Improved Approximation Algorithm for Convex Recoloring of Trees

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 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 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+e)-approximation algorithm for the minimum convex recoloring problem, whose running time is O(n 2+n(1/e)241/e ). 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 $O(n^{2}+n\cdot n^{*}\cdot \Delta^{n^{*}+1})$, where n * is the number of colors that violate convexity in the input tree, and Δ is the maximum degree of vertices in the tree. The parameterized complexity of this algorithm is O(n2+n⋅k⋅2k).