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

We describe a kernel of size $9k-8$ for the NP-hard problem of computing the Tree Bisection and Reconnection (TBR) distance k between two unrooted binary phylogenetic trees. To achieve this, we extend the existing portfolio of reduction rules with three new reduction rules. Two of these 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 neighborhood approach introduced in [20] to more global structures, allowing new situations to be identified when the 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 between two unrooted binary phylogenetic trees. We anticipate that our results are widely applicable for computing agreement-forest based dissimilarity measures.