An analogy with the way ant colonies function has suggested the definition of a new computational paradigm, which we call Ant System. We propose it as a viable new approach to stochastic combinatorial optimization. The main characteristics of this model are positive feedback, distributed computation, and the use of a constructive greedy heuristic. Positive feedback accounts for rapid discovery of good solutions, distributed computation avoids premature convergence, and the greedy heuristic helps find acceptable solutions in the early stages of the search process. We apply the proposed methodology to the classical Traveling Salesman Problem (TSP), and report simulation results. We also discuss parameter selection and the early setups of the model, and compare it with tabu search and simulated annealing using TSP. To demonstrate the robustness of the approach, we show how the Ant System (AS) can be applied to other optimization problems like the asymmetric traveling salesman, the quadrat...

We consider the following scheduling problem. There are m parallel machines and n independent.jobs. Each job is to be assigned to one of the machines. The processing of.job j on machine i requires time Pip The objective is to lind a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than ~ unless P = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

Important classical scheduling theory results for real-time computing are identified. Implications of these results from the perspective of a real-time systems designer are discussed. Uni-processor and multiprocessor results are addressed as well as important issues such as future release times, precedence constraints, shared resources, task value, overloads, static versus dynamic scheduling, preemption versus non-preemption, multiprocessing anomalies, and metrics. Examples of what scheduling algorithms are used in actual applications are given.

We consider the on-line version of the original m-machine scheduling problem: given m machines and n positive real jobs, schedule the n jobs on the m machines so as to minimize the makespan, the completion time of the last job. In the on-line version, as soon as job j arrives, it must be assigned immediately to one of the m machines. We present two main results. The first is a (2 - ffl)-competitive deterministic algorithm for all m. The competitive ratio of all previous algorithms approaches 2 as m !1. Indeed, the problem of improving the competitive ratio for large m had been open since 1966, when the first algorithm for this problem appeared. The second result is an optimal randomized algorithm for the case m = 2. To the best of our knowledge, our 4/3-competitive algorithm is the first specifically randomized algorithm for the original, m-machine, on-line scheduling problem.

In this chapter, we summarize research efforts on several different problems that fall under the rubric of online scheduling. In online scheduling, the scheduler receives jobs that arrive over time, and generally must schedule the jobs without any knowledge of the future. The lack of knowledge of the future generally precludes the scheduler from guaranteeing optimal schedules. Thus much research has been focused on finding scheduling algorithms that guarantee schedules that are in some way not too far from optimal. We focus on problems that arise within the ubiquitous client-server setting. In a client-server system, there are many clients and one server (or a perhaps a few servers). Clients submit requests for service to the server(s) over time. In the language of scheduling, a server is a processor, and a request is a job. Applications that motivate the research we survey include multiuser operating systems such as Unix and Windows, web servers, database servers, name servers, and load...

Scheduling a collection of tasks on a multiprocessor, consisting of p processors, that minimizes the maximum completion time has attracted a lot of attention in the literature [12]. In this paper, we introduce a new problem of scheduling a collection of independent tasks on a multiprocessor, called the partitionable independent task scheduling problem. Associated with each task, we are given the time it takes to run on a uniprocessor, and speedup that can be obtained by running it on i processors, 1i p . We present an approximate algorithm that guarantees a solution within (1+1/p ) 2 ####### of the optimal solution, under a reasonable assumption on the speedup functions. 1.INTRODUCTION The problem that we are interested will be called the partitionable independent task scheduling problem. This problem arises when a parallel algorithm is designed for the circuit extraction problem, where the input is specified hierarchically [3]. It involves scheduling n independent tasks T 1 , . . ....

We consider the on-line version of the original m-machine scheduling problem: given m machines and n positive real jobs, schedule the n jobs on the m machines so as to minimize the makespan, the completion time of the last job. In the on-line version, as soon as a job arrives, it must be assigned immediately to one of the m machines. We study the competitive ratio of the best algorithm for m-machine scheduling. The largest prior lower bound was that if m 4, then every algorithm has a competitive ratio at least 1+1= p 2 1.707. We show that if m is large enough, the competitive ratio of every algorithm exceeds 1.837. The best upper bound on the competitive ratio is now 1.945.

We examine the problem of scheduling 2-machine flowshops in order to minimize makespan, using a limited amount of intermediate storage buffers. Although there are efficient algorithms for the extreme cases of zero and infinite buffer capacities, we show that all the intermediate (finite capacity) cases are NP-complete. We prove exact bounds for the relative improvement of execution times when a given buffer capacity is used. We also analyze an efficient heuristic for solving the 1-buffer problem, showing that it has a 3/2 worst-case performance. Furthermore, we show that the "no-wait " (i.e., zero buffer) flowshop scheduling problem with 4 machines is NP-complete. This partly settles a well-known open question, although the 3-machine case is left open here. *Research supported by NSF Grant MCS77-01192 +Research supported by NSF/RANN grant APR76-12036

: 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...

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