The \textsc{Co-Path Packing} (resp., \textsc{Co-Path Set}) problem asks whether a given graph can be edited to a collection of induced paths by deleting at most $k$ vertices (resp., $k$ edges). Both are fundamental problems with significant applications in bioinformatics and have been extensively studied within the framework of exact and parameterized algorithms. Currently, the state-of-the-art approach utilizes the randomized ``Cut \& Count'' technique, which solves \textsc{Co-Path Set} in $O^*(4^{\mathbf{tw}})$ time and \textsc{Co-Path Packing} in $O^*(5^{\mathbf{pw}})$ time, where $\mathbf{tw}$ is treewidth and $\mathbf{pw}$ is pathwidth. However, as there is no known method to derandomize the ``Cut \& Count'' technique, the existence of deterministic single exponential time algorithms for these problems parameterized by treewidth has remained an open question. In this paper, we resolve this gap by providing deterministic single exponential time algorithms for both problems when parameterized by treewidth.

翻译：\textsc{Co-Path Packing}（分别地，\textsc{Co-Path Set}）问题询问：给定图是否可以通过删除至多 $k$ 个顶点（分别地，$k$ 条边）编辑为一条诱导路径的集合。这两个均是基础问题，在生物信息学中具有重要应用，并在精确算法与参数化算法的框架下得到了广泛研究。目前，最先进的方法采用随机化的“Cut & Count”技术，该技术能在 $O^*(4^{\mathbf{tw}})$ 时间内解决 \textsc{Co-Path Set}，并在 $O^*(5^{\mathbf{pw}})$ 时间内解决 \textsc{Co-Path Packing}，其中 $\mathbf{tw}$ 为树宽，$\mathbf{pw}$ 为路径宽。然而，由于尚无已知方法可以对“Cut & Count”技术进行去随机化，因此以树宽为参数化该问题的确定性单指数时间算法是否存在仍是一个悬而未决的问题。在本文中，我们通过提供以树宽为参数化的两个问题的确定性单指数时间算法，解决了这一空白。