Abraham, D.J., Biro, P. and Manlove, D.F. (2006) "Almost stable" matchings in the Roommates problem. In: Proceedings of WAOA 2005: the 3rd International Workshop on Approximation and Online Algorithms, Palma, Mallorca, 67 October, 2005, pp. 114. ISBN 3540322078 (doi:10.1007/11671411_1)

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Abstract
An instance of the classical Stable Roommates problem (SR) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an SR instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NPhard and not approximable within n^{\frac{1}{2}\varepsilon}, for any \varepsilon>0, unless P=NP. If the preference lists contain ties, we improve this result to n^{1\varepsilon}. Also, we show that, given an integer K and an SR instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NPcomplete. By contrast, if K is constant, we give a polynomialtime algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an SR instance I.
Item Type:  Conference Proceedings 

Status:  Published 
Refereed:  Yes 
Glasgow Author(s) Enlighten ID:  Manlove, Professor David 
Authors:  Abraham, D.J., Biro, P., and Manlove, D.F. 
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 
Publisher:  Springer Verlag 
ISSN:  03029743 
ISBN:  3540322078 
Copyright Holders:  Copyright © 2006 Springer Verlag 
First Published:  First published in the Lecture Notes in Computer Science 3341:315 
Publisher Policy:  Reproduced in accordance with the copyright policy of the publisher. 
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