Stochastic Matching on Uniformly Sparse Graphs.

In this paper, we consider the following stochastic matching problem: We are given a graph G = (V, E) where each edge e is an element of E is realized independently with some constant probability p and the goal is to find a constant degree subgraph R of G whose expected realized matching size is close to that of G. This model of stochastic matching has attracted significant attention over the past few years for its various applications in kidney exchange and recommendation systems. The main open question of the area is whether a (1-epsilon) approximation can be achieved. Currently, the best known bounds are close to 0.66 due to algorithms of Assadi and Bernstein [SOSA'19] and Behnezhad et al. [SODA'19]. We show that indeed this bound can be improved to (1-epsilon) if the graph G has small arboricity. This includes a large family of graphs such as planar or minor-free graphs, bounded treewidth graphs, or arguably any sparse graph that is of interest. Finally, we also practically study a number of natural algorithms on the dataset of a major online freelancing company.