A Parameterized Route to Exact Puzzles: Breaking the 2n-Barrier for Irredundance

(Extended Abstract)
  • Daniel Binkele-Raible
  • Ljiljana Brankovic
  • Henning Fernau
  • Joachim Kneis
  • Dieter Kratsch
  • Alexander Langer
  • Mathieu Liedloff
  • Peter Rossmanith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)

Abstract

The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G) respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph parameters. It is a long-standing open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time less than the trivial Ω(2n) enumeration barrier. We solve this open problem by devising parameterized algorithms for the duals of the natural parameterizations of the problems with running times faster than \(\mathcal{O}^*(4^{k})\). For example, we present an algorithm running in time \(\mathcal{O}^*(3.069^{k}))\) for determining whether IR(G) is at least n − k. Although the corresponding problem has been shown to be in FPT by kernelization techniques, this paper offers the first parameterized algorithms with an exponential dependency on the parameter in the running time. Furthermore, these seem to be the first examples of a parameterized approach leading to a solution to a problem in exponential time algorithmics where the natural interpretation as exact exponential-time algorithms fails.

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References

  1. 1.
    Favaron, O., Haynes, T.W., Hedetniemi, S.T., Henning, M.A., Knisley, D.J.: Total irredundance in graphs. Discrete Mathematics 256(1-2), 115–127 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Allan, R.B., Laskar, R.: On domination and independent domination numbers of a graph. Discrete Mathematics 23(2), 73–76 (1978)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Favaron, O.: Two relations between the parameters of independence and irredundance. Discrete Mathematics 70(1), 17–20 (1988)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Fellows, M.R., Fricke, G., Hedetniemi, S.T., Jacobs, D.P.: The private neighbor cube. SIAM J. Discrete Math. 7(1), 41–47 (1994)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hedetniemi, S.T., Laskar, R., Pfaff, J.: Irredundance in graphs: a survey. Congr. Numer. 48, 183–193 (1985)MathSciNetGoogle Scholar
  6. 6.
    Bollobás, B., Cockayne, E.J.: Graph-theoretic parameters concerning domination, independence, and irredundance. J. Graph Theory 3, 241–250 (1979)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cockayne, E.J., Grobler, P.J.P., Hedetniemi, S.T., McRae, A.A.: What makes an irredundant set maximal? J. Combin. Math. Combin. Comput. 25, 213–224 (1997)MATHMathSciNetGoogle Scholar
  8. 8.
    Chang, M.S., Nagavamsi, P., Rangan, C.P.: Weighted irredundance of interval graphs. Information Processing Letters 66, 65–70 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cockayne, E.J., Hedetniemi, S.T., Miller, D.J.: Properties of hereditary hypergraphs and middle graphs. Canad. Math. Bull. 21(4), 461–468 (1978)MATHMathSciNetGoogle Scholar
  10. 10.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 208. Marcel Dekker, New York (1998)Google Scholar
  11. 11.
    Colbourn, C.J., Proskurowski, A.: Concurrent transmissions in broadcast networks. In: Paredaens, J. (ed.) ICALP 1984. LNCS, vol. 172, pp. 128–136. Springer, Heidelberg (1984)Google Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  13. 13.
    Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theoretical Computer Science 351(3), 446–458 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Downey, R.G., Fellows, M.R., Raman, V.: The complexity of irredundant sets parameterized by size. Discrete Applied Mathematics 100, 155–167 (2000)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. Journal of the ACM 56(5) (2009)Google Scholar
  16. 16.
    Fomin, F.V., Iwama, K., Kratsch, D., Kaski, P., Koivisto, M., Kowalik, L., Okamoto, Y., van Rooij, J., Williams, R.: 08431 Open problems – Moderately exponential time algorithms. In: Moderately Exponential Time Algorithms. Dagstuhl Seminar Proceedings, vol. 08431 (2008)Google Scholar
  17. 17.
    Cygan, M., Pilipczuk, M., Wojtaszczyk, J.O.: Irredundant set faster than O(2n). In: These proceedingsGoogle Scholar
  18. 18.
    Chor, B., Fellows, M., Juedes, D.: 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)Google Scholar
  19. 19.
    Fellows, M.R.: Blow-ups, win/win’s, and crown rules: Some new directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)Google Scholar
  20. 20.
    Telle, J.A.: Complexity of domination-type problems in graphs. Nordic. J. of Comp. 1, 157–171 (1994)MathSciNetGoogle Scholar
  21. 21.
    Telle, J.A.: Vertex Partitioning Problems: Characterization, Complexity and Algorithms on Partial k-Trees. PhD thesis, Department of Computer Science, University of Oregon, USA (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Daniel Binkele-Raible
    • 1
  • Ljiljana Brankovic
    • 2
  • Henning Fernau
    • 1
  • Joachim Kneis
    • 3
  • Dieter Kratsch
    • 4
  • Alexander Langer
    • 3
  • Mathieu Liedloff
    • 5
  • Peter Rossmanith
    • 3
  1. 1.FB 4—Abteilung InformatikUniversität TrierTrierGermany
  2. 2.The University of NewcastleCallaghanAustralia
  3. 3.Department of Computer ScienceRWTH Aachen UniversityGermany
  4. 4.Laboratoire d’Informatique Théorique et AppliquéeUniversité Paul Verlaine - MetzMetz Cedex 01France
  5. 5.Laboratoire d’Informatique Fondamentale d’OrléansUniversité d’OrléansOrléans Cedex 2France

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