A simple fixed parameter tractable algorithm for computing the hybridization number of two (not necessarily binary) trees

Here we present a new fixed parameter tractable algorithm to compute the hybridization number r of two rooted, not necessarily binary phylogenetic trees on taxon set X in time (6^r.r!).poly(n)$, where n=|X|. The novelty of this approach is its use of terminals, which are maximal elements of a natural partial order on X, and several insights from the softwired clusters literature. This yields a surprisingly simple and practical bounded-search algorithm and offers an alternative perspective on the underlying combinatorial structure of the hybridization number problem

Kernelizations for the hybridization number problem on multiple nonbinary trees

Given a finite set $X$, a collection $\mathcal{T}$ of rooted phylogenetic trees on $X$ and an integer $k$, the Hybridization Number problem asks if there exists a phylogenetic network on $X$ that displays all trees from $\mathcal{T}$ and has reticulation number at most $k$. We show two kernelization algorithms for Hybridization Number, with kernel sizes $4k(5k)^t$ and $20k^2(\Delta^+-1)$ respectively, with $t$ the number of input trees and $\Delta^+$ their maximum outdegree. Experiments on simulated data demonstrate the practical relevance of these kernelization algorithms. In addition, we present an $n^{f(k)}t$-time algorithm, with $n=|X|$ and $f$ some computable function of $k$

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

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.Comment: 28 pages, 8 Figure

A practical approximation algorithm for solving massive instances of hybridization number for binary and nonbinary trees

Reticulate events play an important role in determining evolutionary relationships. The problem of computing the minimum number of such events to explain discordance between two phylogenetic trees is a hard computational problem. Even for binary trees, exact solvers struggle to solve instances with reticulation number larger than 40-50. Here we present CycleKiller and NonbinaryCycleKiller, the first methods to produce solutions verifiably close to optimality for instances with hundreds or even thousands of reticulations. Using simulations, we demonstrate that these algorithms run quickly for large and difficult instances, producing solutions that are very close to optimality. As a spin-off from our simulations we also present TerminusEst, which is the fastest exact method currently available that can handle nonbinary trees: this is used to measure the accuracy of the NonbinaryCycleKiller algorithm. All three methods are based on extensions of previous theoretical work and are publicly available. We also apply our methods to real data

On Computing the Maximum Parsimony Score of a Phylogenetic Network

Phylogenetic networks are used to display the relationship of different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as Maximum Parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of Maximum Parsimony on networks, "hardwired" and "softwired", and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multicut, we show that computing the hardwired parsimony score for 2-state characters is polynomial-time solvable, while for characters with more states this problem becomes NP-hard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixed-parameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixed-parameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and softwired parsimony score can be computed efficiently using Integer Linear Programming. The software has been made freely available