Communication Requirements for Stable Marriages

We study the stable marriage problem in a distributed environment, in which there are 2n players, n men and n women, each holding a private ranking of the n persons of the opposite set, and there is a server who communicates with the players and finds a matching for them. We restrict our attention on two communication models: the sketch model and the query model. In the sketch model, each player compresses his/her ranking into a sketch and sends it to the server, while in the query model, the server itself adaptively queries individual bits on each player’s ranking. We show that for the server to output even a slightly stable matching, in which a small constant fraction of matched pairs are stable, it must receive Ω(n log n) bits from the players in the sketch model, and it must query Ω(n log n) bits on their rankings in the query model. This implies that even to find a slightly stable matching, it is impossible to have an algorithm which compresses the input into a sketch of sub-linear size or to have an algorithm which runs in sub-linear time.