In train networks, carefully-chosen delays may be beneficial for certain passengers, who would otherwise miss some connection. Given a simple temporal graph and a set of passengers (each specifying a starting vertex, an ending vertex, and a desired arrival time), we ask whether it is possible to delay some of the edges of the temporal graph to realize all the passengers' demands. We call this problem DelayBetter (DB), and study it along with two variants: in $\delta$-DelayBetter, each delay must be of at most $\delta$; in Path DB, passengers fully specify the vertices they should visit on their journey. On the positive side, we give a polynomial-time algorithm for Path DB, and obtain as a corollary a polynomial-time algorithm for DB and $\delta$-DB on trees. We also provide an fpt algorithm for both problems parameterized by the size of the graph's Feedback Edge Set together with the number of passengers. On the negative side, we show NP-completeness of ($1$-)DB on bounded-degree temporal graphs even when the lifetime is $2$, and of ($10$-)DB on bounded-degree planar temporal graphs of lifetime $19$. Our results complement previous work studying reachability problems in temporal graphs with delaying operations. This is to our knowledge the first such problem in which the aim is to facilitate travel between specific points (as opposed to facilitating or impeding a broadcast from one or many sources).

翻译：在铁路网络中，精心选择的延迟可能对某些乘客有益，否则他们将错过某些换乘。给定一个简单的时序图和一组乘客（每位乘客指定起点、终点和期望到达时间），我们探讨是否可能通过延迟时序图中的某些边来实现所有乘客的需求。我们称此问题为延迟优化问题（DB），并研究其两个变体：在$\delta$-延迟优化问题中，每条边的延迟不得超过$\delta$；在路径延迟优化问题中，乘客需完整指定其行程中应访问的顶点。在积极方面，我们给出了路径延迟优化问题的多项式时间算法，并由此推导出在树结构上求解DB和$\delta$-DB的多项式时间算法。我们还提供了针对这两个问题的固定参数可解算法，其参数为图的反馈边集大小与乘客数量之和。在消极方面，我们证明了即使在生命周期为2的情况下，（1-）DB在有限度时序图上是NP完全的；且生命周期为19时，（10-）DB在有限度平面时序图上也是NP完全的。我们的研究结果补充了先前关于在具有延迟操作的时序图中研究可达性问题的文献。据我们所知，这是首个以促进特定点间通行为目标的问题（而非促进或阻碍来自单个或多个源的广播）。