Subquadratic Algorithms for Succinct Stable Matching

TL;DR

This paper introduces subquadratic algorithms for stable matching problems with succinct preference representations, showing computational advantages in certain models but also inherent complexity for large dimensions.

Contribution

It provides the first subquadratic algorithms for specific succinct models and establishes complexity bounds for large dimensions under ETH.

Findings

Subquadratic algorithms for $d$-attribute, $d$-list, geometric, and single-peaked models.

Verification algorithms for stable matchings in the same models.

Quadratic time complexity is necessary for large $d$ under ETH.

Abstract

We consider the stable matching problem when the preference lists are not given explicitly but are represented in a succinct way and ask whether the problem becomes computationally easier and investigate other implications. We give subquadratic algorithms for finding a stable matching in special cases of natural succinct representations of the problem, the $d$-attribute, $d$-list, geometric, and single-peaked models. We also present algorithms for verifying a stable matching in the same models. We further show that for $d = \omega(\log n)$ both finding and verifying a stable matching in the $d$-attribute and $d$-dimensional geometric models requires quadratic time assuming the Strong Exponential Time Hypothesis. This suggests that these succinct models are not significantly simpler computationally than the general case for sufficiently large $d$.