Hybridization Number on Three Trees

Phylogenetic networks are leaf-labelled directed acyclic graphs that are used to describe nontreelike evolutionary histories and are thus a generalization of phylogenetic trees. The hybridization number of a phylogenetic network is the sum of all indegrees minus the number of nodes plus one. The Hybridization Number problem takes as input a collection of phylogenetic trees and asks to construct a phylogenetic network that contains an embedding of each of the input trees and has a smallest possible hybridization number. We present an algorithm for the Hybridization Number problem on three binary phylogenetic trees on n leaves, which runs in time O(c k poly(n)), with k the hybridization number of an optimal network and c some positive constant. For the case of two trees, an algorithm with running time O(3.18 k n) was proposed before whereas an algorithm with running time O(c k poly(n)) for more than two trees had prior to this article remained elusive. The algorithm for two trees uses the close connection to acyclic agreement forests to achieve a linear exponent in the running time, while previous algorithms for more than two trees (explicitly or implicitly) relied on a brute force search through all possible underlying network topologies, leading to running times that are not O(c k poly(n)), for any c. The connection to acyclic agreement forests is much weaker for more than two trees, so even given the right agreement forest, the reconstruction of the network poses major challenges. We prove novel structural results that allow us to reconstruct a network without having to guess the underlying topology. Our techniques generalize to more than three input trees with the exception of one key lemma that maps nodes in the network to tree nodes and, thus,