Given a weighted graph $G=(V,E)$. For each edge $e\in E$, we are interested in finding a maximum weight matching over all matchings that contains edge $e$.
If $G$ is bipartite, then this can be done in $O(n^3)$ time:
Apply the standard reduction to min-cost flow.
Find a min-cost flow.
Compute all-pairs shortest path in the residual graph, and use the augmenting paths to extract the desired matchings.
Is there an $O(n^3)$ time algorithm in general graphs?