Manlove, D.F., Irving, R. W., Iwama, K., Miyazaki, S. and Morita, Y. (2002) Hard variants of stable marriage. Theoretical Computer Science, 276(12), pp. 261279. (doi: 10.1016/S03043975(01)002067)

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Publisher's URL: http://dx.doi.org/doi:10.1016/S03043975(01)002067
Abstract
The Stable Marriage Problem and its many variants have been widely studied in the literature (Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989; Roth and Sotomayor, Twosided matching: a study in gametheoretic modeling and analysis, Econometric Society Monographs, vol. 18, Cambridge University Press, Cambridge, 1990; Knuth, Stable Marriage and its Relation to Other Combinatorial Problems, CRM Proceedings and Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 1997), partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program (Roth, J. Political Economy 92(6) (1984) 991) and similar largescale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stable––even if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an `egalitarian' and a `minimum regret' stable matching. However, we give a 2approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes.
Item Type:  Articles 

Additional Information:  Postprint provided by the author 
Keywords:  Stable marriage problem; Indifference; Ties; NPcompleteness; Approximation algorithms 
Status:  Published 
Refereed:  Yes 
Glasgow Author(s) Enlighten ID:  Manlove, Professor David 
Authors:  Manlove, D.F., Irving, R. W., Iwama, K., Miyazaki, S., and Morita, Y. 
Subjects:  Q Science > QA Mathematics > QA75 Electronic computers. Computer science 
College/School:  College of Science and Engineering > School of Computing Science 
Research Group:  Formal Analysis, Theory and Algorithms 
Journal Name:  Theoretical Computer Science 
Publisher:  Elsevier 
ISSN:  03043975 
Copyright Holders:  Copyright ©2002 Elsevier Science B.V. 
First Published:  First published in Theoretical Computer Science 276(12):261279 
Publisher Policy:  Reproduced in accordance with the copyright policy of the publisher. 
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