Optimizing Phylogenetic Diversity with Ecological Constraints

Given an edge-weighted tree T with leaf set X, define the weight of a subset S of X as the sum of the edge-weights of the minimal subtree of T connecting the elements in S. It is known that the problem of selecting subsets of X of a given size to maximize this weight can be solved using a greedy algorithm. This optimization problem arises in conservation biology where the weight is referred to as the phylogenetic diversity of a taxa set S. Here, we consider the extension of this problem whereby we are only interested in selecting subsets of the taxa set that are ecologically ‘viable’. Such subsets are specified by an acyclic digraph which represents, for example, a food web. This additional constraint makes the problem computationally hard. In this paper, we analyze the complexity of different variations of the extended problem.