On the Computational Complexity of the Rooted Subtree Prune and Regraft Distance

Abstract

Abstract. The graph-theoretic operation of rooted subtree prune and regraft is increasingly being used as a tool for understanding and modelling reticulation events in evolutionary biology. In this paper, we show that computing the rooted subtree prune and regraft distance between two rooted binary phylogenetic trees on the same label set is NP-hard. This resolves a longstanding open problem. Furthermore, we show that this distance is fixed parameter tractable when parameterised by the distance between the two trees.

Efficiently Calculating Evolutionary Tree Measures Using SAT

“…Roughly, each measure corresponds to the size of the appropriately defined maximum agreement forest. For a more thorough treatment, see [5,18,26].…”

“…When working with rooted trees, we refer to this distance as rooted SPR or rSPR. Bordewich and Semple [5] showed that the rSPR distance of two trees is the same as the size of an appropriately defined maximum agreement forest for rooted trees of the two trees. This number is related to another measure between trees that we next define.…”

“…This number is related to another measure between trees that we next define. Hybridization Number: A hybridization network on a leaf set X [5,26] is a rooted acyclic directed graph with root ρ in which -X is the set of leaves (vertices of outdegree zero);…”

“…rSPR captures individual hybridization events but misses an important acyclicity condition that taxa cannot have themselves as ancestors. Further, while often similar in size, there exist instances where the difference between the rSPR and hybridization number are arbitrarily large [5].…”

“…Agreement forests were developed for evolutionary tree metrics in the pioneering work of Hein et al [14] and Allen and Steel [1] that linked the tree distance to the size of the maximum agreement forest (MAF). With the development of a MAF for the rooted subtree-prune-and-reconnect (rSPR) distance [5] (see Figure 2), Bonet et al [4] showed these algorithms are a 5-approximation for rSPR distance. Algorithms for biologically relevant restricted cases of rSPR were also developed by Hallett and Lagergren [13] and Beiko and Hamilton [3].…”