A Matroid Approach to Stable Matchings with Lower Quotas

Abstract

In SODA'10, Huang introduced the laminar classified stable matching problem (LCSM for short) that is motivated by academic hiring. This problem is an extension of the wellknown hospitals/residents problem in which a hospital has laminar classes of residents and it sets lower and upper bounds on the number of residents that it would hire in that class. Against the intuition that stable matching problems with lower quotas are difficult in general, Huang proved that this problem can be solved in polynomial time. In this paper, we propose a matroid-based approach to this problem and we obtain the following results. (i) We solve a generalization of the LCSM problem. (ii) We exhibit a polyhedral description for stable assignments of the LCSM problem, which gives a positive answer to Huang's question. (iii) We prove that the set of stable assignments of the LCSM problem has a lattice structure similarly to the ordinary stable matching model.• We solve a generalization of the LCSM problem.• By exhibiting a polyhedral description for stable assignments of the LCSM problem, we give a positive answer to Huang's question in [9].• We prove that similarly to the ordinary stable matchings, the set of stable assignments of the LCSM problem has a natural lattice structure.for any class C of C Ci . Hence, I i , J i ∈ I Ci follows from I, J ∈ I C . Moreover, by the condition in the lemma,for any class C of C Ci on which I i ∩ C is deficient. So, by the induction hypothesis,

Controlled School Choice with Soft Bounds and Overlapping Types

“…Biró et al (2010), Sönmez and Switzer (2013), Monte and Tumennasan (2013), , and Fragiadakis et al (2016) discuss how to handle minimum quotas. In the literature of computer science, the complexity of checking the existence of a stable matching has also been discussed when various distributional constraints are imposed (Huang, 2010;Fleiner & Kamiyama, 2016;Hamada, Iwama, & Miyazaki, 2016;Kamiyama, 2013). One fundamental difference between these works and ours is that they consider distributional constraints as hard constraints, which must be completely satisfied.…”

Matching for the Israeli "Mechinot" Gap-Year Programs

“…In contrast, we are concerned not with inter-institution constraints on the number of applicants allowed in a set of institutions, but rather with intra-institution constraints for the number of applicants of a specific population within a specific institution. Such constraints in the form of minimum and maximum quotas were studied by Huang (2010) and subsequently Fleiner and Kamiyama (2012). Both of these papers consider a model with hard maximum and minimum quotas for every population in every institution.…”

“…• Sure thing: many candidates knew that they have a "guaranteed slot" in a certain PMA, i.e., they were assured that the PMA did not rank above them more candidates than it has slots Fleiner and Kamiyama (2012) for a proof, under strict minimum quotas in a generalization of the model of Huang (2010), of this property in particular. 11 This argument, which as noted above led us to choose Algorithm 1 as the choice function of the PMAs is indeed somewhat less convincing in a context (unlike ours) in which the set of minimum-target populations is in fact a union of pairwise-disjoint chains.…”

Envy-Free Matchings with Lower Quotas

“…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.…”

“…Huang showed that it is NP-complete in general to decide the existence of a stable matching, and proved that it is solvable in polynomial time if classes form a laminar family. For this tractable special case, Fleiner and Kamiyama [7] gave a concise explanation in terms of matroids, and their framework is generalized by Yokoi [34] to a framework with generalized matroids.…”