Phylogenetic Network Diversity Parameterized by Reticulation Number and Beyond

TL;DR

This paper studies the computational problem of selecting species to maximize diversity in phylogenetic networks, providing fixed-parameter tractable algorithms for binary networks and proving NP-hardness for level-1 networks.

Contribution

It introduces an FPT algorithm for Max-Network-PD based on reticulation number and proves NP-hardness for level-1 networks, highlighting computational limits.

Findings

FPT algorithm runs in O(2^r log(k)(n + r)) time for binary networks.

Max-Network-PD is NP-hard for level-1 networks.

Reticulation number is a key parameter for algorithmic tractability.

Abstract

Network Phylogenetic Diversity (Network-PD) is a measure for the diversity of a set of species based on a rooted phylogenetic network (with branch lengths and inheritance probabilities on the reticulation edges) describing the evolution of those species. We consider the Max-Network-PD problem: Given such a network, find k species with maximum Network-PD score. We show that this problem is fixed-parameter tractable (FPT) for binary networks, by describing an optimal algorithm running in O(2r log(k)(n + r)) time, with n the total number of species in the network and r its reticulation number. Furthermore, we show that Max-Network-PD is NP-hard for level-1 networks, proving that, unless P=NP, the FPT approach cannot be extended by using the level as parameter instead of the reticulation number.