### Abstract

The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To this end we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the "root uncertain" variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees.

Original language | English |
---|---|

Pages (from-to) | 2993-3022 |

Number of pages | 30 |

Journal | Algorithmica |

Volume | 80 |

Issue number | 11 |

DOIs | |

Publication status | Published - 1 Nov 2018 |

### Keywords

- Binary trees
- Fixed parameter tractability
- Kernelization
- APX-hardness
- NP-completeness
- Phylogenetic networks
- FIXED-PARAMETER ALGORITHMS
- MAXIMUM AGREEMENT FOREST
- HYBRIDIZATION NUMBER
- APPROXIMATION
- TREES
- EVENTS

### Cite this

*Algorithmica*,

*80*(11), 2993-3022. https://doi.org/10.1007/s00453-017-0366-5

}

*Algorithmica*, vol. 80, no. 11, pp. 2993-3022. https://doi.org/10.1007/s00453-017-0366-5

**On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems.** / Van Iersel, Leo; Kelk, Steven; Stamoulis, Georgios; Stougie, Leen; Boes, Olivier.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems

AU - Van Iersel, Leo

AU - Kelk, Steven

AU - Stamoulis, Georgios

AU - Stougie, Leen

AU - Boes, Olivier

PY - 2018/11/1

Y1 - 2018/11/1

N2 - The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To this end we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the "root uncertain" variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees.

AB - The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To this end we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the "root uncertain" variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees.

KW - Binary trees

KW - Fixed parameter tractability

KW - Kernelization

KW - APX-hardness

KW - NP-completeness

KW - Phylogenetic networks

KW - FIXED-PARAMETER ALGORITHMS

KW - MAXIMUM AGREEMENT FOREST

KW - HYBRIDIZATION NUMBER

KW - APPROXIMATION

KW - TREES

KW - EVENTS

U2 - 10.1007/s00453-017-0366-5

DO - 10.1007/s00453-017-0366-5

M3 - Article

VL - 80

SP - 2993

EP - 3022

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 11

ER -