An $\Omega(\log n)$ Lower Bound for Online Matching on the Line

TL;DR

This paper establishes a fundamental lower bound of logarithmic order for online matching algorithms on the line, showing that no algorithm can do better than this bound, thus confirming the optimality of existing solutions.

Contribution

It proves a new lower bound of ( ext{log} n) for online matching on the line, improving previous bounds and matching the known upper bound.

Findings

Lower bound of ( ext{log} n) for online matching algorithms.

No online algorithm can surpass this approximation ratio.

The bound confirms the optimality of existing algorithms.

Abstract

For online matching with the line metric, we present a lower bound of $\Omega(\log n)$ on the approximation ratio of any online (possibly randomized) algorithm. This beats the previous best lower bound of $\Omega(\sqrt{\log n})$ and matches the known upper bound of $O(\log n)$.