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We study the problem of finding a temporal hybridization network for a set ofphylogenetic trees that minimizes the number of reticulations. First, weintroduce an FPT algorithm for this problem on an arbitrary set of $m$ binarytrees with $n$ leaves each with a running time of $O(5^k\cdot n\cdot m)$, where$k$ is the minimum temporal hybridization number. We also present the conceptof temporal distance, which is a measure for how close a tree-child network isto being temporal. Then we introduce an algorithm for computing a tree-childnetwork with temporal distance at most $d$ and at most $k$ reticulations in$O((8k)^d5^ k\cdot n\cdot m)$ time. Lastly, we introduce a $O(6^kk!\cdot k\cdotn^2)$ time algorithm for computing a minimum temporal hybridization network fora set of two nonbinary trees. We also provide an implementation of allalgorithms and an experimental analysis on their performance.

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