A Stable Marriage Requires Communication

TL;DR

This paper proves that finding a stable marriage requires quadratic communication complexity, even approximately, by reducing the problem to the disjointness problem, thus establishing fundamental lower bounds in various models.

Contribution

It introduces a simplified, powerful communication complexity lower bound for stable marriage, extending to randomized algorithms, variants, and approximate stability.

Findings

Quadratic lower bound on Boolean queries for stable marriage

Lower bounds apply to randomized and preprocessing models

Efficient algorithm for computing distance to stability

Abstract

The Gale-Shapley algorithm for the Stable Marriage Problem is known to take $\Theta(n^2)$ steps to find a stable marriage in the worst case, but only $\Theta(n \log n)$ steps in the average case (with $n$ women and $n$ men). In 1976, Knuth asked whether the worst-case running time can be improved in a model of computation that does not require sequential access to the whole input. A partial negative answer was given by Ng and Hirschberg, who showed that $\Theta(n^2)$ queries are required in a model that allows certain natural random-access queries to the participants' preferences. A significantly more general - albeit slightly weaker - lower bound follows from Segal's general analysis of communication complexity, namely that $\Omega(n^2)$ Boolean queries are required in order to find a stable marriage, regardless of the set of allowed Boolean queries. Using a reduction to the communication complexity of the disjointness problem, we give a far simpler, yet significantly more powerful argument showing that $\Omega(n^2)$ Boolean queries of any type are indeed required for finding a stable - or even an approximately stable - marriage. Notably, unlike Segal's lower bound, our lower bound generalizes also to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's preferences profile and of the men's preferences profile, (C) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage, and (D) determining whether a proposed marriage is stable or far from stable. In order to analyze "approximately stable" marriages, we introduce the notion of "distance to stability" and provide an efficient algorithm for its computation.