Algorithmica

, Volume 77, Issue 1, pp 152–172 | Cite as

Maximizing a Submodular Function with Viability Constraints

  • Wolfgang Dvořák
  • Monika Henzinger
  • David P. Williamson
Article

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 \(\mathsf{NP}\)-hard, we investigate approximation algorithms. We present the first constant factor approximation algorithm if the depth is constant. Its approximation ratio is \((1-\frac{1}{\sqrt{e}})\). 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 \((1-1/e+\epsilon )\)-approximation algorithm for our problem setting (even for additive functions) and that there is no approximation algorithm for a slight extension of this setting.

Keywords

Approximation algorithms Submodular functions Phylogenetic diversity Viability constraints 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Fakultät für InformatikUniversität WienViennaAustria
  2. 2.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

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