Speeding up Dynamic Programming for Some NP-Hard Graph Recoloring Problems

  • Oriana Ponta
  • Falk Hüffner
  • Rolf Niedermeier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4978)

Abstract

A vertex coloring of a tree is called convex if each color induces a connected component. The NP-hard Convex Recoloring problem on vertex-colored trees asks for a minimum-weight change of colors to achieve a convex coloring. For the non-uniformly weighted model, where the cost of changing a vertex v to color c depends on both v and c, we improve the running time on trees from O(Δκ·κn) to O(3κ·κn), where Δ is the maximum vertex degree of the input tree T, κ is the number of colors, and n is the number of vertices in T. In the uniformly weighted case, where costs depend only on the vertex to be recolored, one can instead parameterize on the number of bad colors β ≤ κ, which is the number of colors that do not already induce a connected component. Here, we improve the running time from O(Δβ·βn) to O(3β·βn). For the case where the weights are integers bounded by M, using fast subset convolution, we further improve the running time with respect to the exponential part to O(2κ·κ4n2M log2(nM)) and O(2β·β4n2M log2(nM)), respectively. Finally, we use fast subset convolution to improve the exponential part of the running time of the related 1-Connected Coloring Completion problem.

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References

  1. 1.
    Bachoore, E.H., Bodlaender, H.L.: Convex recoloring of leaf-colored trees. In: Proc. 3rd ACiD. Texts in Algorithmics, vol. 9, pp. 19–33. College Publications, London (2007)Google Scholar
  2. 2.
    Bar-Yehuda, R., Feldman, I., Rawitz, D.: Improved approximation algorithm for convex recoloring of trees. Theory of Computing Systems, (to appear, 2007)Google Scholar
  3. 3.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: fast subset convolution. In: Proc. 39th STOC, pp. 67–74. ACM Press, New York (2007)Google Scholar
  4. 4.
    Blum, C.: Revisiting dynamic programming for finding optimal subtrees in trees. European Journal of Operational Research 177(1), 102–115 (2007)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bodlaender, H.L., Weyer, M.: Convex and connected recolorings of trees and graphs (unpublished manuscript, 2005)Google Scholar
  6. 6.
    Bodlaender, H.L., Fellows, M.R., Langston, M.A., Ragan, M.A., Rosamond, F.A., Weyer, M.: Kernelization for convex recoloring. In: Proc. 2nd ACiD. Texts in Algorithmics, vol. 7, pp. 23–35. College Publications, London (2006)Google Scholar
  7. 7.
    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. LNCS, vol. 4598, pp. 86–96. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Chor, B., Fellows, M.R., Ragan, M.A., Razgon, I., Rosamond, F.A., Snir, S.: Connected coloring completion for general graphs: Algorithms and complexity. In: Lin, G. (ed.) COCOON. LNCS, vol. 4598, pp. 75–85. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  10. 10.
    Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1(3), 195–207 (1972)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  12. 12.
    Fürer, M.: Faster integer multiplication. In: Proc. 39th STOC, pp. 57–66. ACM Press, New York (2007)Google Scholar
  13. 13.
    Lingas, A., Wahlen, M.: On exact complexity of subgraph homeomorphism. In: Cai, J.-Y., Cooper, S.B., Zhu, H. (eds.) TAMC 2007. LNCS, vol. 4484, pp. 256–261. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Maffioli, F.: Finding a best subtree of a tree. Technical Report 91.041, Politecnico di Milano, Dipartimento di Elettronica, Italy (1991)Google Scholar
  15. 15.
    Moran, S., Snir, S.: Convex recolorings of strings and trees: Definitions, hardness results and algorithms. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 218–232. Springer, Heidelberg (2005) (to appear in Journal of Computer and System Sciences)Google Scholar
  16. 16.
    Moran, S., Snir, S.: Efficient approximation of convex recolorings. Journal of Computer and System Sciences 73(7), 1078–1089 (2007)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Moran, S., Snir, S., Sung, W.-K.: Partial convex recolorings of trees and galled networks: Tight upper and lower bounds (February 2007) (manuscript)Google Scholar
  18. 18.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications, vol. 31. Oxford University Press, Oxford (2006)MATHGoogle Scholar
  19. 19.
    Ponta, O.: The Fixed-Parameter Approach to the Convex Recoloring Problem. Diplomarbeit, Mathematisches Institut, Ruprecht-Karls-Universität. Springer, Heidelberg (2007)Google Scholar
  20. 20.
    Razgon, I.: A 2O(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 2008

Authors and Affiliations

  • Oriana Ponta
    • 1
  • Falk Hüffner
    • 2
  • Rolf Niedermeier
    • 2
  1. 1.Mathematisches InstitutRuprecht-Karls-Universität HeidelbergHeidelbergGermany
  2. 2.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

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