Hybridization Number on Three Rooted Binary Trees is EPT

Abstract

Phylogenetic networks are leaf-labelled directed acyclic graphs that are used to describe non-treelike evolutionary histories and are thus a generalization of phylogenetic trees. The hybridization number of a phylogenetic network is the sum of all in-degrees minus the number of nodes plus one. The HYBRIDIZATION NUMBER problem takes as input a collection of rooted binary phylogenetic trees and asks to construct a phylogenetic network that contains an embedding of each of the input trees and has the smallest possible hybridization number. We present an algorithm for the HYBRIDIZATION NUMBER problem on three binary phylogenetic trees on n leaves that runs in time O(c k poly(n)), with k the hybridization number of an optimal network and c some (astronomical) constant. For the case of two trees, an algorithm with running time O(3.18 k n) was proposed before whereas an algorithm with running time O(c k poly(n)), also called an EPT algorithm, had prior to this article remained elusive for more than two trees. The algorithm for two trees uses the close connection to acyclic agreement forests to achieve a linear exponent in the running time, while previous algorithms for more than two trees (explicitly or implicitly) relied on a brute force search through all possible underlying network topologies, leading to running times that are not O(c k poly(n)), for any c. The connection to acyclic agreement forests is much weaker for more than two trees, so even given the right agreement forest, the reconstruction of the network poses major challenges. We prove novel structural results that allow us to reconstruct a network without having to guess the underlying topology. Our techniques generalize to more than three input trees with the exception of one key lemma that maps nodes in the network to tree nodes in order to minimize the amount of guessing involved in constructing the network. The main open problem therefore is to prove results that establish such a mapping for more than three trees. the topology of the "true" phylogenetic tree, given only information about the taxa, often DNA data. This is a challenging computational problem and has been the topic of intensive research during the last 40 years [9]. Recently our understanding of evolutionary mechanisms has deepened and there is growing awareness that evolution is not always treelike [1]. In particular, due to reticulate phenomena such as hybridization and horizontal gene transfer [19], the evolution of a set of species is sometimes better modelled as a rooted phylogenetic network [10], essentially a generalization of phylogenetic trees to directed acyclic graphs (DAGs). In such graphs, nodes with indegree two or higher, known as reticulations, represent the points at which two or more lineages merge, rather than diversify.The study of rooted phylogenetic networks is comparatively new and has given rise to many novel and hard combinatorial optimization problems [10]. In this article we focus on the HYBRIDIZATION NUMBER problem, originally introduced in [2, 3], which is on...

Attaching leaves and picking cherries to characterise the hybridisation number for a set of phylogenies

“…To see that these two minima can be different for a fixed set of phylogenetic trees, consider the set P of trees presented in Figure 2. It was shown in [13,15] that h(P) = 3 and that there are six phylogenetic networks each of which displays P and has a hybridisation number of three. However, none of these six phylogenetic networks is tree-child.…”

“…To illustrate, consider the two sets P and P of phylogenetic trees shown in Figure is a cherry-picking sequence for P of weight w(σ) = 12 − 8 = 4. In fact, it follows from [13,15] that h tc (P) = 4 (see Section 6 for details). On the other hand, σ = (5, z), (5,8), (4, z), (4, 1), (z, 3), (z, 6), (2, 3), (3, 1), (6, 7), (7,8), (1,8),…”

Kernelizations for the hybridization number problem on multiple nonbinary trees

“…For this, a fast exponential-time exact algorithm is needed, or a good heuristic. Although we have presented an O (n f (k) t) time algorithm for Hybridization Number, with n = |X| and t = |T |, it is not known if there exists an O (c n )-algorithm for some constant c. While O (c k n O (1) ) algorithms have been developed for instances consisting of two binary trees [28] and very recently for three binary trees [14], it is not clear if they exist for four or more binary trees, or for two or more nonbinary trees. Note that, for practical applications, the kernelization can also be combined with an efficient heuristic.…”

Kernelizations for the Hybridization Number Problem on Multiple Nonbinary Trees

“…While O(c k n O(1) ) algorithms have been developed for instances consisting of two binary trees [28] and very recently for three binary trees [14], it is not clear if they exist for four or more binary trees, or for two or more nonbinary trees. Note that, for practical applications, the kernelization can also be combined with an efficient heuristic.…”