Theory Comput Syst (2008) 43: 3–18 DOI 10.1007/s00224-007-9069-7 Improved Approximation Algorithm for Convex Recoloring of Trees Reuven Bar-Yehuda · Ido Feldman · Dror Rawitz Published online: 18 October 2007 © Springer Science+Business Media, LLC 2007 Abstract A pair (T,C) of a tree T and a coloring C is called a colored tree. Given a colored tree (T,C) any coloring C ′ of T is called a recoloring of T . Given a weight function on the vertices of the tree the recoloring distance of a recoloring is the total weight of recolored vertices. A coloring of a tree is 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. In the 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 phylo- genetic 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 corresponds to 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 + ε)-approximation algorithm for the minimum convex recol- oring problem, whose running time is O(n 2 + n(1/ε) 2 4 1/ε ). This result improves the previously known 3-approximation algorithm for this NP-hard problem. We also present an algorithm for computing an optimal convex recoloring whose running time is O(n 2 + n · n ∗ ·  n ∗ +1 ), where n ∗ is the number of colors that violate convexity in R. Bar-Yehuda · I. Feldman Department of Computer Science, Technion, Haifa 32000, Israel R. Bar-Yehuda e-mail: reuven@cs.technion.ac.il I. Feldman e-mail: idofeld@cs.technion.ac.il D. Rawitz ( ) Caesarea Rothschild Institute, University of Haifa, Haifa 31905, Israel e-mail: rawitz@cri.haifa.ac.il