Stable marriage with ties and bounded length preference lists

Irving, R.W., Manlove, D.F. and O'Malley, G. (2009) Stable marriage with ties and bounded length preference lists. Journal of Discrete Algorithms, 7(2), pp. 213-219. (doi: 10.1016/j.jda.2008.09.003)

Abstract

We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard (even if each women's list is of length at most 3) and not approximable within some δ>1 (even if each woman's list is of length at most 4).

• Item Type: Articles
• Keywords: Stable marriage problem, ties, incomplete lists, NP-hardness, polynomial-time algorithm
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Manlove, Professor David and O'Malley, Mr Gregg and Irving, Dr Robert
• Authors: Irving, R.W., Manlove, D.F., and O'Malley, G.
• Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
• College/School: College of Science and Engineering > School of Computing Science
• Journal Name: Journal of Discrete Algorithms
• Publisher: Elsevier
• ISSN: 1570-8667
• ISSN (Online): 1570-8675
• Published Online: 20 September 2008
• Copyright Holders: Copyright © 2009 Elsevier
• First Published: First published in Journal of Discrete Algorithms 7(2):213-219
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.

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	COM - COMPUTING SCIENCE