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 correspond 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)2 41/ε). This result improves the previously known 3-approximation algorithm for this NP-hard problem.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bodlaender, H.L., Fellows, M.R., Warnow, T.J.: Two strikes against perfect phylogeny. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 273–283. Springer, Heidelberg (1992)
Gusfield, D.: Efficient algorithms for inferring evolutionary trees. Networks 21, 19–28 (1991)
Kannan, S., Warnow, T.: Inferring evolutionary history from DNA sequences. SIAM Journal on Computing 23, 713–737 (1994)
Kannan, S., Warnow, T.: A fast algorithm for the computation and enumeration of perfect phylogenies. SIAM Journal on Computing 26, 1749–1763 (1997)
Semple, C., Steel, M.: Phylogenetics. Mathematics and its Applications series, vol. 22. Oxford University Press, Oxford (2003)
Sankoff, D.: Minimal mutation trees of sequences. SIAM Journal on Applied Mathematics 28, 35–42 (1975)
Goldberg, L.A., Goldberg, P.W., Phillips, C.A., Sweedyk, E., Warnow, T.: Minimizing phylogenetic number to find good evolutionary trees. Discrete Applied Mathematics 71, 111–136 (1996)
Moran, S., Snir, S.: Convex recoloring of trees and strings: definitions, hardness results, and algorithms. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 218–232. Springer, Heidelberg (2005)
Moran, S., Snir, S.: Efficient approximation of convex recoloring. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 192–208. Springer, Heidelberg (2005)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)
Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics 12, 289–297 (1999)
Bar-Yehuda, R.: One for the price of two: A unified approach for approximating covering problems. Algorithmica 27, 131–144 (2000)
Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. Journal of the ACM 48, 1069–1090 (2001)
Bar-Yehuda, R., Bendel, K., Freund, A., Rawitz, D.: Local ratio: a unified framework for approximation algorithms. ACM Computing Surveys 36, 422–463 (2004)
Er, M.: On generating the n-ary reflected gray codes. IEEE Transactions on Computers 33, 739–741 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bar-Yehuda, R., Feldman, I., Rawitz, D. (2006). Improved Approximation Algorithm for Convex Recoloring of Trees. In: Erlebach, T., Persinao, G. (eds) Approximation and Online Algorithms. WAOA 2005. Lecture Notes in Computer Science, vol 3879. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11671411_5
Download citation
DOI: https://doi.org/10.1007/11671411_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32207-8
Online ISBN: 978-3-540-32208-5
eBook Packages: Computer ScienceComputer Science (R0)