Parameterized Inapproximability of Target Set Selection and Generalizations

Abstract

In this paper, we consider the TARGET SET SELECTION problem: given a graph and a threshold value thr(v) for each vertex v of the graph, find a minimum size vertex-subset to “activate” such that all vertices of the graph are activated at the end of the propagation process. A vertex v is activated during the propagation process if at least thr(v) of its neighbors are activated. This problem models several practical issues like faults in distributed networks or word-to-mouth recommendations in social networks. We show that for any functions f and ρ this problem cannot be approximated within a factor of ρ(k) in f(k)·n^O(1) time, unless FPT=W[P], even for restricted thresholds (namely constant and majority thresholds), where k is the number of vertices to activate in the beginning. We also study the cardinality constraint maximization and minimization versions of the problem for which we prove similar hardness results.