Temporal Separators with Deadlines

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. The (s,z)-Temporal Separator problem is known to be NP-hard in general, although some special cases (such as bounded treewidth) admit efficient algorithms [Fluschnik et al., 2020]. 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 τ denote the number of time steps over which the temporal graph is defined (we consider discrete time steps). We characterize the set of parameters τ and t when the problem is NP-hard and when it is polynomial time solvable. Then we present a τ-approximation algorithm for the (s,z)-Temporal Separator problem and convert it to a τ²-approximation algorithm for the (s,z,t)-Temporal Separator problem. We also present an inapproximability lower bound of Ω(ln(n) + ln(τ)) for the (s,z,t)-Temporal Separator problem assuming that NP ⊄ 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.