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.
CITATION STYLE
Dvořák, W., Henzinger, M., & Williamson, D. P. (2017). Maximizing a Submodular Function with Viability Constraints. Algorithmica, 77(1), 152–172. https://doi.org/10.1007/s00453-015-0066-y
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