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Envy-freeness and relaxed stability: hardness and approximation algorithms


We consider the problem of computing matchings under two-sided preferences in the presence of lower as well as upper-quota requirements for the resources. In the presence of lower-quotas a feasible matching may not exist and hence we focus on critical matchings. Informally, a critical matching achieves the smallest deficiency. We first consider two well-studied notions of optimality with respect to preferences, namely stability and envy-freeness. There exist instances that do not admit a critical stable matching or a critical envy-free matching. When critical matching satisfying the optimality criteria does not exist, we focus on computing a minimum-deficiency matching among all stable or envy-free matchings. To ensure guaranteed existence of an optimal critical matching, we introduce and study a new notion of optimality, namely relaxed stability. We show that every instance admits a critical relaxed stable matching and it can be efficiently computed. We then investigate the computational complexity of computing maximum size optimal matchings with smallest deficiency. Our results show that computing a maximum size minimum-deficiency envy-free matching and a maximum size critical relaxed stable matching are both hard to approximate within \(\frac{21}{19}-\epsilon \), for any \(\epsilon > 0\) unless P = NP. For envy-free matchings, we present an approximation algorithm for general instances and a polynomial time exact algorithm for a special case. For relaxed stable matchings, we present a constant factor approximation algorithm for general instances.

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  1. We refer the reader to Fig. 11 in Appendix A for an example in which a maximum size popular matching is not relaxed stable.

  2. The conference version of our work (Krishnaa et al. 2020) uses a weaker notion of relaxed stability and the \(\frac{3}{2}\)-approximation algorithm to the MAXRSM problem is completely different from the one given here.

  3. We remark that a minor typo in line 21 Algorithm 1 Krishnapriya et al. (2018) is corrected in Algorithm 2. The typo is present only in the pseudo-code in Krishnapriya et al. (2018) and not in the description of algorithm.


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We thank anonymous reviewers for their helpful comments that improved the presentation of the paper. Special thanks for suggesting the stronger notion of relaxed stability.


This work was partially supported by the Grant CRG/2019/004757.

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All authors contributed to the results related to envy-freeness. GL, MN and PN contributed to the results related to relaxed stability. The first draft of the manuscript was written by PK and GL and all authors edited the manuscript. All authors have read and approved the final manuscript.

Corresponding author

Correspondence to Girija Limaye.

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The authors declare that they have no conflict of interest.

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A preliminary version of this work appeared in 13th Symposium on Algorithmic Game Theory, SAGT’ 20 (Krishnaa et al. 2020). This work was partially supported by the Grant CRG/2019/004757.

Part of this work was done when the first author was a student at IIT Madras.

A Example from Nasre and Nimbhorkar (2017) illustrating the difference between popularity and relaxed stability

A Example from Nasre and Nimbhorkar (2017) illustrating the difference between popularity and relaxed stability

See Fig. 11.

Fig. 11
figure 11

An HRLQ instance from Nasre and Nimbhorkar (2017) (Fig. 2, Appendix). The maximum cardinality popular matching output by the algorithm in Nasre and Nimbhorkar (2017) is \(M = \{(r_1 , h_5 ), (r_2 , h_1 ), (r_3 , h_2 )\}\). The matching M is not relaxed stable since \((r_2, h_2)\) blocks M and the hospital to which \(r_2\) is assigned, that is \(h_1\) has surplus in M. The algorithm presented in this paper outputs the matching \(M' = \{(r_1, h_5), (r_2, h_2), (r_3, h_1)\}\) which is relaxed stable

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Krishnaa, P., Limaye, G., Nasre, M. et al. Envy-freeness and relaxed stability: hardness and approximation algorithms. J Comb Optim 45, 41 (2023).

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  • Matchings under preferences
  • Lower quota
  • Envy-freeness
  • Relaxed stability
  • Approximation
  • Critical matchings