Given a connected directed acyclic graph $G(V, E)$, is there an algorithm for changing a spanning tree to a minimum spanning tree through a series of edge swaps?
We can use Prim's or Kruskal's algorithm to produce the minimum spanning tree from scratch, but I think we can find an asymptotically more efficient algorithm for finding a minimum spanning tree if we know for instance that we have some constant $k$ more edges than we have vertices, so $|E| = |V| + k$.