We study temporal analogues of the Unrestricted Vertex Separator problem from the static world. An $(s,z)$-temporal separator is a set of vertices whose removal disconnects vertex $s$ from vertex $z$ for every time step in a temporal graph. The $(s,z)$-Temporal Separator problem asks to find the minimum size of an $(s,z)$-temporal separator for the given temporal graph. We introduce a generalization of this problem called the $(s,z,t)$-Temporal Separator problem, where the goal is to find a smallest subset of vertices whose removal eliminates all temporal paths from $s$ to $z$ which take less than $t$ time steps. Let $\tau$ denote the number of time steps over which the temporal graph is defined (we consider discrete time steps). We characterize the set of parameters $\tau$ and $t$ when the problem is $\mathcal{NP}$-hard and when it is polynomial time solvable. Then we present a $\tau$-approximation algorithm for the $(s,z)$-Temporal Separator problem and convert it to a $\tau^2$-approximation algorithm for the $(s,z,t)$-Temporal Separator problem. We also present an inapproximability lower bound of $\Omega(\ln(n) + \ln(\tau))$ for the $(s,z,t)$-Temporal Separator problem assuming that $\mathcal{NP}\not\subset\mbox{\sc Dtime}(n^{\log\log n})$. Then we consider three special families of graphs: (1) graphs of branchwidth at most $2$, (2) graphs $G$ such that the removal of $s$ and $z$ leaves a tree, and (3) graphs of bounded pathwidth. We present polynomial-time algorithms to find a minimum $(s,z,t)$-temporal separator for (1) and (2). As for (3), we show a polynomial-time reduction from the Discrete Segment Covering problem with bounded-length segments to the $(s,z,t)$-Temporal Separator problem where the temporal graph has bounded pathwidth.

翻译：我们研究了静态世界中无限制顶点分隔符问题的时间类比。一个$(s,z)$-时间分隔符是一个顶点集合，其移除使得在时间图的每个时间步中，顶点$s$与顶点$z$都不连通。$(s,z)$-时间分隔符问题的目标是：对于给定的时间图，找出最小规模的$(s,z)$-时间分隔符。我们引入该问题的一个推广，称为$(s,z,t)$-时间分隔符问题，其目标是找出一个最小顶点子集，使得移除该子集后，所有从$s$到$z$且耗时少于$t$个时间步的时间路径均被消除。设$\tau$表示时间图定义的时间步总数（我们考虑离散时间步）。我们刻画了问题为$\mathcal{NP}$-难解或可在多项式时间内求解的参数$\tau$和$t$的取值集合。随后，我们为$(s,z)$-时间分隔符问题提出了一个$\tau$-近似算法，并将其转化为$(s,z,t)$-时间分隔符问题的$\tau^2$-近似算法。我们还给出了$(s,z,t)$-时间分隔符问题的一个不可近似性下界：$\Omega(\ln(n) + \ln(\tau))$，该下界基于假设$\mathcal{NP}\not\subset\mbox{\sc Dtime}(n^{\log\log n})$。接着，我们考虑三类特殊图族：(1) 分支宽度至多为$2$的图；(2) 移除$s$和$z$后剩余部分为树的图$G$；(3) 有界路径宽度的图。我们对(1)和(2)中的图族给出了多项式时间算法以求解最小$(s,z,t)$-时间分隔符。至于(3)，我们将带界长线段的有界离散段覆盖问题多项式归约到时间图具有有界路径宽度的$(s,z,t)$-时间分隔符问题。