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.

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The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. The stable matching problem has many applications to real world markets and ecient centralized algorithms are known. However, little is known about the decentralized case. Several natural randomized algorithmic models for this setting have been proposed but they have worst case exponential time in expectation. We present a novel structure Abstract. The stable matching problem has many applications to real world markets and efficient centralized algorithms are known. However, little is known about the decentralized case. Several natural randomized algorithmic models for this setting have been proposed but they have worst case exponential time in expectation. We present a novel structure associated with a stable matching on a matching market. Using this structure, we are able to provide a finer analysis of the complexity of a subclass of decentralized matching markets.