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On the power of randomization for job shop scheduling with k-units length tasks

Abstract
In the job shop scheduling problem k-units-J_m, there are m machines and each machine has an integer processing time of at most k time units. Each job consists of a permutation of m tasks corresponding to all machines and thus all jobs have an identical dilation D. The contribution of this paper are the following results; (i) for jobs and every fixed k, the makespan of an optimal schedule is at most D+ o(D), which extends the result of [CITE] for k=1; (ii) a randomized on-line approximation algorithm for k-units-J_m is presented. This is the on-line algorithm with the best known competitive ratio against an oblivious adversary for and k > 1; (iii) different processing times yield harder instances than identical processing times. There is no 5/3 competitive deterministic on-line algorithm for k-units-J_m, whereas the competitive ratio of the randomized on-line algorithm of (ii) still tends to 1 for .