  Consider the following coloring process in a simple directed graph $G(V,E)$
with positive indegrees. Initially, a set $S$ of vertices are white, whereas
all the others are black. Thereafter, a black vertex is colored white whenever
more than half of its in-neighbors are white. The coloring process ends when no
additional vertices can be colored white. If all vertices end up white, we call
$S$ an irreversible dynamic monopoly (or dynamo for short) under the
strict-majority scenario. An irreversible dynamo under the simple-majority
scenario is defined similarly except that a black vertex is colored white when
at least half of its in-neighbors are white. We derive upper bounds of
$(2/3)\,|\,V\,|$ and $|\,V\,|/2$ on the minimum sizes of irreversible dynamos
under the strict and the simple-majority scenarios, respectively. For the
special case when $G$ is an undirected connected graph, we prove the existence
of an irreversible dynamo with size at most $\lceil |\,V\,|/2 \rceil$ under the
strict-majority scenario. Let $\epsilon>0$ be any constant. We also show that,
unless $\text{NP}\subseteq \text{TIME}(n^{O(\ln \ln n)}),$ no polynomial-time,
$((1/2-\epsilon)\ln |\,V\,|)$-approximation algorithms exist for finding the
minimum irreversible dynamo under either the strict or the simple-majority
scenario. The inapproximability results hold even for bipartite graphs with
diameter at most 8.
