Dynamic Relaxations for Online Bipartite Matching

TL;DR

This paper introduces dynamic relaxations for the online bipartite matching problem, providing stronger theoretical bounds and practical policies, with empirical validation demonstrating improved performance over previous methods.

Contribution

It develops dynamic relaxations that outperform static ones in theory and practice, offering new insights and heuristics for online bipartite matching.

Findings

Dynamic relaxations dominate static relaxations in theoretical strength.

Heuristic policies derived from dual prices improve matching performance.

Empirical results show the new approach outperforms existing methods.

Abstract

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 i.i.d. OBM problem, where one side of the bipartition is fixed and known in advance, while 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.