Approximating Matches Made in Heaven

Abstract

Motivated by applications in online dating and kidney exchange, we study a stochastic matching problem in which we have a random graph G given by a node set V and probabilities p(i,j) on all pairs i,j ∈ V representing the probability that edge (i,j) exists. Additionally, each node has an integer weight t(i) called its patience parameter. Nodes represent agents in a matching market with dichotomous preferences, i.e., each agent finds every other agent either acceptable or unacceptable and is indifferent between all acceptable agents. The goal is to maximize the welfare, or produce a matching between acceptable agents of maximum size. Preferences must be solicited based on probabilistic information represented by p(i,j), and agent i can be asked at most t(i) questions regarding his or her preferences.

A stochastic matching algorithm iteratively probes pairs of nodes i and j with positive patience parameters. With probability p(i,j), an edge exists and the nodes are irrevocably matched. With probability 1 − p(i,j), the edge does not exist and the patience parameters of the nodes are decremented. We give a simple greedy strategy for selecting probes which produces a matching whose cardinality is, in expectation, at least a quarter of the size of this optimal algorithm’s matching. We additionally show that variants of our algorithm (and our analysis) can handle more complicated constraints, such as a limit on the maximum number of rounds, or the number of pairs probed in each round.