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Fixed-Parameter and Approximation Algorithms for Maximum Agreement Forests of Multifurcating Trees


We present efficient fixed-parameter and approximation algorithms for the NP-hard problem of computing a maximum agreement forest (MAF) of a pair of multifurcating (nonbinary) rooted trees. Multifurcating trees arise naturally as a result of statistical uncertainty in current tree construction methods. The size of an MAF corresponds to the subtree prune-and-regraft distance of the two trees and is intimately connected to their hybridization number. These distance measures are essential tools for understanding reticulate evolution, such as lateral gene transfer, recombination, and hybridization. Our algorithms nearly match the running times of the currently best algorithms for the binary case. This is achieved using a combination of efficient branching rules (similar to but more complex than in the binary case) and a novel edge protection scheme that further reduces the size of the search space the algorithms need to explore.

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  1. This is similar to the generalization of the hybridization number used by Linz and Semple [18].

  2. In fact, using the same ideas as in the proof of Lemma 3, it is not difficult to see that this expansion never precludes obtaining the same forest \(F \div E\) by cutting a different set of \(|E|\) edges. We discuss the importance of this to hybridization and reticulate analysis in Sect. 6.

  3. We excluded this case from the statement of the lemma, in order to keep the cases covered by the different lemmas disjoint, but the lemma also holds for \(s = 1\). A similar comment applies to Lemma 12.

  4. This is a fairly loose bound on \(I(k,t)\), but it is easy to manipulate.


  1. Albrecht, B., Scornavacca, C., Cenci, A., Huson, D.H.: Fast computation of minimum hybridization networks. Bioinformatics 28(2), 191–197 (2012)

    Article  Google Scholar 

  2. Allen, B.L., Steel, M.: Subtree transfer operations and their induced metrics on evolutionary trees. Ann. Comb. 5(1), 1–15 (2001)

    Article  MathSciNet  Google Scholar 

  3. Baroni, M., Grünewald, S., Moulton, V., Semple, C.: Bounding the number of hybridisation events for a consistent evolutionary history. J. Math. Biol. 51(2), 171–182 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beiko, R.G., Hamilton, N.: Phylogenetic identification of lateral genetic transfer events. BMC Evol. Biol. 6(1), 15 (2006)

    Article  Google Scholar 

  5. Bonet, M.L., St. John, K., Mahindru, R., Amenta, N.: Approximating subtree distances between phylogenies. J. Comput. Biol. 13(8), 1419–1434 (2006)

    Article  MathSciNet  Google Scholar 

  6. Bordewich, M., McCartin, C., Semple, C.: A 3-approximation algorithm for the subtree distance between phylogenies. J. Discret. Algorithms 6(3), 458–471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bordewich, M., Semple, C.: On the computational complexity of the rooted subtree prune and regraft distance. Ann. Combin. 8(4), 409–423 (2005)

    Article  MathSciNet  Google Scholar 

  8. Bordewich, M., Semple, C.: Computing the minimum number of hybridization events for a consistent evolutionary history. Discret. Appl. Math. 155(8), 914–928 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Z.-Z., Wang, L.: Hybridnet: a tool for constructing hybridization networks. Bioinformatics 26(22), 2912–2913 (2010)

    Article  Google Scholar 

  10. Chen, Z.-Z., Wang, L.: Algorithms for reticulate networks of multiple phylogenetic trees. IEEE/ACM Trans. Comput. Biol. Bioinform. 9, 372–384 (2012)

    Article  Google Scholar 

  11. Chen, Z.-Z., Wang, L.: Faster exact computation of rSPR distance. In: Frontiers in Algorithmics and Algorithmic Aspects in Information and Management, pp. 36–47. Springer (2013)

  12. Chen, Z.-Z., Wang, L.: An ultrafast tool for minimum reticulate networks. J. Comput. Biol. 20(1), 38–41 (2013)

    Article  MathSciNet  Google Scholar 

  13. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill, New York (2001)

    MATH  Google Scholar 

  14. Hein, J., Jiang, T., Wang, L., Zhang, K.: On the complexity of comparing evolutionary trees. Discrete Appl. Math. 71(1–3), 153–169 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hickey, G., Dehne, F., Rau-Chaplin, A., Blouin, C.: SPR distance computation for unrooted trees. Evolut. Bioinform. 4, 17–27 (2008)

    Google Scholar 

  16. Hillis, D.M., Moritz, C., Mable, B.K. (eds.): Molecular Systematics. Sinauer Associates, Sunderland (1996)

    Google Scholar 

  17. van Iersel, L., Kelk, S., Lekić, N., Stougie, L.: Computing nonbinary agreement forests. SIAM J. Discret. Math. 28(1), 49–66 (2014)

    Article  MATH  Google Scholar 

  18. Linz, S., Semple, C.: Hybridization in nonbinary trees. IEEE/ACM Trans. Comput. Biol. Bioinform. 6, 30–45 (2009)

    Article  Google Scholar 

  19. Maddison, W.: Reconstructing character evolution on polytomous cladograms. Cladistics 5(4), 365–377 (1989)

    Article  Google Scholar 

  20. Rodrigues, E.M., Sagot, M.-F., Wakabayashi, Y.: The maximum agreement forest problem: approximation algorithms and computational experiments. Theor. Comput. Sci. 374(1–3), 91–110 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rosas-Magallanes, V., Deschavanne, P., Quintana-Murci, L., Brosch, R., Gicquel, B., Neyrolles, O.: Horizontal transfer of a virulence operon to the ancestor of mycobacterium tuberculosis. Mol. Biol. Evol. 23(6), 1129–1135 (2006)

    Article  Google Scholar 

  22. Whidden, C.: RSPR software.

  23. Whidden, C.: SPRSupertrees software.

  24. Whidden, C., Beiko, R.G., Zeh, N.: Fast FPT algorithms for computing rooted agreement forests: theory and experiments. In: Proceedings of the 9th International Symposium on Experimental Algorithms, SEA 2010, vol. 6049 of Lecture Notes in Computer Science, pp. 141–153. Springer (2010)

  25. Whidden, C., Beiko, R.G., Zeh, N.: Fixed-parameter algorithms for maximum agreement forests. SIAM J. Comput. 42(4), 1431–1466 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Whidden, C., Zeh, N.: A unifying view on approximation and FPT of agreement forests. In: Proceedings of the 9th International Workshop, WABI 2009, volume 5724 of Lecture Notes in Bioinformatics, pp. 390–401. Springer (2009)

  27. Whidden, C., Zeh, N., Beiko, R.G.: Supertrees based on the subtree prune-and-regraft distance. Syst. Biol. 63(4), 566–581 (2014)

    Article  Google Scholar 

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Correspondence to Norbert Zeh.

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This work was done while Chris Whidden was a PhD student at Dalhousie University. Chris Whidden was supported by a Killam predoctoral scholarship and the Tula Foundation, Robert G. Beiko is a Canada Research Chair and was supported by NSERC, Genome Atlantic, and the Canada Foundation for Innovation, and Norbert Zeh is a Canada Research Chair and was supported by NSERC and the Canada Foundation for Innovation.

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Whidden, C., Beiko, R.G. & Zeh, N. Fixed-Parameter and Approximation Algorithms for Maximum Agreement Forests of Multifurcating Trees. Algorithmica 74, 1019–1054 (2016).

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  • Phylogenetics
  • Fixed-parameter tractability
  • Approximation algorithms