TY - JOUR
T1 - Improved approximation algorithm for convex recoloring of trees
AU - Bar-Yehuda, Reuven
AU - Feldman, Ido
AU - Rawitz, Dror
PY - 2008/7
Y1 - 2008/7
N2 - 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+ε)-approximation algorithm for the minimum convex recoloring problem, whose running time is O(n 2+n(1/ε)241/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(n2+n.n*Δn+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(n 2+n2 k ).
AB - 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+ε)-approximation algorithm for the minimum convex recoloring problem, whose running time is O(n 2+n(1/ε)241/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(n2+n.n*Δn+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(n 2+n2 k ).
KW - Approximation algorithms
KW - Convex recoloring
KW - Local ratio
KW - Phylogenetic trees
UR - http://www.scopus.com/inward/record.url?scp=40849114749&partnerID=8YFLogxK
U2 - 10.1007/s00224-007-9069-7
DO - 10.1007/s00224-007-9069-7
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:40849114749
SN - 1432-4350
VL - 43
SP - 3
EP - 18
JO - Theory of Computing Systems
JF - Theory of Computing Systems
IS - 1
ER -