Generalizing linear interpolation to posets
In a recent paper, we propose such a scheme. The scheme is illustated in a specific crowdsourcing application setting, but the idea is fairly simple. We just see the order constraint $\mu(u) \leq \mu(v)$ of each DAG edge $(u, v)$ as a linear inequality constraint, we consider the admissible convex polytope of assignments to these linear inequalities, and we choose the center of mass of this polytope as our interpolation result.

The paper explains the scheme in more detail, includes an algorithm, computational hardness results, an approximation scheme that derives from existing work, a tractable case when the DAG is a tree, and a discussion of alternate schemes.