Efficient FPT Algorithms for (Strict) Compatibility of Unrooted Phylogenetic Trees

  • Julien Baste
  • Christophe Paul
  • Ignasi Sau
  • Celine Scornavacca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9778)


In phylogenetics, a central problem is to infer the evolutionary relationships between a set of species X; these relationships are often depicted via a phylogenetic tree – a tree having its leaves univocally labeled by elements of X and without degree-2 nodes – called the “species tree”. One common approach for reconstructing a species tree consists in first constructing several phylogenetic trees from primary data (e.g. DNA sequences originating from some species in X), and then constructing a single phylogenetic tree maximizing the “concordance” with the input trees. The so-obtained tree is our estimation of the species tree and, when the input trees are defined on overlapping – but not identical – sets of labels, is called “supertree”. In this paper, we focus on two problems that are central when combining phylogenetic trees into a supertree: the compatibility and the strict compatibility problems for unrooted phylogenetic trees. These problems are strongly related, respectively, to the notions of “containing as a minor” and “containing as a topological minor” in the graph community. Both problems are known to be fixed-parameter tractable in the number of input trees k, by using their expressibility in Monadic Second Order Logic and a reduction to graphs of bounded treewidth. Motivated by the fact that the dependency on k of these algorithms is prohibitively large, we give the first explicit dynamic programming algorithms for solving these problems, both running in time \(2^{O(k^2)}\,\cdot n\), where n is the total size of the input.


Phylogenetics Compatibility Unrooted phylogenetic trees Parameterized complexity FPT algorithm Dynamic programming 


  1. 1.
    Aho, A.V., Sagiv, Y., Szymanski, T.G., Ullman, J.D.: Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM J. Comput. 10(3), 405–421 (1981)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bininda-Emonds, O.R.P. (ed.): Phylogenetic Supertrees: Combining Information to Reveal the Tree of Life. Computational Biology, vol. 4. Springer, Netherlands (2004)MATHGoogle Scholar
  3. 3.
    Bininda-Emonds, O.R., Gittleman, J.L., Steel, M.A.: The (super) tree of life: procedures, problems, and prospects. Ann. Rev. Ecol. Syst. 33, 265–289 (2002)CrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: An \(O(c^k n)\) 5-approximation algorithm for treewidth. In: Proceedings of the IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pp. 499–508 (2013)Google Scholar
  5. 5.
    Bryant, D., Lagergren, J.: Compatibility of unrooted phylogenetic trees is FPT. Theor. Comput. Sci. 351(3), 296–302 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 150–159 (2011)Google Scholar
  7. 7.
    Delsuc, F., Brinkmann, H., Philippe, H.: Phylogenomics and the reconstruction of the tree of life. Nat. Rev. Genet. 6(5), 361–375 (2005)CrossRefGoogle Scholar
  8. 8.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)CrossRefMATHGoogle Scholar
  9. 9.
    Felsenstein, J.: Inferring Phylogenies. Sinauer Associates, Incorporated, Sunderland (2004)Google Scholar
  10. 10.
    Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic 130(1–3), 3–31 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gordon, A.D.: Consensus supertrees: the synthesis of rooted trees containing overlapping sets of labeled leaves. J. Classif. 3(2), 335–348 (1986)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kloks, T.: Treewidth, Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994)MATHGoogle Scholar
  13. 13.
    Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bull. EATCS 105, 41–72 (2011)MathSciNetMATHGoogle Scholar
  14. 14.
    Maddison, W.: Reconstructing character evolution on polytomous cladograms. Cladistics 5(4), 365–377 (1989)CrossRefGoogle Scholar
  15. 15.
    Ng, M., Wormald, N.C.: Reconstruction of rooted trees from subtrees. Discrete Appl. Math. 69(1–2), 19–31 (1996)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Scornavacca, C.: Supertree methods for phylogenomics. Ph.D. thesis, Université Montpellier II-Sciences et Techniques du Languedoc (2009)Google Scholar
  17. 17.
    Scornavacca, C., van Iersel, L., Kelk, S., Bryant, D.: The agreement problem for unrooted phylogenetic trees is FPT. J. Graph Algorithms Appl. 18(3), 385–392 (2014)CrossRefMATHGoogle Scholar
  18. 18.
    Steel, M.: The complexity of reconstructing trees from qualitative characters and subtrees. J. Classif. 9, 91–116 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Julien Baste
    • 1
  • Christophe Paul
    • 1
  • Ignasi Sau
    • 1
  • Celine Scornavacca
    • 2
  1. 1.CNRS, LIRMM, Université de MontpellierMontpellierFrance
  2. 2.Institut des Sciences de l’Evolution (Université de Montpellier, CNRS, IRD, EPHE)MontpellierFrance

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