Online Matching with Stochastic Rewards: : Optimal Competitive Ratio via Path-Based Formulation

In the paper “Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path-Based Formulation,” the authors develop a novel algorithm analysis approach to address stochastic elements in online matching. The approach leads to several new results that were previously out of reach for a fundamental generalization of online matching. More generally, the approach is useful for analyzing the performance of online algorithms for matching in settings with stochastic uncertainty that manifests after matching decisions are made.

The problem of online matching with stochastic rewards is a generalization of the online bipartite matching problem where each edge has a probability of success. When a match is made it succeeds with the probability of the corresponding edge. We consider the more general vertex-weighted version of the problem and give two new results. First, we show that a natural generalization of the perturbed-greedy algorithm is (1−1/e) competitive when probabilities decompose as a product of two factors, one corresponding to each vertex of the edge. This is the best achievable guarantee as it includes the case of identical probabilities and, in particular, the classical online bipartite matching problem. Second, we give a deterministic 0.596 competitive algorithm for the previously well-studied case of fully heterogeneous but vanishingly small edge probabilities. A key contribution of our approach is the use of novel path-based formulations and a generalization of the primal-dual scheme. These allow us to compare against the natural benchmarks of adaptive offline algorithms that know the sequence of arrivals and the edge probabilities in advance but not the outcomes of potential matches. These ideas may be of independent interest in other online settings with postallocation stochasticity.

Funding: This research was partially supported by the National Science Foundation [Grant CMMI 1636046 and Award CMMI 1351838].

Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.2345.