Efficient approximation of convex recolorings

A coloring of a tree is convex if the vertices that pertain to any color induce a connected subtree; a partial coloring (which assigns colors to some of the vertices) is convex if it can be completed to a convex (total) coloring. Convex coloring of trees arises in areas such as phylogenetics, linguistics, etc., e.g., a perfect phylogenetic tree is one in which the states of each character induce a convex coloring of the tree. Research on perfect phylogeny is usually focused on finding a tree so that few predetermined partial colorings of its vertices are convex. When a coloring of a tree is not convex, it is desirable to know "how far" it is from a convex one. In [S. Moran, S. Snir, Convex recoloring of strings and trees: Definitions, hardness results and algorithms, in: WADS, 2005, pp. 218-232; J. Comput. System Sci., submitted for publication], a natural measure for this distance, called the recoloring distance was defined: the minimal number of color changes at the vertices needed to make the coloring convex. This can be viewed as minimizing the number of "exceptional vertices" w.r.t. a closest convex coloring. The problem was proved to be NP-hard even for colored strings. In this paper we continue the work of [S. Moran, S. Snir, Convex recoloring of strings and trees: Definitions, hardness results and algorithms, in: WADS, 2005, pp. 218-232; J. Comput. System Sci., submitted for publication], and present a 2-approximation algorithm of convex recoloring of strings whose running time O (c n), where c is the number of colors and n is the size of the input, and an O (c n²) 3-approximation algorithm for convex recoloring of trees.