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

Better Practical Algorithms for rSPR Distance and Hybridization Number

The problem of computing the rSPR distance of two phylogenetic trees (denoted by RDC) is NP-hard and so is the problem of computing the hybridization number of two phylogenetic trees (denoted by HNC). Since they are important problems in phylogenetics, they have been studied extensively in the literature. Indeed, quite a number of exact or approximation algorithms have been designed and implemented for them. In this paper, we design and implement one exact algorithm for HNC and several approximation algorithms for RDC and HNC. Our experimental results show that the resulting exact program is much faster (namely, more than 80 times faster for the easiest dataset used in the experiments) than the previous best and its superiority in speed becomes even more significant for more difficult instances. Moreover, the resulting approximation programs output much better results than the previous bests; indeed, the outputs are always nearly optimal and often optimal. Of particular interest is the usage of the Monte Carlo tree search (MCTS) method in the design of our approximation algorithms. Our experimental results show that with MCTS, we can often solve HNC exactly within short time

Computing hybridization networks using agreement forests

Rooted phylogenetic trees are widely used in biology to represent the evolutionary history of certain species. Usually, such a tree is a simple binary tree only containing internal nodes of in-degree one and out-degree two representing specific speciation events. In applied phylogenetics, however, trees can contain nodes of out-degree larger than two because, often, in order to resolve some orderings of speciation events, there is only insufficient information available and the common way to model this uncertainty is to use nonbinary nodes (i.e., nodes of out-degree of at least three), also denoted as polytomies. Moreover, in addition to such speciation events, there exist certain biological events that cannot be modeled by a tree and, thus, require the more general concept of rooted phylogenetic networks or, more specifically, of hybridization networks. Examples for such reticulate events are horizontal gene transfer, hybridization, and recombination. Nevertheless, in order to construct hybridization networks, the less general concept of a phylogenetic tree can still be used as building block. More precisely, often, in a first step, phylogenetic trees for a set of species, each based on a distinctive orthologous gene, are constructed. In a second step, specific sets containing common subtrees of those trees, known as maximum acyclic agreement forests, are calculated, which are then glued together to a single hybridization network. In such a network, hybridization nodes (i.e., nodes of in-degree larger than or equal to two) can exist representing potential reticulate events of the underlying evolutionary history. As such events are considered as rare phenomena, from a biological point of view, especially those networks representing a minimum number of reticulate events, which is denoted as hybridization number, are of high interest. Consequently, in a mathematical aspect, the problem of calculating hybridization networks can be briefly described as follows. Given a set T of rooted phylogenetic trees sharing the same set of taxa, compute a hybridization network N displaying T with minimum hybridization number. In this context, we say that such a network N displays a phylogenetic tree T, if we can obtain T from N by removing as well as contracting some of its nodes and edges. Unfortunately, this is a computational hard problem (i.e., it is NP-hard), even for the simplest case given just two binary input trees. In this thesis, we present several methods tackling this NP-hard problem. Our first approach describes how to compute a representative set of minimum hybridization networks for two binary input trees. For that purpose, our approach implements the first non-naive algorithm - called allMAAFs - calculating all maximum acyclic agreement forests for two rooted binary phylogenetic trees on the same set of taxa. In a subsequent step, in order to maximize the efficiency of the algorithm allMAAFs, we have developed additionally several modifications each reducing the number of computational steps and, thus, significantly improving its practical runtime. Our second approach is an extension of our first approach making the underlying algorithm accessible to more than two binary input trees. For this purpose, our approach implements the algorithm allHNetworks being the first algorithm calculating all relevant hybridization networks displaying a set of rooted binary phylogenetic trees on the same set of taxa, which is a preferable feature when studying hybridization events. Lastly, we have developed a generalization of our second approach that can now deal with multiple nonbinary input trees. For that purpose, our approach implements the first non-naive algorithm - called allMulMAAFs - calculating a relevant set of nonbinary maximum acyclic agreement forests for two rooted (nonbinary) phylogenetic trees on the same set of taxa. Each of the algorithms above is integrated into our user friendly Java-based software package Hybroscale, which is freely available and platform independent, so that it runs on all major operating systems. Our program provides a graphical user interface for visualizing trees and networks. Moreover, it facilitates the interpretation of computed hybridization networks by adding specific features to its graphical representation and, thus, supports biologists in investigating reticulate evolution. In addition, we have implemented a method using a user friendly SQL-style modeling language for filtering the usually large amount of reported networks

Efficiently Calculating Evolutionary Tree Measures Using SAT

We develop techniques to calculate important measures in evolutionary biology by encoding to CNF formulas and using powerful SAT solvers. Comparing evolutionary trees is a necessary step in tree reconstruction algorithms, locating recombination and lateral gene transfer, and in analyzing and visualizing sets of trees. We focus on two popular comparison measures for trees: the hybridization number and the rooted subtree-prune-and-regraft (rSPR) distance. Both have recently been shown to be NP-hard, and effcient algorithms are needed to compute and approximate these measures. We encode these as a Boolean formula such that two trees have hybridization number k (or rSPR distance k) if and only if the corresponding formula is satisfiable. We use state-of-the-art SAT solvers to determine if the formula encoding the measure has a satisfying assignment. Our encoding also provides a rich source of real-world SAT instances, and we include a comparison of several recent solvers (minisat, adaptg2wsat, novelty+p, Walksat, March KS and SATzilla).Postprint (author’s final draft

Phylogenetic incongruence through the lens of Monadic Second Order logic

International audienceWithin the field of phylogenetics there is growing interest in measures for summarising the dissimilarity, or incongruence, of two or more phylogenetic trees. Many of these measures are NP-hard to compute and this has stimulated a considerable volume of research into fixed parameter tractable algorithms. In this article we use Monadic Second Order logic to give alternative, compact proofs of fixed parameter tractability for several well-known incongruence measures. In doing so we wish to demonstrate the considerable potential of MSOL - machinery still largely unknown outside the algorithmic graph theory community - within phylogenetics. A crucial component of this work is the observation that many measures, when bounded, imply the existence of an agreement forest of bounded size, which in turn implies that an auxiliary graph structure, the display graph, has bounded treewidth. It is this bound on treewidth that makes the machinery of MSOL available for proving fixed parameter tractability