Semi-online Version Research Articles

Semi-Online Hierarchical Scheduling on Two Machines for lp-Norm Load Balancing

This paper investigates semi-online hierarchical scheduling problems on two identical machines, with the purpose of minimizing the [Formula: see text]-norm of the machines’ loads. We consider two semi-online versions with knowing the total processing time [Formula: see text] of all jobs, or knowing the total processing time [Formula: see text] of the jobs of hierarchy [Formula: see text] for [Formula: see text] in advance. For the two semi-online versions, the best possible online algorithms are designed with competitive ratios of [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is the maximum value of the function [Formula: see text] in [Formula: see text]. When [Formula: see text], our results cover the known results for minimizing the makespan.

Tight upper bounds for semi-online scheduling on two uniform machines with known optimum

We consider a semi-online version of the problem of scheduling a sequence of jobs of different lengths on two uniform machines with given speeds 1 and s. Jobs are revealed one by one (the assignment of a job has to be done before the next job is revealed), and the objective is to minimize the makespan. In the considered variant the optimal offline makespan is known in advance. The most studied question for this online-type problem is to determine the optimal competitive ratio, that is, the worst-case ratio of the solution given by an algorithm in comparison to the optimal offline solution. In this paper, we make a further step towards completing the answer to this question by determining the optimal competitive ratio for s between frac{5 + sqrt{241}}{12} approx 1.7103 and sqrt{3} approx 1.7321, one of the intervals that were still open. Namely, we present and analyze a compound algorithm achieving the previously known lower bounds.

Semi-online hierarchical scheduling for $$l_p$$ l p -norm load balancing with buffer or rearrangements

In this paper, we consider the semi-online hierarchical scheduling for load balancing on two identical machines. In the problem, the jobs are available online over list and the objective is to minimize the \(l_p\)-norm of the two machines’ loads. Two semi-online versions are investigated: the buffer version and the rearrangement version. We design a unified optimal semi-online algorithm for both models.

Semi-online hierarchical load balancing problem with bounded processing times

In this paper, we consider the online hierarchical scheduling problem on two parallel machines, with the objective of maximizing the minimum machine load. Since no competitive algorithm exists for this problem, we consider the semi-online version with bounded processing times, in which the processing times are bounded by an interval [1,α] where α≥1. We prove that no algorithm can have a competitive ratio less than 1+α and present an optimal algorithm. Moreover, if we further know the sum of jobs' processing times in advance, we prove that no algorithm can have a competitive ratio less than α where 1≤α<2, and propose an α-competitive algorithm for this case.

Semi-Online Scheduling on Two Machines withGoSLevels and Partial Information of Processing Time

This paper investigates semi-online scheduling problems on two parallel machines under a grade of service (GoS) provision subject to minimize the makespan. We consider three different semi-online versions with knowing the total processing time of the jobs with higher GoS level, knowing the total processing time of the jobs with lower GoS level, or knowing both in advance. Respectively, for the three semi-online versions, we develop algorithms with competitive ratios of 3/2, 20/13, and 4/3 which are shown to be optimal.

On-line and semi-online scheduling for flow shop problems on two machines

We investigate the on-line and semi-online scheduling for flow shop problems on two machines with the objective to minimize the makespan. Several lower bounds for the problems with and without preemption are derived. For some special cases of semi-online version, we provide two on-line algorithms with a competitive ratio of . Finally we give a lower bound for the problem with unavailable time interval on the first machine. And for the problem with the interval on the second machine, we show that no on-line algorithm has a constant competitive ratio.

Online Clustering on the Line with Square Cost Variable Sized Clusters

In the online clustering problems, the classification of points into sets (called clusters) is done in an online fashion. Points arrive one by one at arbitrary locations, to be assigned to clusters at the time of arrival without any information about the further points. A point can be assigned to an existing cluster, or a new cluster can be opened for it. Existing clusters cannot be merged or split. We study one-dimensional variants. The cost of a cluster is the sum of a fixed setup cost scaled to 1 and the square of the diameter of the cluster. The goal is to minimize the sum of costs of the clusters used by the algorithm. In this paper we investigate the problem on the line. We examine two versions, both maintaining the properties that a point which was assigned to a given cluster must remain assigned to this cluster, and clusters cannot be merged. In the strict variant, the size and the exact location of the cluster must be fixed when it is initialized. In the flexible variant, the algorithm can shift the cluster or expand it, as long as it contains all points assigned to it. We consider the online and the semi-online (the input is sorted according to their coordinates from smallest to largest i.e., from left to right) versions of the above two variants. We present the first online algorithms for the solution of the problem. We describe algorithms for the strict and the flexible variant both for the online and semi-online versions. We also give lower bounds on the possible competitive ratio in all of the cases.

Semi-Online Machine Covering under a Grade of Service Provision

Two semi-online scheduling problems on two parallel identical machines under a grade of service (GoS) provision were studied. The goal is to maximize the minimum machine load. For the semi-online version where the largest processing time of all jobs is known in advance, we show that no competitive algorithm exists. For the semi-online version where the optimal offline value is known in advance, we propose an optimal algorithm with competitive ratio 2.

Optimal semi-online scheduling algorithms on two parallel identical machines under a grade of service provision

This paper investigates semi-online scheduling problems on two parallel identical machines under a grade of service (GoS) provision. We consider two different semi-online versions where the optimal offline value of the instance is known in advance or the largest processing time of all jobs is known in advance. Respectively for the two semi-online models, two algorithms with competitive ratios of 3/2 and (5+1)/2 are shown to be optimal.

ONLINE AND SEMI-ONLINE SCHEDULING ON CAPACITATED TWO-PARALLEL MACHINES

In this paper, we investigate the capacitated two-parallel machines scheduling problem, where one machine is only available for a special period of time after which it can no longer process any job while the other machine is continuously available. Our objective is to minimize the completion time of the machine which is continuously available. The offline version of the problem is equivalent to the minimization version of the Subset-Sum problem. We first show the lower bound of the online version is infinite. We also consider the semi-online version with known the total job processing time in advance, for which both lower bound and semi-online algorithms are given.

Online variable-sized bin packing with conflicts

We study a new kind of online bin packing with conflicts, motivated by a problem arising when scheduling jobs on the Grid. In this bin packing problem, the set of items is given at the beginning, together with a set of conflicts on pairs of items. A conflict on a pair of items implies that they cannot be assigned to a common bin. The online scenario is realized as follows. Variable-sized bins arrive one by one, and items need to be assigned to each bin before the next bin arrives. We analyze the online problem as well as semi-online versions of it, which are the variant where the sizes of the arriving bins are monotonically non-increasing as well as the variant where they are monotonically non-decreasing.

Semi on-line scheduling on three processors with known sum of the tasks

We consider a semi on-line version of the multiprocessor scheduling problem on three processors, where the total size of the tasks is known in advance. We prove a lower bound $1+\frac{\sqrt{129}-9}{6}>1.3929$ on the competitive ratio of any algorithm and propose a simple algorithm with competitive ratio equal to 1.5. The performance is improved to $1+\frac{8}{19}<1.4211$ by a preprocessing strategy. The latter algorithm is only 2% away from the lower bound.

SEMI-ONLINE MACHINE COVERING

This paper investigates two different semi-online versions of the machine covering, which is the problem of assigning a set of jobs to a system of m(m ≥ 3) identical parallel machines so as to maximize the earliest machine completion time. In the first case, we assume that the largest processing times is known in advance. In the second case, we assume that the total processing times of all jobs is known in advance. For each version we propose a semi-online algorithm and investigate its competitive ratio. The competitive ratio of each algorithm is [Formula: see text], which is shown to be the best possible competitive ratio for each semi-online problem.

Optimal semi-online algorithms for machine covering

This paper investigates the semi-online machine covering problems on m ≥ 3 parallel identical machines. Three different semi-online versions are studied and optimal algorithms are proposed. We prove that if the total processing time of all jobs or the largest processing time of all jobs is known in advance, the competitive ratios of the optimal algorithms are both m − 1 . If the total processing time and the largest processing time of all jobs are both known in advance, the competitive ratios of the optimal algorithms are 3 2 when m = 3 , and m − 2 when m ≥ 4 .

Extension of algorithm list scheduling for a semi-online scheduling problem

A general algorithm, called ALG, for online and semi-online scheduling problem Pm||Cmax with m ≥ 2 is introduced. For the semi-online version, it is supposed that all job have their processing times within the interval [p, rp], where p > 0,1 < r ≤ m/m − 1. ALG is a generalization of LS and is optimal in the sense that there is not an algorithm with smaller competitive ratio than that of ALG.

Semi On-Line Scheduling Problem for Maximizing the Minimum Machine Completion Time on Two Uniform Machines

In this paper, we consider a semi on-line version on two uniform machines Mi, i = 1,2, where the processing time of the largest job is known in advance. A speed si(s1 = 1, 1≤ s2 = s) is associated with machine Mi. Our goal is to maximize the Cmin. We give a Cmin 2 algorithm and prove its competitive ratio is at most \(\frac{2s+1}{s+1}\). We also claim the Cmin 2 algorithm is tight and the gap between the competitive ratio of Cmin 2 algorithm and the optimal value is not greater than 0.555. It is obvious that our result coincides with that given by He for s = 1.

New bounds and algorithms for on-line scheduling: two identical processors, known sum and upper bound on the tasks

In this paper we study a semi on-line version of the classical multiprocessor scheduling problem on two identical processors. We assume that the sum of the tasks and an upper bound gamma on the size of each task are known. Each task has to be assigned upon arrival and the assignment cannot be changed later. The objective is the minimization of the maximum completion time on the processors. In this paper we propose new algorithms and improve known lower and upper bounds on the competitive ratio. Algorithms and bounds depend on the value of gamma. An optimal algorithm is obtained for gamma in the interval [ 1/n,2(n+1)/n(2n+1) ] and gamma = (2n-1)/2n(n-1), where n is any integer value larger or equal 2.

Semi-online scheduling jobs with tightly-grouped processing times on three identical machines

This paper investigates the semi-online version of scheduling problem P | | C max on a three-machine system. We assume that all jobs have their processing times between p and rp ( p > 0 , r ⩾ 1 ). We give a comprehensive competitive ratio of LS algorithm which is a piecewise function on r ⩾ 1 . It shows that LS is an optimal semi-online algorithm for every r ∈ [ 1 , 1.5 ] , [ 3 , 2 ] and [ 6 , + ∞ ) . We further present an optimal algorithm for every r ∈ [ 2 , 2.5 ] , and an almost optimal algorithm for every r ∈ ( 2.5 , 3 ] where the largest gap between its competitive ratio and the lower bound of the problem is at most 0.01417. We also present an improved algorithm with smaller competitive ratio than that of LS for every r ∈ ( 3 , 6 ) .

Semi-Online Algorithms for Parallel Machine Scheduling Problems

This paper considers two semi-online versions of scheduling problem P2||Cmax where one type of partial information is available and one type of additional algorithmic extension is allowed simultaneously. For the semi-online version where a buffer of length 1 is available and the total size of all jobs is known in advance, we present an optimal algorithm with competitive ratio 5/4. We also show that it does not help that the buffer length is greater than 1. For the semi-online version where two parallel processors are available and the total size of all jobs is known in advance, we present an optimal algorithm with Competitive ratio 6/5.

Semi-online scheduling with machine cost

For most scheduling problems the set of machines is fixed initially and remains unchanged for the duration of the problem. Recently Imreh and Nogaproposed to add the concept of machine cost to scheduling problems and considered the so-called List Model problem. An online algorithm with a competitive ratio 1.618 was given while the lower bound is 4/3. In this paper, two different semi-online versions of this problem are studied. In the first case, it is assumed that the processing time of the largest job is known a priori. A semi-online algorithm is presented with the competitive ratio at most 1.5309 while the lower bound is 4/3. In the second case, it is assumed that the total processing time of all jobs is known in advance. semi-online algorithm is presented with the competitive ratio at most 1.414 while the lower bound is 1.161. It is shown that the additional partial available information about the jobs leads to the possibility of constructing a schedule with a smaller competitive ratio than that of online algorithms.