ONLINE MINIMUM MAKESPAN SCHEDULING WITH A BUFFER

2014 ◽  
Vol 25 (05) ◽  
pp. 525-536 ◽  
Author(s):  
NING DING ◽  
YAN LAN ◽  
XIN CHEN ◽  
GYÖRGY DÓSA ◽  
HE GUO ◽  
...  
Keyword(s):  
Competitive Ratio ◽  
Online Algorithm ◽  
Buffer Size ◽  
Scheduling Problem ◽  
Uniform Machines ◽  
Minimum Makespan ◽  
Identical Machines ◽  
Better Than

In this paper we study an online minimum makespan scheduling problem with a reordering buffer. We obtain the following results: (i) for m > 51 identical machines, we give a 1.5-competitive online algorithm with a buffer of size ⌈1.5m⌉; (ii) for three identical machines, we give an optimal online algorithm with a buffer size six, better than the previous nine; (iii) for m uniform machines, using a buffer of size m, we improve the competitive ratio from 2 + ε to 2 − 1/m+ ε, where ε > 0 is sufficiently small and m is a constant.

ONLINE SCHEDULING OF MIXED CPU-GPU JOBS

2014 ◽  
Vol 25 (06) ◽  
pp. 745-761 ◽  
Author(s):  
LIN CHEN ◽  
DESHI YE ◽  
GUOCHUAN ZHANG
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Online Scheduling ◽  
Scheduling Problem ◽  
Competitive Algorithm ◽  
Processing Times ◽  
Special Cases ◽  
Identical Machines ◽  
The One

We consider the online scheduling problem in a CPU-GPU cluster. In this problem there are two sets of processors, the CPU processors and the GPU processors. Each job has two distinct processing times, one for the CPU processor and the other for the GPU processor. Once a job is released, a decision should be made immediately about which processor it should be assigned to. The goal is to minimize the makespan, i.e., the largest completion time among all the processors. Such a problem could be seen as an intermediate model between the scheduling problem on identical machines and unrelated machines. We provide a 3.85-competitive online algorithm for this problem and show that no online algorithm exists with competitive ratio strictly less than 2. We also consider two special cases of this problem, the balanced case where the number of CPU processors equals to that of GPU processors, and the one-sided case where there is only one CPU or GPU processor. For the balanced case, we first provide a simple 3-competitive algorithm, and then a better algorithm with competitive ratio of 2.732 is derived. For the one-sided case, a 3-competitive algorithm is given.

ONLINE ALGORITHMS FOR SCHEDULING WITH MACHINE ACTIVATION COST

2007 ◽  
Vol 24 (02) ◽  
pp. 263-277 ◽  
Author(s):  
YONG HE ◽  
SHUGUANG HAN ◽  
YIWEI JIANG
Keyword(s):  
Lower Bound ◽  
Online Algorithms ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Machine Scheduling ◽  
Parallel Machine Scheduling ◽  
Parallel Machine ◽  
Scheduling Problem ◽  
Identical Machines ◽  
Parallel Machine Scheduling Problem

In this paper, we consider a variant of the classical parallel machine scheduling problem. For this problem, we are given m potential identical machines to non-preemptively process a sequence of independent jobs. Machines need to be activated before starting to process, and each machine activated incurs a fixed machine activation cost. No machines are initially activated, and when a job is revealed the algorithm has the option to activate new machines. The objective is to minimize the sum of the makespan and activation cost of machines. We first present two optimal online algorithms with competitive ratios of 3/2 and 5/3 for m = 2, 3 cases, respectively. Then we present an online algorithm with a competitive ratio of at most 2 for general m ≥ 4, while the lower bound is 1.88.

scholarly journals Online Makespan Scheduling with Job Migration on Uniform Machines

Algorithmica ◽  
2021 ◽  
Author(s):  
Matthias Englert ◽  
David Mezlaf ◽  
Matthias Westermann
Keyword(s):  
Competitive Ratio ◽  
Online Algorithm ◽  
Parallel Machines ◽  
Scheduling Algorithm ◽  
Input Sequence ◽  
Job Migration ◽  
Average Completion Time ◽  
Uniform Machine ◽  
Uniform Machines ◽  
Completion Times

AbstractIn 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.

scholarly journals Online Clique Clustering

Algorithmica ◽  
2019 ◽  
Vol 82 (4) ◽  
pp. 938-965
Author(s):  
Marek Chrobak ◽  
Christoph Dürr ◽  
Aleksander Fabijan ◽  
Bengt J. Nilsson
Keyword(s):  
Objective Function ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Optimal Solution ◽  
Deterministic Algorithm ◽  
Input Graph ◽  
Online Clustering ◽  
Objective Value ◽  
Asymptotic Competitive Ratio ◽  
Better Than

Abstract Clique clustering is the problem of partitioning the vertices of a graph into disjoint clusters, where each cluster forms a clique in the graph, while optimizing some objective function. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The goal is to maintain a clustering with an objective value close to the optimal solution. For the variant where we want to maximize the number of edges in the clusters, we propose an online algorithm based on the doubling technique. It has an asymptotic competitive ratio at most 15.646 and a strict competitive ratio at most 22.641. We also show that no deterministic algorithm can have an asymptotic competitive ratio better than 6. For the variant where we want to minimize the number of edges between clusters, we show that the deterministic competitive ratio of the problem is $$n-\omega (1)$$n-ω(1), where n is the number of vertices in the graph.

OPTIMAL PREEMPTIVE SEMI-ONLINE ALGORITHM FOR SCHEDULING TIGHTLY-GROUPED JOBS ON TWO UNIFORM MACHINES

2006 ◽  
Vol 23 (01) ◽  
pp. 77-88 ◽  
Author(s):  
YIWEI JIANG ◽  
YONG HE
Keyword(s):  
Online Algorithm ◽  
Size Ratio ◽  
Speed Ratio ◽  
Preemptive Scheduling ◽  
Scheduling Problem ◽  
Time Zero ◽  
Uniform Machines ◽  
Machine Speed

In this paper, we consider a semi-online preemptive scheduling problem on two uniform machines, where we assume that all jobs have sizes between p and rp for some p > 0 and r ≥ 1. The goal is to maximize the continuous period of time (starting from time zero) when both machines are busy. We present an optimal semi-online algorithm for any combination of the job size ratio r and machine speed ratio s.

SEMI-ON-LINE SCHEDULING PROBLEM FOR MAXIMIZING THE MINIMUM MACHINE COMPLETION TIME ON THREE SPECIAL UNIFORM MACHINES

2005 ◽  
Vol 22 (02) ◽  
pp. 229-237 ◽  
Author(s):  
RUN-ZI LUO ◽  
SHI-JIE SUN
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Processing Time ◽  
Scheduling Problem ◽  
Uniform Machines ◽  
On Line ◽  
Special Case

In this paper, we investigate a semi-on-line version for a special case of three machines M1, M2, M3 where the processing time of the largest job is known in advance. A speed si(s1 = s2 = 1, 1 ≤ s3 = s) is associated with machine Mi. Our goal is to maximize the C min — the minimum workload of three machines. We give a C min 3 algorithm and prove its competitive ratio is [Formula: see text] and the algorithm is the best possible for 1 ≤ s ≤ 2. We also claim the competitive ratio of algorithm C min 3 is tight.

A SEMI-ON-LINE SCHEDULING PROBLEM OF TWO PARALLEL MACHINES WITH COMMON MAINTENANCE TIME

2012 ◽  
Vol 29 (04) ◽  
pp. 1250020 ◽  
Author(s):  
YUHUA CAI ◽  
QI FENG ◽  
WENJIE LI
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Parallel Machines ◽  
Time Interval ◽  
Scheduling Problem ◽  
Maintenance Time ◽  
Identical Machines ◽  
On Line

In this paper, we consider a semi-on-line scheduling problem of two identical machines with common maintenance time interval and nonresumable availability. We prove a lower bound of 2.79129 on the competitive ratio and give an on-line algorithm with competitive ratio 2.79633 for this problem.

scholarly journals A Competitive Online Algorithm for Minimizing Total Weighted Completion Time on Uniform Machines

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Xuyang Chu ◽  
Jiping Tao
Keyword(s):  
Completion Time ◽  
Online Algorithm ◽  
Total Weighted Completion Time ◽  
Scheduling Problem ◽  
Uniform Machines ◽  
Online Setting ◽  
Weighted Completion Time ◽  
Waiting Strategy ◽  
Machine Speed ◽  
Over Time

We consider the classic online scheduling problem on m uniform machines in the online setting where jobs arrive over time. Preemption is not allowed. The objective is to minimize total weighted completion time. An online algorithm based on the directly waiting strategy is proposed. Its competitive performance is proved to be max2smax1−1/2∑si,2smax/1+smax2.5−1/2m by the idea of instance reduction, where sm is the fastest machine speed after being normalized by the slowest machine speed.

Improved Algorithms for Online Scheduling of Malleable Parallel Jobs on Two Identical Machines

2015 ◽  
Vol 32 (05) ◽  
pp. 1550034
Author(s):  
Hao Zhou ◽  
Ping Zhou ◽  
Yiwei Jiang
Keyword(s):  
Execution Time ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Setup Time ◽  
Online Scheduling ◽  
Parallel Jobs ◽  
Identical Machines ◽  
Maximum Completion Time ◽  
One Machine ◽  
Two Machines

This paper addresses online scheduling of malleable parallel jobs to minimize the maximum completion time, i.e., makespan. It is assumed that the execution time of a job Jj with processing time pj is pj/k + (k-1)c if the job is assigned to k machines, where c > 0 is a constant setup time. We consider online algorithms for the scheduling problem on two identical machines. Namely, the job Jj can be processed on one machine with execution time pj or alternatively two machines in parallel with execution time pj/2+c. For the asymptotical competitive ratio, we provide an improved online algorithm with makespan no more than (3/2)C* +c/2, where C* is the optimal makespan. For the strict competitive ratio, we propose an online algorithm with competitive ratio of 1.54, which is close to the lower bound of 1.5.

Online Single Machine Scheduling to Minimize the Maximum Starting Time

2017 ◽  
Vol 34 (05) ◽  
pp. 1750022
Author(s):  
Lingfa Lu ◽  
Liqi Zhang
Keyword(s):  
Single Machine ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Optimal Schedule ◽  
Machine Scheduling ◽  
Single Machine Scheduling ◽  
Scheduling Problem ◽  
Starting Time

In this paper, we consider the online single machine scheduling problem to minimize the maximum starting time of the jobs. For the non-preemptive model, we show that there is no determined or randomized online algorithm with a bounded competitive ratio. For the preemption-resume model, we show that the well-known SRPT rule yields an optimal schedule. For the preemption-restart model, we show that any determined online algorithm has a competitive ratio of at least 2 and present an online algorithm with the best-possible competitive ratio of 2.

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