  Let $G = (V,E)$ be a graph on $n$ vertices, where $d_v$ denotes the degree of
vertex $v$, and $t_v$ is a threshold associated with $v$. We consider a process
in which initially a set $S$ of vertices becomes active, and thereafter, in
discrete time steps, every vertex $v$ that has at least $t_v$ active neighbors
becomes active as well. The set $S$ is contagious if eventually all $V$ becomes
active. The target set selection problem TSS asks for the smallest contagious
set. TSS is NP-hard and moreover, notoriously difficult to approximate.
  In the conservative special case of TSS, $t_v > \frac{1}{2}d_v$ for every $v
\in V$. In this special case, TSS can be approximated within a ratio of
$O(\Delta)$, where $\Delta = \max_{v \in V}[d_v]$. In this work we introduce a
more general class of TSS instances that we refer to as conservative on average
(CoA), that satisfy the condition $\sum_{v\in V} t_v > \frac{1}{2}\sum_{v \in
V} d_v$. We design approximation algorithms for some subclasses of CoA. For
example, if $t_v \geq \frac{1}{2}d_v$ for every $v \in V$, we can find in
polynomial time a contagious set of size $\tilde{O}\left(\Delta \cdot OPT^2
\right)$, where $OPT$ is the size of a smallest contagious set in $G$. We also
provide several hardness of approximation results. For example, assuming the
unique games conjecture, we prove that TSS on CoA instances with $\Delta \le 3$
cannot be approximated within any constant factor.
  We also present results concerning the fixed parameter tractability of CoA
TSS instances, and approximation algorithms for a related problem, that of TSS
with partial incentives.
