Envy-Freeness and Relaxed Stability: Hardness and Approximation Algorithms

Krishnaa, Prem; Limaye, Girija; Nasre, Meghana; Nimbhorkar, Prajakta

doi:10.1007/978-3-030-57980-7_13

Cited by 11 publications

(22 citation statements)

References 17 publications

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“…In other papers, they impose the Complete List restriction (CL-restriction for short). There always exists envy-free matchings in CL-restriction instances, and a maximum-size one can be found in polynomial time [1,4].…”

Section: Preliminariesmentioning

confidence: 99%

“…[3] gives stronger results showing 0-1 Min-BP HRLQ is hard to approximate within the ratio of (|H|+|R|) 1−ǫ for any positive constant ǫ even if all preference lists are complete. While, the reduction can not be used for 0-1 Min-EP HRLQ because there exists envy-free matchings when all preference lists are complete [1] (or a weaker requirement such that all hospitals with positive lower quotas has complete preference lists over all residents [4]) and a maximum-size envy-free matching can be found in linear time.…”

Section: Theorem 5 0-1 Min-ep Hrlq Is Np-hardmentioning

confidence: 99%

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Minimal Envy Matchings in the Hospitals/Residents Problem with Lower Quotas

Hu¹,

Garg²

2021

Preprint

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In the Hospitals/Residents problem, every hospital has an upper quota that limits the number of residents assigned to it. While, in some applications, each hospital also has a lower quota for the number of residents it receives. In this setting, a stable matching may not exist. Envy-freeness is introduced as a relaxation of stability that allows blocking pairs involving a resident and an empty position of a hospital. While, envy-free matching might not exist either when lower quotas are introduced. We consider the problem of finding a feasible matching that satisfies lower quotas and upper quotas and minimizes envy in terms of envy-pairs and envyresidents in the Hospitals/Resident problem with Lower Quota. We show that the problem is NP-hard with both envy measurement. We also give a simple exponential-time algorithm for the Minimum-Envy-Pair HRLQ problem.

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Section: Preliminariesmentioning

confidence: 99%

Section: Theorem 5 0-1 Min-ep Hrlq Is Np-hardmentioning

confidence: 99%

Minimal Envy Matchings in the Hospitals/Residents Problem with Lower Quotas

Hu¹,

Garg²

2021

Preprint

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“…In such scenarios, a resource may set a lower-quota in addition to the upper-quota.. When at least one hospital (resource) has a non-zero lower-quota, the setting is referred to as the Hospital-Residents with Lower Quota (HRLQ) setting [23,15].…”

Section: Introductionmentioning

confidence: 99%

“…Fig. 1 (taken from [15]) gives an example. Thus, different notions of optimality for the HRLQ instances [20,16,23,7,15,15,11] are proposed and studied.…”

Section: Introductionmentioning

confidence: 99%

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Envy-freeness and Relaxed Stability for Lower-Quotas : A Parameterized Perspective

Limaye¹

2021

Preprint

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We consider the problem of assigning agents to resources in the two-sided preference model with upper and lower-quota requirements on resources. This setting (known as the HRLQ setting) models real-world applications like assigning students to colleges or courses, resident doctors to hospitals and so on. In presence of lower-quotas, an instance may not admit a stable matching that fulfils the lower-quotas. Prem Krishnaa et al. [15] study two alternative notions of optimality for the HRLQ instances -envy-freeness and relaxed stability. They investigate the complexity of computing a maximum size envy-free matching (MAXEFM) and a maximum size relaxed stable matching (MAXRSM) that fulfils the lower-quotas. They show that both these optimization problems are NP-hard and not approximable within a constant factor unless P = NP.In this work, we investigate the parameterized complexity of MAXEFM and MAXRSM. We consider natural parameters derived from the instance -the number of lower-quota hospitals, deficiency of the instance, size of a maximum matching, size of a stable matching, length of the preference list of a lower-quota hospital, to name a few. We show that MAXEFM problem is W [1]-hard for several interesting parameters but admits a polynomial size kernel for a combination of parameters. We show that MAXRSM problem does not admit an FPT algorithm unless P = NP for two natural parameters but admits a polynomial size kernel for a combination of parameters in a special case. We also show that both these problems admit FPT algorithms on a set of parameters. ACM Subject ClassificationMathematics of computing → Graph theory; Theory of computation → Design and analysis of algorithms

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Refined Computational Complexities of Hospitals/Residents Problem with Regional Caps

Hamada¹,

Miyazaki

2022

Lecture Notes in Computer Science

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Envy-Freeness and Relaxed Stability: Hardness and Approximation Algorithms

Cited by 11 publications

References 17 publications

Minimal Envy Matchings in the Hospitals/Residents Problem with Lower Quotas

Minimal Envy Matchings in the Hospitals/Residents Problem with Lower Quotas

Envy-freeness and Relaxed Stability for Lower-Quotas : A Parameterized Perspective

Refined Computational Complexities of Hospitals/Residents Problem with Regional Caps

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