Max-min Online Allocations with a Reordering Buffer

Epstein, Leah; Levin, Asaf; Stee, Rob van

doi:10.1137/100794006

Cited by 17 publications

(4 citation statements)

References 26 publications

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“…Another semionline model provides the online algorithm with a reordering buffer, which is used to rearrange the input sequence "on the fly". Epstein et al [16] provide a (H m−1 + 1)-competitive algorithm using a buffer of size m − 1, and show that this ratio cannot be improved for any sensible buffer size. These and many more semi-online models have also been studied for Makespan Minimization, see the survey in [15] and references therein.…”

Section: Related Resultsmentioning

confidence: 99%

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Machine Covering in the Random-Order Model

Albers¹,

Gálvez²,

Janke³

2021

Preprint

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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. Unfortunately, the classical model allows only fairly pessimistic performance guarantees: The best possible deterministic ratio of m is achieved by the Greedystrategy, and the best known randomized algorithm has competitive ratio Õ( √ m) which cannot be improved by more than a logarithmic factor. Modern results try to mitigate this by studying semi-online models, where additional information about the job sequence is revealed in advance or extra resources are provided to the online algorithm. 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 Θ m log(m) which is asymptotically tight. Then, as our main result, we present an improved Õ( 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 log(m) log log(m)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.

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Section: Related Resultsmentioning

confidence: 99%

“…Such restrictive facts have motivated the study of different semi-online models that provide extra information [7,14,34,37] or extra features [41,42,22,16] to the online algorithm.…”

Section: Introductionmentioning

confidence: 99%

Machine Covering in the Random-Order Model

Albers¹,

Gálvez²,

Janke³

2021

Preprint

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show abstract

“…Related Machines: Buffer. Epstein et al [95] investigated the setup Q m |reB(k)|C min and proposed a m-competitive algorithm, where m ≥ 2 and k=m + 1. The algorithm keeps initial m + 1 incoming jobs in the buffer.…”

Section: Recent Work In Semi-online Schedulingmentioning

confidence: 99%

Semi-online scheduling: A survey

Dwibedy

Mohanty

2022

Computers & Operations Research

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“…Epstein, Levin, and van Stee [16] study the objective to maximize the minimum completion time. For m identical machines, they present an upper bound on the competitive ratio of H m−1 + 1 for a buffer of size m and a lower bound of H m for any fixed buffer size.…”

Section: Related Workmentioning

confidence: 99%

Online Makespan Scheduling with Job Migration on Uniform Machines

2021

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In the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and $$\approx 1.4659$$ ≈ 1.4659 . They show that $$k = O(m)$$ k = O ( m ) is sufficient to achieve this bound and no $$k = o(n)$$ k = o ( n ) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a $$\delta = \varTheta (1)$$ δ = Θ ( 1 ) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than $$1.4659 + \delta $$ 1.4659 + δ with $$k = o(n)$$ k = o ( n ) . We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and $$\approx 1.7992$$ ≈ 1.7992 with $$k = O(m)$$ k = O ( m ) . We also show that $$k = \varOmega (m)$$ k = Ω ( m ) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.

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Max-min Online Allocations with a Reordering Buffer

Cited by 17 publications

References 26 publications

Machine Covering in the Random-Order Model

Machine Covering in the Random-Order Model

Semi-online scheduling: A survey

Online Makespan Scheduling with Job Migration on Uniform Machines

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