Seeing the trees and their branches in the network is hard

Abstract

Phylogenetic networks are a restricted class of directed acyclic graphs that model evolutionary histories in the presence of reticulate evolutionary events, such as horizontal gene transfer, hybrid speciation, and recombination. Characterizing a phylogenetic network as a collection of trees and their branches has long been the basis for several methods of reconstructing and evaluating phylogenetic networks. Further, these characterizations have been used to understand molecular sequence evolution on phylogenetic networks. In this paper, we address theoretical questions with regard to phylogenetic networks, their characterizations, and sequence evolution on them. In particular, we prove that the problem of deciding whether a given tree is contained inside a network is NP-complete. Further, we prove that the problem of deciding whether a branch of a given tree is also a branch of a given network is polynomially equivalent to that of deciding whether the evolution of a molecular character (site) on a network is governed by the infinite site model. Exploiting this equivalence, we establish the NP-completeness of both problems, and provide a parameterized algorithm that runs in time O(2 k/2 n 2), where n is the total number of nodes and k is the number of recombination nodes in the network, which significantly improves upon the trivial brute-force O(2 k n) time algorithm for the problem. This reduction in time is significant, particularly when analyzing recombination hotspots.

Computational approaches to species phylogeny inference and gene tree reconciliation

“…For example, the four trees induced by the network in Figure 5 are (((A,D),C),B), (A,(B,(C,D))), ((A,(C,D)),B), ((A,D),(B,C)). Reconciling a gene tree with a phylogenetic network, excluding ILS and DL, is related to testing whether the gene tree is one of the trees induced by the network, which has been shown to be computationally a very hard problem [28]. …”

A decomposition theorem and two algorithms for reticulation-visible networks

“…The TCP and CCP were both proven to be NP-complete even for binary networks [19]. Therefore, it is interesting to investigate whether there are polynomial time algorithms for the TCP and CCP on certain subclasses of phylogenetic networks.…”

“…The cluster containment problem (CCP) is related algorithmic problem that asks whether or not a subset of taxa is a cluster in a tree displayed by a network [18]. Both the TCP and CCP have also been investigated in the development of metrics for phylogenetic networks [3,19].…”

“…The TCP and CCP are NP-complete [19], even on a very restricted class of networks [26]. Because of their importance, a lot of effort has been made to find network classes for which the TCP and CCP are solvable in polynomial time.…”

On normal networks

“…The next step should be to see if this is still true for certain classes of networks. Galled networks are very nice in that tree containment is determinable in polynomial time, as shown in (9). In fact, the cluster containment problem, determining whether a given network displays a tree which has a given cluster, is even polynomial for galled networks (6).…”