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Optimal and Online Preemptive Scheduling on Uniformly Related Machines
"... We consider the problem of preemptive scheduling on uniformly related machines. We present a semionline algorithm which, if the optimal makespan is given in advance, produces an optimal schedule. Using the standard ..."
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We consider the problem of preemptive scheduling on uniformly related machines. We present a semionline algorithm which, if the optimal makespan is given in advance, produces an optimal schedule. Using the standard
The power of reordering for online minimum makespan scheduling
 In Proc. 49th FOCS
"... In the classic minimum makespan scheduling problem, we are given an input sequence of jobs with processing times. 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, w ..."
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In the classic minimum makespan scheduling problem, we are given an input sequence of jobs with processing times. 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 do not require that each arriving job has to be assigned immediately to one of the machines. A reordering buffer with limited storage capacity can be used to reorder the input sequence in a restricted fashion so as to schedule the jobs with a smaller makespan. This is a natural extension of lookahead. We present an extensive study of the power and limits of online reordering for minimum makespan scheduling. As main result, we give, for m identical machines, tight and, in comparison to the problem without reordering, much improved bounds on the competitive ratio for minimum makespan scheduling with reordering buffers. Depending on m, the achieved competitive ratio lies between 4/3 and 1.4659. This optimal ratio is achieved with a buffer of size Θ(m). We show that larger buffer sizes do not result in an additional advantage and that a buffer of size Ω(m) is necessary to achieve this competitive ratio. Further, we present several algorithms for different buffer sizes. Among others, we introduce, for every buffer size k ∈ [1,(m + 1)/2], a (2 − 1/(m − k + 1))competitive algorithm, which nicely generalizes the wellknown result of Graham. For m uniformly related machines, we give a scheduling algorithm that achieves a competitive ratio of 2 with a reordering buffer of size m. Considering that the best known ∗ Supported by DFG grant WE 2842/1. competitive ratio for uniformly related machines without reordering is 5.828, this result emphasizes the power of online reordering further more. 1.
Optimal semionline algorithms for preemptive scheduling problems with inexact partial information
, 2007
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SemiOnline Preemptive Scheduling: One Algorithm for All Variants
, 2008
"... We present a unified optimal semionline algorithm for preemptive scheduling on uniformly related machines with the objective to minimize the makespan. This algorithm works for all types of semionline restrictions, including the ones studied before, like sorted (decreasing) jobs, known sum of proc ..."
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We present a unified optimal semionline algorithm for preemptive scheduling on uniformly related machines with the objective to minimize the makespan. This algorithm works for all types of semionline restrictions, including the ones studied before, like sorted (decreasing) jobs, known sum of processing times, known maximal processing time, their combinations, and so on. Based on the analysis of this algorithm, we derive some global relations between various semionline restrictions and tight bounds on the approximation ratios for a small number of machines.
Optimal online algorithms for the uniform machine scheduling problem with ordinal data
 Inf. Comput
"... In this paper, we consider an ordinal online scheduling problem. A sequence of n independent jobs has to be assigned nonpreemptively to two uniformly related machines. We study two objectives which are maximizing the minimum machine completion time, and minimizing the lp norm of the completion tim ..."
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In this paper, we consider an ordinal online scheduling problem. A sequence of n independent jobs has to be assigned nonpreemptively to two uniformly related machines. We study two objectives which are maximizing the minimum machine completion time, and minimizing the lp norm of the completion times. It is assumed that the values of the processing times of jobs are unknown at the time of assignment. However it is known in advance that the processing times of arriving jobs are sorted in a nonincreasing order. We are asked to construct an assignment of all jobs to the machines at time zero, by utilizing only ordinal data rather than actual magnitudes of jobs. For the problem of maximizing the minimum completion time we first present a comprehensive lower bound on the competitive ratio, which is a piecewise function of machine speed ratio s. Then we propose an algorithm which is optimal for any s ≥ 1. For minimizing the lp norm we study the case of identical machines (s = 1) and present tight bounds as a function of p.
Optimal Preemptive Online Algorithms for Scheduling with Known Largest Size on two Uniform Machines
 Acta Mathematica Sinica
, 2007
"... Abstract In this paper, we consider the semionline preemptive scheduling problem with known largest job sizes on two uniform machines. Our goal is to maximize the continuous period of time (starting from time zero) when both machines are busy, which is equivalent to maximizing the minimum machine c ..."
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Abstract In this paper, we consider the semionline preemptive scheduling problem with known largest job sizes on two uniform machines. Our goal is to maximize the continuous period of time (starting from time zero) when both machines are busy, which is equivalent to maximizing the minimum machine completion time if idle time is not introduced. We design optimal deterministic semionline algorithms for every machine speed ratio s ∈ [1, ∞), and show that idle time is required to achieve the optimality during the assignment procedure of the algorithm for any s>(s 2 +3s +1)/(s 2 +2s +1). The competitive ratio of the algorithms is (s 2 +3s +1)/(s 2 +2s + 1), which matches the randomized lower bound for every s ≥ 1. Hence randomization does not help for the discussed preemptive scheduling problem.
Optimal online algorithms to minimize makespan on two machines with resource augmentation
, 2003
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Barcelona Aarhus Barcelona
, 2002
"... This is the second annual progress report for the ALCOMFT project, supported by the European ..."
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This is the second annual progress report for the ALCOMFT project, supported by the European
Matematickofyzikální fakulta
"... Combinatorial algorithms for online problems: Semionline scheduling on related machines ..."
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Combinatorial algorithms for online problems: Semionline scheduling on related machines