The stable marriage problem: structure and algorithms
The stable marriage problem: structure and algorithms
The Geometry of Fractional Stable Matchings and its Applications
Mathematics of Operations Research
Hard variants of stable marriage
Theoretical Computer Science
A fixed-point approach to stable matchings and some applications
Mathematics of Operations Research
Two algorithms for the Student-Project Allocation problem
Journal of Discrete Algorithms
The hospitals/residents problem with quota lower bounds
ESA'11 Proceedings of the 19th European conference on Algorithms
A matroid approach to stable matchings with lower quotas
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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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.