David Shmoys
580 California St., Suite 400
San Francisco, CA, 94104
Parallel machines - Minmax criteria, no preemption
2019
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Abstract
Four authors wish to write a book, which will consist of fifteen chapters. Each chapter is to be written by a single author. The chapters differ in length, and the authors differ in expertise. The speed at which an author will be able to complete a chapter depends on his, or her, familiarity with its subject matter. The completion date of the manuscript is to be minimized. Who should do what? Irrespective of the practical relevance of this model, it is one of the central problems in scheduling theory. In its general form, the authors are unrelated parallel machines, the chapters are jobs, and the problem is R||C max . The reader should recall the definitions of unrelated, uniform, and identical parallel machines (see Section 1.2). We have seen that, for these machine environments, there is a drastic difference in complexity between minimizing maximum and total completion time: R|| ∑Cj is solvable in polynomial time (see Section 8.1), but P2||C max is already NP-hard (see Section 2.4). As long as the number of machines is a constant, dynamic programming can be applied to solve Rm||C max in pseudopolynomial time (see Section 8.3); however, when m is an input, even P||C max is NP-hard in the strong sense (see Section 2.4). This chapter, then, is about hard problems. The design of efficient approximation algorithms and of enumerative optimization methods is called for. We will emphasize the performance analysis of approximation algorithms. The empirical analysis of approximation and optimization procedures will not be discussed in detail. We have
References (8)
- /2-approximation algorithm implied by Theorem 9.10 and the polynomial approx- imation scheme for Q||C max mentioned at the end of the section are due to Hochbaum and Shmoys [1988].
- Williamson (private communication) observed the algorithm given in Exercise 9.11 and its analysis.
- Unrelated machines: performance guarantees. The first paper on this subject was by Ibarra and Kim [1977], who considered the greedy method of Theorem 9.12 and Exercise 9.12 and other m-approximation algorithms. Davis and Jaffe [1981] analyzed a number of algorithms and obtained the bound for the RS rule given in Theorem 9.13. The linear programming approach of Theorem 9.14 is due to Potts [1985A];
- it was extended by Lenstra, Shmoys, and Tardos [1990] to the algorithm LP of Theorem 9.15 and Exercise 9.13. Trick [1990] proposed the model described in Exercise 9.15; Tardos (private communication) observed the results given in Ex- ercises 9.14 and 9.15. For Rm||C max , the fully polynomial approximation scheme of Theorem 9.16 is due to Horowitz and Sahni [1976], and the polynomial approx- imation scheme derived in Exercises 9.16 and 9.17 is due to Lenstra, Shmoys, and Tardos [1990].
- Hariri and Potts [1991] performed computational tests with approximation algo- rithms for R||C max on problems with up to 50 machines and 100 jobs. Among five constructive heuristics, an implementation of the Lenstra-Shmoys-Tardos algorithm produced the best solutions but required nontrivial running times; a simple itera- tive improvement scheme, applying job reassignments and interchanges, was still able to achieve substantial improvements, in almost negligible amounts of time. Van de Velde [1993] developed optimization and approximation algorithms for R||C max , based on surrogate duality relaxation of the constraints (9.2). His branch-and-bound method performs reasonably well. His 'duality-based heuristic search' algorithm outperforms the constructive heuristics but, again, the method should be supple- mented with some form of iterative improvement.
- 5. Unrelated machines: impossibilities. The results in this section are due to Lenstra, Shmoys, and Tardos [1990].
- Two identical machines: probabilistic analysis. A basic text in probability theory is Feller [1968, 1971]. For the probabilistic analysis of scheduling algorithms, we refer to the monograph by Coffman and Lueker [1991], the surveys by Coffman, Lueker, and Rinnooy Kan [1988] and by Rinnooy Kan and Stougie [1989], and the annotated bibliography by Karp, Lenstra, McDiarmid, and Rinnooy Kan [1985]. The inequality of Hoeffding [1963] was applied by Coffman et al. in their survey paper to bound the rate of convergence of the relative error of the LS algorithm. Frenk and Rinnooy Kan [1987] proved that the absolute error of the LPT rule converges to 0 almost surely and in expectation, even for Q||C max ; see also Coffman, Flatto, and Lueker [1984] and Loulou [1984]. The rate of convergence of this error was investigated by Boxma [1984] and Frenk and Rinnooy Kan [1986]. Karmarkar and 9. Minmax criteria
- Karp [1982] proposed and analyzed the MD method.