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New Results on Optimizing Rooted Triplets Consistency

  • Jaroslaw Byrka
  • Sylvain Guillemot
  • Jesper Jansson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

Abstract

A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomial-time approximability of two related optimization problems called the maximum rooted triplets consistency problem (\(\textsc{MaxRTC}\)) and the minimum rooted triplets inconsistency problem (\(\textsc{MinRTI}\)) in which the input is a set \(\mathcal{R}\) of rooted triplets, and where the objectives are to find a largest cardinality subset of \(\mathcal{R}\) which is consistent and a smallest cardinality subset of \(\mathcal{R}\) whose removal from \(\mathcal{R}\) results in a consistent set, respectively. We first show that a simple modification to Wu’s Best-Pair-Merge-First heuristic [25] results in a bottom-up-based 3-approximation for \(\textsc{MaxRTC}\). We then demonstrate how any approximation algorithm for \(\textsc{MinRTI}\) could be used to approximate \(\textsc{MaxRTC}\), and thus obtain the first polynomial-time approximation algorithm for \(\textsc{MaxRTC}\) with approximation ratio smaller than 3. Next, we prove that for a set of rooted triplets generated under a uniform random model, the maximum fraction of triplets which can be consistent with any tree is approximately one third, and then provide a deterministic construction of a triplet set having a similar property which is subsequently used to prove that both \(\textsc{MaxRTC}\) and \(\textsc{MinRTI}\) are NP-hard even if restricted to minimally dense instances. Finally, we prove that \(\textsc{MinRTI}\) cannot be approximated within a ratio of Ω(logn) in polynomial time, unless P = NP.

Keywords

Approximation Algorithm Binary Tree Internal Node Approximation Ratio Phylogenetic Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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 Journal on Computing 10(3), 405–421 (1981)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ailon, N., Alon, N.: Hardness of fully dense problems. Information and Computation 205(8), 1117–1129 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alon, N.: Ranking Tournaments. SIAM Journal of Discrete Mathematics 20(1), 137–142 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bryant, D.: Building Trees, Hunting for Trees, and Comparing Trees: Theory and Methods in Phylogenetic Analysis. PhD thesis, University of Canterbury, Christchurch, New Zealand (1997)Google Scholar
  5. 5.
    Byrka, J., Gawrychowski, P., Huber, K.T., Kelk, S.: Worst-case optimal approximation algorithms for maximizing triplet consistency within phylogenetic networks (submitted, 2008)Google Scholar
  6. 6.
    Feige, U.: A Threshold of ln n for Approximating Set Cover. Journal of the ACM 45(4), 634–652 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Felsenstein, J.: Inferring Phylogenies. Sinauer Associates, Inc. (2004)Google Scholar
  8. 8.
    Ga̧sieniec, L., Jansson, J., Lingas, A., Östlin, A.: Inferring ordered trees from local constraints. In: Proc. of CATS 1998. Australian Computer Science Communications, vol. 20(3), pp. 67–76. Springer, Singapore (1998)Google Scholar
  9. 9.
    Ga̧sieniec, L., Jansson, J., Lingas, A., Östlin, A.: On the complexity of constructing evolutionary trees. Journal of Combinatorial Optimization 3(2–3), 183–197 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    He, Y.J., Huynh, T.N.D., Jansson, J., Sung, W.-K.: Inferring phylogenetic relationships avoiding forbidden rooted triplets. Journal of Bioinformatics and Computational Biology 4(1), 59–74 (2006)CrossRefGoogle Scholar
  11. 11.
    Henzinger, M.R., King, V., Warnow, T.: Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology. Algorithmica 24(1), 1–13 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM 48(4), 723–760 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    van Iersel, L., Keijsper, J., Kelk, S., Stougie, L., Hagen, F., Boekhout, T.: Constructing level-2 phylogenetic networks from triplets. In: Vingron, M., Wong, L. (eds.) RECOMB 2008. LNCS (LNBI), vol. 4955, pp. 450–462. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    van Iersel, L., Kelk, S., Mnich, M.: Uniqueness, intractability and exact algorithms: reflections on level-k phylogenetic networks (submitted, 2008)Google Scholar
  15. 15.
    Jansson, J.: On the complexity of inferring rooted evolutionary trees. In: Proc. of GRACO 2001. Electronic Notes in Discrete Mathematics, vol. 7, pp. 121–125. Elsevier, Amsterdam (2001)Google Scholar
  16. 16.
    Jansson, J., Lingas, A., Lundell, E.-M.: A triplet approach to approximations of evolutionary trees. In: Poster H15 presented at RECOMB 2004 (2004)Google Scholar
  17. 17.
    Jansson, J., Ng, J.H.-K., Sadakane, K., Sung, W.-K.: Rooted maximum agreement supertrees. Algorithmica 43(4), 293–307 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jansson, J., Nguyen, N.B., Sung, W.-K.: Algorithms for combining rooted triplets into a galled phylogenetic network. SIAM Journal on Computing 35(5), 1098–1121 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jansson, J., Sung, W.-K.: Inferring a level-1 phylogenetic network from a dense set of rooted triplets. Theoretical Computer Science 363(1), 60–68 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Jiang, T., Kearney, P., Li, M.: A polynomial time approximation scheme for inferring evolutionary trees from quartet topologies and its application. SIAM Journal on Computing 30(6), 1942–1961 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ng, M.P., Wormald, N.C.: Reconstruction of rooted trees from subtrees. Discrete Applied Mathematics 69(1–2), 19–31 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Page, R.D.M.: Modified mincut supertrees. In: Guigó, R., Gusfield, D. (eds.) WABI 2002. LNCS, vol. 2452, pp. 537–552. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  23. 23.
    Semple, C., Steel, M.: A supertree method for rooted trees. Discrete Applied Mathematics 105(1–3), 147–158 (2000)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Snir, S., Rao, S.: Using Max Cut to enhance rooted trees consistency. IEEE/ACM Transactions on Computational Biology and Bioinformatics 3(4), 323–333 (2006)CrossRefGoogle Scholar
  25. 25.
    Wu, B.Y.: Constructing the maximum consensus tree from rooted triples. Journal of Combinatorial Optimization 8(1), 29–39 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jaroslaw Byrka
    • 1
  • Sylvain Guillemot
    • 2
  • Jesper Jansson
    • 3
  1. 1.Centrum Wiskunde & Informatica (CWI), Kruislaan 413, NL-1098 SJ Amsterdam, Netherlands and Eindhoven University of TechnologyEindhovenNetherlands
  2. 2.INRIA Lille - Nord EuropeVilleneuve d’AscqFrance
  3. 3.Ochanomizu UniversityTokyoJapan

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