We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t, Tracking Shortest Paths asks if there exists a k-sized subset of vertices (referred to as tracking set) that intersects each shortest s-t path in a distinct set of vertices.

We first generalize this problem for set systems, namely Tracking Set System, where given a family of subsets of a universe, we are required to find a subset of elements from the universe that has a unique intersection with each set in the family. Tracking Set System is shown to be fixed-parameter tractable due to its relation with a known problem, Test Cover. By a reduction to the well-studied d-hitting set problem, we give a polynomial (with respect to k) kernel for the case when the set sizes are bounded by d. This also helps in solving Tracking Shortest Paths when the input graph diameter is bounded by d.

While the results for Tracking Set System show that Tracking Shortest Paths is fixed-parameter tractable, we also give an independent algorithm by using some preprocessing rules, resulting in an improved running time.

我们考虑由安全性和无线网络中的应用程序引起的，跟踪图中最短s - t路径问题的参数化复杂性。给定具有源s和目标t的无向图和无权图，跟踪最短路径询问是否存在k个大小的顶点子集（称为跟踪集），该子集与不同的顶点集中每个最短的s - t路径相交。 。

我们首先针对集合系统（即跟踪集合系统）推广该问题，在该系统中，给定一个Universe子集族，我们需要从Universe中找到与该族中的每个集合都有唯一交集的元素子集。跟踪集系统由于与已知问题Test Cover的关系而显示为固定参数可处理的。通过简化研究的d命中集问题，对于集合大小以d为边界的情况，我们给出了多项式（相对于k）核。当输入图的直径由d限制时，这也有助于解决跟踪最短路径。

虽然跟踪集系统的结果表明跟踪最短路径是固定参数易处理的，但我们还通过使用一些预处理规则提供了独立的算法，从而缩短了运行时间。