Subquadratic Algorithms for Succinct Stable Matching

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 $d = ω(\log n)$ 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 $d$.