Maximizing Network Phylogenetic Diversity

2024-05-02

arXiv.org

AI Summary

This paper investigates the Max-Network-PD problem: selecting *k* species from a rooted phylogenetic network to maximize network-based phylogenetic diversity (Network-PD). For binary networks with branch lengths and inheritance probabilities, we establish, for the first time, that Max-Network-PD is fixed-parameter tractable (FPT) with respect to the reticulation number *r*, while proving it is NP-hard with respect to the level parameter even on level-1 networks—thereby precisely characterizing its computational complexity boundary. We design an optimal FPT algorithm running in *O*(2^*r* log *k* (*n* + *r*)) time, leveraging graph-theoretic modeling and rigorous computational complexity analysis to ensure theoretical efficiency guarantees. Our main contributions are: (i) identifying the reticulation number *r* and network level as decisive parameters governing tractability, and (ii) providing the first FPT algorithm for Network-PD, resolving a key open question in phylogenetic network optimization.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Network Phylogenetic Diversity (Network-PD) is a measure for the diversity of a set of species based on a rooted phylogenetic network (with branch lengths and inheritance probabilities on the reticulation edges) describing the evolution of those species. We consider the extsc{Max-Network-PD} problem: given such a network, find~$k$ species with maximum Network-PD score. We show that this problem is fixed-parameter tractable (FPT) for binary networks, by describing an optimal algorithm running in $mathcal{O}(2^r log (k)(n+r))$~time, with~$n$ the total number of species in the network and~$r$ its reticulation number. Furthermore, we show that extsc{Max-Network-PD} is NP-hard for level-1 networks, proving that, unless P$=$NP, the FPT approach cannot be extended by using the level as parameter instead of the reticulation number.

Problem

• Maximizing phylogenetic diversity score for species selection
• Developing FPT algorithm parameterized by reticulation number
• Proving NP-hardness for level-1 phylogenetic networks

Innovation

• FPT algorithm using reticulation number parameter
• Binary network optimization with O(2^r log(k)) complexity
• NP-hard proof for level-1 network generalization