@article{11676,
  abstract     = {We study the problem of maximizing a monotone submodular function with viability constraints. This problem originates from computational biology, where we are given a phylogenetic tree over a set of species and a directed graph, the so-called food web, encoding viability constraints between these species. These food webs usually have constant depth. The goal is to select a subset of k species that satisfies the viability constraints and has maximal phylogenetic diversity. As this problem is known to be NP-hard, we investigate approximation algorithms. We present the first constant factor approximation algorithm if the depth is constant. Its approximation ratio is (1−1e√). This algorithm not only applies to phylogenetic trees with viability constraints but for arbitrary monotone submodular set functions with viability constraints. Second, we show that there is no (1−1/e+ϵ)-approximation algorithm for our problem setting (even for additive functions) and that there is no approximation algorithm for a slight extension of this setting.},
  author       = {Dvořák, Wolfgang and Henzinger, Monika H and Williamson, David P.},
  issn         = {1432-0541},
  journal      = {Algorithmica},
  keywords     = {Approximation algorithms, Submodular functions, Phylogenetic diversity, Viability constraints},
  number       = {1},
  pages        = {152--172},
  publisher    = {Springer Nature},
  title        = {{Maximizing a submodular function with viability constraints}},
  doi          = {10.1007/s00453-015-0066-y},
  volume       = {77},
  year         = {2017},
}