By Gheorghe Paun, Grzegorz Rozenberg, Arto Salomaa
This publication relies on columns and tutorials released within the Bulletin of the ecu organization for Theoretical laptop technological know-how (EATCS) in the course of the interval 2000-2003.
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Extra info for Current Trends in Theoretical Computer Science: The Challenge of the New Century (Vol 1: Algorithms and Complexity) (Vol 2: Formal Models and Semantics)
Finally, the job of length 6 brings the on-line makespan to 10, whereas the optimal off-line makespan is only 6. Therefore, for m = 3 machine one cannot beat the bound 5/3 — 2 — 1/3. Similarly, for m = 2 machines the job sequence (1,1,2) shows that one cannot beat the bound 3/2 = 2 - 1/2. The 1990s brought a sequence of improvements over the Graham bounds. First, Galambos & Woeginger  showed that for every m > 4 there do exist on-line algorithms with worst case ratios 2 — 1/m — em for values em > 0 depending on m.
What is known on the negative side? 707 for m > 4 machines. 853. D. 88. This (surprisingly strong) result was reached by a carefully designed instance with m = 3600 machines. 9201 for on-line makespan minimization on an arbitrary number of machines. For small values of m, there of course are better on-line algorithms known. Results for 2 < m < 7 are listed in the left half of Table 1. D. thesis of Rudin . 7320508, and intuitively this should also be the exact on-line approximability threshold for m = 4.
G. B. Shmoys. Sequencing and scheduling: Algorithms and complexity. C. G. H. ) Logistics of Production and Inventory, Handbooks in Operations Research and Management Science 4, North-Holland, Amsterdam, 1993, 445-522. 49. K. G. Rinnooy Kan. Complexity of scheduling under precedence constraints. Operations Research 26 (1978), 22-35. 50. K. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Annals of Operations Research 1 (1977), 343 362. 51. K. B. Shmoys, and E. Tardos. Approximation algorithms for scheduling unrelated parallel machines.
Current Trends in Theoretical Computer Science: The Challenge of the New Century (Vol 1: Algorithms and Complexity) (Vol 2: Formal Models and Semantics) by Gheorghe Paun, Grzegorz Rozenberg, Arto Salomaa