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Performance ratios of the Karmarkar-Karp differencing method

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Abstract

We consider the multiprocessor scheduling problem in which one must schedule n independent tasks nonpreemptively on m identical, parallel machines, such that the completion time of the last task is minimal. For this well-studied problem the Largest Differencing Method of Karmarkar and Karp outperforms other existing polynomial-time approximation algorithms from an average-case perspective. For m ≥ 3 the worst-case performance of the Largest Differencing Method has remained a challenging open problem. In this paper we show that the worst-case performance ratio is bounded between \( frac{4}{3}-\frac{1}{3(m-1)}$ and $\frac{4}{3}-\frac{1}{3m}\). For fixed m we establish further refined bounds in terms of n.

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Authors and Affiliations

  1. Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA, Eindhoven, The Netherlands

    Wil Michiels, Jan Korst & Emile Aarts

  2. Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Emile Aarts

  3. Utrecht University, P.O. Box 80.089, 3508 TB, Utrecht, The Netherlands

    Jan van Leeuwen

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  1. Wil Michiels
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  2. Jan Korst
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  3. Emile Aarts
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  4. Jan van Leeuwen
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Correspondence to Wil Michiels.

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Michiels, W., Korst, J., Aarts, E. et al. Performance ratios of the Karmarkar-Karp differencing method. J Comb Optim 13, 19–32 (2007). https://doi.org/10.1007/s10878-006-9010-z

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