Treewidth-Based Algorithms for the Small Parsimony Problem on Networks

Authors Celine Scornavacca, Mathias Weller

Thumbnail PDF


  • Filesize: 0.86 MB
  • 21 pages

Document Identifiers

Author Details

Celine Scornavacca
  • Institut des Sciences de l’Evolution, Université de Montpellier, CNRS, IRD, EPHE, France
Mathias Weller
  • LIGM, CNRS, Université Gustave Eiffel, Paris, France


We thank Christophe Paul for sharing his expertise on treewidth formulations.

Cite AsGet BibTex

Celine Scornavacca and Mathias Weller. Treewidth-Based Algorithms for the Small Parsimony Problem on Networks. In 21st International Workshop on Algorithms in Bioinformatics (WABI 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 201, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Phylogenetic reconstruction is one of the paramount challenges of contemporary bioinformatics. A subtask of existing tree reconstruction algorithms is modeled by the Small Parsimony problem: given a tree T and an assignment of character-states to its leaves, assign states to the internal nodes of T such as to minimize the parsimony score, that is, the number of edges of T connecting nodes with different states. While this problem is polynomial-time solvable on trees, the matter is more complicated if T contains reticulate events such as hybridizations or recombinations, i.e. when T is a network. Indeed, three different versions of the parsimony score on networks have been proposed and each of them is NP-hard to decide. Existing parameterized algorithms focus on combining the number of possible character-states with the number of reticulate events (per biconnected component). Here, we consider the treewidth of the undirected graph underlying the input network as parameter, presenting dynamic programming algorithms for (slight generalizations of) all three versions of the parsimony problem on networks. Our algorithms use a formulation of the treewidth that may facilitate formalizing treewidth-based dynamic programming algorithms on phylogenetic networks for other problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Applied computing → Molecular sequence analysis
  • Phylogenetics
  • parsimony
  • phylogenetic networks
  • parameterized complexity
  • dynamic programming
  • treewidth


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Stefan Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability - a survey. BIT Numerical Mathematics, 25(1):1-23, 1985. Google Scholar
  2. Various Authors. The graph parameter hierarchy. Available at, 2021.
  3. Vincent Berry, Celine Scornavacca, and Mathias Weller. Scanning phylogenetic networks is NP-hard. In Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM'20), pages 519-530. Springer, 2020. Google Scholar
  4. Hans L Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing, 25(6):1305-1317, 1996. Google Scholar
  5. Hans L. Bodlaender. Discovering treewidth. In Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM'05), pages 1-16, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. Google Scholar
  6. Christopher Bryant, Mareike Fischer, Simone Linz, and Charles Semple. On the quirks of maximum parsimony and likelihood on phylogenetic networks. Journal of Theoretical Biology, 417:100-108, 2017. Google Scholar
  7. David Bryant and Jens Lagergren. Compatibility of unrooted phylogenetic trees is FPT. Theoretical Computer Science, 351(3):296-302, 2006. Google Scholar
  8. Laurent Bulteau and Mathias Weller. Parameterized algorithms in bioinformatics: an overview. Algorithms, 12(12):256, 2019. Google Scholar
  9. Bruno Courcelle. The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Information and computation, 85(1):12-75, 1990. Google Scholar
  10. Holger Dell, Christian Komusiewicz, Nimrod Talmon, and Mathias Weller. The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), volume 89 of Leibniz International Proceedings in Informatics (LIPIcs), pages 30:1-30:12, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  11. Nick D. Dendris, Lefteris M. Kirousis, and Dimitrios M. Thilikos. Fugitive-search games on graphs and related parameters. Theoretical Computer Science, 172(1):233-254, 1997. Google Scholar
  12. Joseph Felsenstein. Inferring phylogenies, volume 2. Sinauer associates Sunderland, MA, 2004. Google Scholar
  13. Mareike Fischer, Leo Van Iersel, Steven Kelk, and Celine Scornavacca. On computing the maximum parsimony score of a phylogenetic network. SIAM Journal on Discrete Mathematics, 29(1):559-585, 2015. Google Scholar
  14. Walter M Fitch. Toward defining the course of evolution: minimum change for a specific tree topology. Systematic Biology, 20(4):406-416, 1971. Google Scholar
  15. Jotun Hein. Reconstructing evolution of sequences subject to recombination using parsimony. Mathematical Biosciences, 98(2):185-200, 1990. Google Scholar
  16. Daniel H Huson, Regula Rupp, and Celine Scornavacca. Phylogenetic networks: concepts, algorithms and applications. Cambridge University Press, 2010. Google Scholar
  17. G. Jin, L. Nakhleh, S. Snir, and T. Tuller. Inferring phylogenetic networks by the maximum parsimony criterion: A case study. Molecular Biology and Evolution, 24(1):324-337, 2006. Google Scholar
  18. G. Jin, L. Nakhleh, S. Snir, and T. Tuller. Maximum likelihood of phylogenetic networks. Bioinformatics, 22(21):2604-2611, 2006. Google Scholar
  19. Guohua Jin, L. Nakhleh, S. Snir, and T. Tuller. Parsimony score of phylogenetic networks: Hardness results and a linear-time heuristic. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 6(3):495-505, 2009. Google Scholar
  20. Lavanya Kannan and Ward C. Wheeler. Maximum Parsimony on Phylogenetic networks. Algorithms for Molecular Biology, 7(1):9, 2012. Google Scholar
  21. Lavanya Kannan and Ward C. Wheeler. Exactly computing the parsimony scores on phylogenetic networks using dynamic programming. Journal of Computational Biology, 21(4):303-319, 2014. Google Scholar
  22. Steven Kelk, Fabio Pardi, Celine Scornavacca, and Leo van Iersel. Finding a most parsimonious or likely tree in a network with respect to an alignment. Journal of Mathematical Biology, 78(1-2):527-547, 2019. Google Scholar
  23. Ton Kloks. Treewidth: computations and approximations, volume 842. Springer Science & Business Media, 1994. Google Scholar
  24. Tuukka Korhonen. Single-exponential time 2-approximation algorithm for treewidth. CoRR, abs/2104.07463, 2021. URL:
  25. Guillaume Mescoff, Christophe Paul, and Dimitrios Thilikos. A polynomial time algorithm to compute the connected tree-width of a series-parallel graph, 2021. URL:
  26. Luay Nakhleh, Guohua Jin, Fengmei Zhao, and John Mellor-Crummey. Reconstructing phylogenetic networks using maximum parsimony. In 2005 IEEE Computational Systems Bioinformatics Conference (CSB'05), pages 93-102. IEEE, 2005. Google Scholar
  27. Charles-Elie Rabier, Vincent Berry, Marnus Stoltz, João D. Santos, Wensheng Wang, Glaszmann Jean-Christophe, Fabio Pardi, and Celine Scornavacca. On the inference of complicated phylogenetic networks by Markov Chain Monte-Carlo. Submitted. Google Scholar
  28. Hisao Tamaki. Positive-instance driven dynamic programming for treewidth. Journal of Combinatorial Optimization, 37(4):1283-1311, 2019. Google Scholar
  29. Leo Van Iersel, Mark Jones, and Celine Scornavacca. Improved maximum parsimony models for phylogenetic networks. Systematic Biology, 67(3):518-542, 2018. Google Scholar
  30. Ward C Wheeler. Phylogenetic network analysis as a parsimony optimization problem. BMC Bioinformatics, 16(1):1-9, 2015. Google Scholar
  31. Jiafan Zhu, Dingqiao Wen, Yun Yu, Heidi M Meudt, and Luay Nakhleh. Bayesian inference of phylogenetic networks from bi-allelic genetic markers. PLoS Computational Biology, 14(1):e1005932, 2018. Google Scholar
  32. Jiafan Zhu, Yun Yu, and Luay Nakhleh. In the light of deep coalescence: revisiting trees within networks. BMC Bioinformatics, 17(14):271-282, 2016. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail