Quadratic kernelization for convex recoloring of trees

  • Authors:

  • Hans L. Bodlaender;Michael R. Fellows;Michael A. Langston;Mark A. Ragan;Frances A. Rosamond;Mark Weyer

  • Affiliations:

  • Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands;Parameterized Complexity Research Unit, Office of the DVC(Research), University of Newcastle, Callaghan NSW, Australia and Australian Research Council Centre in Bioinformatics, Oak Ridge National ...;Department of Computer Science, University of Tennessee, Oak Ridge National Laboratory, Oak Ridge, TN;Australian Research Council Centre in Bioinformatics, Oak Ridge National Laboratory, Oak Ridge, TN and Institute for Molecular Bioscience, University of Queensland, Brisbane, QLD, Australia;Parameterized Complexity Research Unit, Office of the DVC(Research), University of Newcastle, Callaghan NSW, Australia and Australian Research Council Centre in Bioinformatics, Oak Ridge National ...;Institut für Informatik, Humboldt-Universität zu Berlin, Berlin, Germany

  • Venue:
  • COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
  • Year:
  • 2007

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Abstract

The Convex Recoloring (CR) problem measures how far a tree of characters differs from exhibiting a so-called "perfect phylogeny". For input consisting of a vertex-colored tree T, the problem is to determine whether recoloring at most k vertices can achieve a convex coloring, meaning by this a coloring where each color class induces a connected subtree. The problem was introduced by Moran and Snir, who showed that CR is NP-hard, and described a search-tree based FPT algorithm with a running time of O(k(k/log k)kn4). The Moran and Snir result did not provide any nontrivial kernelization. Subsequently, a kernelization with a large polynomial bound was established. Here we give the strongest FPT results to date on this problem: (1) We show that in polynomial time, a problem kernel of size O(k2) can be obtained, and (2) We prove that the problem can be solved in linear time for fixed k. The technique used to establish the second result appears to be of general interest and applicability for bounded treewidth problems.