The Parameterized Complexity of Terminal Monitoring Set

TL;DR

This paper investigates the computational complexity of the Terminal Monitoring Set problem, establishing its hardness and fixed parameter tractability under various parameters, and providing algorithms for specific cases.

Contribution

It proves W[2]-hardness for the problem and identifies multiple parameters under which the problem is fixed parameter tractable, also analyzing weighted and relaxed variants.

Findings

TMS is W[2]-hard with respect to solution size.

TMS is fixed parameter tractable with respect to solution size plus distance to cluster.

Weighted TMS is FPT with respect to vertex cover number.

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.