Locating a tree in a phylogenetic network

TL;DR

This paper investigates the computational complexity of locating a phylogenetic tree within a network, showing polynomial solutions for certain classes and NP-completeness for others, advancing understanding of evolutionary history modeling.

Contribution

It provides complexity results for Tree and Cluster Containment problems across various classes of phylogenetic networks, identifying cases with efficient algorithms and those that remain NP-complete.

Findings

Polynomial-time algorithms for normal, binary tree-child, and level-k networks.

NP-completeness results for tree-sibling, time-consistent, and regular networks.

Clarifies computational boundaries for phylogenetic network analysis.

Abstract

Phylogenetic trees and networks are leaf-labelled graphs that are used to describe evolutionary histories of species. The Tree Containment problem asks whether a given phylogenetic tree is embedded in a given phylogenetic network. Given a phylogenetic network and a cluster of species, the Cluster Containment problem asks whether the given cluster is a cluster of some phylogenetic tree embedded in the network. Both problems are known to be NP-complete in general. In this article, we consider the restriction of these problems to several well-studied classes of phylogenetic networks. We show that Tree Containment is polynomial-time solvable for normal networks, for binary tree-child networks, and for level-$k$ networks. On the other hand, we show that, even for tree-sibling, time-consistent, regular networks, both Tree Containment and Cluster Containment remain NP-complete.