Realizing temporal transportation trees

In this paper, we study the complexity of the \textit{periodic temporal graph realization} problem with respect to upper bounds on the fastest path durations among its vertices. This constraint with respect to upper bounds appears naturally in transportation network design applications where, for example, a road network is given, and the goal is to appropriately schedule periodic travel routes, while not exceeding some desired upper bounds on the travel times. This approach is in contrast to verification applications of the graph realization problems, where exact values for the distances (respectively, fastest travel times) are given, following some kind of precise measurement. In our work, we focus only on underlying tree topologies, which are fundamental in many transportation network applications. As it turns out, the periodic upper-bounded temporal tree realization problem (TTR) has a very different computational complexity behavior than both (i) the classic graph realization problem with respect to shortest path distances in static graphs and (ii) the periodic temporal graph realization problem with exact given fastest travel times (which was recently introduced). First, we prove that, surprisingly, TTR is NP-hard, even for a constant period $\Delta$ and when the input tree $G$ satisfies at least one of the following conditions: (a) $G$ has a constant diameter, or (b) $G$ has constant maximum degree. In contrast, when we are given exact values of the fastest travel delays, the problem is known to be solvable in polynomial time. Second, we prove that TTR is fixed-parameter tractable (FPT) with respect to the number of leaves in the input tree $G$, via a novel combination of techniques for totally unimodular matrices and mixed integer linear programming.