Conflict Resolution Algorithms for Deep Coalescence Phylogenetic Networks

We address the problem of inferring an optimal tree displayed by a network, given a gene tree G and a tree-child network N, under the deep coalescence cost. We propose an O(|G||N|)-time dynamic programming algorithm (DP) to compute a lower bound of the optimal displayed tree cost, where |G| and |N| are the sizes of G and N, respectively. This algorithm has the ability to state whether the cost is exact or is a lower bound. In addition, our algorithm provides a set of reticulation edges that correspond to the obtained cost. If the cost is exact, the set induces an optimal displayed tree that yields the cost. If the cost is a lower bound, the set contains pairs of conflicting edges, that is, edges sharing a reticulation node. Next, we show a conflict resolution algorithm that requires 2^{r+1}-1 invocations of DP in the worst case, where r is a number of reticulations. We propose a similar O(2^k|G||N|)-time algorithm for level-k networks and a branch and bound solution to compute lower and upper bounds of optimal costs. We also show how our algorithms can be extended to a broader class of phylogenetic networks. Despite their exponential complexity in the worst case, our solutions perform significantly well on empirical and simulated datasets, thanks to the strategy of resolving internal dissimilarities between gene trees and networks. In particular, experiments on simulated data indicate that the runtime of our solution is Θ(2^{0.543 k}|G||N|) on average. Therefore, our solution is an efficient alternative to enumeration strategies commonly proposed in the literature and enables analyses of complex networks with dozens of reticulations.