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

Abstract

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 $$O(5^k\cdot n\cdot m)$$ 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 $$O((8k)^d5^ k\cdot k\cdot n\cdot m)$$ O ( ( 8 k ) d 5 k · k · n · m ) time. Lastly, we introduce an $$O(6^kk!\cdot k\cdot n^2)$$ 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.

A Practical Fixed-Parameter Algorithm for Constructing Tree-Child Networks from Multiple Binary Trees

“…Can tree-child hybridization networks be computed faster than in O((ck) k • poly(n, t)) time, ideally in O(c k • poly(n, t)) time? For temporal networks, a recent result [8] shows that this is indeed the case. An interesting open question is whether the techniques used in that algorithm can also be used to obtain faster algorithms for computing general tree-child networks.…”

“…Indeed, in experiments on synthetic inputs, the running time grows roughly linearly in the number of trees and taxa. On the other hand, the running time still has a large exponential dependency on the number of reticulation events k. Nevertheless, as long as k is small (at most [7][8][9][10][11][12], our algorithm can solve inputs with up to 100 input trees and 200 taxa. In our experiments on real-world data, we observed that these data sets have substantially more structure than random synthetic data sets, which makes cluster reduction and redundant branch elimination more effective and allowed our algorithm to solve inputs with up to 8 trees and 50 reticulations.…”