Show that Knapsack-like problem on directed graph is NP-hard
$x_i \leq x_j \forall (i,j) \in A$ is a linear constraint that represents your requirement (if $x_i=1$ it would push $x_j$ to take value $1$ as well). You get a constrained knapsack problem. You can easily reduce a knapsack to it by considering a graph that simply has no arrows.