DISPATCH: An Optimal Algorithm for Online Perfect Bipartite Matching with i.i.d. Arrivals

This work presents the first algorithm for the problem of weighted online perfect bipartite matching with i.i.d. arrivals. Previous work only considered adversarial arrival sequences. In this problem, we are given a known set of workers, a distribution over job types, and non-negative utility weights for each worker, job type pair. At each time step, a job is drawn i.i.d. from the distribution over job types. Upon arrival, the job must be irrevocably assigned to a worker. The goal is to maximize the expected sum of utilities after all jobs are assigned. Our work is motivated by the application of ride-hailing, where jobs represent passengers and workers represent drivers. We introduce DISPATCH, a 0.5-competitive, randomized algorithm and prove that 0.5-competitive is the best possible. DISPATCH first selects a "preferred worker" and assign the job to this worker if it is available. The preferred worker is determined based on an optimal solution to a fractional transportation problem. If the preferred worker is not available, DISPATCH randomly selects a worker from the available workers. We show that DISPATCH maintains a uniform distribution over the workers even when the distribution over the job types is non-uniform.