In the {\sc Vertex Multicut} problem the input consists of a graph $G$, integer $k$, and a set $\mathbf{T} = \{(s_1, t_1), \ldots, (s_p, t_p)\}$ of pairs of vertices of $G$. The task is to find a set $X$ of at most $k$ vertices such that, for every $(s_i, t_i) \in \mathbf{T}$, there is no path from $s_i$ to $t_i$ in $G - X$. Marx and Razgon [STOC 2011 and SICOMP 2014] and Bousquet, Daligault, and Thomassé [STOC 2011 and SICOMP 2018] independently and simultaneously gave the first algorithms for {\sc Vertex Multicut} with running time $f(k)n^{O(1)}$. The running time of their algorithms is $2^{O(k^3)}n^{O(1)}$ and $2^{O(k^{O(1)})}n^{O(1)}$, respectively. As part of their result, Marx and Razgon introduce the {\em shadow removal} technique, which was subsequently applied in algorithms for several parameterized cut and separation problems. The shadow removal step is the only step of the algorithm of Marx and Razgon which requires $2^{O(k^3)}n^{O(1)}$ time. Chitnis et al. [TALG 2015] gave an improved version of the shadow removal step, which, among other results, led to a $k^{O(k^2)}n^{O(1)}$ time algorithm for {\sc Vertex Multicut}. We give a faster algorithm for the {\sc Vertex Multicut} problem with running time $k^{O(k)}n^{O(1)}$. Our main technical contribution is a refined shadow removal step for vertex separation problems that only introduces an overhead of $k^{O(k)}\log n$ time. The new shadow removal step implies a $k^{O(k^2)}n^{O(1)}$ time algorithm for {\sc Directed Subset Feedback Vertex Set} and a $k^{O(k)}n^{O(1)}$ time algorithm for {\sc Directed Multiway Cut}, improving over the previously best known algorithms of Chitnis et al. [TALG 2015].

翻译：在{\sc 顶点多割}问题中，输入包括一个图$G$、整数$k$以及一组$G$的顶点对集合$\mathbf{T} = \{(s_1, t_1), \ldots, (s_p, t_p)\}$。任务是找到一个最多包含$k$个顶点的集合$X$，使得对于每个$(s_i, t_i) \in \mathbf{T}$，在$G - X$中不存在从$s_i$到$t_i$的路径。Marx与Razgon [STOC 2011与SICOMP 2014]以及Bousquet、Daligault与Thomassé [STOC 2011与SICOMP 2018]分别独立且同时给出了首个运行时间为$f(k)n^{O(1)}$的{\sc 顶点多割}算法。他们算法的运行时间分别为$2^{O(k^3)}n^{O(1)}$和$2^{O(k^{O(1)})}n^{O(1)}$。作为其成果的一部分，Marx与Razgon引入了{\em 影子移除}技术，该技术随后被应用于多个参数化割与分离问题的算法中。影子移除步骤是Marx与Razgon算法中唯一需要$2^{O(k^3)}n^{O(1)}$时间的步骤。Chitnis等人[TALG 2015]给出了影子移除步骤的改进版本，该版本（连同其他成果）催生了一个运行时间为$k^{O(k^2)}n^{O(1)}$的{\sc 顶点多割}算法。我们提出了一个运行时间为$k^{O(k)}n^{O(1)}$的更快的{\sc 顶点多割}算法。我们的主要技术贡献是针对顶点分离问题的一种改进的影子移除步骤，该步骤仅引入$k^{O(k)}\log n$的时间开销。这一新的影子移除步骤意味着{\sc 有向子集反馈顶点集}问题存在一个$k^{O(k^2)}n^{O(1)}$时间算法，而{\sc 有向多路割}问题存在一个$k^{O(k)}n^{O(1)}$时间算法，这改进了Chitnis等人[TALG 2015]先前已知的最佳算法。