Minimum Quartet Inconsistency Is Fixed Parameter Tractable

  • Jens Gramm
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2089)


We study the parameterized complexity of the problem to reconstruct a binary (evolutionary) tree from a complete set of quartet topologies in the case of a limited number of errors. More precisely, we are given n taxa, exactly one topology for every subset of 4 taxa, and a positive integer k (the parameter). Then, the Minimum Quartet In- consistency (MQI) problem is the question of whether we can find an evolutionary tree inducing a set of quartet topologies that differs from the given set in only k quartet topologies. MQI is NP-complete. However, we can compute the required tree in worst case time O(4k. n + n4)— the problem is fixed parameter tractable. Our experimental results show that in practice, also based on heuristic improvements proposed by us, even a much smaller exponential growth can be achieved. We extend the fixed parameter tractability result to weighted versions of the problem. In particular, our algorithm can produce all solutions that resolve at most k errors.


Search Tree Evolutionary Tree Polynomial Time Approximation Scheme Heuristic Improvement Problem Kernel 
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  1. 1.
    J. Alber, J. Gramm, and R. Niedermeier. Faster exact solutions for hard problems: a parameterized point of view. Discrete Mathematics, 229(1-3):3–27, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    H.-J. Bandelt and A. Dress. Reconstructing the shape of a tree from observed dissimilarity data. Advances in Applied Mathematics, 7:309–343, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    A. Ben-Dor, B. Chor, D. Graur, R. Ophir, and D. Pelleg. Constructing phylogenies from quartets: elucidation of eutherian superordinal relationships. Journal of Computational Biology, 5:377–390, 1998.CrossRefGoogle Scholar
  4. 4.
    V. Berry and O. Gascuel. Inferring evolutionary trees with strong combinatorial evidence. Theoretical Computer Science, 240:271–298, 2000. Software available through Scholar
  5. 5.
    P. Buneman. The recovery of trees from measures of dissimilarity. In Hodson et al., eds, Anglo-Romanian Conference on Mathematics in the Archaeological and Historical Sciences, pages 387–395, 1971. Edinburgh University Press.Google Scholar
  6. 6.
    B. Chor. From quartets to phylogenetic trees. In Proceedings of the 25th SOFSEM, number 1521 in LNCS, pages 36–53, 1998. Springer.Google Scholar
  7. 7.
    R.G. Downey and M.R. Fellows. Parameterized Complexity. 1999. Springer.Google Scholar
  8. 8.
    M.R. Fellows. Parameterized complexity: new developments and research frontiers. In Downey and Hirschfeldt, eds, Aspects of Complexity, to appear, 2001. De Gruyter.Google Scholar
  9. 9.
    J. Felsenstein. PHYLIP (Phylogeny Inference Package) version 3.5c. Distributed by the author. Department of Genetics, University of Washington, Seattle. 1993. Available through
  10. 10.
    J. Gramm and R. Niedermeier. Minimum Quartet Inconsistency is fixed parameter tractable. Technical Report WSI-2001-3, WSI für Informatik, Universität Tübingen, Fed. Rep. of Germany, January 2001. Report available through
  11. 11.
    T. Jiang, P. Kearney, and M. Li. Some open problems in computational molecular biology. Journal of Algorithms, 34:194–201, 2000.MathSciNetCrossRefGoogle Scholar
  12. 12.
    T. Jiang, P. Kearney, and M. Li. A polynomial time approximation scheme for inferring evolutionary trees from quartet topologies and its application. To appear in SIAM Journal on Computing, 2001.Google Scholar
  13. 13.
    R. Niedermeier and P. Rossmanith. An efficient fixed parameter algorithm for 3-Hitting Set. Technical Report WSI-99-18, WSI für Informatik, Universität Tübingen, October 1999. To appear in Journal of Discrete Algorithms.Google Scholar
  14. 14.
    R. Niedermeier and P. Rossmanith. On efficient fixed parameter algorithms for Weighted Vertex Cover. In Proceedings of the 11th International Symposium on Algorithms and Computation, number 1969 in LNCS, pages 180–191, 2000. Springer.CrossRefGoogle Scholar
  15. 15.
    K. St. John, T. Warnow, B.M.E. Moret, and L. Vawter. Performance study of phylogenetic methods: (unweighted) quartet methods and neighbor-joining. In Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms, pages 196–205, 2001. SIAM Press.Google Scholar
  16. 16.
    M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification, 9:91–116, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    K. Strimmer and A. von Haeseler. Quartet puzzling: a quartet maximum-likelihood method for reconstructing tree topologies. Molecular Biology and Evolution, 13(7):964–969, 1996.CrossRefGoogle Scholar
  18. 18.
    M. Weiß, Z. Yang, and F. Oberwinkler. Molecular phylogenetic studies in the genus Amanita. Canadian Journal of Botany, 76:1170–1179, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jens Gramm
    • 1
  • Rolf Niedermeier
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenFed. Rep. of Germany

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