Online Stochastic Matching with Edge Arrivals

TL;DR

This paper introduces prune & greedy algorithms for stochastic bipartite matching with edge arrivals, achieving better competitive ratios than previous worst-case bounds by leveraging probabilistic pruning strategies.

Contribution

It proposes novel prune & greedy algorithms with improved competitive ratios in a Bayesian setting for online bipartite matching, surpassing the known 0.5 limit.

Findings

Achieves 0.552-competitive ratio on 2-regular graphs

Achieves 0.503-competitive ratio on arbitrary graphs

Develops a new analytical framework and optimization approach

Abstract

Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by [Gamlath et al. FOCS 2019], 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 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.