On the Advice Complexity of Online Matching on the Line

TL;DR

This paper investigates the advice complexity of online matching on a line, presenting optimal and near-optimal algorithms with advice complexity bounds, advancing understanding of how advice reduces online matching costs.

Contribution

It introduces a 1-competitive algorithm with advice complexity $n-1$ and proves this bound is tight, also providing algorithms with sublinear advice for near-competitive ratios.

Findings

A 1-competitive algorithm with advice complexity $n-1$

No 1-competitive algorithm with less than $n-1$ advice bits

Near-competitive algorithms with advice complexity $O(k(\log N + \log n))$

Abstract

We consider the matching problem on the line with advice complexity. We give a 1-competitive online algorithm with advice complexity $n-1,$ and show that there is no 1-competitive online algorithm reading less than $n-1$ bits of advice. Moreover, for each $0<k<n$ we present a $c(n/k)$-competitive online algorithm with advice complexity $O(k(\log N + \log n))$ where $n$ is the number of servers, $N$ is the distance of the minimal and maximal servers, and $c(n)$ is the complexity of the best online algorithm without advice.