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Scheduling and packing malleable and parallel tasks with precedence constraints of bounded width

Abstract

We study the problems of non-preemptively scheduling and packing malleable and parallel tasks with precedence constraints to minimize the makespan. In the scheduling variant, we allow the free choice of processors; in packing, each task must be assigned to a contiguous subset. Malleable tasks can be processed on different numbers of processors with varying processing times, while parallel tasks require a fixed number of processors.

For precedence constraints of bounded width, we resolve the complexity status of the problem with any number of processors and any width bound. We present an FPTAS based on Dilworth’s decomposition theorem for the NP-hard problem variants, and exact efficient algorithms for all remaining special cases. For our positive results, we do not require the otherwise common monotonous penalty assumption on the processing times of malleable tasks, whereas our hardness results hold even when assuming this restriction. We complement our results by showing that these problems are all strongly NP-hard under precedence constraints which form a tree.

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Acknowledgements

We thank Rolf H. Möhring and Martin Skutella for fruitful discussions.

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Correspondence to Elisabeth Günther.

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An extended abstract has been published in Proceedings of the 7th Workshop on Approximation and Online Algorithms (WAOA 2009), Springer, 2010 (Günther et al. 2010). The first author is supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.

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Günther, E., König, F.G. & Megow, N. Scheduling and packing malleable and parallel tasks with precedence constraints of bounded width. J Comb Optim 27, 164–181 (2014). https://doi.org/10.1007/s10878-012-9498-3

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  • DOI: https://doi.org/10.1007/s10878-012-9498-3

Keywords

  • Approximation
  • Scheduling
  • Precedence constraints
  • Makespan
  • Malleable
  • Parallel