Directed Temporal Tree Realization for Periodic Public Transport: Easy and Hard Cases

We study the complexity of the directed periodic temporal graph realization problem. This work is motivated by the design of periodic schedules in public transport with constraints on the quality of service. Namely, we require that the fastest path between (important) pairs of vertices is upper bounded by a specified maximum duration, encoded in an upper distance matrix $D$. While previous work has considered the undirected version of the problem, the application in public transport schedule design requires the flexibility to assign different departure times to the two directions of an edge. A problem instance can only be feasible if all values of the distance matrix are at least shortest path distances. However, the task of realizing exact fastest path distances in a periodic temporal graph is often too restrictive. Therefore, we introduce a minimum slack parameter $k$ that describes a lower bound on the maximum allowed waiting time on each path. We concentrate on tree topologies and provide a full characterization of the complexity landscape with respect to the period $Δ$ and the minimum slack parameter $k$, showing a sharp threshold between NP-complete cases and cases which are always realizable. We also provide hardness results for the special case of period $Δ= 2$ for general directed and undirected graphs.