In Terminal Monitoring Set (TMS), the input is an undirected graph $G=(V,E)$, together with a collection $T$ of terminal pairs and the goal is to find a subset $S$ of minimum size that hits a shortest path between every pair of terminals. We show that this problem is W[2]-hard with respect to solution size. On the positive side, we show that TMS is fixed parameter tractable with respect to solution size plus distance to cluster, solution size plus neighborhood diversity, and feedback edge number. For the weighted version of the problem, we obtain a FPT algorithm with respect to vertex cover number, and for a relaxed version of the problem, we show that it is W[1]-hard with respect to solution size plus feedback vertex number.

翻译：在终端监控集问题中，输入是一个无向图$G=(V,E)$及一个终端对集合$T$，目标是找到一个最小规模的子集$S$，使其能够覆盖每对终端之间的最短路径。我们证明该问题相对于解规模是W[2]-难的。在积极方面，我们证明终端监控集相对于解规模加聚类距离、解规模加邻域多样性、以及反馈边数均是固定参数可解的。针对该问题的加权版本，我们获得了相对于顶点覆盖数的固定参数算法；而对于该问题的松弛版本，我们证明其相对于解规模加反馈顶点数是W[1]-难的。