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.

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.

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.

Phylogenetic Diversity (PD) is a measure of the overall biodiversity of a set of present-day species (taxa) within a phylogenetic tree. We consider an extension of PD to phylogenetic networks. Given a phylogenetic network with weighted edges and a subset S of leaves, the all-paths phylogenetic diversity of S is the summed weight of all edges on a path from the root to some leaf in S. The problem of finding a bounded-size set S that maximizes this measure is polynomial-time solvable on trees, but NP-hard on networks. We study the latter from a parameterized perspective. While this problem is W[2]-hard with respect to the size of S (and W[1]-hard with respect to the size of the complement of S), we show that it is FPT with respect to several other parameters, including the phylogenetic diversity of S, the acceptable loss of phylogenetic diversity, the number of reticulations in the network, and the treewidth of the underlying graph.

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.

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.

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)).

In the NP-hard Longest Common Subsequence problem (LCS), given a set of strings, the task is to find a string that can be obtained from every input string using as few deletions as possible. LCS is one of the most fundamental string problems with numerous applications in various areas, having gained a lot of attention in the algorithms and complexity research community. Significantly improving on an algorithm by Irving and Fraser [CPM'92], featured as a research challenge in a 2014 survey paper, we show that LCS is fixed-parameter tractable (FPT) when parameterized by the maximum number of deletions per input string. Given the relatively moderate running time of our algorithm (linear time when the parameter is a constant) and small parameter values to be expected in several applications, we believe that our purely theoretical analysis could finally pave the way to a new, exact and practically useful algorithm for this notoriously hard string problem.

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.

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.

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.

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.

Maximum parsimony distance is a measure used to quantify the dissimilarity of two unrooted phylogenetic trees. It is NP-hard to compute, and very few positive algorithmic results are known due to its complex combinatorial structure. Here we address this shortcoming by showing that the problem is fixed parameter tractable. We do this by establishing a linear kernel i.e., that after applying certain reduction rules the resulting instance has size that is bounded by a linear function of the distance. As powerful corollaries to this result we prove that the problem permits a polynomial-time constant-factor approximation algorithm; that the treewidth of a natural auxiliary graph structure encountered in phylogenetics is bounded by a function of the distance; and that the distance is within a constant factor of the size of a maximum agreement forest of the two trees, a well studied object in phylogenetics.

Phylogenetic networks are important for the study of evolution. The number of methods to find such networks is increasing, but most such methods can only reconstruct small networks. To find bigger networks, one can attempt to combine small networks. In this paper, we study the Network Hybridization problem, a problem of combining networks into another network with low complexity. We characterize this complexity via a restricted problem, Tree-child Network Hybridization, and we present an FPT algorithm to efficiently solve this restricted problem.

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.

Unrooted phylogenetic networks are graphs used to represent reticulate evolutionary relationships. Accurately reconstructing such networks is of great relevance for evolutionary biology. It has recently been conjectured that all unrooted phylogenetic networks for at least five taxa can be uniquely reconstructed from their subnetworks obtained by deleting a single taxon. Here, we show that this conjecture is false, by presenting a counter-example for each possible number of taxa that is at least 4. Moreover, we show that the conjecture is still false when restricted to binary networks. This means that, even if we are able to reconstruct the unrooted evolutionary history of each proper subset of some taxon set, this still does not give us enough information to reconstruct their full unrooted evolutionary history.

Network reconstruction lies at the heart of phylogenetic research. Two well-studied classes of phylogenetic networks include tree-child networks and level-k networks. In a tree-child network, every non-leaf node has a child that is a tree node or a leaf. In a level-k network, the maximum number of reticulations contained in a biconnected component is k. Here, we show that level-k tree-child networks are encoded by their reticulate-edge-deleted subnetworks, which are subnetworks obtained by deleting a single reticulation edge, if k≥ 2. Following this, we provide a polynomial-time algorithm for uniquely reconstructing such networks from their reticulate-edge-deleted subnetworks. Moreover, we show that this can even be done when considering subnetworks obtained by deleting one reticulation edge from each biconnected component with k reticulations.

Perfect phylogenies are fundamental in the study of evolutionary trees because they capture the situation when each evolutionary trait emerges only once in history; if such events are believed to be rare, then by Occam's Razor such parsimonious trees are preferable as a hypothesis of evolution. A classical result states that 2-state characters permit a perfect phylogeny precisely if each subset of 2 characters permits one. More recently, it was shown that for 3-state characters the same property holds but for size-3 subsets. A long-standing open problem asked whether such a constant exists for each number of states. More precisely, it has been conjectured that for any fixed number of states $r$ there exists a constant $f(r)$ such that a set of $r$-state characters $C$ has a perfect phylogeny if and only if every subset of at most $f(r)$ characters has a perfect phylogeny. Informally, the conjecture states that checking fixed-size subsets of characters is enough to correctly determine whether input data permits a perfect phylogeny, irrespective of the number of characters in the input. In this article, we show that this conjecture is false. In particular, we show that for any constant $t$, there exists a set $C$ of $8$-state characters such that $C$ has no perfect phylogeny, but there exists a perfect phylogeny for every subset of at most $t$ characters. Moreover, there already exists a perfect phylogeny when ignoring just one of the characters, independent of which character you ignore. This negative result complements the two negative results ("strikes") of Bodlaender et al. (1992,2000). We reflect on the consequences of this third strike, pointing out that while it does close off some routes for efficient algorithm development, many others remain open.