Scheduling has been studied extensively in many varieties and from many viewpoints. Inspired by applications in practical computer systems, it developed into a theoretical area with many interesting results, both positive and negative. The basic situation we study is the following. We have some sequence of jobs that have to be processed on the machines available to us. In the most basic problem, each job is characterized by its running time and has to be scheduled for that time on one of the machines. In other variants there may be additional restrictions or relaxations specifying which schedules are allowed. We want to schedule the jobs as efficiently as possible, which most often means that the total length of the schedule (the makespan) should be as small as possible, but other objective functions are also considered. The notion of an on-line algorithm is intended to formalize the realistic scenario, where the algorithm does not have the access to the whole inp...

Scheduling has been studied extensively in many varieties and from many viewpoints. Inspired by applications in practical computer systems, it developed into a theoretical area with many interesting results, both positive and negative. The basic situation we study is the following. We have some sequence of jobs

We consider the problem of preemptively scheduling n jobs on a system of m uniform processors. Each job has a release time, a due time, and a memory requirement associated with it. Each processor has a speed and a memory capacity associated with it. Linear programming formulations are obtained for the following optimization criteria: (1) minimize the schedule length and (2) minimize the maximum lateness. We also consider three special cases with the former optimization criteria. For each of these cases, low order polynomial time algorithms are obtained.

. This paper surveys, analyses and establishes new polynomial-time reductions among scheduling problems that connect identical parallel machines with unit-time shops and the preemption facility with chain-like precedence constraints in equal machine environments. The reductions turn out to be unexpectedly fruitful in clarifying the complexity status of many scheduling problems that were open before. New complexity results in the paper are devoted to identical parallel machines, ow shops and open shops. 1. Introduction Recently found mass polynomial-time reductions between scheduling problems on identical parallel machines and shop scheduling problems with nonpreemptive unit processing time operations proved to be very eective in studying their complexity. The latter problems we simply call unit-time shops. Reductions of unit-time shops to identical parallel machines we call parallelizings, and the inverse reductions, i.e., of identical parallel machines to unittime shops, we c...

. We consider the on-line scheduling problem of jobs with precedence constraints on m parallel identical machines. Each job has a time processing requirement, and may depend on other jobs (has to be processed after them). A job arrives only after its predecessors have been completed. The cost of an algorithm is the time that the last job is completed. We show lower bounds on the competitive ratio of on-line algorithms for this problem in several versions. We prove a lower bound of 2 \Gamma 1=m on the competitive ratio of any deterministic algorithm (with or without preemption) and a lower bound of 2 \Gamma 2=(m+ 1) on the competitive ratio of any randomized algorithm (with or without preemption). The lower bounds for the cases that preemption is allowed require arbitrarily long sequences. If we use only sequences of length O(m 2 ), we can show a lower bound of 2 \Gamma 2=(m + 1) on the competitive ratio of deterministic algorithms with preemption, and a lower bound of 2 \Gamma O(1=m)...

: The notion of profile scheduling was first introduced by Ullman in 1975 in the complexity analysis of deterministic scheduling algorithms. In such a model, the number of processors available to a set of tasks may vary in time. Since the last decade, this model has been used to deal with systems subject to processor failures, multiprogrammed systems, or dynamically reconfigured systems. The aim of this paper is to overview optimal polynomial solutions for scheduling a set of partially ordered tasks in these systems. Particular attentions are given to a class of algorithms referred to as list scheduling algorithms. The objective of the scheduling problem is to minimize either the maximum lateness or the makespan. Results on preemptive and nonpreemptive deterministic scheduling, and on preemptive stochastic scheduling, are presented. Keywords: Deterministic Scheduling, Stochastic Scheduling, Profile Scheduling, List Schedule, Priority Schedule, Precedence Constraints, Lateness, Makespan...

We show that no multiprocessor system that contains at least one processor with memory size smaller than at least two other processors can be scheduled nearly on line to minimize the finish time. An efficient nearly on line algorithm to minimize C max is developed for multiprocessor systems that do not satisfy the preceding requirement. Finally, we review the complexity of some other scheduling problems for multiprocessor systems with memories.

Abstract 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.