Matching on the line admits no $o(\sqrt{\log n})$-competitive algorithm

TL;DR

This paper proves a fundamental lower bound on the competitive ratio for randomized online matching algorithms on the line, showing no algorithm can outperform a certain logarithmic threshold.

Contribution

It provides a simple proof establishing a lower bound of rac{rac{{ }log n}{12} for the competitive ratio, highlighting inherent limitations in online matching on the line.

Findings

No randomized online matching algorithm on the line can have a competitive ratio better than rac{rac{{ }log n}{12}.

The proof applies to all instances where n=2^i-1, ia0b1a0b1.

The result sets a fundamental limit on the performance of online algorithms in this setting.

Abstract

We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line is at least $\sqrt{\log_2(n\!+\!1)}/12$ for all $n=2^i\!-\!1: i\in\mathbb{N}$.