[Paper Review] Rooting for phylogenetic networks

Katharina T. Huber, Leo van Iersel+5 more|arXiv (Cornell University)|Jun 18, 2019

Genomics and Phylogenetic Studies3 references17 citations

TL;DR

This paper presents a polynomial-time algorithm to determine whether an undirected binary phylogenetic network can be uniquely oriented into a directed phylogenetic network when root and reticulation locations are known, and generalizes this to nonbinary networks. It further introduces a fixed-parameter tractable (FPT) algorithm for orienting networks of specific classes—such as tree-child, stack-free, and orchard networks—while proving the general problem of orienting as a tree-based network is NP-hard.

ABSTRACT

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected binary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed phylogenetic network. Moreover, we give a mathematical characterization of when this is the case and show that this directed phylogenetic network is then always unique. These results are generalized to the nonbinary case. In addition, we describe an algorithm for deciding whether an undirected binary phylogenetic network can be oriented as a directed phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of the network and is applicable to a wide range of network classes, including tree-child, tree-based, stack-free and orchard networks. It can also be used to decide whether an undirected phylogenetic network is tree-based and whether a partly-directed phylogenetic network can be oriented as a directed phylogenetic network. Finally, we show that, in general, it is NP-hard to decide whether an undirected network can be oriented as a tree-based network.

Motivation & Objective

• To determine when an undirected binary phylogenetic network can be uniquely oriented into a directed network given root and reticulation vertex locations.
• To generalize the orientation conditions to nonbinary phylogenetic networks.
• To develop an FPT algorithm for orienting networks into specific classes, including tree-child, stack-free, and orchard networks.
• To decide whether an undirected network can be oriented as a tree-based network.
• To investigate the computational complexity of orienting networks as tree-based networks.

Proposed method

• Proposes a polynomial-time algorithm that checks whether an undirected binary network with specified root and reticulation vertices can be oriented into a directed phylogenetic network.
• Provides a mathematical characterization of the conditions under which such an orientation exists and is unique.
• Generalizes the orientation conditions to nonbinary networks using structural constraints on network components.
• Develops a fixed-parameter tractable (FPT) algorithm parameterized by the network level to determine if an undirected network can be oriented into a network of a specific class.
• Applies the FPT algorithm to decide orientability for classes such as tree-child, tree-based, stack-free, and orchard networks.
• Uses the algorithm to also determine whether a partly-directed network can be completed to a directed phylogenetic network.

Experimental results

Research questions

• RQ1 Under what conditions can an undirected binary phylogenetic network with known root and reticulation vertices be uniquely oriented into a directed phylogenetic network?
• RQ2 Can the orientation problem be generalized to nonbinary phylogenetic networks, and what are the necessary and sufficient conditions for orientability?
• RQ3 Is there an efficient algorithm to determine whether an undirected network can be oriented into a network of a specific class, such as tree-child or orchard networks?
• RQ4 What is the computational complexity of deciding whether an undirected network can be oriented as a tree-based network?
• RQ5 Can the FPT algorithm be applied to decide orientability of partly-directed phylogenetic networks?

Key findings

• An undirected binary phylogenetic network with specified root and reticulation vertices can be uniquely oriented into a directed network if and only if certain structural conditions are met, and this can be decided in polynomial time.
• The orientation is always unique when it exists, providing a canonical way to root such networks.
• The characterization and algorithm extend to nonbinary phylogenetic networks, maintaining correctness and efficiency.
• An FPT algorithm is developed for orienting networks into classes like tree-child, stack-free, and orchard networks, parameterized by network level.
• The algorithm can also determine whether a partly-directed network can be completed to a directed phylogenetic network.
• The problem of deciding whether an undirected network can be oriented as a tree-based network is NP-hard in general.