Maximizing a Submodular Function with Viability Constraints

TL;DR

This paper introduces the first constant-factor approximation algorithm for maximizing a monotone submodular function with viability constraints, relevant in computational biology, achieving a ratio of (1-1/√e), and establishes hardness results for better approximations.

Contribution

It provides the first constant-factor approximation algorithm for the problem with constant-depth viability constraints and proves hardness results for improved approximation ratios.

Findings

First constant-factor approximation algorithm with ratio (1-1/√e)

No (1-1/e+ε)-approximation exists for the problem

Applicable to arbitrary monotone submodular functions 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 $(1-\frac{1}{\sqrt{e}})$. 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.