Why it matters

Stable matching has real applications: medical residency (Match), school choice. Understanding shapes matching problems.

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The architecture

Proposers (say men) propose to preferred.

Receivers (say women) tentatively accept best; reject others.

Gale-ShapleyPropose to bestunproposedTentative acceptkeep best offerRepeatuntil all matchedProposers get their best possible match; receivers get worst possible
Stable matching.
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How it works end to end

Complexity: O(n²).

Stability: no unmatched pair prefers each other.

Applications: NRMP (medical residency), school choice, kidney exchange.

Variants: many-to-one, with ties, incomplete lists.