CS 321. Algorithm Analysis & Design Lecture 15

Interlude: Stable MatchingsA Valentine Day Special

Disclaimer - People shown on these slides are a part of a fictitious story. Not to be taken seriously!

Input: Two sets of agents (men and women), and some notion of preferences.

The Matching Problem

Output: A pairing of the agents that satisfies some reasonable property.

There is a perfect matching between A and B if, and only if,

there is no congestion*.

Hall’s Marriage Theorem

Let G = (V,E) be a bipartite graph with bipartition V = (A,B); and suppose |A| = |B|.

(i.e, there is no subset S of A for which N(S) is smaller than S.)

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Stability

A marriage between A and B is stable if there is no pair of couples

(a ⟷ p) and (b ⟷ q)

such that a prefers q over p and q prefers a over b.

Goal: Design an Stable Matching Algorithm

The men keep proposing.

The women keep rejecting, as long as it makes sense.

To begin with, every man proposes to the

woman on the top of his list.

A woman always accepts her first proposal,

and accepts the best among multiple proposals.

The first round Proposals on the left

and engagements on the right.

As long as a man is free, he proposes to the next woman on

his list.

A woman accepts a proposal either if she is single, or if the proposal is better than

her current partner.

The second round Proposals on the left

and engagements on the right.

The third round Proposals on the left

and engagements on the right.

The algorithm terminates because we now have a matching.

The total number of proposals made is at most n2.

The algorithm always terminates with a matching.

The matching is stable.

If m is free at some point in the execution of the algorithm, then there is a woman to whom he has

not yet proposed.

The algorithm always terminates with a matching.

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This situation is not possible with the algorithm - RK would have proposed to KK first, and

she has no reason to reject the proposal!

Proof of STABILITY