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

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.

The rigid hybrid number for two phylogenetic trees

“…Indeed, Theorem 2 and the main result in Döcker et al. ( 2019 , Theorem 1) (which states that it is NP-complete to decide whether or not there is a cherry-picking sequence for two phylogenetic trees) immediately imply:…”

“… 2013a ; Linz and Semple 2019 ) and related algorithms/complexity results (Bordewich and Semple 2007a , b ; Döcker et al. 2019 ; Humphries et al. 2013b ; Huson and Linz 2016 ).…”

“…2, Fig. 3) and represent a hypothetical evolutionary scenario tracing the evolution of a genomic region within four species for when collections of trees are displayed by special types of networks (Humphries et al 2013a;Linz and Semple 2019) and related algorithms/complexity results (Bordewich and Semple 2007a, b;Döcker et al 2019;Humphries et al 2013b;Huson and Linz 2016). However, all of these results rely on the fact that the networks display the set of trees in question.…”

Linear Time Algorithm for Tree-Child Network Containment

“…This sequence is minimal for the network, as every element of the sequence reduces either a cherry or a reticulated cherry of the network. An example of a cherry (3, 1) can be seen in the network NS [:3] , and a reticulated cherry (3,4) can be seen in the network N . The reduction of both reducible pairs is carried out as in Subsect.…”

“…2) by considering tree-child sequences. These sequences were developed to tackle the problem of finding a "simple" network that contains a given set of trees [4,11]. Two leaves of a tree form a cherry if they share a common parent-by successively picking cherries (removing one of the leaves in a cherry) from the set of input trees, we obtain a sequence of cherries that ultimately reduce each input tree to a tree on a single leaf.…”