Almost stable matchings in constant time

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.

💡 Research Summary

The paper revisits the classic stable marriage problem through the lens of the Gale‑Shapley algorithm’s iterative propose‑accept rounds. Its first major theoretical contribution is a precise linear relationship between the number of rounds executed (t) and the ratio of matched participants to blocking pairs: after t rounds the number of matched agents is Θ(t) times the total number of blocking pairs that existed initially. This shows that each additional round eliminates a constant fraction of the remaining instability, and the effect is quantifiable rather than merely asymptotic.

A striking aspect of the analysis is that it does not require global knowledge of the preference lists. Each participant only needs to know its own “acceptable” neighbourhood – the set of partners it would ever propose to. Consequently, in a distributed setting where each node communicates only with a constant number of neighbours, the system reaches an “almost stable” matching after a constant number of synchronous communication rounds. “Almost stable” means that the proportion of remaining blocking pairs can be made arbitrarily small (ε) by choosing a suitable constant number of rounds.

The authors extend the result to instances with ties in the preference lists. By adopting the weak‑stability notion (no pair strictly prefers each other over their current matches), they prove that the same linear round‑to‑instability reduction holds, thereby covering many realistic scenarios where agents are indifferent among several partners.

Two concrete applications illustrate the power of the framework. First, they design a distributed algorithm for maximum‑weight matching in bipartite graphs that achieves a (2 + ε)‑approximation. The algorithm assumes each vertex has a constant‑size acceptable set (bounded degree) and runs in a constant number of rounds, dramatically improving over prior O(log n)‑round distributed matching schemes. Second, they present a centralized randomized constant‑time approximation scheme for estimating the size of a stable matching. By sampling a constant‑size neighbourhood and leveraging the linear relationship, the estimator returns an ε‑approximation of the total matching size with high probability, without constructing the full matching.

In the related‑work discussion the paper positions itself against earlier distributed matching research that either required full preference information or incurred logarithmic round complexities. The new “round‑vs‑blocking‑pair” insight enables simultaneous constant‑time communication and constant‑factor approximation, a combination previously unattained.

Overall, the work provides a novel quantitative bridge between algorithmic progress (number of rounds) and solution quality (stability), demonstrates that local information suffices for near‑optimal outcomes, and opens avenues for fast, scalable matching solutions in both distributed systems and large‑scale centralized databases.