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

Leo Van Iersel,Georgios Stamoulis,Steven Kelk,Olivier Boes,Leen Stougie

doi:10.1007/s00453-017-0366-5

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.

Highlights

Within the field of phylogenetics the evolutionary history of a set of contemporary species X, known as taxa, is usually modelled as a tree where the leaves are bijectively labelled by X
In this article we have studied two variations of the classical hybridization number (HN) problem: the root-uncertain variant Root Uncertain Hybridization Number (RUHN) and the unrooted variant Unrooted Hybridization Number (UHN)
We have studied the natural unrooted variant of the tree containment (TC) decision problem, Unrooted Tree Containment (UTC)

Summary

Introduction

Within the field of phylogenetics the evolutionary history of a set of contemporary species X , known as taxa, is usually modelled as a tree where the leaves are bijectively labelled by X. To take a simple example, consider two identical unrooted trees on a set X of n taxa which should, in principle, be rooted in the same place, so the hybridization number should be 0. Speaking, RUHN is the most relevant problem we study because it explicitly acknowledges the fact that the input unrooted trees need to be rooted in some way. This highlights the fact that a root exists, but its location is uncertain and we would like to infer the root locations such that the reticulation number of a network that displays them all is minimized. We begin with a section dedicated to preliminaries in which we formally define all the models studied in this paper and briefly discuss their differences

Preliminaries

The Tree Containment Problem on Unrooted Networks and Trees

Hardness

Parameterized Complexity of RUHN

Conclusions and Open Problems

A Omitted APX-Hardness Proof