  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 \textsc{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 $\mathcal{O}(2^r \log
(k)(n+r))$~time, with~$n$ the total number of species in the network and~$r$
its reticulation number. Furthermore, we show that \textsc{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.
