Paper ID: 9316
Size versus stability in the Marriage problem
Biro,P.
Manlove,D.F.
Mittal,S.
Publication Type:
Journal
Appeared in:
Theoretical Computer Science, volume 411
Page Numbers : 18281841
Publisher: Elsevier Science
Year: 2010
ISBN/ISSN:
URL: This publication is available at this URL.
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 largescale 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 NPhard and not
approximable within n^{1\varepsilon}, for any
\varepsilon>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 NPhard and not
approximable within \delta, for some \delta>1. By contrast,
we give a polynomialtime 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 minimise the number of
agents involved in a blocking pair.
Keywords: Stable marriage problem; Stable matching; Blocking pair; Blocking agent; Inapproximability result; Polynomialtime algorithm
