Tree Containment With Soft Polytomies

Authors Matthias Bentert, Josef Malík, Mathias Weller

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Matthias Bentert
  • TU Berlin, Institut für Softwaretechnik und Theoretische Informatik, Berlin, Germany
Josef Malík
  • Czech Technical University, Prague, Czech Republic
Mathias Weller
  • CNRS, LIGM, Université Paris Est, Marne-la-Vallée, France

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Matthias Bentert, Josef Malík, and Mathias Weller. Tree Containment With Soft Polytomies. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


The Tree Containment problem has many important applications in the study of evolutionary history. Given a phylogenetic network N and a phylogenetic tree T whose leaves are labeled by a set of taxa, it asks if N and T are consistent. While the case of binary N and T has received considerable attention, the more practically relevant variant dealing with biological uncertainty has not. Such uncertainty manifests itself as high-degree vertices ("polytomies") that are "jokers" in the sense that they are compatible with any binary resolution of their children. Contrasting the binary case, we show that this problem, called Soft Tree Containment, is NP-hard, even if N is a binary, multi-labeled tree in which each taxon occurs at most thrice. On the other hand, we reduce the case that each label occurs at most twice to solving a 2-SAT instance of size O(|T|^3). This implies NP-hardness and polynomial-time solvability on reticulation-visible networks in which the maximum in-degree is bounded by three and two, respectively.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Applied computing → Biological networks
  • Phylogenetics
  • Reticulation-Visible Networks
  • Multifurcating Trees


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