Improving the Approximation Ratio of the Maximum Agreement Forest (MAF) on k trees and Estimating the Approximation Ratio of the Acyclic-MAF on k trees

Molecular phylogenetics has long been a well-established field of scientific research where the structure of the phylogenetic tree has been analysed to know about the evolutionary process of the organism. In biology, leaf-labelled trees are widely used to describe the evolutionary relationships. In this setting, the leaves of the tree correspond to extant species, and the internal vertices represent the ancestral species. However, for certain species, evolution is not completely tree-like. Reticulation events such as horizontal gene transfer (HGT), hybridization and recombination play a significant role in the evolution of the species. Suppose we have two phylogenetic trees each of which is for a gene of the same set of species. Due to reticulate evolution the two gene trees, though related, appear different. As a result, instead of the tree like structure, a phylogenetic network is widely viewed as a most suitable tool to represent reticulation. A phylogenetic network contains hybrid nodes for the species evolved from two parents. The distance between two phylogenetic trees can be computed with the help of a Maximum Agreement Forest (MAF) of those trees. The fewer components in MAF, the greater is the similarity between the two trees. This number of components in that agreement forest shows how many edges from each of the two trees need to be cut so that the resulting forest agree after all forced edge contractions. Recent research reveals that the MAF on k trees can be approximated within a ratio of 8. We have given a better approximation ratio for the MAF on k trees and also provide an approximation ratio for Maximum Acyclic Agreement Forest (MAAF) on k (≥2) trees.