, Volume 68, Issue 4, pp 886–915 | Cite as

Constructing Minimal Phylogenetic Networks from Softwired Clusters is Fixed Parameter Tractable

  • Steven Kelk
  • Celine ScornavaccaEmail author


Here we show that, given a set of clusters \({\mathcal{C}}\) on a set of taxa \({\mathcal{X}}\), where \(|{\mathcal{X}}|=n\), it is possible to determine in time f(k)⋅poly(n) whether there exists a level-≤k network (i.e. a network where each biconnected component has reticulation number at most k) that represents all the clusters in \({\mathcal{C}}\) in the softwired sense, and if so to construct such a network. This extends a result from Kelk et al. (in IEEE/ACM Trans. Comput. Biol. Bioinform. 9:517–534, 2012) which showed that the problem is polynomial-time solvable for fixed k. By defining “k-reticulation generators” analogous to “level-k generators”, we then extend this fixed parameter tractability result to the problem where k refers not to the level but to the reticulation number of the whole network.


Phylogenetics Fixed parameter tractability Directed acyclic graphs 


  1. 1.
    Bordewich, M., Semple, C.: Computing the hybridization number of two phylogenetic trees is fixed-parameter tractable. IEEE/ACM Trans. Comput. Biol. Bioinf. 4(3), 458–466 (2007) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bordewich, M., Semple, C.: Computing the minimum number of hybridization events for a consistent evolutionary history. Discrete Appl. Math. 155(8), 914–928 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bordewich, M., Linz, S., John, K.St., Semple, C.: A reduction algorithm for computing the hybridization number of two trees. Evol. Bioinform. 3, 86–98 (2007) Google Scholar
  4. 4.
    Chen, Z.-Z., Wang, L.: Algorithms for reticulate networks of multiple phylogenetic trees. IEEE/ACM Trans. Comput. Biol. Bioinform. 9, 372–384 (2012) CrossRefGoogle Scholar
  5. 5.
    Collins, J., Linz, S., Semple, C.: Quantifying hybridization in realistic time. J. Comput. Biol. 18(10), 1305–1318 (2011) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) CrossRefGoogle Scholar
  7. 7.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  8. 8.
    Gambette, P., Berry, V., Paul, C.: The structure of level-k phylogenetic networks. In: Proceedings of the 20th Annual Symposium on Combinatorial Pattern Matching, CPM ’09, pp. 289–300. Springer, Berlin (2009) CrossRefGoogle Scholar
  9. 9.
    Gascuel, O. (ed.): Mathematics of Evolution and Phylogeny. Oxford University Press, Oxford (2005) zbMATHGoogle Scholar
  10. 10.
    Gascuel, O., Steel, M. (eds.): Reconstructing Evolution: New Mathematical and Computational Advances. Oxford University Press, Oxford (2007) Google Scholar
  11. 11.
    Gramm, J., Nickelsen, A., Tantau, T.: Fixed-parameter algorithms in phylogenetics. Comput. J. 51(1), 79–101 (2008) CrossRefGoogle Scholar
  12. 12.
    Gusfield, D., Bansal, V., Bafna, V., Song, Y.: A decomposition theory for phylogenetic networks and incompatible characters. J. Comput. Biol. 14(10), 1247–1272 (2007) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Gusfield, D., Hickerson, D., Eddhu, S.: An efficiently computed lower bound on the number of recombinations in phylognetic networks: theory and empirical study. Discrete Appl. Math. 155(6–7), 806–830 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Huson, D.H., Scornavacca, C.: A survey of combinatorial methods for phylogenetic networks. Genome Biol. Evol. 3, 23–35 (2011) CrossRefGoogle Scholar
  15. 15.
    Huson, D.H., Rupp, R., Berry, V., Gambette, P., Paul, C.: Computing galled networks from real data. Bioinformatics 25(12), i85–i93 (2009) CrossRefGoogle Scholar
  16. 16.
    Huson, D.H., Rupp, R., Scornavacca, C.: Phylogenetic Networks: Concepts, Algorithms and Applications. Cambridge University Press, Cambridge (2011) Google Scholar
  17. 17.
    Huynh, T.N.D., Jansson, J., Nguyen, N.B., Sung, W.-K.: Constructing a smallest refining galled phylogenetic network. In: Research in Computational Molecular Biology (RECOMB). Lecture Notes in Bioinformatics, vol. 3500, pp. 265–280 (2005) CrossRefGoogle Scholar
  18. 18.
    Jansson, J., Sung, W.-K.: Inferring a level-1 phylogenetic network from a dense set of rooted triplets. Theor. Comput. Sci. 363(1), 60–68 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Jansson, J., Nguyen, N.B., Sung, W.-K.: Algorithms for combining rooted triplets into a galled phylogenetic network. SIAM J. Comput. 35(5), 1098–1121 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Kelk, S., Scornavacca, C., van Iersel, L.: On the elusiveness of clusters. IEEE/ACM Trans. Comput. Biol. Bioinform. 9, 517–534 (2012) CrossRefGoogle Scholar
  21. 21.
    Myers, S.R., Griffiths, R.C.: Bounds on the minimum number of recombination events in a sample history. Genetics 163, 375–394 (2003) Google Scholar
  22. 22.
    Nakhleh, L.: Evolutionary phylogenetic networks: models and issues. In: The Problem Solving Handbook for Computational Biology and Bioinformatics. Springer, Berlin (2009) Google Scholar
  23. 23.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006) CrossRefzbMATHGoogle Scholar
  24. 24.
    Semple, C.: Hybridization networks. In: Reconstructing Evolution—New Mathematical and Computational Advances. Oxford University Press, Oxford (2007) Google Scholar
  25. 25.
    Semple, C., Steel, M.: Phylogenetics. Oxford University Press, Oxford (2003) zbMATHGoogle Scholar
  26. 26.
    To, T.-H., Habib, M.: Level-k phylogenetic networks are constructable from a dense triplet set in polynomial time. In: CPM09. LNCS, vol. 5577, pp. 275–288 (2009) Google Scholar
  27. 27.
    van Iersel, L., Kelk, S.: Constructing the simplest possible phylogenetic network from triplets. Algorithmica 60, 207–235 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    van Iersel, L.J.J., Kelk, S.M.: When two trees go to war. J. Theor. Biol. 269(1), 245–255 (2011) CrossRefGoogle Scholar
  29. 29.
    van Iersel, L.J.J., Keijsper, J.C.M., Kelk, S.M., Stougie, L., Hagen, F., Boekhout, T.: Constructing level-2 phylogenetic networks from triplets. IEEE/ACM Trans. Comput. Biol. Bioinform. 6(4), 667–681 (2009) CrossRefGoogle Scholar
  30. 30.
    van Iersel, L.J.J., Kelk, S.M., Mnich, M.: Uniqueness, intractability and exact algorithms: reflections on level-k phylogenetic networks. J. Bioinform. Comput. Biol. 7(2), 597–623 (2009) CrossRefGoogle Scholar
  31. 31.
    van Iersel, L.J.J., Kelk, S.M., Rupp, R., Huson, D.H.: Phylogenetic networks do not need to be complex: Using fewer reticulations to represent conflicting clusters. Bioinformatics 26, i124–i131 (2010). Special issue: Proceedings of Intelligent Systems for Molecular Biology 2010 (ISMB2010), 10th–13th September (2010) CrossRefGoogle Scholar
  32. 32.
    Whidden, C., Zeh, N.: A unifying view on approximation and fpt of agreement forests. In: Salzberg, S., Warnow, T. (eds.) Algorithms in Bioinformatics. Lecture Notes in Computer Science, vol. 5724, pp. 390–402. Springer, Berlin (2009) CrossRefGoogle Scholar
  33. 33.
    Whidden, C., Beiko, R.G., Zeh, N.: Fixed-parameter and approximation algorithms for maximum agreement forests. arXiv:1108.2664v1 [q-bio.PE]
  34. 34.
    Wu, Y.: Close lower and upper bounds for the minimum reticulate network of multiple phylogenetic trees. Bioinformatics 26, i140–i148 (2010). Special issue: Proceedings of Intelligent Systems for Molecular Biology 2010 (ISMB2010), 10th–13th September (2010) CrossRefGoogle Scholar
  35. 35.
    Wu, Y., Gusfield, D.: A new recombination lower bound and the minimum perfect phylogenetic forest problem. J. Comb. Optim. 16(3), 229–247 (2008) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Knowledge Engineering (DKE)Maastricht UniversityMaastrichtThe Netherlands
  2. 2.ISEM, UMR 5554, CNRSUniv. Montpellier 2MontpellierFrance

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