Two-sided matching provides a model of search processes such as those between firms and workers in labor markets or between buyers and sellers in auctions. This book gives a comprehensive account of recent results concerning the game-theoretic analysis of two-sided matching. The focus of the book is on the stability of outcomes, on the incentives that different rules of organization give to agents, and on the constraints that these incentives impose on the ways such markets can be organized. The results for this wide range of related models and matching situations help clarify which conclusions depend on particular modeling assumptions and market conditions, and which are robust over a wide range of conditions.

Two-Sided Matching

Preliminary Definitions, Results and Motivation Stable Matching Problems: The Stable Marriage Problem: An Update SM and HR with Indifference The Stable Roommates Problem Further Stable Matching Problems Other Optimal Matching Problems: Pareto Optimal Matchings Popular Matchings Profile-Based Optimal Matchings.

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Algorithmics of Matching Under Preferences

Hard variants of stable marriage

NP-complete stable matching problems

NP-completeness in hedonic games

This thesis concerns the complexity analysis of certain matching problems. The description of each of these matching problems consists of a group of agents, and, for each agent, a preference list over the agents with whom the agent can be matched. A solution to the problem is a matching that has the property that no pair of agents that are eligible for each other, prefer each other to their assigned mates. A matching that satisfies this property is called a stable matching.
The variants of this problem that will be dealt with are: (1) The marriage problem: matching of men and women into couples; (2) The roommate problem: matching of students into rooms of two occupants; (3) The intern assignment problem: the assignment of interns to hospital positions; (4) The intern assignment problem with couples: same problem as (3) except that some of the interns are married couples.
These matching problems can be defined to allow or disallow ties in the preference lists. When ties are allowed, stability is defined as the absence of an unmatched pair that strictly prefer each other to their assigned mates.
This thesis will show that the option of allowing ties in the preference lists can significantly affect the computational complexity of the problems. It will be shown that, when ties are allowed, the roommate problem and the intern assignment problem with couples are NP-complete. In contrast, the roommate problem without ties, is solvable in polynomial time. The computational complexity of the intern-hospital assignment problem with couples is unresolved. A variant of the marriage problem that allows ties will be shown to be logspace P-complete. Finally, the intern-hospital assignment problem with couples, but without ties, will be shown to be logspace P-hard.

Eytan Ronn

Papers

On the complexity of stable matchings with and without ties