Approximating transitive reduction of a transitive closure of a dag
Due to [Aho, Garey, and Ullman, The transitive reduction of a directed graph, 1972]; $G^t = G - GG^+$, where $G^t$ is the transitive reduction of $G$. In this particular case, $G = G^+$. So, we can conclude $G^t = G - G^2$. Although this result does not lead to a linear non-approximate algorithm, I hope such an observation would be helpful in this context.