Given two rooted phylogenetic trees on the same set of taxa X, the Maximum Agreement Forest (MAF) problem asks to find a forest that is, in a certain sense, common to both trees and has a minimum number of components. The Maximum Acyclic Agreement Forest (MAAF) problem has the additional restriction that the components of the forest cannot have conflicting ancestral relations in the input trees. There has been considerable interest in the special cases of these problems in which the input trees are required to be binary. However, in practice, phylogenetic trees are rarely binary, due to uncertainty about the precise order of speciation events. Here, we show that the general, nonbinary version of MAF has a polynomial-time 4-approximation and a fixed-parameter tractable (exact) algorithm that runs in O(4(k)poly(n)) time, where n = vertical bar X vertical bar and k is the number of components of the agreement forest minus one. Moreover, we show that a c-approximation algorithm for nonbinary MAF and a d-approximation algorithm for the classical problem Directed Feedback Vertex Set (DFVS) can be combined to yield a d(c+3)-approximation for nonbinary MAAF. The algorithms for MAF have been implemented and made publicly available.
van Iersel, L., Kelk, S., Lekic, N., & Stougie, L. (2014). Approximation algorithms for nonbinary agreement forests. Siam Journal on Discrete Mathematics, 28(1), 49-66. https://doi.org/10.1137/120903567