The stable marriage problem: structure and algorithms
The stable marriage problem: structure and algorithms
NP-complete stable matching problems
Journal of Algorithms
On-line algorithms for weighted bipartite matching and stable marriages
Theoretical Computer Science
Discrete Applied Mathematics
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
The Hospitals/Residents Problem with Ties
SWAT '00 Proceedings of the 7th Scandinavian Workshop on Algorithm Theory
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Ruling Out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Two algorithms for the Student-Project Allocation problem
Journal of Discrete Algorithms
The stable marriage problem with master preference lists
Discrete Applied Mathematics
An improved approximation lower bound for finding almost stable maximum matchings
Information Processing Letters
Size versus stability in the marriage problem
Theoretical Computer Science
Detecting high log-densities: an O(n¼) approximation for densest k-subgraph
Proceedings of the forty-second ACM symposium on Theory of computing
The College Admissions problem with lower and common quotas
Theoretical Computer Science
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
“Almost stable” matchings in the roommates problem
WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
A stab at approximating minimum subadditive join
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
A matroid approach to stable matchings with lower quotas
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
“Almost stable” matchings in the Roommates problem with bounded preference lists
Theoretical Computer Science
Strategy-proof mechanisms for two-sided matching with minimum and maximum quotas
Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems - Volume 3
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The Hospitals/Residents problem is a many-to-one extension of the stable marriage problem. In its instance, each hospital specifies a quota, i.e., an upper bound on the number of positions it provides. It is well-known that in any instance, there exists at least one stable matching, and finding one can be done in polynomial time. In this paper, we consider an extension in which each hospital specifies not only an upper bound but also a lower bound on its number of positions. In this setting, there can be instances that admit no stable matching, but the problem of asking if there is a stable matching is solvable in polynomial time. In case there is no stable matching, we consider the problem of finding a matching that is "as stable as possible", namely, a matching with a minimum number of blocking pairs. We show that this problem is hard to approximate within the ratio of (|H|+|R|)1-ε for any positive constant ε where H and R are the sets of hospitals and residents, respectively. We tackle this hardness from two different angles. First, we give an exponential-time exact algorithm for a special case where all the upper bound quotas are one. This algorithm runs in time O(t2(|H|(|R|+t))t+1) for instances whose optimal cost is t. Second, we consider another measure for optimization criteria, i.e., the number of residents who are involved in blocking pairs. We show that this problem is still NP-hard but has a polynomial-time √|R|-approximation algorithm.