The C-Orientation problem asks whether it is possible to orient an undirected graph to a directed phylogenetic network of a desired network class C. This problem arises, for example, when visualising evolutionary data, as popular methods such as Neighbor-Net are distance-based and 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 exact FPT algorithm for C-Orientation that is applicable to any class C and parameterised by the reticulation number and the maximum size of minimal basic cycles, and a very fast heuristic for Tree-Child Orientation. While the state-of-the-art for C-Orientation is a simple exponential time algorithm whose computational bottleneck lies in searching for appropriate reticulation vertex placements, our methods significantly reduce this search space. Experiments show that, although our FPT algorithm is still exponential, it significantly outperforms the existing method. The heuristic runs even faster but with increasing false negatives as the reticulation number grows. Given this trade-off, we also discuss theoretical directions for improvement and biological applicability of the heuristic approach.

翻译：C-定向问题探讨是否可能将无向图定向为特定网络类C的有向系统发育网络。该问题在可视化进化数据时尤为突出，因为诸如邻接网络等主流方法基于距离度量，不可避免地生成无向图。目前C-定向问题的计算复杂度对许多网络类（包括二叉树子网络）仍未解决，且缺乏实用算法。本文提出一种适用于任意网络类C的精确FPT算法，其参数为网状结构数量与最小基本环的最大规模，同时提出一种针对树子定向问题的极速启发式算法。现有C-定向研究主要依赖简单的指数时间算法，其计算瓶颈在于搜索合适的网状顶点布局，而我们的方法显著压缩了该搜索空间。实验表明，尽管所提FPT算法仍具指数复杂度，但其性能显著优于现有方法。启发式算法运行速度更快，但随着网状结构数量增加，其假阴性率会上升。基于此权衡关系，我们进一步探讨了该启发式算法的理论改进方向及其在生物学领域的应用潜力。