New FPT Algorithms for Finding the Temporal Hybridization Number for Sets of Phylogenetic Trees

We study the problem of finding a temporal hybridization network containing at most k reticulations, for an input consisting of a set of phylogenetic trees. First, we introduce an FPT algorithm for the problem on an arbitrary set of m binary trees with n leaves each with a running time of O(5(k).n.m). We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most d and at most k reticulations in O((8k)(d)5(k).k.n.m) time. Lastly, we introduce an O(6(k)k!.k.n(2)) time algorithm for computing a temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.