Locating a Phylogenetic Tree in a Reticulation-Visible Network in Quadratic Time

TL;DR

This paper proves that locating a phylogenetic tree within a reticulation-visible network can be done in quadratic time, providing efficient algorithms for this NP-complete problem in specific network classes.

Contribution

It introduces a quadratic-time algorithm for the tree containment problem in reticulation-visible networks, solving an open problem and offering new decomposition techniques.

Findings

Tree containment problem is solvable in quadratic time for reticulation-visible networks.

A linear-time algorithm for cluster containment in these networks is developed.

Every galled network with n leaves has at most 2(n-1) reticulation nodes.

Abstract

In phylogenetics, phylogenetic trees are rooted binary trees, whereas phylogenetic networks are rooted arbitrary acyclic digraphs. Edges are directed away from the root and leaves are uniquely labeled with taxa in phylogenetic networks. For the purpose of validating evolutionary models, biologists check whether or not a phylogenetic tree is contained in a phylogenetic network on the same taxa. This tree containment problem is known to be NP-complete. A phylogenetic network is reticulation-visible if every reticulation node separates the root of the network from some leaves. We answer an open problem by proving that the problem is solvable in quadratic time for reticulation-visible networks. The key tool used in our answer is a powerful decomposition theorem. It also allows us to design a linear-time algorithm for the cluster containment problem for networks of this type and to prove that every galled network with n leaves has 2(n-1) reticulation nodes at most.