Average-Tree Phylogenetic Diversity Parameterized by Scanwidth and Invisibility

2026-04-30

AI Summary

This work addresses the computational complexity of calculating Average Phylogenetic Diversity (APD) over rooted phylogenetic networks, a problem generally NP-hard in the presence of non-treelike structures. The study introduces scanwidth as a novel structural parameter and establishes that APD can be computed in polynomial time when scanwidth ≤ 2, while it becomes NP-hard for scanwidth ≥ 3. Furthermore, the authors devise a linear-time algorithm specifically for reticulation-visible networks. By integrating dynamic programming, biconnected component decomposition, and structural properties of the network, they propose a parameterized algorithm with time complexity O(2^sw·n), where sw denotes scanwidth. This approach yields efficient solutions when scanwidth is bounded or when the number of invisible reticulation nodes remains constant.

Abstract

We investigate parameterized algorithms for computing the average-tree phylogenetic diversity (APD) in rooted phylogenetic networks, studying the problem under different structural parameters that capture the deviation of a network from a tree. Our primary parameter is the scanwidth, a measure of the tree-likeness of a given directed acyclic graph. We show that a subset of taxa with maximum APD can be found in polynomial time in phylogenetic networks of scanwidth at most 2, but becomes NP-hard in networks of scanwidth 3. Further, we design an algorithm that computes the APD of a given set of taxa in O(2^sw n) time, where sw denotes the scanwidth and n the number of taxa in the input network. Finally, we give a linear-time algorithm for computing the APD of a given set of taxa if the network induced by these taxa is reticulation-visible. We generalize this algorithm to still run in polynomial time if each biconnected component of the induced network has only constantly many invisible reticulations.

Problem

• phylogenetic diversity
• average-tree
• scanwidth
• reticulation-visible
• parameterized complexity

Innovation

• scanwidth
• phylogenetic diversity
• parameterized algorithm
• reticulation-visible
• NP-hardness