  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.
