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+e)-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.
Note that switching partners along unstable edges can be done in a distributed manner. The Gale--Shapley algorithm is also parallel by its nature: the proposals rejects can be undertaken by all men women simultaneously during synchronised rounds (albeit it can happen that only one man is free at a round [Section A.3] gusfield89stable , @cite_22 ). Lower bounds on the running time of the algorithm [Section 1.5] gusfield89stable show that a linear number of rounds is required to attain stability. But can a nearly stable matching be obtained with fewer rounds?
@cite_22
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We show that the ratio of matched individuals to blocking pairs grows linearly with the number of proposeaccept rounds executed by the GaleShapley 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+e)-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.