Given a DAG (directed acyclic graph) $D$, with sources $S$ and sinks $T$. Find a DAG $D'$, with sources $S$ and sinks $T$, with minimum number of edges such that:
For all pairs $u \in S, v \in T$ there is a path from $u$ to $v$ in $D$ if and only if there is a path from $u$ to $v$ in $D'$.
One application of this is representing a set family by a DAG. For such a representation each source is a variable in the universe and each sink is a set in the set family, and a element u is in a set S if and only if there is a path from the vertex representing u to the vertex representing the set S.
Is this problem well known? Is there a polynomial algorithm for this problem?