Semi-online scheduling with decreasing job sizes

We investigate the problem of semi-online scheduling jobs on m identical parallel machines where the jobs arrive in order of decreasing sizes. We present a complete solution for the preemptive variant of semi-online scheduling with decreasing job sizes. We give matching lower and upper bounds on the competitive ratio for any fixed number m of machines; these bounds tend to (1+3)/2~1.36603, as the number of machines goes to infinity. Our results are also the best possible for randomized algorithms. For the non-preemptive variant of semi-online scheduling with decreasing job sizes, a result of Graham (SIAM J. Appl. Math. 17 (1969) 263-269) yields a (4/3-1/(3m))-competitive deterministic non-preemptive algorithm. For m=2 machines, we prove that this algorithm is the best possible (it is 7/6-competitive). For m=3 machines we give a lower bound of (1+37)/6~1.18046. Finally, we present a randomized non-preemptive 8/7-competitive algorithm for m=2 machines and prove that this is optimal.

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