Online Matching Frameworks under Stochastic Rewards, Product Ranking, and Unknown Patience

We study generalizations of online bipartite matching in which each arriving vertex (customer) views a ranked list of offline vertices (products) and matches to (purchases) the first one they deem acceptable. The number of products that the customer has patience to view can be stochastic and dependent on the products seen. We develop a framework that views the interaction with each customer as an abstract resource consumption process, and derive new results for these online matching problems under the adversarial, non-stationary, and IID arrival models, assuming we can (approximately) solve the product ranking problem for each single customer. To that end, we show new results for product ranking under two cascade-click models: an optimal algorithm for item-dependent hazard rates, and a 1/2-approximate algorithm for general item-independent patience distributions. We also present a constant-factor 0.027-approximate algorithm in a new model where items are not initially available and arrive over time. Finally, we present three negative results of interest: one formalizing the notion of a stochasticity gap exhibited by existing approaches to this problem, an example showing the analysis of SimpleGreedy in existing work to be tight, and another one for the single-customer problem in which any constant-factor approximation is impossible when compared to a benchmark that knows the realization of the patience in advance. A corollary of this last result is that for general single-item online accept/reject problems with IID arrivals, any constant-factor approximation is impossible if the number of arrivals is unknown.