Maximizing a Submodular Function with Viability Constraints

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 −). 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 + ǫ)-approximation algorithm for our problem setting (even for additive functions) and that there is no approximation algorithm for a slight extension of this setting.

Maximizing a Submodular Function with Viability Constraints

“…A preliminary version of this paper has been presented at the 21st European Symposium on Algorithms (ESA'13) [4].…”