Finding a most parsimonious or likely tree in a network with respect to an alignment

Steven Kelk, Fabio Pardi, Celine Scornavacca, Leo van Iersel

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Phylogenetic networks are often constructed by merging multiple conflicting phylogenetic signals into a directed acyclic graph. It is interesting to explore whether a network constructed in this way induces biologically-relevant phylogenetic signals that were not present in the input. Here we show that, given a multiple alignment A for a set of taxa X and a rooted phylogenetic network N whose leaves are labelled by X, it is NP-hard to locate a most parsimonious phylogenetic tree displayed by N (with respect to A) even when the level of N—the maximum number of reticulation nodes within a biconnected component—is 1 and A contains only 2 distinct states. (If, additionally, gaps are allowed the problem becomes APX-hard.) We also show that under the same conditions, and assuming a simple binary symmetric model of character evolution, finding a most likely tree displayed by the network is NP-hard. These negative results contrast with earlier work on parsimony in which it is shown that if A consists of a single column the problem is fixed parameter tractable in the level. We conclude with a discussion of why, despite the NP-hardness, both the parsimony and likelihood problem can likely be well-solved in practice.

Original languageEnglish
Pages (from-to)1-21
Number of pages21
JournalJournal of Mathematical Biology
Publication statusE-pub ahead of print - 2018


  • APX-hardness
  • Maximum likelihood
  • Maximum parsimony
  • NP-hardness
  • Phylogenetic network
  • Phylogenetic tree

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