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Performance Ratios for the Differencing Method Applied to the Balanced Number Partitioning Problem

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STACS 2003 (STACS 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2607))

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Abstract

We consider the problem of partitioning a set of n numbers into m subsets of cardinality k = ⌈n/m⌉ or ⌊n/m⌋, such that the maximum subset sum is minimal. We prove that the performance ratios of the Differencing Method of Karmarkar and Karp for k = 3,4,5, and 6 are precisely 4/3, 19/12, 103/60, and 643/360, respectively, by means of a novel approach in which the ratios are explicitly calculated using mixed integer linear programming. Moreover, we show that for k ≥ 7 the performance ratio lies between 2/3 - 2/k and 2/3 - 1/(k/3 - 1). For the case that m is given instead of k, we prove a performance ratio of precisely 2 - 1/m. The results settle the problem of determining theworst-case performance of the Differencing Method.

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Author information

Authors and Affiliations

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

    Wil Michiels & Emile Aarts

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

    Wil Michiels, Jan Korst, Emile Aarts & Jan van Leeuwen

Authors
  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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Editors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takusstr. 9, 14195, Berlin, Germany

    Helmut Alt

  2. LIRMM, 161, Rue Ada, 34392, Montpellier Cedex 5, France

    Michel Habib

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© 2003 Springer-Verlag Berlin Heidelberg

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Michiels, W., Korst, J., Aarts, E., van Leeuwen, J. (2003). Performance Ratios for the Differencing Method Applied to the Balanced Number Partitioning Problem. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_51

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  • DOI: https://doi.org/10.1007/3-540-36494-3_51

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00623-7

  • Online ISBN: 978-3-540-36494-8

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