Preemptive online algorithms for scheduling with machine cost

For most scheduling problems the set of machines is fixed initially and remains unchanged. Recently Imreh and Noga proposed adding the concept of machine cost to scheduling problems and considered the so-called List Model problem. For this problem, we are given a sequence of independent jobs with positive sizes, which must be processed non-preemptively on a machine. No machines are initially provided, and when a job is revealed the algorithm has the option to purchase new machines. The objective is to minimize the sum of the makespan and cost of machines. In this paper, a modified model of List Model is presented where preemption is allowed. For this model, it is shown that better performance is possible. We present an online algorithm with a competitive ratio of while the lower bound is 4/3. For the semi-online problem with decreasing sizes, we design an optimal algorithm with a competitive ratio of 4/3, which improves the known upper bound of 3/2. The algorithm does not introduce any preemption, and hence is also an optimal semi-online algorithm for the non-preemptive semi-online problem. For the semi-online problem with known largest size, we present an optimal algorithm with a competitive ratio of 4/3.Received: 7 June 2004, Published online: 11 November 2004This research is supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, China, and National Natural Science Foundation of China (10271110, 60021201).     

Preemptive and Non-preemptive On-line Algorithms for Scheduling with Rejection on Two Uniform Machines

In this paper, we consider the problem of on-line scheduling a job sequence on two uniform machines. A job can be either rejected, in which case we pay its penalty, or scheduled on machines, in which case it contributes its processing time to the makspan of the constructed schedule. The objective is to minimize the sum of the makespan of the schedule for all accepted jobs and the penalties of all rejected jobs. Both preemptive and non-preemptive versions are considered. For the preemptive version, we present an optimal on-line algorithm with a competitive ratio for any s≥1, where s is the machine speed ratio. For the non-preemptive version, we present an improved lower bound. Moreover, as an optimal algorithm for s≥1.6180 is known, we present a modified version of the known algorithm, and show that it becomes optimal for any 1.3852≤s<1.6180 and has a smaller competitive ratio than that of original version for any 1≤s<1.3852. The maximum gap between its competitive ratio and the lower bound is 0.0534.     

Preemptive online scheduling with rejection of unit jobs on two uniformly related machines

We consider preemptive online and semi-online scheduling of unit jobs on two uniformly related machines. Jobs are presented one by one to an algorithm, and each job has a rejection penalty associated with it. A new job can either be rejected, in which case the algorithm pays its rejection penalty, or it can be scheduled preemptively on the machines, in which case it may increase the maximum completion time of any machine in the schedule, also known as the makespan of the constructed schedule. The objective is to minimize the sum of the makespan of the schedule of all accepted jobs and the total penalty of all rejected jobs. We study two versions of the problem. The first one is the online problem where the jobs arrive unsorted, and the second variant is the semi-online case, where the jobs arrive sorted by a non-increasing order of penalties. We also show that the variant where the jobs arrive sorted by a non-decreasing order of penalties is equivalent to the unsorted one. We design optimal online algorithms for both cases. These algorithms have smaller competitive ratios than the optimal competitive ratio for the more general problem with arbitrary processing times (except for the case of identical machines), but larger competitive ratios than the optimal competitive ratio for preemptive scheduling of unit jobs without rejection.     

Bin stretching revisited

We study three on-line models of bin stretching on two machines. For the case where the machines are identical and the jobs arrive sorted by non-increasing sizes, we show a tight bound of 10/9 on the competitive ratio. For two related machines, we show a preemptive algorithm with competitive ratio 1 for any speed ratio, and two new non-preemptive algorithms. We prove that the upper bound on the competitive ratio achieved by the non-preemptive algorithms is optimal for almost any speed ratio, and close to optimal for all other speed ratios. Received: 14 February 2002 / 18 November 2002 Research supported in part by the Israel Science Foundation, (grant No. 250/01-1).     

带两个服务等级的3台机半在线算法

This paper studies a semi-online hierarchical scheduling problem on three identical machines. In the problem, there is only one machine with hierarchy 1 and two machines with hierarchy 2, and the goal is to minimize the makespan. When the total size of low-hierarchy is known, an online algorithm with the competitive ratio of 5/3 and the lower bound of 3/2 is given. When the total size of high-hierarchy is known, an online algorithm with the competitive ratio of 9/5 and the lower bound of 3/2 is given. When the total size of each hierarchy is known, an online algorithm with the competitive ratio of 3/2 and the lower bound of 4/3 is given. When the total size of jobs is known, a best possible online algorithm with the competitive ratio of 3/2 is given.     

Scheduling parallel jobs to minimize the makespan

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a prespecified, job-dependent number of machines when being processed. We prove that the makespan of any nonpreemptive list-schedule is within a factor of 2 of the optimal preemptive makespan. This gives the best-known approximation algorithms for both the preemptive and the nonpreemptive variant of the problem. We also show that no list-scheduling algorithm can achieve a better performance guarantee than 2 for the nonpreemptive problem, no matter which priority list is chosen. List-scheduling also works in the online setting where jobs arrive over time and the length of a job becomes known only when it completes; it therefore yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. We show that no list-scheduling algorithm has a constant competitive ratio. Still, we present the first online algorithm for scheduling parallel jobs with a constant competitive ratio in this context. We also prove a new information-theoretic lower bound of 2.25 for the competitive ratio of any deterministic online algorithm for this model. Moreover, we show that 6/5 is a lower bound for the competitive ratio of any deterministic online algorithm of the preemptive version of the model jobs arriving over time.     

Optimal On-Line Algorithms to Minimize Makespan on Two Machines with Resource Augmentation

We study the problem of on-line scheduling on two uniformly related machines where the on-line algorithm has resources different from those of the off-line algorithm. We consider three versions of this problem, preemptive semi-online, non-preemptive on-line and preemptive on-line scheduling. For all these cases we design algorithms with best possible competitive ratios as functions of the machine speeds. This work was submitted as a part of the M.Sc. thesis of the second author. A preliminary version of this paper appeared in the proceedings of The First Workshop on Approximation and Online Algorithms (WAOA’03), pages 109–122.     

Semi-Online Algorithms for Parallel Machine Scheduling Problems

This paper considers two semi-online versions of scheduling problem P2||C_max 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.The second author is supported by TRAPOYT of China, National Natural Science Foundation of China (10271110). Corresponding author: Y. He.     

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 algorthm with a competitive ratio 1.618 was given while the lower boud is 4/3.In this paper,two different semi-onlne 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.A 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.     

Optimal semi-online algorithms for scheduling problems with reassignment on two identical machines

In this paper, we consider semi-online minimum makespan scheduling problem with reassignment on two identical machines. Two versions are discussed. In the first version, one can reassign the last job of one machine that is based on the problem proposed by Tan and Yu (2008) [1], in which case the last job of each machine is allowed to be reassigned. An optimal algorithm which has the same competitive ratio is presented. In the second version we consider the combination of the next two conditions: the total size of all jobs is known in advance and one can reassign the last job of one machine. For this problem an optimal algorithm with competitive ratio is also given.     

Optimal semi-online algorithms for preemptive scheduling problems with inexact partial information

In semi-online scheduling problems, we always assume that some partial additional information is exactly known in advance. This may not be true in some application. This paper considers semi-online problems on identical machines with inexact partial information. Three problems are considered, where we know in advance that the optimal value, or the largest job size are in given intervals, respectively, while their exact values are unknown. We give both lower bounds of the problems and competitive ratios of algorithms as functions of a so-called disturbance parameter r ∈[1, ∞). We establish for which r the inexact partial information is useful to improve the performance of a semi-online algorithm with respect to its pure online problem. Optimal preemptive semi-online algorithms are then obtained. Research supported by Natural Science Foundation of China (10671177) and Natural Science Foundation of Zhejiang Province (Y605316) and its preliminary version appeared in proceedings of ISAAC’05.     

Preemptive scheduling on two parallel machines with a single server

This paper addresses a preemptive scheduling problem on two parallel machines with a single server. Each job has to be loaded (setup) by the server before being processed on the machines. The preemption is allowed in this paper. The goal is to minimize the makespan. We first show that it is no of use to preempt the job during its setup time. Namely, every optimal preemptive schedule can be converted to another optimal schedule where all the setup times are non-preemptively performed on one machine. We then present an algorithm with a tight bound of 4/3 for the general case. Furthermore, we show that the algorithm can produce optimal schedules for two special cases: equal processing times and equal setup times, which are NP-hard in the non-preemptive version.     

Minimizing Makespan and Preemption Costs on a System of Uniform Machines

It is well known that for preemptive scheduling on uniform machines there exist polynomial time exact algorithms, whereas for non-preemptive scheduling there are probably no such algorithms. However, it is not clear how many preemptions (in total, or per job) suffice in order to guarantee an optimal polynomial time algorithm. In this paper we investigate exactly this hardness gap, formalized as two variants of the classic preemptive scheduling problem. In generalized multiprocessor scheduling (GMS) we have a job-wise or total bound on the number of preemptions throughout a feasible schedule. We need to find a schedule that satisfies the preemption constraints, such that the maximum job completion time is minimized. In minimum preemptions scheduling (MPS) the only feasible schedules are preemptive schedules with the smallest possible makespan. The goal is to find a feasible schedule that minimizes the overall number of preemptions. Both problems are NP-hard, even for two machines and zero preemptions. For GMS, we develop polynomial time approximation schemes, distinguishing between the cases where the number of machines is fixed, or given as part of the input. Our scheme for a fixed number of machines has linear running time, and can be applied also for instances where jobs have release dates, and for instances with arbitrary preemption costs. For MPS, we derive matching lower and upper bounds on the number of preemptions required by any optimal schedule. Our results for MPS hold for any instance in which a job, J_j, can be processed simultaneously by ρ_j machines, for some ρ_j ≥ 1.     

Semi-Online Preemptive Scheduling: One Algorithm for All Variants

We present a unified optimal semi-online algorithm for preemptive scheduling on uniformly related machines with the objective to minimize the makespan. This algorithm works for all types of semi-online 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 semi-online restrictions and tight bounds on the approximation ratios for a small number of machines.     

LP rounding and combinatorial algorithms for minimizing active and busy time

We consider fundamental scheduling problems motivated by energy issues. In this framework, we are given a set of jobs, each with a release time, deadline, and required processing length. The jobs need to be scheduled on a machine so that at most g jobs are active at any given time. The duration for which a machine is active (i.e., “on”) is referred to as its active time. The goal is to find a feasible schedule for all jobs, minimizing the total active time. When preemption is allowed at integer time points, we show that a minimal feasible schedule already yields a 3-approximation (and this bound is tight) and we further improve this to a 2-approximation via LP rounding techniques. Our second contribution is for the non-preemptive version of this problem. However, since even asking if a feasible schedule on one machine exists is NP-hard, we allow for an unbounded number of virtual machines, each having capacity of g. This problem is known as the busy time problem in the literature and a 4-approximation is known for this problem. We develop a new combinatorial algorithm that gives a 3-approximation. Furthermore, we consider the preemptive busy time problem, giving a simple and exact greedy algorithm when unbounded parallelism is allowed, i.e., g is unbounded. For arbitrary g, this yields an algorithm that is 2-approximate.     

Machine scheduling with job class setup and delivery considerations

We study machine scheduling problems in which the jobs belong to different job classes and they need to be delivered to customers after processing. A setup time is required for a job if it is the first job to be processed on a machine or its processing on a machine follows a job that belongs to another class. Processed jobs are delivered in batches to their respective customers. The batch size is limited by the capacity of the delivery vehicles and each shipment incurs a transport cost and takes a fixed amount of time. The objective is to minimize the weighted sum of the last arrival time of jobs to customers and the delivery (transportation) cost. For the problem of processing jobs on a single machine and delivering them to multiple customers, we develop a dynamic programming algorithm to solve the problem optimally. For the problem of processing jobs on parallel machines and delivering them to a single customer, we propose a heuristic and analyze its performance bound.     

Scheduling on two identical machines with a speed-up resource

We address a variant of scheduling problem on two identical machines, where we are given an additional speed-up resource. If a job uses the resource, its processing time may decrease. However, at any time the resource can only be used by at most one job. The objective is to minimize the makespan. For the offline version, we present an FPTAS. For the online version where jobs arrive over list, we propose an online algorithm with competitive ratio of 1.781, and show a lower bound of 1.686 for any online algorithm.     

Semi-online scheduling problems on a small number of machines

We consider the semi-online parallel machine scheduling problem of minimizing the makespan given a priori information: the total processing time, the largest processing time, the combination of the previous two or the optimal makespan. We propose a new algorithm that can be applied to the problem with the known total or largest processing time and prove that it has improved competitive ratios for the cases with a small number of machines. Improved lower bounds of the competitive ratio are also provided by presenting adversary lower bound examples.     

Semi-online two-level supply chain scheduling problems

We consider two-level supply chain scheduling problems where customers release jobs to a manufacturer that has to process the jobs and deliver them to the customers. Processed jobs are grouped into batches, which are delivered to the customers as single shipments. The objective is to minimize the total cost which is the sum of the total flow time and the total delivery cost. Such problems have been considered in the off-line environment where future jobs are known, and in the online environment where at any time there is no information about future jobs. It is known that the best possible competitive ratio for an online algorithm is 2. We consider the problem in the semi-online environment, assuming that a lower bound P for all processing times is available a priori, and present a semi-online algorithm with competitive ratio \(\frac{2D}{D+P}\) where D is the cost of a delivery. Also, for the special case where all processing times are equal, we prove that the algorithm is \(1.045\sqrt{\frac{2-u}{u}}\)-competitive, where u is the density of the instance.     

Online scheduling of two job types on a set of multipurpose machines with unit processing times

We study a problem of scheduling a set of n jobs with unit processing times on a set of m multipurpose machines in which the objective is to minimize the makespan. It is assumed that there are two different job types, where each job type can be processed on a unique subset of machines. We provide an optimal offline algorithm to solve the problem in constant time and an online algorithm with a competitive ratio that equals the lower bound. We show that the worst competitive ratio is obtained for an inclusive job-machine structure in which the first job type can be processed on any of the m machines while the second job type can be processed only on a subset of m/2 machines. Moreover, we show that our online algorithm is 1-competitive if the machines are not flexible, i.e., each machine can process only a single job type.