Quadratic Kernelization for Convex Recoloring of Trees

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/logk)^k n ⁴). 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(k ²) 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.

This research has been supported by the Australian Research Council Centre in Bioinformatics, by the U.S. National Science Foundation under grant CCR–0311500, by the U.S. National Institutes of Health under grants 1-P01-DA-015027-01, 5-U01-AA-013512-02 and 1-R01-MH-074460-01, by the U.S. Department of Energy under the EPSCoR Laboratory Partnership Program and by the European Commission under the Sixth Framework Programme. The second and fifth authors have been supported by a Fellowship to the Institute of Advanced Studies at Durham University, and hosted by a William Best Fellowship to Grey College during the preparation of the paper.