We study a family of reachability problems under waiting-time restrictions in temporal and vertex-colored temporal graphs. Given a temporal graph and a set of source vertices, we find the set of vertices that are reachable from a source via a time-respecting path, where the difference in timestamps between consecutive edges is at most a resting time. Given a vertex-colored temporal graph and a multiset query of colors, we find the set of vertices reachable from a source via a time-respecting path such that the vertex colors of the path agree with the multiset query and the difference in timestamps between consecutive edges is at most a resting time. These kind of problems have applications in understanding the spread of a disease in a network, tracing contacts in epidemic outbreaks, finding signaling pathways in the brain network, and recommending tours for tourists, among other. We present an algebraic algorithmic framework based on constrained multi\-linear sieving for solving the restless reachability problems we propose. In particular, parameterized by the length $k$ of a path sought, we show that the proposed problems can be solved in $O(2^k k m \Delta)$ time and $O(n \Delta)$ space, where $n$ is the number of vertices, $m$ the number of edges, and $\Delta$ the maximum resting time of an input temporal graph. In addition, we prove that our algorithms for the restless reachability problems in vertex-colored temporal graphs are optimal under plausible complexity-theoretic assumptions. Finally, with an open-source implementation, we demonstrate that our algorithm scales to large graphs with up to one billion temporal edges, despite the problems being NP-hard. Specifically, we present extensive experiments to evaluate our scalability claims both on synthetic and real-world graphs. Our implementation is efficiently engineered and highly optimized.

翻译：我们研究了一类在时序图及顶点着色时序图中，受等待时间限制的可达性问题。给定一个时序图及一组源顶点，我们找出从源点通过一条时间一致路径可达的顶点集合，其中连续边之间的时间戳差值不超过一个静止时间。给定一个顶点着色时序图和一个颜色的多重集查询，我们找出从源点通过一条时间一致路径可达的顶点集合，要求路径的顶点颜色与多重集查询相匹配，且连续边之间的时间戳差值不超过一个静止时间。这类问题在理解网络中疾病传播、追踪流行病暴发中的接触者、寻找脑网络中的信号通路以及为游客推荐旅行路线等方面具有应用价值。我们提出了一个基于约束多重线性筛法的代数算法框架，用于解决我们提出的无休可达性问题。特别地，以所求路径长度 $k$ 为参数，我们证明了所提出的问题可以在 $O(2^k k m \Delta)$ 时间和 $O(n \Delta)$ 空间内求解，其中 $n$ 是顶点数，$m$ 是边数，$\Delta$ 是输入时序图的最大静止时间。此外，我们证明了在合理的复杂性理论假设下，我们针对顶点着色时序图中无休可达性问题的算法是最优的。最后，通过一个开源实现，我们证明了尽管这些问题都是NP难的，但我们的算法可以扩展到具有多达十亿条时序边的大规模图上。具体而言，我们进行了大量实验，在合成图和真实世界图上评估了我们的可扩展性主张。我们的实现经过了高效工程化和高度优化。