Edge-weighted Online Stochastic Matching: Beating $1-\frac1e$
Shuyi Yan
TL;DR
The paper addresses the edge-weighted online stochastic matching problem and surpasses the longstanding $1-\frac{1}{e}$ barrier by achieving a $0.645$-competitive algorithm under Poisson arrivals. It combines a Jaillet-Lu LP-based preprocessing that discretizes edges into two classes with a multistage version of the Suggested Matching, enabling near-independent, edge-wise analysis. The core novelty lies in per-edge probability accounting: by bounding the unmatched probability of offline vertices and carefully scheduling early and late second-class matches, the authors derive a guaranteed edge-wise ratio of at least $0.645$. This result extends the reach of competitive guarantees in general edge-weighted online stochastic matching and offers a principled framework for LP-guided, edge-centric analysis.
Abstract
We study the edge-weighted online stochastic matching problem. Since Feldman, Mehta, Mirrokni, and Muthukrishnan proposed the $(1-\frac1e)$-competitive Suggested Matching algorithm, there has been no improvement for the general edge-weighted online stochastic matching problem. In this paper, we introduce the first algorithm beating the $1-\frac1e$ barrier in this setting, achieving a competitive ratio of $0.645$. Under the LP proposed by Jaillet and Lu, we design an algorithmic preprocessing, dividing all edges into two classes. Then based on the Suggested Matching algorithm, we adjust the matching strategy to improve the performance on one class in the early stage and on another class in the late stage, while keeping the matching events of different edges highly independent. By balancing them, we finally guarantee the matched probability of every single edge.