"Almost stable" matchings in the Roommates problem

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, 6-7 October, 2005, pp. 1-14. ISBN 3-540-32207-8 (doi: 10.1007/11671411_1)

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 NP-hard 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 NP-complete. By contrast, if K is constant, we give a polynomial-time 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: 0302-9743
• ISBN: 3-540-32207-8
• Copyright Holders: Copyright © 2006 Springer Verlag
• First Published: First published in the Lecture Notes in Computer Science 3341:3-15
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.