Dynamic Stochastic Matching Under Limited Time

In centralized matching markets, such as kidney exchange schemes and car-pooling platforms, new participants constantly enter the market and remain available for potential matches during a limited period of time. To reach an efficient allocation, the "timing" of the matching decisions is a critical aspect of the platform's operations. There is a fundamental tradeoff between increasing market thickness and mitigating the risk that some participants abandon the market. Such dynamic properties of matching markets have been mostly overlooked in the classic algorithmic literature. In this paper, we introduce a dynamic matching model that captures nuanced arrival and abandonment patterns. Specifically, we study an online stochastic matching problem on edge-weighted graphs, where the agents' arrivals and abandonments are stochastic and heterogeneous. While the resulting Markov decision process is computationally intractable, we design simple greedy-like matching algorithms that admit constant-factor approximation guarantees. We devise a 3-approximation algorithm for the cost-minimization problem on graphs satisfying a metric-like property. In contrast, we show that the performance guarantee of widely used batching algorithms can be arbitrarily bad. We further devise a (e-1)/(2e)-approximation algorithm for the reward-maximization problem on arbitrary bipartite graphs. Our analysis synthesizes various novel technical ideas that could be of broader interest, including linear programming benchmarks and approximations of continuous-time Markov chains. In extensive experiments, we simulate the matching operations of a car-pooling platform using real-world taxi demand data. The newly developed algorithms have the potential to improve cost efficiency by 10% in certain realistic market conditions against batching algorithms.