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Minimal Triangulation Algorithms for Perfect Phylogeny Problems

  • Rob Gysel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)

Abstract

In this paper, we show that minimal triangulation techniques similar to those proposed by Bouchitté and Todinca can be applied to a variety of perfect phylogeny (or character compatibility) problems. These problems arise in the context of supertree construction, a critical step in estimating the Tree of Life.

Keywords

perfect phylogeny minimal triangulation 

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References

  1. 1.
    Agarwala, R., Fernández-Baca, D.: A polynomial-time algorithm for the perfect phylogeny problem when the number of character states is fixed. SIAM Journal on Computing 23, 1216–1224 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Berry, A., Bordat, J., Cogis, O.: Generating all the minimal separators of a graph. International Journal of Foundations of Computer Science 11(3), 397–403 (2000)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bininda-Emonds, O.R.: The evolution of supertrees. Trends in Ecology and Evolution 19(6), 315–322 (2004)CrossRefGoogle Scholar
  4. 4.
    Blair, J., Peyton, B.: An introduction to chordal graphs and clique trees. In: George, J., Gilbert, J., Liu, J.H. (eds.) Graph Theory and Sparse Matrix Computations, IMA Volumes in Mathematics and its Applications, vol. 56, pp. 1–27. Springer (1993)Google Scholar
  5. 5.
    Bodlaender, H., Heggernes, P., Villanger, Y.: Faster parameterized algorithms for minimum fill–in. Algorithmica 61, 817–838 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bodlaender, H., Fellows, M., Warnow, T.: Two strikes against perfect phylogeny. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 273–283. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  7. 7.
    Bonet, M., Linz, S., John, K.S.: The complexity of finding multiple solutions to betweenness and quartet compatibility. IEEE/ACM Transactions on Computational Biology and Bioinformatics 9(1), 273–285 (2012)CrossRefGoogle Scholar
  8. 8.
    Bordewich, M., Huber, K., Semple, C.: Identifying phylogenetic trees. Discrete Mathematics 300(1-3), 30–43 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bouchitté, V., Todinca, I.: Treewidth and minimum fill-in: Grouping the minimal separators. SIAM Journal on Computing 31(1), 212–232 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bouchitté, V., Todinca, I.: Listing all potential maximal cliques of a graph. Theoretical Computer Science 276(1-2), 17–32 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Brinkmeyer, M., Griebel, T., Böcker, S.: Polynomial supertree methods revisited. Advances in Bioinformatics (2011)Google Scholar
  12. 12.
    Buneman, P.: A characterisation of rigid circuit graphs. Discrete Mathematics 9(3), 205–212 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Fernández-Baca, D.: The perfect phylogeny problem. In: Cheng, X., Du, D.Z. (eds.) Steiner Trees in Industry, pp. 203–234. Kluwer (2001)Google Scholar
  14. 14.
    Fomin, F., Kratsch, D., Todinca, I., Villanger, Y.: Exact algorithms for treewidth and minimum fill-in. SIAM Journal on Computing 38(3), 1058–1079 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Grünewald, S., Huber, K.: Identifying and defining trees. In: Gascuel, O., Steel, M. (eds.) Reconstructing Evolution: New Mathematical and Computational Advances, pp. 217–246. Oxford University Press (2007)Google Scholar
  16. 16.
    Gusfield, D.: The multi–state perfect phylogeny problem with missing and removable data: solutions via integer–programming and chordal graph theory. Journal of Computational Biology 17(3), 383–399 (2010)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Gysel, R.: Potential maximal clique algorithms for perfect phylogeny problems. Pre-print: arXiv 1303.3931 (2013)Google Scholar
  18. 18.
    Gysel, R., Gusfield, D.: Extensions and improvements to the chordal graph approach to the multistate perfect phylogeny problem. IEEE/ACM Transactions on Computational Biology and Bioinformatics 8(4), 912–917 (2011)CrossRefGoogle Scholar
  19. 19.
    Habib, M., Stacho, J.: Unique perfect phylogeny is intractable. Theoretical Computer Science 476, 47 – 66 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Heggernes, P.: Minimal triangulations of graphs: a survey. Discrete Mathematics 306(3), 297–317 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hudson, R.: Generating samples under a wright-fisher neutral model of genetic variation. Bioinformatics 18(2), 337–338 (2002)CrossRefGoogle Scholar
  22. 22.
    Kloks, T., Kratsch, D., Spinrad, J.: On treewidth and minimum fill-in of asteroidal triple-free graphs. Theoretical Computer Science 175(2), 309–335 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    McMorris, F., Warnow, T., Wimer, T.: Triangulating vertex–colored graphs. SIAM Journal of Discrete Mathematics 7, 296–306 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Meacham, C.: Theoretical and computational considerations of the compatibility of qualitative taxonomic characters. In: Felsenstein, J. (ed.) Numerical Taxonomy. NATO ASI Series G, vol. 1, pp. 304–314. Springer (1983)Google Scholar
  25. 25.
    Parra, A., Scheffler, P.: How to use the minimal separators of a graph for its chordal triangulation. In: Fülöp, Z. (ed.) ICALP 1995. LNCS, vol. 944, pp. 123–134. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  26. 26.
    Parra, A., Scheffler, P.: Characterizations and algorithmic applications of chordal graph embeddings. Discrete Applied Mathematics 79(1-3), 171–188 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Ross, H., Rodrigo, A.: An assessment of matrix representation with compatibility in supertree construction. In: Bininda-Emonds, O. (ed.) Phylogenetic supertrees: Combining information to reveal the Tree of Life, pp. 35–63. Kluwer Academic Publishers (2004)Google Scholar
  28. 28.
    Semple, C., Steel, M.: A characterization for a set of partial partitions to define an X-tree. Discrete Mathematics 247(1-3), 169–186 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Semple, C., Steel, M.: Phylogenetics. In: Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press (2003)Google Scholar
  30. 30.
    Steel, M.: The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification 9(1), 91–116 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM Journal on Algebraic Discrete Methods 2, 77–79 (1981)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Rob Gysel
    • 1
  1. 1.Department of Computer ScienceUniversity of California, DavisDavisUSA

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