实现时间运输树
Realizing temporal transportation trees
AI总结：
本文研究运输网络设计中周期性时间树实现问题（TTR）的复杂度，证明其为NP难（即使常数周期、星型树或常数最大度），并提出基于叶子数量的固定参数可解（FPT）方法。
摘要：
本文研究了周期性时间图实现问题的复杂度，该问题涉及顶点间最快路径持续时间的上界约束。这种上界约束在运输网络设计应用中自然出现，例如给定道路网络时，目标是合理安排周期性出行路线，同时不超过某些期望的旅行时间上界。我们的工作仅关注基础树拓扑结构，这类结构在许多运输网络应用中至关重要。事实证明，周期性上界时间树实现问题（TTR）的计算复杂度行为与以下两者截然不同：（i）静态图中关于最短路径距离的经典图实现问题；（ii）最近提出的具有精确给定最快旅行时间的周期性时间图实现问题。首先，我们证明，令人惊讶的是，即使对于常数周期Δ，且当输入树G满足以下条件之一时，TTR仍然是NP难的：（a）G是星型树；（b）G具有常数最大度。其次，我们证明，通过将全幺模矩阵技术与混合整数线性规划相结合的新方法，TTR关于输入树G的叶子数量是固定参数可解的（FPT）。
英文摘要：
In this paper, we study the complexity of the 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. 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 $Δ$ and when the input tree $G$ satisfies at least one of the following conditions: (a) $G$ is a star, or (b) $G$ has constant maximum degree. 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.
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