The College Admissions problem with lower and common quotas

Abstract

We study two generalised stable matching problems motivated by the current matching scheme used in the higher education sector in Hungary. The first problem is an extension of the College Admissions problem in which the colleges have lower quotas as well as the normal upper quotas. Here, we show that a stable matching may not exist and we prove that the problem of determining whether one does is NP-complete in general. The second problem is a different extension in which, as usual, individual colleges have upper quotas, but in addition, certain bounded subsets of colleges have common quotas smaller than the sum of their individual quotas. Again, we show that a stable matching may not exist and the related decision problem is NP-complete. On the other hand we prove that, when the bounded sets form a nested set system, a stable matching can be found by generalising, in non-trivial ways, both the applicant-oriented and college-oriented versions of the classical Gale-Shapley algorithm. Finally, we present an alternative view of this nested case using the concept of choice functions, and with the aid of a matroid model we establish some interesting structural results for this case.

Efficiency and fair access in Kindergarten allocation policy design

“…In particular, there are regulations determining the minimum number of children needed to start a new group. This feature is similar to the lower quotas used in the Hungarian higher education matching scheme (Biró et al, 2010a), where programmes may be cancelled if there is a lack of students. This is a natural requirement that makes the education service economical, but the theoretical model for college admissions with lower quotas is not always solvable.…”

“…Making decisions on the size and the age composition of the groups in Kindergartens and determining this in an optimal way based on the application data could give an additional boost to the number of families receiving a place at their most preferred Kindergarten. Some of this research has been done in terms of lower quotas for opening groups (Biró et al, 2010a).…”

Matching With Couples: A Multidisciplinary Survey

“…So, no 10 Ackermann et al [2008] showed that the expected time of convergence to a stable matching may be exponential in two-sided matching markets. 11 This application has some special features (see Biró [2008] and Biró, Fleiner, Irving and Manlove [2010]), but the automated mechanism of the scheme is based on the DA algorithm. student has incentives to misrepresent his preferences.…”

“…Apart from the issue of ties (i.e., when there are students with identical scores), there are at least two other special features in the scheme. Universities and colleges not only have different kinds of common (upper) quotas, but they can also set lower quotas for each study (see Biró, Fleiner, Irving and Manlove [2010]). Finally, students can apply for pairs of MSc studies, which are special studies for future teachers.…”

Designing Matching Mechanisms under General Distributional Constraints

“…School districts may need at least a certain number of stu-dents in each school in order for the school to operate, as in college admissions in Hungary Biro, Fleiner, Irving, and Manlove (2010). The cadet-branch matching program organized by United States Military Academy (USMA) imposes minimum quotas on the number of cadets who can be assigned to each branch Sönmez and Switzer (2013).…”