Orienting Unrooted Binary Networks Faster: Focus on the Generator

2026-06-30

AI Summary

This study addresses the NP-hard problem of orienting unrooted binary phylogenetic networks into specific rooted network classes. By analyzing the structure of network generators, the authors propose two fixed-parameter tractable (FPT) algorithmic frameworks: one based on enumerating directed spanning trees over the generator, and another that directly guesses the locations of reticulation nodes. Their approach reveals that the core difficulty of the orientation problem lies in constructing directed spanning trees on the generator. As a result, they achieve the first single-exponential FPT algorithms for tree-based and orchard networks with running times of $O(5.3334^\ell \cdot n)$ and $O(5.3334^\ell \cdot n^2)$, respectively, significantly improving upon existing results. Moreover, their method yields the current best-known single-exponential FPT running times for several other network classes.

Abstract

The problem of orienting an unrooted network to obtain a specific class of rooted phylogenetic networks is known to be NP-hard in many cases. In this paper, we introduce two algorithmic frameworks that yield significantly improved fixed-parameter tractable (FPT) algorithms parameterized by the network level $\ell$. Our first main contribution shows that for several prominent network classes, the core algorithmic difficulty lies in finding a directed spanning tree on the network's undirected generator. By enumerating these spanning trees in $O(5.3334^\ell + \ell)$ time and orienting all remaining edges in polynomial time, we solve the orientation problem in $O(5.3334^\ell \cdot n)$ time for tree-based networks and in $O(5.3334^\ell \cdot n^2)$ time for orchards, where $n$ is the number of vertices of the graph. Extending this approach with further branching yields $O(10.6667^\ell \cdot n^2)$-time algorithms for tree-child and normal networks. Our second technique bypasses spanning trees by directly guessing the placement of reticulations on the generator. This framework provides $O(12.2071^\ell \cdot n^2)$-time algorithms for temporal, reticulation-visible, and tree-sibling networks. Finally, we demonstrate the versatility of the reticulation-guessing framework by showing that even computing an orientation with minimum scanwidth is single-exponential FPT with respect to the level. Together, these results significantly improve the best-known running times for phylogenetic network orientation.

Problem

• phylogenetic network
• network orientation
• unrooted network
• rooted network
• NP-hard

Innovation

• phylogenetic network orientation
• fixed-parameter tractable algorithm
• network generator
• reticulation placement
• spanning tree enumeration