The complexity of finding temporal separators under waiting time constraints

In this work, we investigate the computational complexity of RESTLESS TEMPORAL (s,z)-SEPARATION, where we are asked whether it is possible to destroy all restless temporal paths between two distinct vertices s and z by deleting at most k vertices from a temporal graph. A temporal graph has a fixed vertex set but the edges have (discrete) time stamps. A restless temporal path uses edges with non-decreasing time stamps and the time spent at each vertex must not exceed a given duration Δ. RESTLESS TEMPORAL (s,z)-SEPARATION naturally generalizes the NP-hard TEMPORAL (s,z)-SEPARATION problem. We show that RESTLESS TEMPORAL (s,z)-SEPARATION is complete for Σ₂^P, a complexity class located in the second level of the polynomial time hierarchy. We further provide some insights in the parameterized complexity of RESTLESS TEMPORAL (s,z)-SEPARATION parameterized by the separator size k.