Fast distributed almost stable marriages

TL;DR

This paper introduces new distributed algorithms that find almost stable matchings in significantly fewer rounds than previous methods, applicable to arbitrary preferences and improving efficiency in the stable marriage problem.

Contribution

The paper presents the first sub-polynomial distributed algorithms for the stable marriage problem with unbounded preferences, including deterministic and randomized variants with improved round complexities.

Findings

Deterministic algorithm finds an almost stable matching in O(log^5 n) rounds.

Randomized variant achieves O(log^2 n) rounds.

For almost regular preferences, the runtime reduces to O(1) rounds.

Abstract

In their seminal work on the Stable Marriage Problem, Gale and Shapley describe an algorithm which finds a stable matching in $O(n^2)$ 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 $O(\log^5 n)$ communication rounds for arbitrary preferences. We also present a faster randomized variant which requires $O(\log^2 n)$ 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.