If you have some bipartite graph with an adjacency matrix that represents the weights of the edges of that particular bipartite graph, then the Hungarian algorithm outputs a maximum weight transversal. However if you would apply this to a marriage problem, where you have a bipartite graph with an equal amount of men and women that need to be paired with each other for marriage where the weights would represent the combined 'happiness/utility' between the pair, then when applying the Hungarian algorithm it will give you a maximized sum of these weights, but this matching will not guarantee to be a stable matching. Now I have found that there exists an algorithm called the Gale-Shapely algorithm that always finds a stable matching when you have an equal number of vertices in both classes of a bipartite graph. But if I understood it correctly, a stable matching does not have to mean that the sum of the edge weights in the matching is maximized. Now my question becomes: Does there exist an algorithm that outputs a maximum weight transveral and has a stable matching? Or is this still an open problem in graph theory to find such an algorithm?
Correct me if my explanation above about the Hungarian algorithm and Gale-Shapely algorithm is wrong, I am not a mathematics students.