Almost Stable Matchings by Truncating the Gale–Shapley Algorithm

We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about t...

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Bibliographic Details
Published in:	Algorithmica Vol. 58; no. 1; pp. 102 - 118
Main Authors:	Floréen, Patrik, Kaski, Petteri, Polishchuk, Valentin, Suomela, Jukka
Format:	Journal Article
Language:	English
Published:	New York Springer-Verlag 01.09.2010
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation Distributed stable matching Local algorithms Constant-time randomised algorithms Almost stable matching
ISSN:	0178-4617, 1432-0541
Online Access:	Get full text
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Summary:	We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2+ ε )-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.
ISSN:	0178-4617 1432-0541
DOI:	10.1007/s00453-009-9353-9