Almost stable matchings in constant time

TL;DR

This paper demonstrates that in the stable marriage problem, an almost stable matching can be achieved quickly with limited local information, leading to efficient distributed algorithms and approximation schemes.

Contribution

It introduces a method to find almost stable matchings in constant time using local neighborhood information, applicable to distributed systems and approximation algorithms.

Findings

Ratio of matched individuals to blocking pairs grows linearly with rounds

Almost stable matchings emerge after constant rounds with local info

Applicable to distributed algorithms and approximation schemes

Abstract

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. This holds even if ties are present in the preference lists. We apply our results to give a distributed $(2+\epsilon)$-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.