Orientability of Undirected Phylogenetic Networks to a Desired Class: Practical Algorithms and Application to Tree-Child Orientation

The 𝒞-Orientation problem asks whether it is possible to orient an undirected graph to a directed phylogenetic network of a desired class 𝒞, and to find such an orientation if one exists. The problem can arise when visualising evolutionary data, for example, because popular phylogenetic network reconstruction methods such as Neighbor-Net are distance-based and thus inevitably produce undirected graphs. The complexity of 𝒞-Orientation remains open for many classes 𝒞, including binary tree-child networks, and practical methods are still lacking. In this paper, we propose an exponential but practically efficient FPT algorithm for 𝒞-Orientation, which is parameterised by the reticulation number and the maximum size of minimal basic cycles used in the computation. We also present a very fast heuristic for Tree-Child Orientation. To evaluate the empirical performance of the proposed methods, we compared their accuracy and execution time for Tree-Child Orientation with those of an exponential time 𝒞-orientation algorithm from the literature. Our experiments show that the proposed exact algorithm is significantly faster than the state-of-the-art exponential time algorithm. The proposed heuristic runs even faster but the accuracy decreases as the reticulation number increases.