Stable Matchings with Covering Constraints: A Complete Computational Trichotomy

Stable matching problems with lower quotas are fundamental in academic hiring and ensuring operability of rural hospitals. Only few tractable (polynomial-time solvable) cases of stable matching with lower quotas have been identified; most such problems are NP -hard and also hard to approximate (Hama...

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Veröffentlicht in:Algorithmica Jg. 82; H. 5; S. 1136 - 1188
Hauptverfasser: Mnich, Matthias, Schlotter, Ildikó
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.05.2020
Springer Nature B.V
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Cloning
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Hardness
Hospitals
Matching
Mathematics of Computing
Parameters
Polynomials
Quotas
Theory of Computation
Women
Fixed-parameter algorithms
68R10
Lower quotas
Stable marriage
91B68
ISSN:0178-4617, 1432-0541
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Zusammenfassung:Stable matching problems with lower quotas are fundamental in academic hiring and ensuring operability of rural hospitals. Only few tractable (polynomial-time solvable) cases of stable matching with lower quotas have been identified; most such problems are NP -hard and also hard to approximate (Hamada et al. in Algorithmica 74(1):440–465, 2016). We therefore consider stable matching problems with lower quotas under a relaxed notion of tractability, namely fixed-parameter tractability. By cloning hospitals we focus on the case when all hospitals have upper quota equal to 1, which generalizes the setting of “arranged marriages” first considered by Knuth (Mariages stables et leurs relations avec d’autres problèmes combinatoires, Les Presses de l’Université de Montréal, Montreal, 1976). We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy: for each choice of parameters we either provide a polynomial-time algorithm, or an NP -hardness proof and fixed-parameter algorithm, or NP -hardness proof and W [ 1 ] -hardness proof. As corollary, we negatively answer a question by Hamada et al. (Algorithmica 74(1):440–465, 2016) by showing fixed-parameter intractability parameterized by optimal solution size. We also classify all cases of one-sided constraints where only women may be distinguished.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00636-y