TR#: | MSC-2006-10 |

Class: | MSC |

Title: | 2+epsilon approximation algorithm for convex recoloring of trees |

Authors: | Ido Feldman |

Supervisors: | Reuven Bar-Yehuda |

MSC-2006-10.pdf | |

Abstract: | A coloring of a tree is \emph{convex} if for any two vertices $u$
and $v$ that are colored by the same color $c$, every vertex on the
path from $u$ to $v$ is also colored by $c$. That is, the vertices
that are colored with the same color induce a subtree. Given a
weight function on the vertices of the tree the \emph{recoloring
distance} of a recoloring is the total weight of recolored vertices.
In the \emph{minimum convex recoloring problem} we are given a
colored tree and a weight function and our goal is to find a convex
recoloring of minimum recoloring distance.
The minimum convex recoloring problem naturally arises in the context of \emph{phylogenetic trees}. Given a set of related species the goal of phylogenetic reconstruction is to construct a tree that would best describe the evolution of this set of species. In this context a convex coloring correspond to \emph{perfect phylogeny}. Since perfect phylogeny is not always possible the next best thing is to find a tree which is as close to convex as possible, or, in other words, a tree with minimum recoloring distance. We present a $(2+\eps)$-approximation algorithm for the minimum convex recoloring problem, whose running time is $O(n^2+ (1/\eps)^2 4^{1/\eps})$. This result improves the previously known $3$-approximation algorithm for this NP-hard problem. |

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