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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, USA (2006)zbMATHCrossRefGoogle 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

Personalised recommendations