  Inspired by the famous Target Set Selection problem, we propose a new
discrete model for simultaneously spreading several opinions within a social
network and perform an initial study of its complexity. Here, we are given a
social network, a seed-set of agents for each opinion, and two thresholds for
each agent. The first threshold represents the willingness of an agent to adopt
an opinion if the agent has no opinion at all, while the second threshold
states for willingness to acquire second opinion. The goal is to add as few
agents as possible to the initial seed-sets such that, once the process started
with these seed-sets stabilizes, each agent has either both opinions or none.
  We show that the problem is NP-hard. Further, we investigate the complexity
from the parameterized point-of-view. The problem is W[1]-hard with respect to
the solution size. The problem remains W[1]-hard even for the combination of
parameters the solution size and treewidth of the network even if all
thresholds are at most 3, or the activation process stabilizes within 4 rounds.
On the other hand, the problem is FPT when parameterized by the number of
rounds, maximum threshold, and treewidth. This algorithm applies also for
combined parameter treedepth and maximum threshold. Finally, we show that the
problem is FPT when parameterized by vertex cover number of the input network
alone. Our results also imply that the original Target Set Selection problem is
FPT when parameterized by 3-PVC.
