Yes, the techniques presented in this paper can be extended to other generalizations of the stable marriage problem, including the college admissions problem with common quotas and the classified stable matching problem. The core idea of utilizing a modified instance of the Gale-Shapley algorithm, along with the properties of matroid kernels, can be adapted to accommodate the specific constraints and preferences present in these problems.
For the college admissions problem, where students apply to multiple colleges with common quotas, the algorithm can be modified to account for the capacity constraints of each college while still ensuring stability in the matching process. By treating the colleges as matroids with specific independent sets defined by their quotas, the 1.5-approximation can be achieved similarly to the matroid kernel problem.
In the case of the classified stable matching problem, where each college has upper and lower quotas for subsets of students, the techniques can be adapted to handle the additional complexity of these constraints. The algorithm can be designed to find a stable matching that respects both the preferences of students and the quota requirements of colleges, leveraging the properties of interval orders and matroid structures.