A Multivariate Complexity Analysis of the Generalized Noah's Ark Problem
Christian Komusiewicz, Jannik Schestag
TL;DR
This paper analyzes GNAP, a problem that extends maximizing phylogenetic diversity by incorporating per-taxon conservation actions with costs and survival probabilities under a budget. It develops a parameterized complexity framework and proves tractability when the number of distinct costs and survival probabilities is small, and when the number of taxa is constant, while showing W[1]-hardness for GNAP with respect to the number of taxa. By leveraging reductions to and from Multiple-Choice Knapsack (MCKP), it derives a spectrum of algorithms, including a two-table DP for GNAP on general trees and efficient MCKP-based methods for Star-GNAP, along with explicit hardness results for two-action per taxon cases. The work also analyzes two-node-per-taxon restrictions (0→1 NAP and unit-cost variants), obtaining PFPT/FPT results in some parameters but NP-hardness in others, thereby outlining a comprehensive complexity landscape and guiding future exploration of pseudopolynomial-time algorithms and broader structural parameterizations.
Abstract
In the Generalized Noah's Ark Problem, one is given a phylogenetic tree on a set of species X and a set of conservation projects for each species. Each project comes with a cost and raises the survival probability of the corresponding species. The aim is to select a conservation project for each species such that the total cost of the selected projects does not exceed some given threshold and the expected phylogenetic diversity is as large as possible. We study the complexity of Generalized Noah's Ark Problem and some of its special cases with respect to several parameters related to the input structure, such as the number of different costs, the number of different survival probabilities, or the number of species, |X|.