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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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.
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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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