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The Multiobjective Shortest Path Problem Is NP-Hard, or Is It?

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Evolutionary Multi-Criterion Optimization (EMO 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10173))

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Abstract

To show that multiobjective optimization problems like the multiobjective shortest path or assignment problems are hard, we often use the theory of \(\mathbf {NP} \)-hardness. In this paper we rigorously investigate the complexity status of some well-known multiobjective optimization problems and ask the question if these problems really are \(\mathbf {NP} \)-hard. It turns out, that most of them do not seem to be and for one we prove that if it is \(\mathbf {NP} \)-hard then this would imply \(\mathbf {P} = \mathbf {NP} \) under assumptions from the literature. We also reason why \(\mathbf {NP} \)-hardness might not be well suited for investigating the complexity status of intractable multiobjective optimization problems.

The author has been supported by the Bundesministerium für Wirtschaft und Energie (BMWi) within the research project “Bewertung und Planung von Stromnetzen” (promotional reference 03ET7505) and by DFG GRK 1855 (DOTS).

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Acknowledgements

The author especially thanks Paolo Serafini for a very kind and indepth discussion of this topic. Also many thanks to Kathrin Klamroth and Michael Stiglmayr for many fruitful discussions on the complexity of multiobjective combinatorial optimization problems.

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Authors and Affiliations

  1. Department of Computer Science, TU Dortmund, Dortmund, Germany

    Fritz Bökler

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  1. Fritz Bökler
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Correspondence to Fritz Bökler .

Editor information

Editors and Affiliations

  1. University of Münster, Münster, Germany

    Heike Trautmann

  2. TU Dortmund University, Dortmund, Germany

    Günter Rudolph

  3. University of Wuppertal, Wuppertal, Germany

    Kathrin Klamroth

  4. CINVESTAV-IPN, Mexico City, Mexico

    Oliver Schütze

  5. Clemson University, Clemson, South Carolina, USA

    Margaret Wiecek

  6. University of Surrey, Guildford, United Kingdom

    Yaochu Jin

  7. University of Münster, Münster, Germany

    Christian Grimme

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Bökler, F. (2017). The Multiobjective Shortest Path Problem Is NP-Hard, or Is It?. In: Trautmann, H., et al. Evolutionary Multi-Criterion Optimization. EMO 2017. Lecture Notes in Computer Science(), vol 10173. Springer, Cham. https://doi.org/10.1007/978-3-319-54157-0_6

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  • DOI: https://doi.org/10.1007/978-3-319-54157-0_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-54156-3

  • Online ISBN: 978-3-319-54157-0

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