Edge-weighted Online Stochastic Matching: Beating $1-\frac1e$

TL;DR

This paper introduces a novel algorithm for edge-weighted online stochastic matching that surpasses the longstanding $1-rac1e$ competitive ratio barrier, achieving a ratio of 0.645.

Contribution

It presents the first algorithm to beat the $1-rac1e$ ratio in this setting, using a new preprocessing and adaptive matching strategy based on LP insights.

Findings

Achieved a competitive ratio of 0.645.

Developed a preprocessing method dividing edges into two classes.

Enhanced matching performance by balancing early and late stage strategies.

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.