hand1.pdf

Perfect matching

Def: Match

A matching M is a set of ordered pairs h-s with h ∈ H and s ∈ S s.t. • Each hospital h ∈ H appears in at most one pair of M. • Each student s ∈ S appears in at most one pair of M.

H 和 S都最多出现一次

Def: Perfect Match

A matching M is perfect if |M |=|H |=|S |=n.

一一对应，是Match的一个子集，精化概念：refinement

Unstable pair

Given a perfect matching M, hospital h and student s form an unstable pair if both:

• h prefers s to matched student.
• s prefers h to matched hospital.

原来官方配的是 s-h' s'-h

但是h 更喜欢s s更喜欢h

Yolanda 和 Atlanta就是unstable pair

B-Y , B-X 是unstable pair

Def: Stable matching problem

A stable matching is a perfect matching with no unstable pairs.