Split-or-decompose: Improved FPT branching algorithms for maximum agreement forests

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). 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. 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.