Network Flow

network flow

OR 215 Spring 1998

Network Flows M. Hartmann

STABLE MATCHING PROBLEMS

Stable Marriage Problem

Propose and Reject Algorithm

Bipartite Stable Matching

Application: NRMP

Linear Programming Formulation

Non-Bipartite Stable Matching

STABLE MARRIAGE PROBLEM

A certain community consists of n men and n women. Each person has a strict preference over members of the opposite sex, for example:

Bengt: Anita, Christine, Elena Anita: Peter, Reid, Bengt

Peter: Christine, Elena, Anita Christine: Reid, Bengt, Peter

Reid: Elena, Anita, Christine Elena: Bengt, Peter, Reid

Note: The preferences must be strict, i.e. Elena cannot be indifferent between Bengt and Peter.

For a given matching, a man-woman pair is unstable if they are not married to each other, but prefer each other to their current mates. A perfect matching is said to be stable if it has no unstable pairs.

• Is the matching Bengt-Christine, Peter-Anita, Reid-Elena stable?

(consider Peter-Elena)

• Is the matching Bengt-Christine, Peter-Elena, Reid-Anita stable?

PROPOSE AND REJECT ALGORITHM

We will show constructively that every set of preferences admits a stable perfect matching.

Consider the following algorithm:

• Initially, all men and women are single.

• In each iteration, a man who is single proposes to the woman he prefers most among the women to whom he has not yet proposed.

• The woman that receives the proposal accepts it if she is single or if she prefers the proposing man to her current mate.

• In the latter case, the man disavowed becomes single again, and the algorithm continues to its next iteration.

• This courtship process ends when each single man has proposed to all of the women.

EXAMPLE

Suppose we replace Bengt by Cyrus, with preferences:

Cyrus: Christine, Elena,

References: H.G. Abeledo and U.G. Rothblum, “Courtship and linear programming,” Linear Algebra and Its Applications 216 (1995) 111-124. H.G. Abeledo and U.G. Rothblum, “Stable matchings and linear inequalities,” Discrete Applied Mathematics 54 (1994) 1-27. A.T. Benjamin, C. Converse and H.A. Krieger, “How do I marry thee? Let me count the ways,” Discrete Applied Mathematics 59 (1995) 285-292. D. Gusfield and R.W. Irving, The Stable Marriage Problem: Structure and Algorithms (MIT Press, Cambridge, 1989). A.E. Roth, “New physicians: a natural experiment in market organization,” Science 250 (1990) 1524-1528. A.E. Roth and E. Peranson, “The effects of the change in the NRMP matching algorithm,” Journal of the American Medical Association 278 (1997) 729-732. -----------------------