In the Minimum Consistent Subset (MCS) problem, we are presented with a connected simple undirected graph G, consisting of a vertex set V(G) of size n and an edge set E(G). Each vertex in V(G) is assigned a color from the set {1,2,…, c}. The objective is to determine a subset V' ⊆ V(G) with minimum possible cardinality, such that for every vertex v ∈ V(G), at least one of its nearest neighbors in V' (measured in terms of the hop distance) shares the same color as v. The decision problem, indicating whether there exists a subset V' of cardinality at most l for some positive integer l, is known to be NP-complete even for planar graphs. In this paper, we establish that the MCS problem is NP-complete on trees. We also provide a fixed-parameter tractable (FPT) algorithm for MCS on trees parameterized by the number of colors (c) running in O(2^{6c} n^6) time, significantly improving the currently best-known algorithm whose running time is O(2^{4c} n^{2c+3}). In an effort to comprehensively understand the computational complexity of the MCS problem across different graph classes, we extend our investigation to interval graphs. We show that it remains NP-complete for interval graphs, thus enriching graph classes where MCS remains intractable.

Several works recently focus on monitoring air quality of critical areas using sensors attached to buses. They aim to monitor the maximum number of critical areas using a limited number of sensors. In practice, we may want to have information for all critical areas. We work on the problem of covering all the areas using the minimum number of sensors in this work. We show that, even when the bus routes are not pre-defined, the problem is NP-hard and is significantly harder than the problem of the previous works. Then, we develop two algorithms for the case that the routes are pre-defined. Those algorithms include a fixed parameter tractability and a 2-approximation algorithm for a special case of the problem. Our experiment results show that, although we usually give the similar number of sensors as the algorithm in the previous works, our algorithms have a shorter computation time than the classical greedy algorithm.