Phylogenetic diversity and the maximum coverage problem

For a weighted hypergraph (H, omega), with vertex set X, edge set E, and weighting omega : E --> R(>= 0), the maximum coverage problem is to find a k-element subset Y subset of X that maximiz-es the total weight of those edges that have non-empty intersection with Y among all k-element subsets of X. Such a subset Y is called optimal. Recently, within the field of phylogenetics it has been shown that for certain weighted hypergraphs coming from phylogenetic trees the collection of optimal subsets of X forms a so-called strong greedoid. We call hypergraphs having this latter property strongly greedy. In this note we characterize the r-uniform hypergraphs H with unit edge weights that are strongly greedy in the case where r is a prime number. (C) 2009 Elsevier Ltd. All rights reserved.