Genome rearrangement is a common model for molecular evolution. In this paper, we consider the Pairwise Rearrangement problem, which takes as input two genomes and asks for the number of minimum-length sequences of permissible operations transforming the first genome into the second. In the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010), Pairwise Rearrangement is $\#\textsf{P}$-complete (Bailey, et. al., COCOON 2023), which implies that exact sampling is intractable. In order to cope with this intractability, we investigate the parameterized complexity of this problem. We exhibit a fixed-parameter tractable algorithm with respect to the number of components in the adjacency graph that are not cycles of length $2$ or paths of length $1$. As a consequence, we obtain that Pairwise Rearrangement in the Single Cut-and-Join model is fixed-parameter tractable by distance. Our results suggest that the number of nontrivial components in the adjacency graph serves as the key obstacle for efficient sampling.

翻译：基因组重排是分子演化的常见模型。本文考虑**成对重排问题**，该问题以两个基因组为输入，要求计算将第一个基因组转化为第二个基因组的可允许操作的最短序列数目。在单切割与连接模型（Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010）中，成对重排问题是$\#\textsf{P}$-完全的（Bailey 等, COCOON 2023），这意味着精确采样在计算上难以处理。为应对这一困难，我们研究该问题的参数化复杂度。我们提出一个固定参数可解算法，其参数为邻接图中非长度为$2$的环或长度为$1$的路径的分量数目。由此可得，在单切割与连接模型中，成对重排问题是关于距离固定参数可解的。我们的结果表明，邻接图中非平凡分量的数目是高效采样的关键障碍。