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Performance ratios of the Karmarkar-Karp differencing method

Author

Listed:
  • Wil Michiels

    (Philips Research Laboratories)

  • Jan Korst

    (Philips Research Laboratories)

  • Emile Aarts

    (Philips Research Laboratories
    Eindhoven University of Technology)

  • Jan van Leeuwen

    (Utrecht University)

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.

Suggested Citation

  • Wil Michiels & Jan Korst & Emile Aarts & Jan van Leeuwen, 2007. "Performance ratios of the Karmarkar-Karp differencing method," Journal of Combinatorial Optimization, Springer, vol. 13(1), pages 19-32, January.
    • Handle: RePEc:spr:jcomop:v:13:y:2007:i:1:d:10.1007_s10878-006-9010-z
    DOI: 10.1007/s10878-006-9010-z
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    References listed on IDEAS

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    1. Frenk, J.B.G. & Rinnooy Kan, A.H.G., 1986. "The rate of convergence to optimality of the LPT rule," Econometric Institute Research Papers 11698, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Benjamin Yakir, 1996. "The Differencing Algorithm LDM for Partitioning: A Proof of a Conjecture of Karmarkar and Karp," Mathematics of Operations Research, INFORMS, vol. 21(1), pages 85-99, February.
    3. E. G. Coffman & G. N. Frederickson & G. S. Lueker, 1984. "A Note on Expected Makespans for Largest-First Sequences of Independent Tasks on Two Processors," Mathematics of Operations Research, INFORMS, vol. 9(2), pages 260-266, May.
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    Cited by:

    1. Federico Della Croce & Rosario Scatamacchia, 2020. "The Longest Processing Time rule for identical parallel machines revisited," Journal of Scheduling, Springer, vol. 23(2), pages 163-176, April.

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