The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.

/pdf/optimization-and-approximation-in-deterministic-sequencing-327pfewdb6.pdf

Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

/pdf/sequencing-and-scheduling-algorithms-and-complexity-lnutyrr3kz.pdf

Sequencing and scheduling: algorithms and complexity

Static scheduling of a program represented by a directed task graph on a multiprocessor system to minimize the program completion time is a well-known problem in parallel processing. Since finding an optimal schedule is an NP-complete problem in general, researchers have resorted to devising efficient heuristics. A plethora of heuristics have been proposed based on a wide spectrum of techniques, including branch-and-bound, integer-programming, searching, graph-theory, randomization, genetic algorithms, and evolutionary methods. The objective of this survey is to describe various scheduling algorithms and their functionalities in a contrasting fashion as well as examine their relative merits in terms of performance and time-complexity. Since these algorithms are based on diverse assumptions, they differ in their functionalities, and hence are difficult to describe in a unified context. We propose a taxonomy that classifies these algorithms into different categories. We consider 27 scheduling algorithms, with each algorithm explained through an easy-to-understand description followed by an illustrative example to demonstrate its operation. We also outline some of the novel and promising optimization approaches and current research trends in the area. Finally, we give an overview of the software tools that provide scheduling/mapping functionalities.

http://charm.cs.uiuc.edu/users/arya/docs/6.pdf

Static scheduling algorithms for allocating directed task graphs to multiprocessors

Publisher Summary This chapter discusses different types of sequencing and scheduling problems, and describes different types of algorithms and the concepts of complexity theory. A class of deterministic machine scheduling problems has been introduced in the chapter. The chapter also deals with the single machine, parallel machine and multi-operation problems in this class, respectively. The two generalizations of the deterministic machine-scheduling model have been presented in the chapter. A deterministic scheduling model may give rise to various stochastic counterparts, as there is a choice in the parameters that are randomized, in their distributions, and in the classes of policies that can be applied. A characteristic feature of these models is that the stochastic parameters are regarded as independent random variables with a given distribution and that their realization occurs only after the scheduling decision has been made. In the deterministic model, one has perfect information, and capitalizing on it in minimizing the realization of a performance measure may require exponential time.

Chapter 9 Sequencing and scheduling: Algorithms and complexity

A state-of-the-art review of parallel-machine scheduling research

Coffman and Graham give an algorithm to schedule unit execution time task systems nonpreemptively. On two processors, their algorithm is optimal. We show that in general, if $\omega$ is the length of a schedule produced by their algorithm and $\omega_0$ the length of an optimal schedule, then $\omega / \omega_{0} \leqq 2 - 2/m$, where m is the number of processors. The preemptive equivalent of the above algorithm has been considered by Muntz and Coffman. Again we show that the ratio of the lengths of theirs and an optimal schedule is bounded by $2 - 2/m$. In both the nonpreemptive and the preemptive cases there exist task systems for which the ratio $2 - 2/m$ can be approached arbitrarily closely. On a small number of machines, $2 - 2/m$ is not too far from 1. In particular, as noted above, on two machines the ratio is 1.

Worst Case Analysis of Two Scheduling Algorithms

Muntz and Coffman give a level algorithm that constructs optimal preemptive schedules on identical processors when the task system is a tree or when there are only two processors available. Their algorithm is adapted here to handle processors of different speeds. The new algorithm is optimal for independent tasks on any number of processors and for arbitrary task systems on two processors, but not on three or more processors, even for trees. By taking the algorithm as a heuristic on m processors and using the ratio of the lengths of the constructed and optimal schedules as a measure, an upper bound on its performance is derived in terms of the speeds of the processors. It is further shown that 1.23√m is an upper bound over all possible processor speeds and that the 1.23√m bound can be improved at most by a constant factor, by giving an example of a system for which the bound 0.35√m can be approached asymptotically.

A Level Algorithm for Preemptive Scheduling

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Worst Case Analysis of Two Scheduling Algorithms

A Level Algorithm for Preemptive Scheduling