Scheduling of Independent Dedicated Multiprocessor Tasks

Abstract

We study the off and on-line versions of the well known problem of scheduling a set of n independent multiprocessor tasks with prespecified 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– ∈}, neither for some ∈ > 0, unless P= NP; nor for any ∈ > 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 2√m and 2√m + 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 3√m-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 6√m. Due to the conducted experimental results, we conclude that our algorithms can perform well in practice.

Supported by the bilateral French-German project PROCOPE,

Research supported by EU APPOL II project IST-2001-32007.

Research supported by CNR project CNRG007FC1.

Research supported by EU ARACNE project HPRN-CT-1999-00112, GAČR project 201/99/0242, and by MECR project LN00A056.

Supported by Graduiertenkolleg 357 “Effiziente Algorithmen und Mehrskalenmethoden”.

Research supported by CNR project CNRG007FC1.