Phylogenetic trees are leaf-labelled trees used to model the evolution of species. In practice it is not uncommon to obtain two topologically distinct trees for the same set of species, and this motivates the use of distance measures to quantify dissimilarity. A well-known measure is the maximum agreement forest (MAF): a minimum-size partition of the leaf labels which splits both trees into the same set of disjoint, leaf-labelled subtrees (up to isomorphism after suppressing degree-2 vertices). Computing such a MAF is NP-hard and so considerable effort has been invested in finding FPT algorithms, parameterised by $k$, the number of components of a MAF. The state of the art has been unchanged since 2015, with running times of $O^*(3^k)$ for unrooted trees and $O^*(2.3431^k)$ for rooted trees. In this work we present improved algorithms for both the unrooted and rooted cases, with runtimes $O^*(2.846^k)$ and $O^*(2.3391^k)$ respectively. The key to our improvement is a novel branching strategy in which we show that any overlapping components obtained on the way to a MAF can be `split' by a branching rule with favourable branching factor, and then the problem can be decomposed into disjoint subproblems to be solved separately. We expect that this technique may be more widely applicable to other problems in algorithmic phylogenetics.

翻译：系统发育树是用于模拟物种进化的带叶标签树。在实践中，对同一组物种获得两个拓扑结构不同的树的情况并不罕见，这促使人们使用距离度量来量化相异度。一个著名的度量是最大一致森林（MAF）：一个最小规模的叶标签划分，使得两棵树被分割成相同的一组无交、带叶标签的子树（在抑制二度顶点后同构）。计算这样的MAF是NP困难的，因此大量研究致力于寻找参数化为MAF组件数k的FPT算法。自2015年以来，该领域的最先进水平一直保持不变：无根树的运行时间为$O^*(3^k)$，有根树的运行时间为$O^*(2.3431^k)$。在这项工作中，我们提出了针对无根和有根情况的改进算法，运行时间分别为$O^*(2.846^k)$和$O^*(2.3391^k)$。我们改进的关键是一种新颖的分支策略，其中我们表明，在通往MAF的过程中获得的任何重叠组件都可以通过具有有利分支因子的分支规则进行“分割”，然后问题可以分解为不相交的子问题分别求解。我们预计该技术可能更广泛地适用于算法系统发育学中的其他问题。