On unrooted and root-uncertain variants of several well-known phylogenetic network problems

TL;DR

This paper explores unrooted and root-uncertain variants of phylogenetic network problems, establishing complexity results and fixed-parameter tractability for various cases, with implications for biological inference accuracy.

Contribution

It introduces and analyzes unrooted and root-uncertain variants of hybridization number problems, providing complexity classifications and fixed-parameter algorithms.

Findings

Unrooted phylogenetic network display problem is NP-hard.

Unrooted hybridization number problem is FPT in reticulation number.

Root-uncertain hybridization number problem is APX-hard but FPT in hybridization number.

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 \emph{unrooted} phylogenetic network displays (i.e. embeds) an \emph{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 (TBR) 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.