Parameterized complexity of weighted target set selection - Suzuki, Takahiro; Kimura, Kei; Suzuki, Akira; Tamura, Yuma; Zhou, Xiao (Theoretical computer science, 2025)
Consider a graph G where each vertex has a threshold. A vertex v in G is activated if the number of active vertices adjacent to v is at least as many as its threshold. A vertex subset A0 of G is a target set if eventually all vertices in G are activated by initially activating vertices of A0. The Target Set Selection problem (TSS) involves finding a smallest target set of G. This problem has already been extensively studied and is known to be NP-hard even for very restricted conditions. In this paper, we analyze TSS and its weighted variant, called the Weighted Target Set Selection problem (WTSS), from the perspective of parameterized complexity. Let k be the solution size and let ℓ be the maximum threshold. We first show that TSS is W[1]-hard for split graphs when parameterized by k+ℓ, and W[2]-hard for cographs when parameterized by k. We next prove that WTSS is W[2]-hard for trivially perfect graphs when parameterized by k. On the other hand, we show that WTSS can be solved in O(nlog⁡n) time for complete graphs with n vertices. Additionally, we design FPT algorithms for WTSS when parameterized by nd+ℓ, tw+ℓ, ce, and vc, where nd, tw, ce, and vc are the neighborhood diversity, the treewidth, the cluster editing number, and the vertex cover number of the input graph, respectively.