A tight approximation bound for the stable marriage problem with restricted ties
Paper i proceeding, 2015

© Chien-Chung Huang, Kazuo Iwama, Shuichi Miyazaki, and Hiroki Yanagisawa. The problem of finding a maximum cardinality stable matching in the presence of ties and unacceptable partners, called MAX SMTI, is a well-studied NP-hard problem. The MAX SMTI is NP-hard even for highly restricted instances where (i) ties appear only in women's preference lists and (ii) each tie appears at the end of each woman's preference list. The current best lower bounds on the approximation ratio for this variant are 1.1052 unless P=NP and 1.25 under the unique games conjecture, while the current best upper bound is 1.4616. In this paper, we improve the upper bound to 1.25, which matches the lower bound under the unique games conjecture. Note that this is the first special case of the MAX SMTI where the tight approximation bound is obtained. The improved ratio is achieved via a new analysis technique, which avoids the complicated case-by-case analysis used in earlier studies. As a by-product of our analysis, we show that the integrality gap of natural IP and LP formulations for this variant is 1.25. We also show that the unrestricted MAX SMTI cannot be approximated with less than 1.5 unless the approximation ratio of a certain special case of the minimum maximal matching problem can be improved.

Approximation algorithm

Integer program

Stable marriage with ties and incomplete lists

Linear program relaxation

Integrality gap


Chien-Chung Huang

Chalmers, Data- och informationsteknik, Datavetenskap

K. Iwama

Kyoto University

S. Miyazaki

Kyoto University

H. Yanagisawa

IBM, Japan

Leibniz International Proceedings in Informatics, LIPIcs

1868-8969 (ISSN)

Vol. 40 361-380


Data- och informationsvetenskap