Displaying trees across two phylogenetic networks

Abstract

Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal gene transfer. Indeed, each phylogenetic network embeds a collection of phylogenetic trees. Referring to the collection of trees that a given phylogenetic network N embeds as the display set of N , several questions in the context of the display set of N have recently been analyzed. For example, the widely studied Tree-Containment problem asks if a given phylogenetic tree is contained in the display set of a given network. The focus of this paper are two questions that naturally arise in comparing the display sets of two phylogenetic networks. First, we analyze the problem of deciding if the display sets of two phylogenetic networks have a tree in common. Surprisingly, this problem turns out to be NP-complete even for two temporal normal networks. Second, we investigate the question of whether or not the display sets of two phylogenetic networks are equal. While we recently showed that this problem is polynomial-time solvable for a normal and a tree-child network, it is computationally hard in the general case. In establishing hardness, we show that the problem is contained in the second level of the polynomial-time hierarchy. Specifically, it is Π P 2 -complete. Along the way, we show that two other problems are also Π P 2 -complete, one of which being a generalization of Tree-Containment.Classes of phylogenetic networks. Let N be a phylogenetic network on X with vertex set V . An edge e = (u, v) is a shortcut if there is a directed path from u to v whose set of edges does not contain e. A vertex v of N is called visible if there exists a leaf ∈ X such that each directed path from the root of N to passes through v. Now N is reticulation-visible if each reticulation in N is visible, and N is tree-child if each non-leaf vertex in N has a child that is a leaf or a tree vertex. Lastly, N is normal if it is treechild and does not contain any shortcuts. Clearly, by definition, each normal network is also tree-child. Furthermore, it follows from the next well-known equivalence result [2] that each tree-child network is also reticulation-visible.Lemma 2.1. Let N be a phylogenetic network. Then N is tree-child if and only if each vertex of N is visible.Thus, the class of normal networks is a subclass of tree-child networks. Furthermore, if there exists a map t : V → R + that assigns a time stamp to each vertex of N and satisfies the following two properties:a tree edge, then we say that N is temporal, in which case we call t a temporal labeling of N . Note that, although normal networks have no shortcuts, a normal network need not be temporal. Tree-child, normal, and temporal networks were first introduced by Cardona et al. [2], Willson [17], and Moret et al. [11], respectively.Caterpillars. Let C be a phylogenetic tree with leaf set { 1 , 2 , . . . , n }. Furthermore, for each i ∈ {1, 2...

A QUBO formulation for the Tree Containment problem

“…Furthermore, it would be interesting to explore quantum-classical hybrid approaches, as discussed in [1]. An immediate next step could be the development of a QUBO for problems that are closely related to Tree Containment such as the problem of deciding if a given phylogenetic network contains a given phylogenetic tree as a so-called base tree, or the problem of deciding if two phylogenetic networks display the same set of phylogenetic trees [3,13,18]. How different are QUBO formulations for these problems from the QUBO presented in this paper?…”

Non-essential arcs in phylogenetic networks

“…or it has a cherry in {{x, 0 }, {x, 1 }, {x, 2 }}. Both cases contradict the restriction of T in (8) when considering E ; thereby implying that (u, w) is a shortcut and u lies on the tree path from p 2 to p 1 . Now, let E 1 be an embedding of a phylogenetic X-tree T that uses…”

“…To complete the proof, we establish a reduction from the Π P 2 -complete problem Display-Set-Containment to Non-Essential-Arc. The reduction has a similar flavor to that used in [8,Theorem 4.5]. Throughout the remainder of the proof, we sometimes label internal vertices of a phylogenetic network.…”

Display Sets of Normal and Tree-Child Networks

“…In particular, we consider the problem of deciding whether or not two given phylogenetic networks embed the same set of phylogenetic trees. Called Display-Set-Equivalence, we recently showed that, in general, this problem is Π P 2complete [5], that is, complete for the second-level of the polynomial hierarchy. Problems on the second level of the hierarchy are computationally more difficult than problems on the first level which include all NP-and co-NPcomplete problems.…”

“…It is shown in [5] that, in general, Display-Set-Equivalence is Π P 2complete. In contrast, the main result of this paper shows that this decision problem is solvable in polynomial time if N is normal and N is tree-child.…”