New FPT algorithms for finding the temporal hybridization number for sets of phylogenetic trees

We study the problem of finding a temporal hybridization network for a set of phylogenetic trees that minimizes the number of reticulations. First, we introduce an FPT algorithm for this problem on an arbitrary set of m binary trees with n leaves each with a running time of O(5k· n· m), where k is the minimum temporal hybridization number. 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)d5^ k· n· m) time. Lastly, we introduce a O(6kk!· k· n2) time algorithm for computing a minimum 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.