The Multi-State Perfect Phylogeny Problem with Missing and Removable Data: Solutions via Integer-Programming and Chordal Graph Theory

  • Dan Gusfield
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5541)


The Multi-State Perfect Phylogeny Problem is an extension of the Binary Perfect Phylogeny Problem, allowing characters to take on more than two states. In this paper we consider three problems that extend the utility of the multi-state perfect phylogeny model: The Missing Data (MD) Problem where some entries in the input are missing and the question is whether (bounded) values for the missing data can be imputed so that the resulting data has a multi-state perfect phylogeny; The Character-Removal (CR) Problem where we want to minimize the number of characters to remove from the data so that the resulting data has a multi-state perfect phylogeny; and The Missing-Data Character-Removal (MDCR) Problem where the input has missing data and we want to impute values for the missing data to minimize the solution to the resulting Character-Removal Problem.

We detail Integer Linear Programming (ILP) solutions to these problems for the special case of three permitted states per character and report on extensive empirical testing of these solutions. Then we develop a general theory to solve the MD problem for an arbitrary number of permitted states, using chordal graph theory and results on minimal triangulation of non-chordal graphs. This establishes new necessary and sufficient conditions for the existence of a perfect phylogeny with (or without) missing data. We implement the general theory using integer linear programming, although other optimization methods are possible. We extensively explore the empirical behavior of the general solution, showing that the methods are very practical for data of size and complexity that is characteristic of many current applications in phylogenetics. Some of the empirical results for the MD problem with an arbitrary number of permitted states are very surprising, suggesting the existence of additional combinatorial structure in multi-state perfect phylogenies.


computational biology phylogenetics perfect phylogeny integer programming chordal graphs graph triangulation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwala, R., Fernandez-Baca, D.: A polynomial-time algorithm for the perfect phylogeny problem when the number of character states is fixed. SIAM J. on Computing 23, 1216–1224 (1994)CrossRefGoogle Scholar
  2. 2.
    Berry, A., Bordat, J.-P., Cogis, O.: Generating All the Minimal Separators of a Graph. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, pp. 167–172. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H., Fellows, M., Warnow, T.: Two strikes against perfect phylogeny. In: Proc. of the 19’th Inter. colloquium on Automata, Languages and Programming, pp. 273–283 (1992)Google Scholar
  4. 4.
    Buneman, P.: A characterization of rigid circuit graphs. Discrete Math. 9, 205–212 (1974)CrossRefGoogle Scholar
  5. 5.
    Dress, A., Steel, M.: Convex tree realizations of partitions. Applied Math. Letters 5, 3–6 (1993)CrossRefGoogle Scholar
  6. 6.
    Estabrook, G., Johnson, C., McMorris, F.: An idealized concept of the true cladistic character. Math. Bioscience 23, 263–272 (1975)CrossRefGoogle Scholar
  7. 7.
    Felsenstein, J.: Inferring Phylogenies. Sinauer, Sunderland (2004)Google Scholar
  8. 8.
    Fernandez-Baca, D.: The perfect phylogeny problem. In: Du, D.Z., Cheng, X. (eds.) Steiner Trees in Industries. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  9. 9.
    Fernandez-Baca, D., Lagergren, J.: A polynomial-time algorithm for near-perfect phylogeny. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 670–680. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  10. 10.
    Fitch, W.: Towards finding the tree of maximum parsimony. In: Estabrook, G.F. (ed.) Proceedings of the eighth international conference on numerical taxonomy, pp. 189–230. W.H. Freeman, New York (1975)Google Scholar
  11. 11.
    Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Combinatorial Theory, B 16, 47–56 (1974)CrossRefGoogle Scholar
  12. 12.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)Google Scholar
  13. 13.
    Gusfield, D.: Efficient algorithms for inferring evolutionary history. Networks 21, 19–28 (1991)CrossRefGoogle Scholar
  14. 14.
    Gusfield, D., Frid, Y., Brown, D.: Integer programming formulations and computations solving phylogenetic and population genetic problems with missing or genotypic data. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 51–64. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Gusfield, D., Wu, Y.: The three-state perfect phylogeny problem reduces to 2-SAT (to appear)Google Scholar
  16. 16.
    Heggernes, P.: Minimal triangulations of graphs: A survey. Discrete Mathematics 306, 297–317 (2006)CrossRefGoogle Scholar
  17. 17.
    Hudson, R.: Generating samples under the Wright-Fisher neutral model of genetic variation. Bioinformatics 18(2), 337–338 (2002)CrossRefPubMedGoogle Scholar
  18. 18.
    Kannan, S., Warnow, T.: Inferring evolutionary history from DNA sequences. SIAM J. on Computing 23, 713–737 (1994)CrossRefGoogle Scholar
  19. 19.
    Kannan, S., Warnow, T.: A fast algorithm for the computation and enumeration of perfect phylogenies when the number of character states is fixed. SIAM J. on Computing 26, 1749–1763 (1997)CrossRefGoogle Scholar
  20. 20.
    McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. Siam Monographs on Discrete Mathematics (1999)Google Scholar
  21. 21.
    Parra, A., Scheffler, P.: How to use the minimal separators of a graph for its chordal triangulation. In: Fülöp, Z., Gecseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 123–134. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  22. 22.
    Parra, A., Scheffler, P.: Characterizations and algorithmic applications of chordal graph embeddings. Discrete Applied Mathematics 79, 171–188 (1997)CrossRefGoogle Scholar
  23. 23.
    Pe’er, I., Pupko, T., Shamir, R., Sharan, R.: Incomplete directed perfect phylogeny. SIAM J. on Computing 33, 590–607 (2004)CrossRefGoogle Scholar
  24. 24.
    Semple, C., Steel, M.: Phylogenetics. Oxford University Press, Oxford (2003)Google Scholar
  25. 25.
    Steel, M.: The complexity of reconstructing trees from qualitative characters and subtrees. J. of Classification 9, 91–116 (1992)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Dan Gusfield
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaDavisUSA

Personalised recommendations