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.
翻译:我们用9k-8大小的内核来描述NP的硬性问题,即计算两棵未扎根的双生植物树之间的树分和重新连接(TRI)距离 k。我们通过三个新的新的削减规则扩大现有的削减规则组合来实现这一点。其中两个规则基于以远距离保护方式对树木进行地形改变以保证执行先前的削减规则的理念。第三项规则将本地邻里办法(克尔克和林茨,24(3)、2020)扩大到更多的全球结构,允许在删除叶片时确定新的情况,从而肯定将TBR距离降低一个。内核的界限紧凑到一个添加词。我们的结果也适用于计算两个未扎根的二生植物树之间最大森林协议(MAF)的同等问题。我们预计,我们的结果将更广泛地适用于计算基于协议-森林的脱异性措施。