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Interesting! I knew that med school applicants were accepted on some sort of ranking and matching process but didn't realize there was a real algorithm behind it. This, or one similar, seems to be what's behind it.

There's no matching process at the beginning of medical school, but there is for assigning med school graduates to residencies. This variant of the problem is NP-hard, so there's no exact solution, but matching still works pretty well. https://web.stanford.edu/~alroth/papers/rothperansonaer.PDF has all the details.

When you are referring to this "variant", are you referring to maximal matching, or bipartite matching when the number of people and schools don't equal?

Not the parent, but I think they refer to the fact that with admissions, matches are permanent (once a student accepts a school's offer, said offer can't be rescinded), while Gale-Shapely requires provisional pairings until all the rounds have been gone through and a final result is available.

The issue is that couples want to be placed together. That makes everything trickier than if people just placed independently.

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