NP-hardness of finding a subset of vertices in a vertex-weighted graph
The problem is polynomial-time solvable, suggested in Lemma 1 of the paper Computational Complexity of Some Maximum Average Weight Problems with Precedence
Constraints.

Basically the idea is to write a linear program on variables $x_i\in[0,1]$, where $x_u-x_v\leq 0$ if $(u,v)\in E$. The signed adjacency matrix is totally unimodular, so we can compute the integral optimum.