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On Strong NP-Completeness of Rational Problems

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Computer Science – Theory and Applications (CSR 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10846))

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Abstract

The computational complexity of the partition, 0-1 subset sum, unbounded subset sum, 0-1 knapsack and unbounded knapsack problems and their multiple variants were studied in numerous papers in the past where all the weights and profits were assumed to be integers. We re-examine here the computational complexity of all these problems in the setting where the weights and profits are allowed to be any rational numbers. We show that all of these problems in this setting become strongly NP-complete and, as a result, no pseudo-polynomial algorithm can exist for solving them unless P = NP. Despite this result we show that they all still admit a fully polynomial-time approximation scheme.

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Acknowledgments

We would like to thank the anonymous reviewers whose comments helped to improve this paper. This work was partially supported by the EPSRC through grants EP/M027287/1 (Energy Efficient Control) and EP/P020909/1 (Solving Parity Games in Theory and Practice).

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Authors and Affiliations

  1. University of Liverpool, Liverpool, UK

    Dominik Wojtczak

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  1. Dominik Wojtczak
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Correspondence to Dominik Wojtczak .

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Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Fedor V. Fomin

  2. Steklov Mathematical Institute, Moscow, Russia

    Vladimir V. Podolskii

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Wojtczak, D. (2018). On Strong NP-Completeness of Rational Problems. In: Fomin, F., Podolskii, V. (eds) Computer Science – Theory and Applications. CSR 2018. Lecture Notes in Computer Science(), vol 10846. Springer, Cham. https://doi.org/10.1007/978-3-319-90530-3_26

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  • DOI: https://doi.org/10.1007/978-3-319-90530-3_26

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-90530-3

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