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A Lagrangian Relaxation for Golomb Rulers

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Part of the Lecture Notes in Computer Science book series (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.

Keywords

  • Constraint Programming
  • Lagrangian Formulation
  • Lagrangian Relaxation
  • Search Node
  • Subgradient Optimization

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

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