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)


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(Δ κ ·κn) to O(3 κ ·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(Δ β ·βn) to O(3 β ·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(2 κ ·κ 4 n 2 M log2(nM)) and O(2 β ·β 4 n 2 M 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.


Dynamic Programming Dynamic Programming Algorithm Good Color Weighted Case Exponential Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© 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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