Shared Processor Scheduling of Multiprocessor Jobs (original) (raw)
Related papers
Shared multi-processor scheduling
European Journal of Operational Research, 2017
We study shared multi-processor scheduling problem where each job can be executed on its private processor and simultaneously on one of many processors shared by all jobs in order to reduce the job's completion time due to processing time overlap. The total weighted overlap of all jobs is to be maximized. The problem models subcontracting scheduling in supply chains and divisible load scheduling in computing. We show that synchronized schedules that complete each job at the same time on its private and shared processor, if any is actually used by the job, include optimal schedules. We prove that the problem is NP-hard in the strong sense for jobs with arbitrary weights, and we give an efficient, polynomial-time algorithm for the problem with equal weights.
Journal of Scheduling
We study the shared processor scheduling problem with a single shared processor to maximize total weighted overlap, where an overlap for a job is the amount of time it is processed on its private and shared processor in parallel. A polynomial-time optimization algorithm has been given for the problem with equal weights in the literature. This paper extends that result by showing an O(n log n)-time optimization algorithm for a class of instances in which non-decreasing order of jobs with respect to processing times provides a non-increasing order with respect to weights-this instance generalizes the unweighted case of the problem. This algorithm also leads to a 1 2-approximation algorithm for the general weighted problem. The complexity of the weighted problem remains open.
Scheduling with Interjob Communication on Parallel Processors
Lecture Notes in Computer Science, 2016
Consider a scheduling problem in which a set of tasks needs to be scheduled on m parallel processors. Each task T i consists of a set of jobs with interjob communication demands, represented by a weighted, undirected graph G i. The processors are assumed to be interconnected by a shared communication channel, which can be used by jobs to communicate among each other while being processed in parallel. In each time step, the scheduler assigns jobs to the processors and allows any processed job to use a certain capacity of the channel in order to satisfy (parts of) its communication demands to adjacent jobs processed in the same step. The goal is to find a schedule with minimum length in which the communication demands of all jobs are satisfied. We show that this problem is NP-hard in the strong sense even if the number of processors is constant and the underlying graph is a single path or a forest with arbitrary constant maximum degree. Consequently, we design and analyze approximation algorithms with asymptotic approximation ratio min{1.8, 1.5 m m−1 } + 1 if the underlying graph G, the union of the G i , is a forest. For general graphs it is min 1.8, 1.5m m−1 • arb(G) + 5 3 , where arb(G) denotes the arboricity of G.
On the Approximability of Related Machine Scheduling Under Arbitrary Precedence
IEEE Transactions on Network and Service Management, 2021
Distributed computing systems often need to consider the scheduling problem involving a collection of highly dependent data-processing tasks that must work in concert to achieve mission-critical objectives. This paper considers the unrelated machine scheduling problem for minimizing weighted sum completion time under arbitrary precedence constraints and on heterogeneous machines with different processing speeds. The problem is known to be strongly NP-hard even in the single machine setting. By making use of Queyranne's constraint set and constructing a novel Linear Programming relaxation for the scheduling problem under arbitrary precedence constraints, our results in this paper advance the state of the art. We develop a 2(1 + (m − 1)/D)-approximation algorithm (and 2(1 + (m − 1)/D) + 1-approximation) for the scheduling problem with zero release time (and arbitrary release time), where m is the number of servers and D is the task-skewness product. The algorithm can be efficiently computed in polynomial time using the Ellipsoid method and achieves nearly optimal performance in practice as D > O(m) when the number of tasks per job to schedule is sufficiently larger than the number of machines available. Our implementation and evaluation using a heterogeneous testbed and real-world benchmarks confirms significant improvement in weighted sum completion time for dependent computing tasks.
Scheduling in the presence of processor networks : complexity and approximation
RAIRO - Operations Research, 2012
In this paper, we study the problem of makespan minimization for the multiprocessor scheduling problem in the presence of communication delays. The communication delay between two tasks i and j depends on the distance between the two processors on which these two tasks are executed. Lahlou shows that a simple polynomial-time algorithm exists when the length of the schedule is at most two (the problem becomes N P-complete when the length of the schedule is at most three). We prove that there is no polynomial-time algorithm with a performance guarantee of less than 4/3 (unless P = N P) to minimize the makespan when the network topology is a chain or ring and the precedence graph is a bipartite graph of depth one. We also develop two polynomial-time approximation algorithms with constant ratio dedicated to cases where the processor network admits a limited or unlimited number of processors.
A Polynomial Algorithm for Multiprocessor Scheduling with Two Job Lengths
Mathematics of Operations Research, 2001
The following multiprocessor scheduling problem was motivated by scheduling maintenance periods for aircraft. Each maintenance period is a job, and the maintenance facilities are machines. In this context, there are very few different types of maintenances performed, so it is natural to consider the problem with only a small, fixed number C of different types of jobs. Each job type has a processing time, and each machine is available for the same length of time. A machine can handle at most one job at a time, all jobs are released at time zero, there are no due dates or precedence constraints, and preemption is not allowed. The question is whether it is possible to finish all jobs. We call this problem the Multiprocessor Scheduling Problem with C job lengths (MSPC).
2007
Scheduling theory is concerned with the optimal allocation of scarce resources (for instance, machines, processors, robots, operators, etc.) to activities over time, with the objective of optimizing one or several performance measures. The study of scheduling started about fifty years ago, being initiated by seminal papers by and Bellman (1956). Since then machine scheduling theory have received considerable development. As a result, a great diversity of scheduling models and optimization techniques have been developed that found wide applications in industry, transport and communications. Today, scheduling theory is an integral, generally recognized and rapidly evolving branch of operations research, fruitfully contributing to computer science, artificial intelligence, and industrial engineering and management. The interested reader can find many nice pearls of scheduling theory in textbooks, monographs and handbooks by Tanaev et al. (1994a,b), Pinedo (2001), Leung (2001), Brucker (2007, and Blazewicz et al. (2007). This book is the result of an initiative launched by Prof. Vedran Kordic, a major goal of which is to continue a good tradition -to bring together reputable researchers from different countries in order to provide a comprehensive coverage of advanced and modern topics in scheduling not yet reflected by other books. The virtual consortium of the authors has been created by using electronic exchanges; it comprises 50 authors from 18 different countries who have submitted 23 contributions to this collective product. In this sense, the volume in your hands can be added to a bookshelf with similar collective publications in scheduling, started by Coffman (1976) and successfully continued by Chretienne et al. (1995), Gutin and Punnen (2002), and Leung (2004). This volume contains four major parts that cover the following directions: the state of the art in theory and algorithms for classical and non-standard scheduling problems; new exact optimization algorithms, approximation algorithms with performance guarantees, heuristics and metaheuristics; novel models and approaches to scheduling; and, last but least, several real-life applications and case studies. The brief outline of the volume is as follows. Part I presents tutorials, surveys and comparative studies of several new trends and modern tools in scheduling theory. Chapter 1 is a tutorial on theory of cyclic scheduling. It is included for those readers who are unfamiliar with this area of scheduling theory. Cyclic scheduling models are traditionally used to control repetitive industrial processes and enhance the performance of robotic lines in many industries. A brief overview of cyclic scheduling models arising in manufacturing systems served by robots is presented, started with a discussion of early works appeared in the 1960s. Although the considered scheduling problems are, in general, NP-hard, a graph approach presented in this chapter permits to reduce some special cases to the parametric critical path problem in a graph and solve them in polynomial time. Chapter 2 describes the so-called multi-agent scheduling models applied to the situations in which the resource allocation process involves different stakeholders ("agents"), each having his/her own set of jobs and interests, and there is no central authority which can VI solve possible conflicts in resource usage over time. In this case, standard scheduling models become invalid, since rather than computing "optimal solutions", the model is asked to provide useful elements for the negotiation process, which eventually should lead to a stable and acceptable resource allocation. The chapter does not review the whole scope in detail, but rather concentrates on combinatorial models and their applications. Two major mechanisms for generating schedules, auctions and bargaining models, corresponding to different information exchange scenarios, are considered. Known results are reviewed and venues for future research are pointed out. Chapter 3 considers a class of scheduling problems under unavailability constraints associated, for example, with breakdown periods, maintenance durations and/or setup times. Such problems can be met in different industrial environments in numerous real-life applications. Recent algorithmic approaches proposed to solve these problems are presented, and their complexity and worst-case performance characteristics are discussed. The main attention is devoted to the flow-time minimization in the weighted and unweighted cases, for single-machine and parallel machine scheduling problems. Chapter 4 is devoted to the analysis of scheduling problems with communication delays. With the increasing importance of parallel computing, the question of how to schedule a set of precedence-constrained tasks on a given computer architecture, with communication delays taken into account, becomes critical. The chapter presents the principal results related to complexity, approximability and non-approximability of scheduling problems in presence of communication delays. Part II comprising eight chapters is devoted to the design of scheduling algorithms. Here the reader can find a wide variety of algorithms: exact, approximate with performance guarantees, heuristics and meta-heuristics; most algorithms are supplied by the complexity analysis and/or tested computationally. Chapter 5 deals with a batch version of the single-processor scheduling problem with batch setup times and batch delivery costs, the objective being to find a schedule which minimizes the sum of the weighted number of late jobs and the delivery costs. A new dynamic programming (DP) algorithm which runs in pseudo-polynomial time is proposed. By combining the techniques of binary range search and static interval partitioning, the DP algorithm is converted into a fully polynomial time approximation scheme for the general case. The DP algorithm becomes polynomial for the special cases when jobs have equal weights or equal processing times. Chapter 6 studies on-line approximation algorithms with performance guarantees for an important class of scheduling problems defined on identical machines, for jobs with arbitrary release times. Chapter 7 presents a new hybrid metaheuristic for solving the jobshop scheduling problem that combines augmented-neural-networks with genetic algorithm based search. In Chapter 8 heuristics based on a combination of the guided search and tabu search are considered to minimize the maximum completion time and maximum tardiness in the parallel-machine scheduling problems. Computational characteristics of the proposed heuristics are evaluated through extensive experiments. Chapter 9 presents a hybrid meta-heuristics based on a combination of the genetic algorithm and the local search aimed to solve the re-entrant flowshop scheduling problems. The hybrid method is compared with the optimal solutions generated by the integer programming technique, and the near optimal solutions generated by a pure genetic algorithm. Computational experiments are performed to illustrate the effectiveness and efficiency of the proposed algorithm.
Parallel machine scheduling with a common server
Discrete Applied Mathematics, 2000
This paper considers the nonpreemptive scheduling of a given set of jobs on several identical, parallel machines. Each job must be processed on one of the machines. Prior to processing, a job must be loaded (setup) by a single server onto the relevant machine. The paper considers a number of classical scheduling objectives in this environment, under a variety of assumptions about setup and processing times. For each problem considered, the intention is to provide either a polynomial-or pseudo-polynomial-time algorithm, or a proof of binary or unary NP-completeness. The results provide a mapping of the computational complexity of these problems.
Minimizing the number of machines for scheduling jobs with equal processing times
European Journal of Operational Research, Vol. 199, No. 2, 2009, 595 - 600, 2009
In this paper, we consider a parallel machine environment when all jobs have the same processing time and arbitrary release dates and deadlines of the jobs are given. We suppose that the available number of machines, which can be used simultaneously, may vary over time. The aim is to construct a feasible schedule in such a way that the maximal number of simultaneously used machines is minimal. We give a polynomial algorithm for this problem.
Decentralized Subcontractor Scheduling with Divisible Jobs
Subcontracting allows manufacturer agents to reduce completion times of their jobs and thus obtain savings. This paper addresses the coordination of decentralized scheduling systems with a single subcontractor and several agents having divisible jobs. Assuming complete information, we design parametric pricing schemes that strongly coordinate this decentralized system, i.e., the agents' choices of subcontracting intervals always result in efficient schedules. The subcontractor's revenue under the pricing schemes depends on a single parameter which can be chosen to make the revenue as close to the total savings as required. Also, we give a lower bound on the subcontractor's revenue for any coordinating pricing scheme. Allowing private information about processing times, we prove that the pivotal mechanism is coordinating, i.e., agents are better off by reporting their true processing times, and by participating in the subcontracting. We show that the subcontractor's maximum revenue with any coordinating mechanism under private information equals the lower bound of that with coordinating pricing schemes under complete information. Finally, we address the asymmetric case where agents obtain savings at different rates per unit reduction in completion times. We show that coordinating pricing schemes do not always exist in this case.