Kernelizations for the Hybridization Number Problem on Multiple Nonbinary Trees

  • Leo van IerselEmail author
  • Steven Kelk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)


A well-studied problem in phylogenetics is to determine the minimum number of hybridization events necessary to explain conflicts among several evolutionary trees, e.g. from different genes. An evolutionary history with hybridization events (or, more generally, reticulations) can be described by a rooted leaf-labelled directed acyclic graph, which is called a phylogenetic network. The reticulation number of such a phylogenetic network can be defined as the sum of all indegrees minus the number of vertices plus one. The considered problem can now formally be stated as follows. Given a finite set \(X\), a collection \(\mathcal {T}\) of rooted phylogenetic trees on \(X\) and \(k\in \mathbb {N}^{+}\), the Hybridization Number problem asks if there exists a rooted phylogenetic network on \(X\) that displays all trees from \(\mathcal {T}\) and has reticulation number at most \(k\). We show that Hybridization Number admits a kernel of size \(4k(5k)^t\) if \(\mathcal {T}\) contains \(t\) (not necessarily binary) rooted phylogenetic trees. In addition, we show a slightly different kernel of size \(20k^2(\varDelta ^+-1)\) with \(\varDelta ^+\) the maximum outdegree of the input trees.


  1. 1.
    Bapteste, E., van Iersel, L., Janke, A., Kelchner, S., Kelk, S., McInerney, J.O., Morrison, D.A., Nakhleh, L., Steel, M., Stougie, L., Whitfield, J.: Networks: expanding evolutionary thinking. Trends Genet. 29(8), 439–441 (2013)CrossRefGoogle Scholar
  2. 2.
    Baroni, M., Grünewald, S., Moulton, V., Semple, C.: Bounding the number of hybridisation events for a consistent evolutionary history. Math. Biol. 51, 171–182 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bordewich, M., Semple, C.: Computing the minimum number of hybridization events for a consistent evolutionary history. Discrete Appl. Math. 155(8), 914–928 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, Z.-Z., Wang, L.: Algorithms for reticulate networks of multiple phylogenetic trees. IEEE/ACM Trans. Comput. Biol. Bioinf. 9(2), 372–384 (2012)CrossRefGoogle Scholar
  7. 7.
    Chen, Z.-Z., Wang, L.: An ultrafast tool for minimum reticulate networks. J. Comput. Biol. 20(1), 38–41 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefGoogle Scholar
  9. 9.
    Huson, D.H., Rupp, R., Scornavacca, C.: Phylogenetic Networks: Concepts, Algorithms and Applications. Cambridge University Press, Cambridge (2011)Google Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kelk, S., Scornavacca, C.: Towards the fixed parameter tractability of constructing minimal phylogenetic networks from arbitrary sets of nonbinary trees (2012). arXiv:1207.7034 [q-bio.PE]
  12. 12.
    Kelk, S., Scornavacca, C.: Constructing minimal phylogenetic networks from softwired clusters is fixed parameter tractable. Algorithmica 68(4), 886–915 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kelk, S., van Iersel, L., Lekić, N., Linz, S., Scornavacca, C., Stougie, L.: Cycle killer.. qu’est-ce que c’est? on the comparative approximability of hybridization number and directed feedback vertex set. SIAM J. Discrete Math. 26(4), 1635–1656 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Linz, S., Semple, C.: Hybridization in non-binary trees. IEEE/ACM Trans. Comput. Biol. Bioinf. 6(1), 30–45 (2009)CrossRefGoogle Scholar
  15. 15.
    Morrison, D.: Introduction to Phylogenetic Networks. RJR Productions, Uppsala (2011)Google Scholar
  16. 16.
    Nakhleh, L., Ringe, D., Warnow, T.: Perfect phylogenetic networks: a new methodology for reconstructing the evolutionary history of natural languages. Language 81(2), 382–420 (2005)CrossRefGoogle Scholar
  17. 17.
    Piovesan, T., Kelk, S.: A simple fixed parameter tractable algorithm for computing the hybridization number of two (not necessarily binary) trees. IEEE/ACM Trans. Comput. Biol. Bioinf. 10(1), 18–25 (2013)CrossRefGoogle Scholar
  18. 18.
    Semple, C., Steel, M.: Phylogenetics. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  19. 19.
    van Iersel, L., Kelk, S., Lekić, N., Stougie, L.: Approximation algorithms for nonbinary agreement forests. SIAM J. Discrete Math. 28(1), 49–66 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    van Iersel, L., Linz, S.: A quadratic kernel for computing the hybridization number of multiple trees. Inf. Process. Lett. 113(9), 318–323 (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Whidden, C., Beiko, R.G., Zeh, N.: Fixed-parameter algorithms for maximum agreement forests. SIAM J. Comput. 42(4), 1431–1466 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yufeng, W.: Close lower and upper bounds for the minimum reticulate network of multiple phylogenetic trees. Bioinformatics 26, i140–i148 (2010)CrossRefGoogle Scholar
  23. 23.
    Yufeng, W.: An algorithm for constructing parsimonious hybridization networks with multiple phylogenetic trees. J. Comput. Biol. 20(10), 792–804 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Centrum Wiskunde and Informatica (CWI)AmsterdamThe Netherlands
  2. 2.Department of Knowledge Engineering (DKE)Maastricht UniversityMaastrichtThe Netherlands

Personalised recommendations