Résumé en anglais

In this paper, we consider the Target Set Selection problem: given a graph and a threshold value https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-08019-2_2/978-3-319-08019-2_2_IEq1_HTML.gif for each vertex v of the graph, find a minimum size vertex-subset to “activate” s.t. all the 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 https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-08019-2_2/978-3-319-08019-2_2_IEq2_HTML.gif 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). We also study the cardinality constraint maximization and minimization versions of the problem for which we prove similar hardness results.