In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it is NP-complete to decide whether a tree-child network with m roots is proper forest-based, for each m≥2. Moreover, for binary networks the problem remains NP-complete when m≥3 but becomes polynomial-time solvable for m=2. We also give a fixed parameter tractable (FPT) algorithm, with parameters the maximum outdegree of a vertex, the number of roots, and the number of indegree 2 vertices, for deciding if a semi-binary network is proper forest-based. A key element in proving our results is a new characterization for when a network with m roots is proper forest-based in terms of certain m-colorings.

The Hybridization problem asks to reconcile a set of conflicting phylogenetic trees into a single phylogenetic network with the smallest possible number of reticulation nodes. This problem is computationally hard and previous solutions are limited to small and/or severely restricted data sets, for example, a set of binary trees with the same taxon set or only two non-binary trees with non-equal taxon sets. Building on our previous work on binary trees, we present FHyNCH, the first algorithmic framework to heuristically solve the Hybridization problem for large sets of multifurcating trees whose sets of taxa may differ. Our heuristics combine the cherry-picking technique, recently proposed to solve the same problem for binary trees, with two carefully designed machine-learning models. We demonstrate that our methods are practical and produce qualitatively good solutions through experiments on both synthetic and real data sets.

How many reticulations are needed for a phylogenetic network to display a given set of k phylogenetic trees on n leaves? For k = 2, Baroni et al. [Ann. Comb. 8, 391-408 (2005)] showed that the answer is n − 2. Here, we show that, for k ≥ 3 the answer is at least (3 /2 − ε)n. Concretely, we prove that, for each ε > 0, there is some n ∈ N such that three n-leaf caterpillar trees can be constructed in such a way that any network displaying these caterpillars contains at least (3 /2 − ε)n reticulations. The case of three trees is interesting since it is the easiest case that cannot be equivalently formu-lated in terms of agreement forests. Instead, we base the result on a surprising lower bound for multilabelled trees (MUL-trees) displaying the caterpillars. Indeed, we show that one cannot do (more than an ε) better than the trivial MUL-tree resulting from a simple concatenation of the given caterpillars. The results are relevant for the development of methods for the Hybridization Number problem on more than two trees. This fundamental problem asks to construct a binary phylogenetic network with a minimum number of reticulations displaying a given set of phylogenetic trees.

The maximum parsimony distance d_MP(T₁,T₂) and the bounded-state maximum parsimony distance d_MP^t(T₁,T₂) measure the difference between two phylogenetic trees T₁,T₂ in terms of the maximum difference between their parsimony scores for any character (with t a bound on the number of states in the character, in the case of d_MP^t(T₁,T₂)). While computing d_MP(T₁,T₂) was previously shown to be fixed-parameter tractable with a linear kernel, no such result was known for d_MP^t(T₁,T₂). In this paper, we prove that computing d_MP^t(T₁,T₂) is fixed-parameter tractable for all t. Specifically, we prove that this problem has a kernel of size O(klg⁡k), where k=d_MP^t(T₁,T₂). As the primary analysis tool, we introduce the concept of leg-disjoint incompatible quartets, which may be of independent interest.

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network N can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of N and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.

Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariants defined thus far for phylogenetic networks that are not trees require vertex labels and either contain a large number of variables, or they have exponentially many terms in the number of reticulations. This can make it difficult to compute these polynomials and to use them to analyse unlabelled networks. In this paper, we shall show how to circumvent some of these difficulties for rooted cactuses and cactuses. As well as being important in other areas such as operations research, rooted cactuses contain some common classes of phylogenetic networks such phylogenetic trees and level-1 networks. More specifically, we define a polynomial F that is a complete invariant for the class of rooted cactuses without vertices of indegree 1 and outdegree 1 that has 5 variables, and a polynomial Q that is a complete invariant for the class of rooted cactuses that has 6 variables whose degree can be bounded linearly in terms of the size of the rooted cactus. We also explain how to extend the Q polynomial to define a complete invariant for leaf-labelled rooted cactuses as well as (unrooted) cactuses.

Phylogenetic networks are used to represent the evolutionary history of species. Recently, the new class of orchard networks was introduced, which were later shown to be interpretable as trees with additional horizontal arcs. This makes the network class ideal for capturing evolutionary histories that involve horizontal gene transfers. Here, we study the minimum number of additional leaves needed to make a network orchard. We demonstrate that computing this proximity measure for a given network is NP-hard and describe a tight upper bound. We also give an equivalent measure based on vertex labellings to construct a mixed integer linear programming formulation. Our experimental results, which include both real-world and synthetic data, illustrate the efficiency of our implementation.

Background: Combining a set of phylogenetic trees into a single phylogenetic network that explains all of them is a fundamental challenge in evolutionary studies. Existing methods are computationally expensive and can either handle only small numbers of phylogenetic trees or are limited to severely restricted classes of networks. Results: In this paper, we apply the recently-introduced theoretical framework of cherry picking to design a class of efficient heuristics that are guaranteed to produce a network containing each of the input trees, for practical-size datasets consisting of binary trees. Some of the heuristics in this framework are based on the design and training of a machine learning model that captures essential information on the structure of the input trees and guides the algorithms towards better solutions. We also propose simple and fast randomised heuristics that prove to be very effective when run multiple times. Conclusions: Unlike the existing exact methods, our heuristics are applicable to datasets of practical size, and the experimental study we conducted on both simulated and real data shows that these solutions are qualitatively good, always within some small constant factor from the optimum. Moreover, our machine-learned heuristics are one of the first applications of machine learning to phylogenetics and show its promise.

We study the problem of finding a temporal hybridization network containing at most k reticulations, for an input consisting of a set of phylogenetic trees. First, we introduce an FPT algorithm for the problem on an arbitrary set of m binary trees with n leaves each with a running time of O(5 ^k· n· m). We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most d and at most k reticulations in O((8 k) ^d5 ^k· k· n· m) time. Lastly, we introduce an O(6 ^kk! · k· n²) time algorithm for computing a temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.

We present the first fixed-parameter algorithm for constructing a tree-child phylogenetic network that displays an arbitrary number of binary input trees and has the minimum number of reticulations among all such networks. The algorithm uses the recently introduced framework of cherry picking sequences and runs in O((8 k) ^kpoly (n, t)) time, where n is the number of leaves of every tree, t is the number of trees, and k is the reticulation number of the constructed network. Moreover, we provide an efficient parallel implementation of the algorithm and show that it can deal with up to 100 input trees on a standard desktop computer, thereby providing a major improvement over previous phylogenetic network construction methods.

Evolutionary histories for species that cross with one another or exchange genetic material can be represented by leaf-labelled, directed graphs called phylogenetic networks. A major challenge in the burgeoning area of phylogenetic networks is to develop algorithms for building such networks by amalgamating small networks into a single large network. The level of a phylogenetic network is a measure of its deviation from being a tree; the higher the level of a network, the less treelike it becomes. Various algorithms have been developed for building level-1 networks from small networks. However, level-1 networks may not be able to capture the complexity of some data sets. In this paper, we present a polynomial-time algorithm for constructing a rooted binary level-2 phylogenetic network from a collection of 3-leaf networks or trinets. Moreover, we prove that the algorithm will correctly reconstruct such a network if it is given all of the trinets in the network as input. The algorithm runs in time O(t⋅n+n⁴) with t the number of input trinets and n the number of leaves. We also show that there is a fundamental obstruction to constructing level-3 networks from trinets, and so new approaches will need to be developed for constructing level-3 and higher level-networks.

Phylogenetic networks are used in biology to represent evolutionary histories. The class of orchard phylogenetic networks was recently introduced for their computational benefits, without any biological justification. Here, we show that orchard networks can be interpreted as trees with additional horizontal arcs. Therefore, they are closely related to tree-based networks, where the difference is that in tree-based networks the additional arcs do not need to be horizontal. Then, we use this new characterization to show that the space of orchard networks on n leaves with k reticulations is connected under the rNNI rearrangement move with diameter O(kn+ nlog (n)).

Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called Tree Containment, arises when validating networks constructed by phylogenetic inference methods. We present the first algorithm for (rooted) Tree Containment using the treewidth t of the input network N as parameter, showing that the problem can be solved in 2O(t2) |N| time and space.

Combining a set of phylogenetic trees into a single phylogenetic network that explains all of them is a fundamental challenge in evolutionary studies. In this paper, we apply the recently-introduced theoretical framework of cherry picking to design a class of heuristics that are guaranteed to produce a network containing each of the input trees, for practical-size datasets. The main contribution of this paper is the design and training of a machine learning model that captures essential information on the structure of the input trees and guides the algorithms towards better solutions. This is one of the first applications of machine learning to phylogenetic studies, and we show its promise with a proof-of-concept experimental study conducted on both simulated and real data consisting of binary trees with no missing taxa.

Phylogenetic networks can represent evolutionary events that cannot be described by phylogenetic trees. These networks are able to incorporate reticulate evolutionary events such as hybridization, introgression, and lateral gene transfer. Recently, network-based Markov models of DNA sequence evolution have been introduced along with model-based methods for reconstructing phylogenetic networks. For these methods to be consistent, the network parameter needs to be identifiable from data generated under the model. Here, we show that the semi-directed network parameter of a triangle-free, level-1 network model with any fixed number of reticulation vertices is generically identifiable under the Jukes–Cantor, Kimura 2-parameter, or Kimura 3-parameter constraints.

Background: Rooted phylogenetic networks are used to display complex evolutionary history involving so-called reticulation events, such as genetic recombination. Various methods have been developed to construct such networks, using for example a multiple sequence alignment or multiple phylogenetic trees as input data. Coronaviruses are known to recombine frequently, but rooted phylogenetic networks have not yet been used extensively to describe their evolutionary history. Here, we created a workflow to compare the evolutionary history of SARS-CoV-2 with other SARS-like viruses using several rooted phylogenetic network inference algorithms. This workflow includes filtering noise from sets of phylogenetic trees by contracting edges based on branch length and bootstrap support, followed by resolution of multifurcations. We explored the running times of the network inference algorithms, the impact of filtering on the properties of the produced networks, and attempted to derive biological insights regarding the evolution of SARS-CoV-2 from them. Results: The network inference algorithms are capable of constructing rooted phylogenetic networks for coronavirus data, although running-time limitations require restricting such datasets to a relatively small number of taxa. Filtering generally reduces the number of reticulations in the produced networks and increases their temporal consistency. Taxon bat-SL-CoVZC45 emerges as a major and structural source of discordance in the dataset. The tested algorithms often indicate that SARS-CoV-2/RaTG13 is a tree-like clade, with possibly some reticulate activity further back in their history. A smaller number of constructed networks posit SARS-CoV-2 as a possible recombinant, although this might be a methodological artefact arising from the interaction of bat-SL-CoVZC45 discordance and the optimization criteria used. Conclusion: Our results demonstrate that as part of a wider workflow and with careful attention paid to running time, rooted phylogenetic network algorithms are capable of producing plausible networks from coronavirus data. These networks partly corroborate existing theories about SARS-CoV-2, and partly produce new avenues for exploration regarding the location and significance of reticulate activity within the wider group of SARS-like viruses. Our workflow may serve as a model for pipelines in which phylogenetic network algorithms can be used to analyse different datasets and test different hypotheses.

Phylogenetic networks are used to represent evolutionary relationships between species in biology. Such networks are often categorized into classes by their topological features, which stem from both biological and computational motivations. We study two network classes in this paper: tree-based networks and orchard networks. Tree-based networks are those that can be obtained by inserting edges between the edges of an underlying tree. Orchard networks are a recently introduced generalization of the class of tree-child networks. Structural characterizations have already been discovered for tree-based networks; this is not the case for orchard networks. In this paper, we introduce cherry covers—a unifying characterization of both network classes—in which we decompose the edges of the networks into so-called cherry shapes and reticulated cherry shapes. We show that cherry covers can be used to characterize the class of tree-based networks as well as the class of orchard networks. Moreover, we also generalize these results to non-binary networks.

A phylogenetic network is a graph-theoretical tool that is used by biologists to represent the evolutionary history of a collection of species. One potential way of constructing such networks is via a distance-based approach, where one is asked to find a phylogenetic network that in some way represents a given distance matrix, which gives information on the evolutionary distances between present-day taxa. Here, we consider the following question. For which k are unrooted level-k networks uniquely determined by their distance matrices? We consider this question for shortest distances as well as for the case that the multisets of all distances is given. We prove that level-1 networks and level-2 networks are reconstructible from their shortest distances and multisets of distances, respectively. Furthermore we show that, in general, networks of level higher than 1 are not reconstructible from shortest distances and that networks of level higher than 2 are not reconstructible from their multisets of distances.

A common problem in phylogenetics is to try to infer a species phylogeny from gene trees. We consider different variants of this problem. The first variant, called Unrestricted Minimal Episodes Inference, aims at inferring a species tree based on a model with speciation and duplication where duplications are clustered in duplication episodes. The goal is to minimize the number of such episodes. The second variant, Parental Hybridization, aims at inferring a species network based on a model with speciation and reticulation. The goal is to minimize the number of reticulation events. It is a variant of the well-studied Hybridization Number problem with a more generous view on which gene trees are consistent with a given species network. We show that these seemingly different problems are in fact closely related and can, surprisingly, both be solved in polynomial time, using a structure we call 'beaded trees'. However, we also show that methods based on these problems have to be used with care because the optimal species phylogenies always have a restricted form. To mitigate this problem, we introduce a new variant of Unrestricted Minimal Episodes Inference that minimizes the duplication episode depth. We prove that this new variant of the problem can also be solved in polynomial time.