A Kernel for Convex Recoloring of Weighted Forests

  • Hans L. Bodlaender
  • Marc Comas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5901)


In this paper, we show that the following problem has a kernel of quadratic size: given is a tree T whose vertices have been assigned colors and a non-negative integer weight, and given is an integer k. In a recoloring, the color of some vertices is changed. We are looking for a recoloring such that each color class induces a subtree of T and such that the total weight of all recolored vertices is at most k. Our result generalizes a result by Bodlaender et al. [3] who give quadratic size kernel for the case that all vertices have unit weight.


  1. 1.
    Bachoore, E.H., Bodlaender, H.L.: Convex recoloring of leaf-colored trees. Technical Report UU-CS-2006-010, Department of Information and Computing Sciences, Utrecht University (2006)Google Scholar
  2. 2.
    Bar-Yehuda, R., Feldman, I., Rawitz, D.: Improved Approximation Algorithm for Convex Recoloring of Trees. Theory Comput. Syst. 43(1), 3–18 (2008)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bodlaender, H.L., Fellows, M.R., Langston, M.A., Ragan, M.A., Rosamond, F.A., Weyer, M.: Quadratic Kernelization for Convex Recoloring of Trees. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 86–96. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  5. 5.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Secaucus (2006)Google Scholar
  6. 6.
    Gramm, J., Nickelsen, A., Tantau, T.: Fixed-Parameter Algorithms in Phylogenetics. The Computer Journal 51(1), 79–101 (2008)CrossRefGoogle Scholar
  7. 7.
    Guo, J., Niedermeier, R.: Invitation to Data Reduction and Problem Kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  8. 8.
    Moran, S., Snir, S.: Efficient Approximation of Convex Recolorings. Journal of Computer and System Sciences 73(7), 1078–1089 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Moran, S., Snir, S.: Convex Recolorings of Strings and Trees: Definitions, Hardness Results and Algorithms. Journal of Computer and System Sciences 74(5), 850–869 (2008)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, USA (2006)MATHCrossRefGoogle Scholar
  11. 11.
    Ponta, O., Hüffner, F., Niedermeier, R.: Speeding up Dynamic Programming for Some np-Hard Graph Recoloring Problems. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 490–501. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Razgon, I.: A \(2^{\mbox{o(k)}}\)Poly(n) Algorithm for the Parameterized Convex Recoloring Problem. Information Processing Letters 104(2), 53–58 (2007)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  • Marc Comas
    • 2
  1. 1.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  2. 2.Departament de Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelonaSpain

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