Online Stochastic Matching with Edge Arrivals

Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by Gamlath et al., who showed that no online policy is better than the straightforward greedy algorithm, i.e., no online algorithm has a worst-case competitive ratio better than 0.5. In this work, we consider the bipartite matching problem with edge arrivals in a natural stochastic framework, i.e., Bayesian setting where each edge of the graph is independently realized according to a known probability distribution. We focus on a natural class of prune & greedy online policies motivated by practical considerations from a multitude of online matching platforms. Any prune & greedy algorithm consists of two stages: first, it decreases the probabilities of some edges in the stochastic instance and then runs greedy algorithm on the pruned graph. We propose prune & greedy algorithms that are 0.552-competitive on the instances that can be pruned to a 2-regular stochastic bipartite graph, and 0.503-competitive on arbitrary stochastic bipartite graphs. The algorithms and our analysis significantly deviate from the prior work. We first obtain analytically manageable lower bound on the size of the matching, which leads to a non-linear optimization problem. We further reduce this problem to a continuous optimization with a constant number of parameters that can be solved using standard software tools

Improved Competitive Ratio for Edge-Weighted Online Stochastic Matching

We consider the edge-weighted online stochastic matching problem, in which an edge-weighted bipartite graph G=(I\cup J, E) with offline vertices J and online vertex types I is given. The online vertices have types sampled from I with probability proportional to the arrival rates of online vertex types. The online algorithm must make immediate and irrevocable matching decisions with the objective of maximizing the total weight of the matching. For the problem with general arrival rates, Feldman et al. (FOCS 2009) proposed the Suggested Matching algorithm and showed that it achieves a competitive ratio of 1-1/e \approx 0.632. The ratio has recently been improved to 0.645 by Yan (2022), who proposed the Multistage Suggested Matching (MSM) algorithm. In this paper, we propose the Evolving Suggested Matching (ESM) algorithm, and show that it achieves a competitive ratio of 0.650.Comment: To appear in WINE202