Phylogenetic diversity and the maximum coverage problem

Abstract

For a weighted hypergraph ( H , ω ) , with vertex set X , edge set E , and weighting ω : E → R ≥ 0 , the maximum coverage problem is to find a k -element subset Y ⊆ X that maximizes 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.