Approximate Maximum Parsimony and Ancestral Maximum Likelihood

We explore the maximum parsimony (MP) and ancestral maximum likelihood (AML) criteria in phylogenetic tree reconstruction. Both problems are NP-hard, so we seek approximate solutions. We formulate the two problems as Steiner tree problems under appropriate distances. The gist of our approach is the succinct characterization of Steiner trees for a small number of leaves for the two distances. This enables the use of known Steiner tree approximation algorithms. The approach leads to a 16/9 approximation ratio for AML and asymptotically to a 1.55 approximation ratio for MP.

[1]  T. Jukes.  CHAPTER 24 – Evolution of Protein Molecules , 1969 .

[2]  J. Hartigan,et al.  Statistical Analysis of Hominoid Molecular Evolution , 1987 .

[3]  W. Fitch.  Toward Defining the Course of Evolution: Minimum Change for a Specific Tree Topology , 1971 .

[4]  Piotr Berman,et al.  Improved approximations for the Steiner tree problem , 1992, SODA '92.

[5]  Alessandro Panconesi,et al.  Ancestral Maximum Likelihood of Evolutionary Trees Is Hard , 2003, WABI.

[6]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[7]  Nick Goldman,et al.  MAXIMUM LIKELIHOOD INFERENCE OF PHYLOGENETIC TREES, WITH SPECIAL REFERENCE TO A POISSON PROCESS MODEL OF DNA SUBSTITUTION AND TO PARSIMONY ANALYSES , 1990 .

[8]  J. Neyman.  MOLECULAR STUDIES OF EVOLUTION: A SOURCE OF NOVEL STATISTICAL PROBLEMS* , 1971 .

[9]  Alex Zelikovsky,et al.  Tighter Bounds for Graph Steiner Tree Approximation , 2005, SIAM J. Discret. Math..

[10]  R. Shamir,et al.  A fast algorithm for joint reconstruction of ancestral amino acid sequences. , 2000, Molecular biology and evolution.

[11]  D Penny,et al.  Parsimony, likelihood, and the role of models in molecular phylogenetics. , 2000, Molecular biology and evolution.

[12]  M. Kimura.  Estimation of evolutionary distances between homologous nucleotide sequences. , 1981, Proceedings of the National Academy of Sciences of the United States of America.

[13]  J. Felsenstein.  Evolutionary trees from DNA sequences: A maximum likelihood approach , 2005, Journal of Molecular Evolution.

[14]  Hans Jürgen Prömel,et al.  A 1.598 approximation algorithm for the Steiner problem in graphs , 1999, SODA '99.

[15]  R. Graham,et al.  The steiner problem in phylogeny is NP-complete , 1982 .

[16]  Alexander Zelikovsky.  An 11/6-Approximation Algorithm for the Steiner Problem on Graphs , 1992 .