Scheduling in the Random-Order Model

TL;DR

This paper improves the competitive ratio for online makespan minimization on identical machines in the random-order model, achieving a ratio of 1.8478 with high probability, and provides lower bounds for such algorithms.

Contribution

It introduces a new deterministic algorithm with improved guarantees and a novel analysis approach based on properties of random permutations.

Findings

Achieved a competitive ratio of 1.8478 in the random-order model.

Proved no deterministic algorithm can do better than 4/3 in this setting.

Established lower bounds of 3/2 with high probability for deterministic algorithms.

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.