Size versus stability in the marriage problem

Biro, P., Manlove, D.F. and Mittal, S. (2010) Size versus stability in the marriage problem. Theoretical Computer Science, 411(16-18), pp. 1828-1841. (doi:10.1016/j.tcs.2010.02.003)

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Abstract

Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many large-scale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is “as stable as possible”). We show that this problem is NP-hard and not approximable within n1−ε, for any ε>0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NP-hard and not approximable within δ, for some δ>1. By contrast, we give a polynomial-time algorithm for the case where the preference lists of one sex are of length at most 2. We also extend these results to the cases where (i) preference lists may include ties, and (ii) we seek to minimize the number of agents involved in a blocking pair.

Item Type:Articles
Additional Information:An earlier version of this paper is available via the Related URL link.
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Manlove, Professor David and Biro, Dr Peter
Authors: Biro, P., Manlove, D.F., and Mittal, S.
Subjects:Q Science > QA Mathematics > QA75 Electronic computers. Computer science
College/School:College of Science and Engineering > School of Computing Science
Journal Name:Theoretical Computer Science
Publisher:Elsevier
ISSN:0304-3975
ISSN (Online):1879-2294
Published Online:12 February 2010
Copyright Holders:Copyright © 2010 Elsevier
First Published:First published in Theoretical Computer Science 411(16-18):1828-1841
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher.
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
320021Algorithmics of Stable Matching Problems with IndifferenceDavid ManloveEngineering & Physical Sciences Research Council (EPSRC)GR/R84597/01Computing Science
436361MATCH-UP - matching under preferences - algorithms and complexityRobert IrvingEngineering & Physical Sciences Research Council (EPSRC)EP/E011993/1Computing Science