Symmetry Exploitation for Online Machine Covering with Bounded Migration

Gálvez, Waldo; Soto, José A.; Verschae, José

doi:10.1145/3397535

Cited by 14 publications

(9 citation statements)

References 23 publications

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“…Sanders et al [41] provide a 2-competitive algorithm for β = 1. Later results [42,22] study the interplay between improved competitive ratios and larger values of β. Another semionline model provides the online algorithm with a reordering buffer, which is used to rearrange the input sequence "on the fly".…”

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

Self Cite

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“…[14,23,31,36,37], and robust algorithms were designed and analyzed for various scheduling problems and other packing problems (see e.g. [6,7,15,16,17,18,19,21,25,26,35]).…”

Section: Related Workmentioning

confidence: 99%

Cardinality Constrained Scheduling in Online Models

Epstein¹,

Lassota²,

Levin³

et al. 2022

Preprint

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Makespan minimization on parallel identical machines is a classical and intensively studied problem in scheduling, and a classic example for online algorithm analysis with Graham's famous list scheduling algorithm dating back to the 1960s. In this problem, jobs arrive over a list and upon an arrival, the algorithm needs to assign the job to a machine. The goal is to minimize the makespan, that is, the maximum machine load. In this paper, we consider the variant with an additional cardinality constraint: The algorithm may assign at most k jobs to each machine where k is part of the input. While the offline (strongly NP-hard) variant of cardinality constrained scheduling is well understood and an EPTAS exists here, no non-trivial results are known for the online variant. We fill this gap by making a comprehensive study of various different online models. First, we show that there is a constant competitive algorithm for the problem and further, present a lower bound of 2 on the competitive ratio of any online algorithm. Motivated by the lower bound, we consider a semi-online variant where upon arrival of a job of size p, we are allowed to migrate jobs of total size at most a constant times p. This constant is called the migration factor of the algorithm. Algorithms with small migration factors are a common approach to bridge the performance of online algorithms and offline algorithms. One can obtain algorithms with a constant migration factor by rounding the size of each incoming job and then applying an ordinal algorithm to the resulting rounded instance. With this in mind, we also consider the framework of ordinal algorithms and characterize the competitive ratio that can be achieved using the aforementioned approaches. More specifically, we show that in both cases, one can get a competitive ratio that is strictly lower than 2, which is the bound from the standard online setting. On the other hand, we prove that no PTAS is possible.

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“…A class of the online max-min fair allocation problem with identical agents (i.e., v 1 = • • • = v n ) has also been studied as the online machine covering problem in the context of scheduling [9,18,21,22,29,30]. Here, an agent's utility corresponds to a machine load.…”

Section: Related Workmentioning

confidence: 99%

Online Max-min Fair Allocation

Kawase

Sumita

2021

Preprint

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We study an online version of the max-min fair allocation problem for indivisible items. In this problem, items arrive one by one, and each item must be allocated irrevocably on arrival to one of n agents, who have additive valuations for the items. Our goal is to make the least happy agent as happy as possible. In research on the topic of online allocation, this is a fundamental and natural problem. Our main result is to reveal the asymptotic competitive ratios of the problem for both the adversarial and i.i.d. input models. We design a polynomial-time deterministic algorithm that is asymptotically 1/n-competitive for the adversarial model, and we show that this guarantee is optimal. To this end, we present a randomized algorithm with the same competitive ratio first and then derandomize it. A natural derandomization fails to achieve the competitive ratio of 1/n. We instead build the algorithm by introducing a novel technique. When the items are drawn from an unknown identical and independent distribution, we construct a simple polynomial-time deterministic algorithm that outputs a nearly optimal allocation. We analyze the strict competitive ratio and show almost tight bounds for the solution. We further mention some implications of our results on variants of the problem.

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Symmetry Exploitation for Online Machine Covering with Bounded Migration

Cited by 14 publications

References 23 publications

Machine Covering in the Random-Order Model

Machine Covering in the Random-Order Model

Cardinality Constrained Scheduling in Online Models

Online Max-min Fair Allocation

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