Machine Covering in the Random-Order Model

TL;DR

This paper studies the online Machine Covering problem in the random-order model, showing improved algorithms and bounds by leveraging the randomness in job arrivals, with implications for online scheduling and secretary problems.

Contribution

It introduces an improved $ ilde{O}( oot4 ext{th})$-competitive algorithm and establishes a new lower bound, advancing understanding of online scheduling in the random-order setting.

Findings

Graham's greedy strategy has a competitive ratio of $ heta(m/\log m)$ in the random-order model.

The paper presents an $ ilde{O}( oot4 ext{th})$-competitive algorithm exploiting random job order.

A new lower bound shows no algorithm can achieve $O(rac{\log m}{\log\log m})$ competitiveness.

Abstract

In the Online Machine Covering problem jobs, defined by their sizes, arrive one by one and have to be assigned to $m$ parallel and identical machines, with the goal of maximizing the load of the least-loaded machine. In this work, we study the Machine Covering problem in the recently popular random-order model. Here no extra resources are present, but instead the adversary is weakened in that it can only decide upon the input set while jobs are revealed uniformly at random. It is particularly relevant to Machine Covering where lower bounds are usually associated to highly structured input sequences. We first analyze Graham's Greedy-strategy in this context and establish that its competitive ratio decreases slightly to $\Theta\left(\frac{m}{\log(m)}\right)$ which is asymptotically tight. Then, as our main result, we present an improved $\tilde{O}(\sqrt[4]{m})$-competitive algorithm for the problem. This result is achieved by exploiting the extra information coming from the random order of the jobs, using sampling techniques to devise an improved mechanism to distinguish jobs that are relatively large from small ones. We complement this result with a first lower bound showing that no algorithm can have a competitive ratio of $O\left(\frac{\log(m)}{\log\log(m)}\right)$ in the random-order model. This lower bound is achieved by studying a novel variant of the Secretary problem, which could be of independent interest.