Matchings with Lower Quotas: Algorithms and Complexity

Arulselvan, Ashwin; Cseh, Ágnes; Groß, Martin; Manlove, David F.; Matuschke, Jannik

doi:10.1007/s00453-016-0252-6

Cited by 14 publications

(49 citation statements)

References 36 publications

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“…The only exception is the combination (stable)-greedy max that exhibits highly varying running times, reaching 5157 seconds for the instance of 2015. 4 • Comparing the combinations (minimax)- * and (stable+minimax)- * we observe that some stability is lost if only the minimax constraint is included. On the other hand, looking at the column "Utility", which gives the deviation from the best utility, we see instead that the loss in global utility is very small if also stability is included among the constraints.…”

Section: Combinations Of Criteriamentioning

confidence: 93%

“…Consequently, they conclude that the matching problem is NP-complete, already when all vertices in U have size 1 or 2 and each vertex in V has capacity 2. Arulselvan et al [4] have recently shown that the existence problem remains NP-complete even without group registrations but with lower bounds on the number of students per team and with closed teams allowed. More precisely, they have shown that the optimization problem asking to maximize the total weight of the assignment can be solved efficiently if the number of applicant students for every team is at most 2 while it becomes NP-hard as soon as there are teams with at least 3 applicant students and students with preference lists of at least 2 projects or, alternatively, as soon as there are projects with upper bound of at least 3.…”

Section: Problem Definitionmentioning

confidence: 99%

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Handling preferences in student-project allocation

2017

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Section: Combinations Of Criteriamentioning

confidence: 93%

Section: Problem Definitionmentioning

confidence: 99%

Handling preferences in student-project allocation

2017

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“…Our work also fits into the line of research that addresses computationally hard problems in the area of stable matchings by focusing on instances with bounded preference lists [6,27,29,32,41] or by applying the more flexible approach of parameterized complexity [1,3,5,37,38].…”

Section: Related Workmentioning

confidence: 99%

“…There is a natural Turing reduction from SMC to the variant considered by Knuth. Coarse analysis into polynomial-time solvable and NP-hard cases has been studied by several researchers [1].…”

Section: Introductionmentioning

confidence: 99%

Stable Matchings with Covering Constraints: A Complete Computational Trichotomy

Mnich

Schlotter

2020

Algorithmica

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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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“…Related Works Recently, the study of matching models with lower quotas has developed substantially [1,7,13,15,16,18,21,22]. The Hospitals/Residents problem with lower quotas (HR-LQ) was first studied by Hamada et al [15,16], who considered the minimization of the number of blocking pairs subject to upper and lower quotas.…”

Section: Introductionmentioning

confidence: 99%

Envy-Free Matchings with Lower Quotas

Yokoi

2018

Algorithmica

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While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-free matching, which is a relaxation of a stable matching preserving a kind of fairness property.In this paper, we investigate the existence of an envy-free matching in several settings, in which hospitals have lower quotas and not all doctor-hospital pairs are acceptable. We first provide an algorithm that decides whether a given HR-LQ instance has an envy-free matching or not. Then, we consider envy-freeness in the Classified Stable Matching model due to Huang (2010), i.e., each hospital has lower and upper quotas on subsets of doctors. We show that, for this model, deciding the existence of an envy-free matching is NP-hard in general, but solvable in polynomial time if quotas are paramodular.

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Matchings with Lower Quotas: Algorithms and Complexity

Abstract: We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = (A∪ P, E) with weights on the edges in E, and with lower and upper quotas on the vertices in P.

Cited by 14 publications

References 36 publications

Handling preferences in student-project allocation

Handling preferences in student-project allocation

Stable Matchings with Covering Constraints: A Complete Computational Trichotomy

Envy-Free Matchings with Lower Quotas

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