Abstract
We consider the problem of matchings under two-sided preferences in the presence of maximum as well as minimum quota requirements for the agents. When there are no minimum quotas, stability is the de-facto notion of optimality. In the presence of minimum quotas, ensuring stability and simultaneously satisfying lower quotas is not an attainable goal in many instances.
To address this, a relaxation of stability known as envy-freeness, is proposed in literature. In our work, we thoroughly investigate envy-freeness from a computational view point. Our results show that computing envy-free matchings that match maximum number of agents is computationally hard and also hard to approximate up to a constant factor. Additionally, it is known that envy-free matchings satisfying lower-quotas may not exist. To circumvent these drawbacks, we propose a new notion called relaxed stability. We show that relaxed stable matchings are guaranteed to exist even in the presence of lower-quotas. Despite the computational intractability of finding a largest matching that is feasible and relaxed stable, we give an efficient algorithm that computes a constant factor approximation to this matching in terms of size.
This work was partially supported by the grant CRG/2019/004757.
P. Krishnaa—Part of this work was done when the author was a student at IIT Madras.
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Notes
- 1.
Our initial idea was to allow them to participate in envy-pairs. We thank anonymous reviewer for suggesting this modification which is stricter than our earlier notion.
References
Biró, P., Fleiner, T., Irving, R.W., Manlove, D.: The college admissions problem with lower and common quotas. Theoret. Comput. Sci. 411(34–36), 3136–3153 (2010). https://doi.org/10.1016/j.tcs.2010.05.005
Biró, P., Manlove, D., Mittal, S.: Size versus stability in the marriage problem. Theoret. Comput. Sci. 411(16–18), 1828–1841 (2010). https://doi.org/10.1016/j.tcs.2010.02.003
Ehlers, L., Hafalir, I.E., Yenmez, M.B., Yildirim, M.A.: School choice with controlled choice constraints: hard bounds versus soft bounds. J. Econ. Theory 153, 648–683 (2014). https://doi.org/10.1016/j.jet.2014.03.004
Fleiner, T., Kamiyama, N.: A matroid approach to stable matchings with lower quotas. Math. Oper. Res. 41(2), 734–744 (2016). https://doi.org/10.1287/moor.2015.0751
Fragiadakis, D., Iwasaki, A., Troyan, P., Ueda, S., Yokoo, M.: Strategyproof matching with minimum quotas. ACM Trans. Econ. Comput. 4(1), 6:1–6:40 (2015). http://doi.acm.org/10.1145/2841226
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962). http://www.jstor.org/stable/2312726
Goto, M., Iwasaki, A., Kawasaki, Y., Kurata, R., Yasuda, Y., Yokoo, M.: Strategyproof matching with regional minimum and maximum quotas. Artif. Intell. 235, 40–57 (2016). https://doi.org/10.1016/j.artint.2016.02.002
Halldórsson, M.M., Iwama, K., Miyazaki, S., Yanagisawa, H.: Improved approximation results for the stable marriage problem. ACM Trans. Algorithms 3(3), 30 (2007). https://doi.org/10.1145/1273340.1273346
Hamada, K., Iwama, K., Miyazaki, S.: The hospitals/residents problem with lower quotas. Algorithmica 74(1), 440–465 (2016). https://doi.org/10.1007/s00453-014-9951-z
Huang, C.: Classified stable matching. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 1235–1253 (2010). https://doi.org/10.1137/1.9781611973075.99
Kamada, Y., Kojima, F.: Efficient matching under distributional constraints: theory and applications. Am. Econo. Rev. 105(1), 67–99 (2015). https://www.aeaweb.org/articles?id=10.1257/aer.20101552
Kamada, Y., Kojima, F.: Stability concepts in matching under distributional constraints. J. Econ. Theory 168, 107–142 (2017). https://doi.org/10.1016/j.jet.2016.12.006
Király, Z.: Linear time local approximation algorithm for maximum stable marriage. Algorithms 6(3), 471–484 (2013). https://doi.org/10.3390/a6030471
Kleinberg, J., Tardos, E.: Algorithm Design. Addison-Wesley Longman Publishing Co. Inc., Boston (2005)
Krishnaa, P., Limaye, G., Nasre, M., Nimbhorkar, P.: Envy-freeness and relaxed stability under lower quotas. CoRR abs/1910.07159 (2019). http://arxiv.org/abs/1910.07159
Krishnapriya, A.M., Nasre, M., Nimbhorkar, P., Rawat, A.: How good are popular matchings? In: 17th International Symposium on Experimental Algorithms, SEA 2018, pp. 9:1–9:14 (2018). https://doi.org/10.4230/LIPIcs.SEA.2018.9
Nasre, M., Nimbhorkar, P.: Popular matchings with lower quotas. In: 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, pp. 44:1–44:15 (2017). https://doi.org/10.4230/LIPIcs.FSTTCS.2017.44
Roth, A.E.: On the allocation of residents to rural hospitals: a general property of two-sided matching markets. Econometrica 54(2), 425–427 (1986). http://www.jstor.org/stable/1913160
Wu, Q., Roth, A.E.: The lattice of envy-free matchings. Games Econ. Behav. 109, 201–211 (2018). https://doi.org/10.1016/j.geb.2017.12.016
Yokoi, Y.: Envy-free matchings with lower quotas. Algorithmica 82(2), 188–211 (2020). https://doi.org/10.1007/s00453-018-0493-7
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We thank the anonymous reviewers for their useful comments which has improved the presentation of the paper.
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Krishnaa, P., Limaye, G., Nasre, M., Nimbhorkar, P. (2020). Envy-Freeness and Relaxed Stability: Hardness and Approximation Algorithms. In: Harks, T., Klimm, M. (eds) Algorithmic Game Theory. SAGT 2020. Lecture Notes in Computer Science(), vol 12283. Springer, Cham. https://doi.org/10.1007/978-3-030-57980-7_13
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