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Approximate Maximum Parsimony and Ancestral Maximum Likelihood
January-March 2010 (vol. 7 no. 1)
pp. 183-187
Noga Alon, Tel Aviv University, Tel Aviv
Benny Chor, Tel Aviv University, Tel Aviv
Fabio Pardi, EMBL-European Bioinformatics Institute and University of Cambridge, Cambridge
Anat Rapoport, Tel Aviv University, Tel Aviv
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] L. Addario-Berry, B. Chor, M. Hallett, J. Lagergren, A. Panconesi, and T. Wareham, "Ancestral Maximum Likelihood of Evolutionary Trees Is Hard," J. Bioinformatics and Computational Biology, vol. 2, no. 2, pp. 257-271, 2004.
[2] D. Barry and J.A. Hartigan, "Statistical Analysis of Hominoid Molecular Evolution," Statistical Science, vol. 2, pp. 191-210, 1987.
[3] P. Berman and V. Ramaiyer, "Improved Approximations for the Steiner Tree Problem," J. Algorithms, vol. 17, pp. 381-408, 1994.
[4] T. Cover and J. Thomas, Elements of Information Theory. John Wiley & Sons, 1991.
[5] J. Felsenstein, "Evolutionary Trees from DNA Sequences: A Maximum Likelihood Approach," J. Molecular Evolution, vol. 17, pp. 368-376, 1981.
[6] W.M. Fitch, "Toward Defining the Course of Evolution: Minimum Change for Specified Tree Topology," Systematic Zoology, vol. 20, pp. 406-416, 1971.
[7] L.R. Foulds and R.L. Graham, "The Steiner Problem in Phylogeny Is NP-Complete," Advances in Applied Math., vol. 3, pp. 43-49, 1982.
[8] N. Goldman, "Maximum Likelihood Inference of Phylogenetic Trees, with Special Reference to a Poisson Process Model of DNA Substitution and to Parsimony Analyses," Systematic Zoology, vol. 39, pp. 345-361, 1990.
[9] S. Hougardy and H.J. Prömel, "A 1.598 Approximation Algorithm for the Steiner Problem in Graphs," Proc. 10th ACM-SIAM Symp. Discrete Algorithms (SODA '99), pp. 448-453, 1999.
[10] T.H. Jukes and C.R. Cantor, "Evolution of Protein Molecules," Mammalian Protein Metabolism, H.N. Munro, ed., pp. 21-132, Academic Press, 1969.
[11] R. Karp, "Reducibility among Combinatorial Problems," Complexity of Computer Computations, R.E. Miller and J.W. Thatcher, eds., pp. 85-103, Plenum Press, 1972.
[12] M. Kimura, "Estimation of Evolutionary Sequences between Homologous Nucleotide Sequences," Proc. Nat'l Academy of Sciences USA, vol. 78, pp. 454-458, 1981.
[13] J. Neyman, "Molecular Studies of Evolution: A Source of Novel Statistical Problems," Statistical Decision Theory and Related Topics, S. Gupta and Y. Jackel, eds., pp. 1-27, Academic Press, 1971.
[14] T. Pupko, I. Pe'er, R. Shamir, and D. Graur, "A Fast Algorithm for Joint Reconstruction of Ancestral Amino-Acid Sequences," Molecular Biology and Evolution, vol. 17, no. 6, pp. 890-896, 2000.
[15] G. Robins and A. Zelikovsky, "Tighter Bounds for Graph Steiner Tree Approximation," SIAM J. Discrete Math., vol. 19, no. 1, pp. 122-134, 2005.
[16] M. Steel and D. Penny, "Parsimony, Likelihood, and the Role of Models in Molecular Phylogenetics," Molecular Biology and Evolution, vol. 17, no. 6, pp. 839-850, 2000.
[17] H. Takahashi and A. Matsuyama, "An Approximate Solution for the Steiner Problem in Graphs," Math. Japonica 4, pp. 573-577, 1980.
[18] A.Z. Zelikovsky, "An 11/6-Approximation Algorithm for the Steiner Problem on Graphs," Proc. Fourth Czechoslovakian Symp. Combinatorics, Graphs and Complexity, J. Nesetril and M. Fiedler, eds., Annals of Discrete Math., vol. 51, pp. 351-354, 1992.

Index Terms:
Phylogenetic reconstruction, ancestral maximum likelihood, maximum parsimony, Steiner trees, approximation algorithms.
Noga Alon, Benny Chor, Fabio Pardi, Anat Rapoport, "Approximate Maximum Parsimony and Ancestral Maximum Likelihood," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 7, no. 1, pp. 183-187, Jan.-March 2010, doi:10.1109/TCBB.2008.13
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