Computing the Arc-Deletion Distance to Orchard Networks is NP-hard

2026-05-19

AI Summary

This study addresses the arc deletion distance problem for transforming a phylogenetic network into an orchard network by removing the minimum number of reticulation arcs. By constructing a polynomial-time reduction from the Degree-3 Vertex Cover problem, the authors establish for the first time that computing this distance is NP-hard. This result fills a critical gap in the complexity theory of phylogenetic network transformations, delineating a fundamental theoretical boundary for algorithm design in this domain. The hardness proof provides a rigorous complexity foundation for the development of future approximation or heuristic approaches aimed at practical instances of the problem.

Abstract

Phylogenetic networks generalize phylogenetic trees by allowing reticulate evolutionary events such as horizontal gene transfer and hybridization. Among the many subclasses of phylogenetic networks, orchard networks have attracted increasing attention due to their structural and algorithmic properties. In this paper, we study the arc-deletion distance to orchard networks, defined as the minimum number of reticulate arcs whose deletion transforms a phylogenetic network into an orchard network. We prove that computing this distance is NP-hard via a polynomial-time reduction from the Degree-3 Vertex Cover problem. Our result establishes the computational intractability of this proximity measure and contributes to the complexity theory of phylogenetic network transformations.

Problem

• phylogenetic networks
• orchard networks
• arc-deletion distance
• NP-hard
• reticulate arcs

Innovation

• arc-deletion distance
• orchard networks
• NP-hard
• phylogenetic networks
• computational complexity