Efficient algorithm for merging two DAGs

Question

I have two weighted DAGs (directed acyclic graphs) and need to merge them into one, so I can get a topological ordering (it could be more than two in some cases). The problem is

Answer 1

One issue is that there is may be more than one solution.

Consider for instance the DAGs {(a,b),(a,c)} and {(b,a),(b,c)}. You can "merge" these in two different ways:

1. {(a,b),(a,c),(b,c)}
2. {(a,c),(b,a),(b,c)}

The number of solutions grows combinatorially with the number of cycles in the union of the two graphs, so for your big graphs there is probably a huge number of ways you can "merge" them.

Do you have a criterion which could help you decide which DAG is "better" than another?