Online-List Scheduling on a Single Bounded Parallel-Batch Machine to Minimize Makespan

2015 ◽  
Vol 32 (04) ◽  
pp. 1550028
Author(s):  
Wenhua Li ◽  
Jie Gao ◽  
Jinjiang Yuan
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Positive Root ◽  
List Scheduling ◽  
Batch Machine

In this paper, we consider the online-list scheduling on a single bounded parallel-batch machine to minimize makespan. In the problem, the jobs arrive online over list. The first unassigned job in the list should be assigned to a batch before the next job is released. Each batch can accommodate up to b jobs. For b = 2, we establish a lower bound 1 + γ of competitive ratio and provide an online algorithm with a competitive ratio of [Formula: see text], where γ is the positive root of γ(γ + 1)2 = 1. For b = 3, we establish a lower bound 1 + α of competitive ratio and provide an online algorithm with a competitive ratio of 2, where α is the positive root of the equation (1 + α)(1 + α2) = 2.

A BEST POSSIBLE ONLINE ALGORITHM FOR SCHEDULING TO MINIMIZE MAXIMUM FLOW-TIME ON BOUNDED BATCH MACHINES

2014 ◽  
Vol 31 (04) ◽  
pp. 1450030 ◽  
Author(s):  
CHENGWEN JIAO ◽  
WENHUA LI ◽  
JINJIANG YUAN
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Unit Length ◽  
Maximum Flow ◽  
Flow Time ◽  
Release Time ◽  
Batch Machine ◽  
The Difference ◽  
Over Time

We consider online scheduling of unit length jobs on m identical parallel-batch machines. Jobs arrive over time. The objective is to minimize maximum flow-time, with the flow-time of a job being the difference of its completion time and its release time. A parallel-batch machine can handle up to b jobs simultaneously as a batch. Here, the batch capacity is bounded, that is b < ∞. In this paper, we provide a best possible online algorithm for the problem with a competitive ratio of [Formula: see text].

scholarly journals COMPETITIVE ALGORITHMS FOR ONLINE PRICING

2012 ◽  
Vol 04 (02) ◽  
pp. 1250015 ◽  
Author(s):  
YONG ZHANG ◽  
YUXIN WANG ◽  
FRANCIS Y. L. CHIN ◽  
HING-FUNG TING
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Value Function ◽  
Unit Price ◽  
Competitive Algorithms ◽  
Online Pricing

Given a seller with m items, a sequence of users {u1, u2, …} come one by one, the seller must set the unit price and assign some items to each user on his/her arrival. Items can be sold fractionally. Each ui has his/her value function vi(⋅) such that vi(x) is the highest unit price ui is willing to pay for x items. The objective is to maximize the revenue by setting the price and number of items for each user. In this paper, we have the following contributions: if the highest value h among all vi(x) is known in advance, we first show the lower bound of the competitive ratio is ⌊ log h⌋/2, then give an online algorithm with competitive ratio 4⌊ log h⌋ + 6; if h is not known in advance, we give an online algorithm with competitive ratio 2⋅h log -1/2 h + 8⋅h3 log -1/2 h.

An Improved Online Algorithm for the Online Preemptive Scheduling of Equal-Length Intervals on a Single Machine with Lookahead

2015 ◽  
Vol 32 (06) ◽  
pp. 1550047
Author(s):  
Wenjie Li ◽  
Jinjiang Yuan
Keyword(s):  
Lower Bound ◽  
Single Machine ◽  
Operations Research ◽  
Competitive Ratio ◽  
Processing Time ◽  
Online Algorithm ◽  
Equal Length ◽  
Preemptive Scheduling ◽  
Time Segment ◽  
Time Point

This paper studies the online preemptive scheduling of equal-length intervals on a single machine with lookahead. Let [Formula: see text] be the length (processing time) of all intervals. In the problem, at every time point [Formula: see text], online algorithms can foresee all the intervals that will arrive in the time segment [Formula: see text] for a certain [Formula: see text]. When [Formula: see text], Zheng et al. [Comput- ers & Operations Research, 2013] established a lower bound of [Formula: see text] and provided an online algorithm with a competitive ratio of 3. In this paper, we provide for this problem an improved online algorithm with a competitive ratio of 2.

scholarly journals Online Scheduling with Delivery Time on a Bounded Parallel Batch Machine with Limited Restart

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Hailing Liu ◽  
Long Wan ◽  
Zhigang Yan ◽  
Jinjiang Yuan
Keyword(s):  
Positive Solution ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Online Scheduling ◽  
Equal Length ◽  
Delivery Time ◽  
Batch Machine ◽  
Time Scheduling ◽  
Work Done ◽  
Over Time

We consider the online (over time) scheduling of equal length jobs on a bounded parallel batch machine with batch capacitybto minimize the time by which all jobs have been delivered with limited restart. Here, “restart” means that a running batch may be interrupted, losing all the work done on it, and jobs in the interrupted batch are then released and become independently unscheduled jobs, called restarted jobs. “Limited restart” means that a running batch which contains some restarted jobs cannot be restarted again. Whenb=2, we propose a best possible online algorithmH(b=2)with a competitive ratio of1+α, whereαis the positive solution of2α(1+α)=1. Whenb≥3, we present a best possible online algorithmH(b≥3)with a competitive ratio of1+β, whereβis the positive solution ofβ(1+β)2=1.

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.

Online Machine Scheduling with Family Setups

2016 ◽  
Vol 33 (04) ◽  
pp. 1650027
Author(s):  
Lele Zhang ◽  
Andrew Wirth
Keyword(s):  
Lower Bound ◽  
Single Machine ◽  
Competitive Analysis ◽  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Machine Scheduling ◽  
Processing Times ◽  
Family Setups ◽  
Special Case

We consider the problem of online scheduling a single machine with family setups under job availability. A setup must be scheduled when the next job comes from a different family from the last completed one, if any. The aim is to minimize the total completion time of all jobs. For the special case of identical processing times, we provide a lower bound for the competitive ratio and an online algorithm with its competitive analysis.

Online Algorithms for Scheduling Unit Length Jobs on Unbounded Parallel-Batch Machines with Linearly Lookahead

2019 ◽  
Vol 36 (05) ◽  
pp. 1950024
Author(s):  
Chengwen Jiao ◽  
Jinjiang Yuan ◽  
Qi Feng
Keyword(s):  
Online Algorithms ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Positive Root ◽  
Unit Length ◽  
Online Scheduling ◽  
Time Interval ◽  
Steady Growth ◽  
Scheduling Model

In this paper, we propose a new online scheduling model with linear lookahead intervals, which has the character that at any time [Formula: see text], one can foresee the jobs that will coming in the time interval [Formula: see text] in which [Formula: see text]. In this new lookahead model, the length of the lookahead intervals are variable as the time going on and the number of jobs increasing, and has the tend of steady growth. In this paper, we consider online scheduling of unit length jobs on [Formula: see text] identical parallel-batch machines under this new lookahead model to minimize makespan. The batch capacity is unbounded, that is [Formula: see text]. We present an optimal online algorithm for [Formula: see text], and provide a best possible online algorithm of competitive ratio [Formula: see text] for [Formula: see text], where [Formula: see text] is the positive root of [Formula: see text].

Online Scheduling of Incompatible Family Jobs with Equal Length on an Unbounded Parallel-Batch Machine with Job Delivery

2018 ◽  
Vol 35 (04) ◽  
pp. 1850026
Author(s):  
Qijia Liu ◽  
Jinjiang Yuan
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Online Scheduling ◽  
Equal Length ◽  
Batch Machine ◽  
Job Delivery ◽  
Over Time

In this paper, we consider the online scheduling of incompatible family jobs with equal length on an unbounded parallel-batch machine with job delivery. The jobs arrive online over time and belong to [Formula: see text] incompatible job families, where [Formula: see text] is known in advance. The jobs are first processed in batches on an unbounded parallel-batch machine and then the completed jobs are delivered in batches by a vehicle with infinite capacity to their customers. The jobs from distinct families cannot be processed and delivered in the same batch. The objective is to minimize the maximum delivery completion time of the jobs. For this problem, we present an online algorithm with the best competitive ratio of [Formula: see text].

Online Minimum Spanning Tree with Advice

2018 ◽  
Vol 29 (04) ◽  
pp. 505-527
Author(s):  
Maria Paola Bianchi ◽  
Hans-Joachim Böckenhauer ◽  
Tatjana Brülisauer ◽  
Dennis Komm ◽  
Beatrice Palano
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Competitive Ratio ◽  
Spanning Tree ◽  
Online Algorithm ◽  
Minimum Spanning Tree ◽  
Optimal Solution ◽  
Bounded Degree ◽  
Advice Complexity ◽  
Graph Classes

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size [Formula: see text], we show an asymptotically tight bound of [Formula: see text] on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., [Formula: see text] advice bits allow to compute an optimal result if the weight function equals the Euclidean distance; if the graph is complete and has two different edge weights, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal’s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem, even on graphs with three different edge weights.

An Optimal Preemptive Algorithm for Online MapReduce Scheduling on Two Parallel Machines

2018 ◽  
Vol 35 (03) ◽  
pp. 1850013 ◽  
Author(s):  
Yiwei Jiang ◽  
Wei Zhou ◽  
Ping Zhou
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Parallel Machines ◽  
Online Scheduling ◽  
Scheduling Problem ◽  
Mapreduce Scheduling

In this paper, we study an online scheduling on two parallel machines in MapReduce-like system where each job contains two kinds of tasks: map tasks and reduce tasks. A job’s reduce tasks can only be processed after all its map tasks are finished. We assume that the map tasks are fractional and the reduce tasks are preemptive. Our objective is to minimize makespan. We show that the lower bound for this MapReduce scheduling problem is [Formula: see text]. We then present an online algorithm with competitive ratio of [Formula: see text] and thus it is optimal.

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