Scheduling in the Secretary Model
This paper studies online makespan minimization in the secretary model. Jobs, specified by their processing times, are presented in a uniformly random order. The input size n is known in advance. An online algorithm has to non-preemptively assign each job permanently and irrevocably to one of m parallel and identical machines such that the expected time it takes to process them all, the makespan, is minimized.
We give two deterministic algorithms. First, a straightforward adaptation of the semi-online strategy Light Load [Albers and Hellwig, 2012] provides a very simple approach retaining its competitive ratio of 1.75. A new and sophisticated algorithm is 1.535-competitive. These competitive ratios are not only obtained in expectation but, in fact, for all but a very tiny fraction of job orders.
Classically, online makespan minimization only considers the worst-case order. Here, no competitive ratio below 1.885 for deterministic algorithms and 1.581 using randomization is possible. The best randomized algorithm so far is 1.916-competitive. Our results show that classical worst-case orders are quite rare and pessimistic for many applications.
We complement our results by providing first lower bounds. A competitive ratio obtained on nearly all possible job orders must be at least 1.257. This implies a lower bound of 1.043 for both deterministic and randomized algorithms in the general model.
Scheduling
makespan minimization
online algorithm
competitive analysis
lower bound
random-order
secretary problem
Theory of computation~Online algorithms
6:1-6:22
Regular Paper
Work Supported by the European Research Council, Grant Agreement No. 691672, Project APEG.
https://arxiv.org/abs/2103.16340
Susanne
Albers
Susanne Albers
Department of Computer Science, Technische Universität München, Germany
Maximilian
Janke
Maximilian Janke
Department of Computer Science, Technische Universität München, Germany
10.4230/LIPIcs.FSTTCS.2021.6
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Susanne Albers and Maximilian Janke
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