Abstract

Makespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to m identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is (2-1/m)-competitive. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201. No deterministic online strategy can obtain a competitiveness smaller than 1.88. In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds, for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability.

Highlights

  • We study one of the most basic scheduling problems

  • In this paper we investigate online makespan minimization in the random-order model

  • We present an in-depth study of online makespan minimization in the randomorder model

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Summary

Introduction

We study one of the most basic scheduling problems. Consider a sequence of jobs J = J1, ... , Jn that has to be assigned to m identical parallel machines. The goal is to minimize the makespan, i.e. the maximum completion time of any job in the constructed schedule. Both the offline and online variants of this problem have been studied extensively, see e.g. Given a job sequence J , let A(J) denote the makespan of an online algorithm A on J. In this paper we investigate online makespan minimization in the random-order model. We present an in-depth study of online makespan minimization in the randomorder model. After almost 20 years this is the first progress for the pure online setting, where an algorithm does not resort to extra resources in handling a job sequence

Previous Work
Our Contribution
Strong Competitiveness in the Random‐Order Model
Description of the New Algorithm
How Random‐Order Arrival Helps
Formal Definition
Analysis Basics
Stable Job Sequences
An Adversarial Analysis
Findings
Lower Bounds
Full Text
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