  In this paper, we consider the problem of maximizing the spread of influence
through a social network. Given a graph with a threshold value~$thr(v)$
attached to each vertex~$v$, the spread of influence is modeled as follows: A
vertex~$v$ becomes "active" (influenced) if at least $thr(v)$ of its neighbors
are active. In the corresponding optimization problem the objective is then to
find a fixed number of vertices to activate such that the number of activated
vertices at the end of the propagation process is maximum. We show that this
problem is strongly inapproximable in fpt-time with respect to (w.r.t.)
parameter $k$ even for very restrictive thresholds. In the case that the
threshold of each vertex equals its degree, we prove that the problem is
inapproximable in polynomial time and it becomes $r(n)$-approximable in
fpt-time w.r.t. parameter $k$ for any strictly increasing function $r$.
  Moreover, we show that the decision version is W[1]-hard w.r.t. parameter $k$
but becomes fixed-parameter tractable on bounded degree graphs.
