NAPX: A Polynomial Time Approximation Scheme for the Noah's Ark Problem

Abstract

The Noah's Ark Problem (NAP) is an NP-Hard optimization problem with relevance to ecological conservation management. It asks to maximize the phylogenetic diversity (PD) of a set of taxa given a fixed budget, where each taxon is associated with a cost of conservation and a probability of extinction. NAP has received renewed interest with the rise in availability of genetic sequence data, allowing PD to be used as a practical measure of biodiversity. However, only simplified instances of the problem, where one or more parameters are fixed as constants, have as of yet been addressed in the literature. We present NAPX, the first algorithm for the general version of NAP that returns a $1 - \epsilon$ approximation of the optimal solution. It runs in $O(\frac{n B^2 h^2 \log^2n}{\log^2(1 - \epsilon)})$ time where $n$ is the number of species, and $B$ is the total budget and $h$ is the height of the input tree. We also provide improved bounds for its expected running time.