Abstract
An instance of the stable marriage problem is an undirected bipartite graph G = (X ∪ W, E) with linearly ordered adjacency lists; ties are allowed. A matching M is a set of edges no two of which share an endpoint. An edge \(e = (a,b) \in E \ M\) is a blocking edge for M if a is either unmatched or strictly prefers b to its partner in M, and b is either unmatched or strictly prefers a to its partner in M or is indifferent between them. A matching is strongly stable if there is no blocking edge with respect to it. We give an O(nm) algorithm for computing strongly stable matchings, where n is the number of vertices and m is the number of edges. The previous best algorithm had running time O(m 2).
We also study this problem in the hospitals-residents setting, which is a many-to-one extension of the above problem. We give an \(O(m(|R| + \sum_{h \in H^{Ph}}))\) algorithm for computing a strongly stable matching in the hospitals-residents problem, where |R| is the number of residents and p h is the quota of a hospital h. The previous best algorithm had running time O(m 2).
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References
Canadian Resident Matching Scheme. How the matching algorithm works, http://www.carms.ca/matching/algorith.htm
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. American Mathematical Monthly 69, 9–15 (1962)
Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Boston (1989)
Irving, R.W., Manlove, D.F., Scott, S.: Strong stability of the hospitals/residents problem. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 439–450. Springer, Heidelberg (2003)
Irving, R.W.: Matching medical students to pairs of hospitals: a new variation of a well-known theme. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 381–392. Springer, Heidelberg (1998)
Irving, R.W.: Stable marriage and indifference. Discrete Applied Mathematics, 261–272 (1994)
Iwama, K., Manlove, D., Miyazaki, S., Morita, Y.: Stable Marriage with incomplete lists and ties. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 443–452. Springer, Heidelberg (1999)
Manlove, D.F.: Stable marriage with ties and unacceptable partners. Technical report, University of Glasgow (1999)
Roth, A.E.: The evolution of the labor market for medical interns and residents: a case study in game theory. Journal of Political Economy 92(6), 991–1016 (1984)
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© 2004 Springer-Verlag Berlin Heidelberg
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Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K. (2004). Strongly Stable Matchings in Time O(nm) and Extension to the Hospitals-Residents Problem. In: Diekert, V., Habib, M. (eds) STACS 2004. STACS 2004. Lecture Notes in Computer Science, vol 2996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24749-4_20
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DOI: https://doi.org/10.1007/978-3-540-24749-4_20
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