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 $\Delta$, referred to as $\Delta$-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.
翻译:计算两个顶点之间的( 短) 路径是图形算法中最基本的原始值之一 。 最近几年, 在时间图中研究路径, 也就是在时间图中研究路径, 也就是说, 图表中, 顶点设置固定, 边缘设置随时间变化而变化, 越来越引起更多的关注。 路径是尊重时间的路径, 或者是时间的路径, 如果它使用非递减时间印记的边緣。 我们调查一个时间路径的基本限制, 每个顶点所花的时间不能超过一定的时间长度 $\Delta$, 称为$\Delta$- reset 时间路径。 最近几年里, 也就是在时间图中, 时间图设置中, 顶点是螺旋组合的路径, 时间图里程里是自然的路径, 直线路里是自然的路径。 我们发现, 问题“ 淡变变数”, 即使在非常严格的设置中, 问题也很难计算 。 例如, 新的数字是W[ 1] 硬,, 由直径直线路路路段到直到直径直径直路路段的直径, 。 因此, 直路路的直径直径直路的直路路的直路的直路段 。 [ 1) 。 [ 。 [1] 。