### Abstract

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 language | English |
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Pages (from-to) | 527-547 |

Number of pages | 21 |

Journal | Journal of Mathematical Biology |

Volume | 78 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Keywords

- Phylogenetic tree
- Phylogenetic network
- Maximum parsimony
- Maximum likelihood
- NP-hardness
- APX-hardness
- MAXIMUM-LIKELIHOOD
- BAYESIAN-INFERENCE
- RECOMBINATION
- EVOLUTION
- MODEL

### Cite this

*Journal of Mathematical Biology*,

*78*(1-2), 527-547. https://doi.org/10.1007/s00285-018-1282-2