1. Stable Matchings
-- Consider two node sets U and V ("men" and "women")
-- For simplicity: Assume |U| = |V| = n.
-- Each node has a ranked order of the nodes on the other side. (different for different nodes)
Examples: Hospitals & residents, colleges & applicants.
-- Stable matching: A perfect matching (i.e., matches each node of U to a distinct node of V) such that: if u in U and v in V are not
matched, then either u likes its mate v' better than v, or v likes its mate u' better than u.
2. Gale-Shapley Proposal Algorithm
-- While there is an unattached man u
-- u proposes to the top woman v on his preference list who hasn't rejected him yet
-- Each woman entertains only the best proposal received so far
[Invariant: current engagements = a matching]
-- Terminates with a stable matching after <= n^2 iterations.
-- Each man makes <= n proposals ==> <= n^2 iterations.
-- Terminates with a perfect matching.
-- If not, some man rejected by all women. ==> All n women engaged at conclusion of algorithm ==> All n men engaged at end, as well [contradiction]
-- Terminates with a stable matching. Consider some u, v not matched to each other.
-- Case 1: u never proposed to v. ==> u matched to someone he prefers to v.
-- Case 2: u proposed to v. ==> v got a better offer, ends up matched to someone she prefers to u.