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)

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.

Related URLs

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Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
32002	1	Algorithmics of Stable Matching Problems with Indifference	David Manlove	Engineering & Physical Sciences Research Council (EPSRC)	GR/R84597/01	Computing Science
43636	1	MATCH-UP - matching under preferences - algorithms and complexity	Robert Irving	Engineering & Physical Sciences Research Council (EPSRC)	EP/E011993/1	Computing Science