Deciding the existence of a cherry-picking sequence is hard on two trees

TL;DR

This paper proves that determining the existence of a cherry-picking sequence for two rooted binary phylogenetic trees is NP-complete, impacting the understanding of temporal phylogenetic networks and related optimization problems.

Contribution

It establishes NP-completeness for two trees, strengthening previous results and linking cherry-picking sequences to the complexity of temporal phylogenetic networks.

Findings

Deciding cherry-picking sequences is NP-complete for two trees.

NP-completeness extends to the existence of certain temporal phylogenetic networks.

The result impacts inapproximability of the minimum temporal-hybridization number.

Abstract

Here we show that deciding whether two rooted binary phylogenetic trees on the same set of taxa permit a cherry-picking sequence, a special type of elimination order on the taxa, is NP-complete. This improves on an earlier result which proved hardness for eight or more trees. Via a known equivalence between cherry-picking sequences and temporal phylogenetic networks, our result proves that it is NP-complete to determine the existence of a temporal phylogenetic network that contains topological embeddings of both trees. The hardness result also greatly strengthens previous inapproximability results for the minimum temporal-hybridization number problem. This is the optimization version of the problem where we wish to construct a temporal phylogenetic network that topologically embeds two given rooted binary phylogenetic trees and that has a minimum number of indegree-2 nodes, which represent events such as hybridization and horizontal gene transfer. We end on a positive note, pointing out that fixed parameter tractability results in this area are likely to ensure the continued relevance of the temporal phylogenetic network model.