Multiway Online Correlated Selection

TL;DR

This paper introduces multiway online correlated selection algorithms that improve the competitive ratio for online bipartite matching from 0.5086 to 0.5368, surpassing previous methods by generalizing the selection process.

Contribution

It develops multiway OCS algorithms that achieve higher competitive ratios and simplifies the reduction process for online bipartite matching, demonstrating the superiority over prior two-way OCS methods.

Findings

Achieved a 0.5368-competitive ratio for edge-weighted online bipartite matching.

Multiway OCSes outperform two-way OCS, with a proven upper bound of 0.5239 for the latter.

Simplified the reduction from online bipartite matching to OCS using multiway formulations.

Abstract

We give a $0.5368$-competitive algorithm for edge-weighted online bipartite matching. Prior to our work, the best competitive ratio was $0.5086$ due to Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020). They achieved their breakthrough result by developing a subroutine called \emph{online correlated selection} (OCS) which takes as input a sequence of pairs and selects one item from each pair. Importantly, the selections the OCS makes are negatively correlated. We achieve our result by defining \emph{multiway} OCSes which receive arbitrarily many elements at each step, rather than just two. In addition to better competitive ratios, our formulation allows for a simpler reduction from edge-weighted online bipartite matching to OCSes. While Fahrbach et al. used a factor-revealing linear program to optimize the competitive ratio, our analysis directly connects the competitive ratio to the parameters of the multiway OCS. Finally, we show that the formulation of Farhbach et al. can achieve a competitive ratio of at most $0.5239$, confirming that multiway OCSes are strictly more powerful.