Parameterized Complexity of Weighted Target Set Selection

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 the smallest target set of G with vertex thresholds. 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 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 also 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(nlogn) time for complete graphs. Additionally, we design FPT algorithms for WTSS when parameterized by nd+ℓ, tw+ℓ, ce, and vc, where nd is the neighborhood diversity, tw is the treewidth, ce is the cluster editing number, and vc is the vertex cover number of the input graph.