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.


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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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