Scheduling of independent dedicated multiprocessor tasks

We study the off and on-line versions of the well known problem of scheduling a set of n independent multiprocessor tasks with pre-specified processor allocations on a set of identical processors in order to minimize the makespan. Recently, in [12], it has been proven that in the case when all tasks have unit processing time the problem cannot be approximated within a factor of m(1/2-epsilon) neither for some epsilon > 0, unless P= NP; nor for any epsilon > 0, unless NP=ZPP. For this special case we give a simple algorithm based on the classical first-fit technique. We analyze the algorithm for both tasks arrive over time and tasks arrive over list on-line scheduling versions, and show that its competitive ratio is bounded by 2rootm and 2rootm + 1, respectively. Here we also use some preliminary results on (vertex) coloring of k-tuple graphs. For the case of arbitrary processing times, we show that any algorithm which uses the first-fit technique cannot be better than m competitive. Then, by using our split-round technique, we give a 3rootm-approximation algorithm for the off-line version of the problem. Finally, by using some ideas from [20], we adapt the algorithm to the on-line case, in the paradigm of tasks arriving over time in which the existence of a task is unknown until its release date, and show that its competitive ratio is bounded by 6rootm. Due to the conducted experimental results, we conclude that our algorithms can perform well in practice.