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

TL;DR

This paper introduces improved fixed-parameter tractable algorithms for computing maximum agreement forests in phylogenetic trees, achieving faster runtimes by novel branching strategies that decompose complex overlaps into simpler subproblems.

Contribution

It presents the first improvements since 2015 on FPT algorithms for MAF, with new branching rules that enhance efficiency for both rooted and unrooted trees.

Findings

Unrooted MAF algorithm runtime improved to O*(2.846^k)

Rooted MAF algorithm runtime improved to O*(2.3391^k)

Novel branching strategy effectively decomposes overlapping components

Abstract

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.