The Power of Multiple Choices in Online Stochastic Matching

TL;DR

This paper advances online stochastic matching algorithms by leveraging multiple choices, achieving improved competitive ratios and breaking previous barriers in certain weighted matching settings.

Contribution

It introduces two novel approaches for multiple choices in online stochastic matching, enhancing competitive ratios and analyzing complex weighted matching problems.

Findings

Improved competitive ratio to 0.716 for unweighted and vertex-weighted matching.

Achieved a 0.706 competitive ratio for edge-weighted matching with free disposal, breaking the 1-1/e barrier.

Proved no algorithm can surpass 0.703 competitiveness in edge-weighted matching without free disposal.

Abstract

We study the power of multiple choices in online stochastic matching. Despite a long line of research, existing algorithms still only consider two choices of offline neighbors for each online vertex because of the technical challenge in analyzing multiple choices. This paper introduces two approaches for designing and analyzing algorithms that use multiple choices. For unweighted and vertex-weighted matching, we adopt the online correlated selection (OCS) technique into the stochastic setting, and improve the competitive ratios to $0.716$, from $0.711$ and $0.7$ respectively. For edge-weighted matching with free disposal, we propose the Top Half Sampling algorithm. We directly characterize the progress of the whole matching instead of individual vertices, through a differential inequality. This improves the competitive ratio to $0.706$, breaking the $1-\frac{1}{e}$ barrier in this setting for the first time in the literature. Finally, for the harder edge-weighted problem without free disposal, we prove that no algorithms can be $0.703$ competitive, separating this setting from the aforementioned three.