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A Lagrangian Relaxation for Golomb Rulers

  • Conference paper
Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CPAIOR 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7874))

Abstract

The Golomb Ruler Problem asks to position n integer marks on a ruler such that all pairwise distances between the marks are distinct and the ruler has minimum total length. It is a very challenging combinatorial problem, and provably optimal rulers are only known for n up to 26. Lower bounds can be obtained using Linear Programming formulations, but these are computationally expensive for large n. In this paper, we propose a new method for finding lower bounds based on a Lagrangian relaxation. We present a combinatorial algorithm that finds good bounds quickly without the use of a Linear Programming solver. This allows us to embed our algorithm into a constraint programming search procedure. We compare our relaxation with other lower bounds from the literature, both formally and experimentally. We also show that our relaxation can reduce the constraint programming search tree considerably.

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, USA

    Marla R. Slusky

  2. Tepper School of Business, Carnegie Mellon University, USA

    Willem-Jan van Hoeve

Authors
  1. Marla R. Slusky
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  2. Willem-Jan van Hoeve
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Cornell University, Ithaca, NY, USA

    Carla Gomes

  2. Thomas J. Watson Research Center, Yorktown Heights, NY, USA

    Meinolf Sellmann

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Slusky, M.R., van Hoeve, WJ. (2013). A Lagrangian Relaxation for Golomb Rulers. In: Gomes, C., Sellmann, M. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2013. Lecture Notes in Computer Science, vol 7874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38171-3_17

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  • DOI: https://doi.org/10.1007/978-3-642-38171-3_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38170-6

  • Online ISBN: 978-3-642-38171-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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