Three remarks on the many-to-many stable matching problem

The National Resident Matching program seeks a stable matching of medical students to teaching hospitals. With couples, stable matchings need not exist. Nevertheless, for any student preferences, we show that each instance of a matching problem has a “nearby” instance with a stable matching. The nearby instance is obtained by perturbing the capacities of the hospitals. In this perturbation, aggregate capacity is never reduced and can increase by at most four. The capacity of each hospital never changes by more than two. (JEL C78, D47, I11, J41, J44)

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On the Approximability of the Stable Matching Problem with Ties of Size Two

Algorithmica ◽

10.1007/s00453-020-00703-9 ◽

2020 ◽

Vol 82 (9) ◽

pp. 2668-2686

Author(s):

Robert Chiang ◽

Kanstantsin Pashkovich

Keyword(s):

Stable Matching ◽

Matching Problem

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Two-Sided Matching with Endogenous Preferences

American Economic Journal Microeconomics ◽

10.1257/mic.20130272 ◽

2015 ◽

Vol 7 (3) ◽

pp. 241-258 ◽

Cited By ~ 5

Author(s):

Yair Antler

Keyword(s):

Stable Matching ◽

The Other ◽

Matching Theory ◽

Endogenous Preferences ◽

Matching Problem ◽

Matching Mechanism

We modify the stable matching problem by allowing agents' preferences to depend on the endogenous actions of agents on the other side of the market. Conventional matching theory results break down in the modified setup. In particular, every game that is induced by a stable matching mechanism (e.g., the Gale-Shapley mechanism) may have equilibria that result in matchings that are not stable with respect to the agents' endogenous preferences. However, when the Gale-Shapley mechanism is slightly modified, every equilibrium of its induced game results in a pairwise stable matching with respect to the endogenous preferences as long as they satisfy a natural reciprocity property. (JEL C78, D82)

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Three-sided stable matching problem with two of them as cooperative partners

Journal of Combinatorial Optimization ◽

10.1007/s10878-017-0224-z ◽

2017 ◽

Vol 37 (1) ◽

pp. 286-292 ◽

Cited By ~ 6

Author(s):

Liwei Zhong ◽

Yanqin Bai

Keyword(s):

Stable Matching ◽

Matching Problem

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The Maximum-Weight Stable Matching Problem: Duality and Efficiency

SIAM Journal on Discrete Mathematics ◽

10.1137/120864866 ◽

2012 ◽

Vol 26 (3) ◽

pp. 1346-1360 ◽

Cited By ~ 4

Author(s):

Xujin Chen ◽

Guoli Ding ◽

Xiaodong Hu ◽

Wenan Zang

Keyword(s):

Stable Matching ◽

Maximum Weight ◽

Matching Problem

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Stable Matchings with Diversity Constraints: Affirmative Action is beyond NP

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2020/21 ◽

2020 ◽

Author(s):

Jiehua Chen ◽

Robert Ganian ◽

Thekla Hamm

Keyword(s):

Affirmative Action ◽

Stable Matching ◽

Comprehensive Analysis ◽

Complexity Class ◽

Np Hard ◽

Matching Problem ◽

Stable Matchings ◽

Student College ◽

Total Capacity ◽

New Algorithms

We investigate the following many-to-one stable matching problem with diversity constraints (SMTI-DIVERSE): Given a set of students and a set of colleges which have preferences over each other, where the students have overlapping types, and the colleges each have a total capacity as well as quotas for individual types (the diversity constraints), is there a matching satisfying all diversity constraints such that no unmatched student-college pair has an incentive to deviate? SMTI-DIVERSE is known to be NP-hard. However, as opposed to the NP-membership claims in the literature [Aziz et al., AAMAS 2019; Huang,SODA 2010], we prove that it is beyond NP: it is complete for the complexity class Σ^P_2. In addition, we provide a comprehensive analysis of the problem’s complexity from the viewpoint of natural restrictions to inputs and obtain new algorithms for the problem.

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