Is the weighted transitive reduction problem NP-hard?
Consider $G=(V,E)$. Let $w_{ij} = 1$ if $e_{ij} \in E$ and let $w_{ij} \gg 1$ if $e_{ij} \notin E$. This choice of weights correspond exactly to the constraint $E' \subseteq E$. Thus, if the weighted transitive reduction is polynomial time solvable, then so is the minimum equivalent graph problem.