Average-Tree Phylogenetic Diversity Parameterized by Scanwidth and Invisibility

TL;DR

This paper explores algorithms for calculating average-tree phylogenetic diversity in rooted networks, focusing on structural parameters like scanwidth, and presents efficient algorithms for specific cases.

Contribution

It introduces parameterized algorithms based on scanwidth for APD computation, including polynomial-time solutions for networks with low scanwidth and reticulation-visible networks.

Findings

Polynomial-time algorithm for scanwidth ≤ 2 networks.

NP-hardness of APD in networks with scanwidth 3.

Linear-time algorithm for reticulation-visible induced networks.

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.