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Fast Distributed Almost Stable Matchings

In their seminal work on the Stable Marriage Problem, Gale and Shapley describe an algorithm which finds a stable matching in O(n2) communication rounds. Their algorithm has a natural interpretation as a distributed algorithm where each player is represented by a single processor. In this distributed model, Floreen, Kaski, Polishchuk, and Suomela recently showed that for bounded preference lists, terminating the Gale-Shapley algorithm after a constant number of rounds results in an almost stable matching. In this paper, we describe a new deterministic distributed algorithm which finds an almost stable matching in O(log5 n) communication rounds for arbitrary preferences. We also present a faster randomized variant which requires O(log2 n) rounds. This run-time can be improved to O(1) rounds for "almost regular" (and in particular complete) preferences. To our knowledge, these are the first sub-polynomial round distributed algorithms for any variant of the stable marriage problem with unbounded preferences.