 51(3), 363–384 (2004)MathSciNetMATHGoogle Scholar
  3. 3.
    Alber, J., Betzler, N., Niedermeier, R.: Experiments on data reduction for optimal domination in networks. Ann. Oper. Res. 146(1), 105–117 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)MathSciNetMATHGoogle Scholar
  5. 5.
    Bateni, M., Hajiaghayi, M., Marx, D.: Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth. In. In: Proc. 42th STOC, pp. 211–220. ACM Press (2010)Google Scholar
  6. 6.
    Bodlaender, H.L.: Kernelization: New Upper and Lower Bound Techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) kernelization. In: Proc. 50th FOCS, pp. 629–638. IEEE (2009)Google Scholar
  8. 8.
    Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: Lower bounds and upper bounds on kernel size. SIAM J. Comput. 37(4), 1077–1106 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chor, B., Fellows, M., Juedes, D.W.: Linear Kernels in Linear Time, or How to Save k Colors in O(n 2) Steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Dorn, F., Fomin, F.V., Thilikos, D.M.: Subexponential parameterized algorithms. Computer Science Review 2(1), 29–39 (2008)CrossRefMATHGoogle Scholar
  11. 11.
    Fellows, M.R., Rosamond, F.A., Fomin, F.V., Lokshtanov, D., Saurabh, S., Villanger, Y.: Local search: Is brute-force avoidable? In: Proc. 21st IJCAI, pp. 486–491 (2009)Google Scholar
  12. 12.
    Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: Proc. 21st SODA, pp. 503–510. ACM/SIAM (2010)Google Scholar
  13. 13.
    Guo, J., Niedermeier, R.: Linear Problem Kernels for NP-Hard Problems on Planar Graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 375–386. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  15. 15.
    Hagerup, T.: Linear-time kernelization for planar dominating set. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 181–193. Springer, Heidelberg (2012)Google Scholar
  16. 16.
    Wang, J., Yang, Y., Guo, J., Chen, J.: Linear Problem Kernels for Planar Graph Problems with Small Distance Property. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 592–603. Springer, Heidelberg (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • René van Bevern
    • 1
  • Sepp Hartung
    • 1
  • Frank Kammer
    • 2
  • Rolf Niedermeier
    • 1
  • Mathias Weller
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany
  2. 2.Institut für InformatikUniversität AugsburgAugsburgGermany

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