Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration \(\varDelta \), referred to as \(\varDelta \)-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the “restless variant” of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

计算两个顶点之间的（短）路径是图算法中最基本的原语之一。近年来，对时间图中路径的研究，即顶点集固定但边集随时间变化的图，越来越受到关注。如果路径使用具有非递减时间戳的边，则该路径是时间尊重的或时间性的。我们研究了时间路径的基本约束，其中每个顶点花费的时间不得超过给定的持续时间 \(\varDelta \)，称为\(\varDelta \) -不安分的时间路径. 这种限制自然出现在现实世界过程的建模中，例如通信网络中的数据包路由和疾病的感染传播路径，恢复赋予持久的抵抗力。虽然在多项式时间内找到没有等待时间限制的时间路径是可行的，但我们表明，即使在非常严格的设置中，这个问题的“不稳定变体”在计算上也变得困难。例如，当通过与底层图的不相交路径的距离进行参数化时，它是 W[1]-hard ，这意味着许多其他参数（如反馈顶点数和路径宽度）的 W[1]-hardness。因此，一个自然的问题是问题在某些自然环境中是否变得易于处理。我们探索了几个自然参数化，展示了三个的 FPT 算法参数种类：（1）输出相关参数（这里是路径的最大长度），（2）应用于底层图的经典参数（例如，反馈边数），以及（3）称为定时反馈的新参数vertex number，它捕获输入时间图的更细粒度的时间特征，并且在这项工作之外可能会引起人们的兴趣。