Average-Tree Phylogenetic Diversity of Networks
Leo van Iersel, Mark Jones, Jannik Schestag, , Mathias Weller
Phylogenetic diversity is a measure used to quantify the biodiversity of a set of species. Here, we introduce the “average-tree” phylogenetic diversity score in rooted binary phylogenetic networks and consider algorithms for computing and maximizing the score on a given network. Basically, the score is the weighted average of the phylogenetic diversity scores in all trees displayed by the network, with the weights determined by the inheritance probabilities on the reticulation edges used in the embeddings. We show that computing the score of a given set of taxa in a given network is #P-hard, directly implying #P-hardness of finding a subset of $k$ taxa achieving maximum diversity score and, thereby, ruling out polynomial-time algorithms for these problems unless the polynomial hierarchy collapses. However, we show that both problems can be solved efficiently if the input network is close to being a tree in the sense that its reticulation number is small. More precisely, we prove that we can solve the optimization problem in networks with $n$ leaves and $r$ reticulations in $2^{O(r)} nk$ time. Using experiments on data produced by simulating a reticulate-evolution process, we show that our algorithm runs within 5 minutes on a laptop computer on networks with up to 500 taxa and 55 reticulations, for $k \leq 5$.