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A Linearly-Growing Conversion from the Set Splitting Problem to the Directed Hamiltonian Cycle Problem

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Optimization and Control Methods in Industrial Engineering and Construction

Part of the book series: Intelligent Systems, Control and Automation: Science and Engineering ((ISCA,volume 72))

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Abstract

We consider a direct conversion of the, classical, set splitting problem to the directed Hamiltonian cycle problem. A constructive procedure for such a conversion is given, and it is shown that the input size of the converted instance is a linear function of the input size of the original instance. A proof that the two instances are equivalent is given, and a procedure for identifying a solution to the original instance from a solution of the converted instance is also provided. We conclude with two examples of set splitting problem instances, one with solutions and one without, and display the corresponding instances of the directed Hamiltonian cycle problem, along with a solution in the first example.

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Notes

  1. 1.

    In fact, due to the nature of the construction, if the element appears in \(q\) subsets, then exactly \(q\) type 2 forced paths must occur here.

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Acknowledgments

The authors gratefully acknowledge useful conversations with Dr. Richard Taylor that helped in the development of this chapter.

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Authors and Affiliations

  1. Flinders University, Adelaide, Australia

    Michael Haythorpe & Jerzy A. Filar

Authors
  1. Michael Haythorpe
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  2. Jerzy A. Filar
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Corresponding author

Correspondence to Michael Haythorpe .

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Editors and Affiliations

  1. Dept of Mathematics & Statistics, Curtin University, Perth, West Australia, Australia

    Honglei Xu

  2. School of Built Environment, Curtin University, Perth, West Australia, Australia

    Xiangyu Wang

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Haythorpe, M., Filar, J.A. (2014). A Linearly-Growing Conversion from the Set Splitting Problem to the Directed Hamiltonian Cycle Problem. In: Xu, H., Wang, X. (eds) Optimization and Control Methods in Industrial Engineering and Construction. Intelligent Systems, Control and Automation: Science and Engineering, vol 72. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8044-5_3

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  • DOI: https://doi.org/10.1007/978-94-017-8044-5_3

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  • Print ISBN: 978-94-017-8043-8

  • Online ISBN: 978-94-017-8044-5

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