Time complexity of art gallery-like problem
Your problem is a special case of finding a dominating set of size at most $w$ for directed graphs $G$, where we require that there are no isolated vertices in $G=(V,E)$ and that $w\geq |V|/2$. 

Unfortunately, this problem is still NP-hard, since there is a polynomial time reduction from the general directed dominating set problem: 

Given an arbitrary directed graph $G=(V,E)$ and an integer $w$, construct the instance $(G',w)$ as follows:


Set $G_0 \leftarrow G$ and $w_0\leftarrow w$.
Remove all $i$ isolated vertices from $G_0$ and 'place guards on them': $w_0\leftarrow w_0 - i$. If $w_0\leq 0$, simply construct some instance that yields NO and we're done.
If $w_0\geq |V|/2$, we are done. Otherwise, 'add two guards' for every cycle we will introduce: $w'\leftarrow w_0+2\cdot(|V_0|-2w_0)$ and add $|V_0|-2w_0$ directed 3-cycles to $G_0$. Set $G'\leftarrow G_0$.


Observe that now $w'\geq |V|/2$, since if we have added the three-cycles, we get $2w'= 2w_0+4(|V_0|-2w_0) \geq |V_0| + 3(|V_0|-2w_0) = |V'|$. The reduced instance is a YES-instance if and only if the original instance is, since 'disconnected' components do not influence the rest of the graph and at least 2 guards are required to watch a 3-cycle.

So a polynomial time algorithm most likely doesn't exist, even for this special case.