Maximizing Network Phylogenetic Diversity

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(2^r 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.