Fast distributed almost stable marriages

In their seminal work on the Stable Marriage Problem, Gale and Shapley describe an algorithm which finds a stable marriage in $\mathcal{O}(n^2)$ communication rounds. Their algorithm has a natural interpretation as a distributed algorithm where each player is represented by a single processor. The complexity measure of such a distributed algorithm is typically measured by the round complexity, assuming all processors can communicate simultaneously in each round, or in terms of synchronous running time. Recently, Flor\'een, Kaski, Polishchuk and Suomela showed that in the special case of bounded preference lists, terminating the Gale-Shapley algorithm after a constant number of rounds results in an almost stable (partial) marriage. Lifting such an approximation to unbounded preference lists remained open. In this paper, we describe a new distributed algorithm which computes an almost stable marriage in $\mathcal{O}(1)$ communication rounds for unbounded preference lists, so long as the ratio of the lengths of longest to shortest preference lists is bounded by a constant. The synchronous run-time of our algorithm is $\mathcal{O}(n)$ for complete preference lists. To our knowledge, this is the first sub-polynomial round and sub-quadratic time distributed algorithm for any variant of the stable marriage problem with complete preferences.