  In this paper we consider the Target Set Selection problem. The problem
naturally arises in many fields like economy, sociology, medicine. In the
Target Set Selection problem one is given a graph $G$ with a function
$\operatorname{thr}: V(G) \to \mathbb{N} \cup \{0\}$ and integers $k, \ell$.
The goal of the problem is to activate at most $k$ vertices initially so that
at the end of the activation process there is at least $\ell$ activated
vertices. The activation process occurs in the following way: (i) once
activated, a vertex stays activated forever; (ii) vertex $v$ becomes activated
if at least $\operatorname{thr}(v)$ of its neighbours are activated. The
problem and its different special cases were extensively studied from
approximation and parameterized points of view. For example, parameterizations
by the following parameters were studied: treewidth, feedback vertex set,
diameter, size of target set, vertex cover, cluster editing number and others.
  Despite the extensive study of the problem it is still unknown whether the
problem can be solved in $\mathcal{O}^*((2-\epsilon)^n)$ time for some
$\epsilon >0$. We partially answer this question by presenting several
faster-than-trivial algorithms that work in cases of constant thresholds,
constant dual thresholds or when the threshold value of each vertex is bounded
by one-third of its degree. Also, we show that the problem parameterized by
$\ell$ is W[1]-hard even when all thresholds are constant.
