The stable marriage problem: structure and algorithms
The stable marriage problem: structure and algorithms
Characterization of stable matchings as extreme points of a polytope
Mathematical Programming: Series A and B
Stable matchings, optimal assignments, and linear programming
Mathematics of Operations Research
Stable matchings and linear inequalities
Discrete Applied Mathematics
The Geometry of Fractional Stable Matchings and its Applications
Mathematics of Operations Research
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
Many-to-One Stable Matching: Geometry and Fairness
Mathematics of Operations Research
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
Note: Blockers and antiblockers of stable matchings
Theoretical Computer Science
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We introduce the classified stable matching problem, a problem motivated by academic hiring. Suppose that a number of institutes are hiring faculty members from a pool of applicants. Both institutes and applicants have preferences over the other side. An institute classifies the applicants based on their research areas (or any other criterion), and, for each class, it sets a lower bound and an upper bound on the number of applicants it would hire in that class. The objective is to find a stable matching from which no group of participants has reason to deviate. Moreover, the matching should respect the upper/lower bounds of the classes. In the first part of the paper, we study classified stable matching problems whose classifications belong to a fixed set of "order types." We show that if the set consists entirely of downward forests, there is a polynomial-time algorithm; otherwise, it is NP-complete to decide the existence of a stable matching. In the second part, we investigate the problem using a polyhedral approach. Suppose that all classifications are laminar families and there is no lower bound. We propose a set of linear inequalities to describe stable matching polytope and prove that it is integral. This integrality result allows us to find optimal stable matchings in polynomial time using Ellipsoid algorithm; furthermore, it gives a description of the stable matching polytope for the many-to-many (unclassified) stable matching problem, thereby answering an open question posed by Sethuraman, Teo and Qian.