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)


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.


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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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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