Dynamic Relaxations for Online Bipartite Matching

Online bipartite matching (OBM) is a fundamental model underpinning many important applications, including search engine advertisement, website banner and pop-up ads, and ride hailing. We study the independent and identically distributed (i.i.d.) OBM problem, in which one side of the bipartition is fixed and known in advance, whereas nodes from the other side appear sequentially as i.i.d. realizations of an underlying distribution and must immediately be matched or discarded. We introduce dynamic relaxations of the set of achievable matching probabilities; show how they theoretically dominate lower dimensional, static relaxations from previous work; and perform a polyhedral study to theoretically examine the new relaxations’ strength. We also discuss how to derive heuristic policies from the relaxations’ dual prices in a similar fashion to dynamic resource prices used in network revenue management. We finally present a computational study to demonstrate the empirical quality of the new relaxations and policies. Summary of Contribution: Online bipartite matching (OBM) is one of the fundamental problems in the area of online decision analysis with a wide variety of applications in operations research and computer science, for example, online advertising, ride sharing, and general resource allocation. Over the last decades, both communities have been interested in the design and analysis of new approaches. Our main contribution is to provide a polyhedral study that considers the problem’s sequential nature. Specifically, we achieve this via dynamic relaxations. We also discuss how to derive heuristic policies from the relaxations’ dual prices. We support our theoretical findings with a detailed computational study.