New Reduction Rules for the Tree Bisection and Reconnection Distance

Abstract

Recently it was shown that, if the subtree and chain reduction rules have been applied exhaustively to two unrooted phylogenetic trees, the reduced trees will have at most $$15k-9$$ 15 k - 9 taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees, and that this bound is tight. Here, we propose five new reduction rules and show that these further reduce the bound to $$11k-9$$ 11 k - 9 . The new rules combine the “unrooted generator” approach introduced in Kelk and Linz (SIAM J Discrete Math 33(3):1556–1574, 2019) with a careful analysis of agreement forests to identify (i) situations when chains of length 3 can be further shortened without reducing the TBR distance, and (ii) situations when small subtrees can be identified whose deletion is guaranteed to reduce the TBR distance by 1. To the best of our knowledge these are the first reduction rules that strictly enhance the reductive power of the subtree and chain reduction rules.

Reflections on kernelizing and computing unrooted agreement forests

“…Our notation closely follows (Kelk and Linz 2020). Throughout this paper, X denotes a finite set of taxa.…”

“…In this article we adopt an experimental approach to answering the following question: do the new reduction rules from Kelk and Linz (2020) produce smaller kernels in practice than, say, when only the subtree and chain reductions are applied? This mirrors several recent articles in the algorithmic graph theory literature where the practical effectiveness of kernelization has also been analyzed (Fellows et al 2018;Ferizovic et al 2020;Henzinger et al 2020;Mertzios et al 2020;Alber et al 2006).…”

“…A final contribution of this article is our new solver Tubro. This was initially intended simply as a vehicle for the new reduction rules from Kelk and Linz (2020). However, during development we reasoned that it would be useful for Tubro to incorporate its own solver.…”

“…Tubro additionally incorporates a number of extra features that are possibly of independent interest, such as a cluster reduction (Bordewich et al 2017;Li and Zeh 2017). It is also able to significantly reduce the number of variables in the ILP by specifying that even in fully kernelized instances certain chains should never be cut, leveraging an insight used in the proofs of correctness in Kelk and Linz (2020). An executable version of Tubro is available upon request.…”

Parameterized Algorithms in Bioinformatics: An Overview

“…For two trees, deciding uMAF is NP-hard [184], but fixed-parameter tractable in k. More precisely, the problem can be solved in O * (k O (k)) time [185], O(4 k k 5 + n O (1) ) time [186], and O(4 k k + n O(1) ) time [187]. These results make use of the known kernelizations with 15k [185,188] and 11k taxa [189]. For t > 2 binary trees, Shi et al [190] presented an O(4 k nt)-time algorithm.…”