A Stronger Impossibility for Fully Online Matching

TL;DR

This paper improves the theoretical lower bound on the competitive ratio for fully online matching algorithms, showing it cannot exceed approximately 0.6297, even for fractional algorithms on bipartite graphs.

Contribution

It introduces new ideas and parameter optimization techniques to establish a stronger impossibility result for fully online matching.

Findings

Improved impossibility bound of 0.6297 for fully online matching.

The bound applies to fractional algorithms on bipartite graphs.

The result advances understanding of the limitations of online matching algorithms.

Abstract

We revisit the fully online matching model (Huang et al., J.\ ACM, 2020), an extension of the classic online matching model due to Karp, Vazirani, and Vazirani (STOC 1990), which has recently received a lot of attention (Huang et al., SODA 2019 and FOCS 2020), partly due to applications in ride-sharing platforms. It has been shown that the fully online version is harder than the classic version for which the achievable competitive ratio is at most $0.6317$, rather than precisely $1-\frac{1}{e}\approx 0.6321$. We introduce two new ideas to the construction. By optimizing the parameters of the modified construction numerically, we obtain an improved impossibility result of $0.6297$. Like the previous bound, the new bound even holds for fractional (rather than randomized) algorithms on bipartite graphs.