The paper focuses on the problem of maximizing network phylogenetic diversity (Network-PD), which is a measure of the diversity of a set of species based on a rooted phylogenetic network describing their evolution.
The key insights are:
The authors present an optimal algorithm for the Max-Network-PD problem on binary networks, which runs in O(2^r log(k)(n + r)) time, where n is the total number of species and r is the reticulation number of the network. This shows that the problem is fixed-parameter tractable with respect to the reticulation number.
The authors prove that Max-Network-PD is NP-hard for level-1 networks, which are networks without overlapping cycles. This shows that the fixed-parameter tractability result cannot be extended by using the level as a parameter instead of the reticulation number.
Along the way, the authors also show that the unit-cost version of the Network Augmentation Problem (unit-cost-NAP) is NP-hard, answering an open question.
The hardness results are shown via reductions from the Subset Product problem, which is proven to be NP-hard.
Overall, the paper provides a comprehensive analysis of the computational complexity of the Max-Network-PD problem, establishing both positive and negative results.
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by Leo van Iers... at arxiv.org 05-03-2024
https://arxiv.org/pdf/2405.01091.pdfDeeper Inquiries