The Parameterized Complexity of Terminal Monitoring Set
N. R. Aravind, Roopam Saxena
TL;DR
This work introduces Terminal Monitoring Set (TMS), a problem of placing monitoring vertices to intersect all shortest paths between given terminal pairs. It establishes NP-hardness and $W[2]$-hardness with respect to the solution size, and then provides several fixed-parameter tractable algorithms: (i) by solution size plus distance to cluster, (ii) by solution size plus neighborhood diversity, (iii) the weighted variant by vertex cover, and (iv) by the feedback edge number. It also studies a relaxed variant, showing $W[1]$-hardness with respect to solution size plus feedback vertex number on planar graphs. The core technique is reducing TMS to Hitting Set and applying sunflowers and kernelization to bound instance sizes under structural parameterizations, enabling efficient FPT algorithms for practical graph families. The results connect shortest-path hitting with classical parameterized problems and extend known FPT approaches to a new, practically motivated monitoring setting. Mathematical formulations are employed to formalize distances and path properties, yielding precise, implementable parameterized algorithms with implications for network monitoring and incremental deployment scenarios.
Abstract
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.