Deep kernelization for the Tree Bisection and Reconnnect (TBR) distance in phylogenetics

TL;DR

This paper introduces a new kernelization method with a size bound of 9k-8 for the NP-hard TBR distance problem in phylogenetics, using novel reduction rules and transformations.

Contribution

It extends existing reduction rules with three new rules, improving kernel size bounds for TBR distance computation in phylogenetics.

Findings

Kernel size bound of 9k-8 achieved

Three novel reduction rules introduced

Results applicable to Maximum Agreement Forest problem

Abstract

We describe a kernel of size 9k-8 for the NP-hard problem of computing the Tree Bisection and Reconnect (TBR) distance k between two unrooted binary phylogenetic trees. We achieve this by extending the existing portfolio of reduction rules with three novel new reduction rules. Two of the rules are based on the idea of topologically transforming the trees in a distance-preserving way in order to guarantee execution of earlier reduction rules. The third rule extends the local neighbourhood approach introduced in (Kelk and Linz, Annals of Combinatorics 24(3), 2020) to more global structures, allowing new situations to be identified when deletion of a leaf definitely reduces the TBR distance by one. The bound on the kernel size is tight up to an additive term. Our results also apply to the equivalent problem of computing a Maximum Agreement Forest (MAF) between two unrooted binary phylogenetic trees. We anticipate that our results will be more widely applicable for computing agreement-forest based dissimilarity measures.