I don't think the "matchmaking algorithm" in the blog post actually matches the colloquial meaning of the word "Algorithm".
Actually a relatively interesting algorithm, but I don't think the author explains it fully. The men must also rank the females. The key point is that for a match between people A and B, A has no person ranked higher than B who also ranks A higher than their current partner. it is optimal for every participant.
I'm not too familiar with residency- I wonder if hospitals submit a list of potential students they accept? How does the length of a hospital's acceptable candidates list compare to the number of hospitals a student will apply to? In the Stable Marriage Problem, each set member must rank every member of the other set.
The article does mention that, "Med students and hospitals are currently ranking their top choices from these interviews." I would assume that the hospital only ranks students they interview and would consider acceptable for the program.
> Each set member must rank every member of the other set.
I would assume that they just fill in the blanks with some value that prevents matching. I don't think a hospital or a resident would be very happy with what would amount to random assignment of the leftovers.