Minimum equivalent digraph with respect to sources and sinks 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?
Let's assume that $D$ contains only sources and sinks, since the input can be translated into an equivalent such input easily.

Then, note that, in any solution $D'$ for $D$, each vertex $v$ corresponds to a biclique in the underlying undirected graph $G$ of $D$ (the biclique between all sources that reach $v$ in $D'$ and all sinks that are reached from $v$ in $D'$).

I conjecture that, if $D'$ is optimal, then it contains a vertex-cut that 1:1 corresponds to an optimal biclique covering of $G$. Then, any minimum vertex-cut in $D'$ corresponds to an optimal biclique covering in $G$. However, since BICLIQUE COVER (aka BIPARTITE DIMENSION) is NP-complete, your problem is unlikely to admit a polynomial-time algorithm unless my conjecture fails.

Note that, even if my conjecture holds, technically this argument does not prove NP-hardness of your problem, since the reduction is not a Karp- but a Cook reduction.