Exploiting a Hypergraph Model for Finding Golomb Rulers

  • Manuel Sorge
  • Hannes Moser
  • Rolf Niedermeier
  • Mathias Weller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)


Golomb rulers are special rulers where for any two marks it holds that the distance between them is unique. They find applications in positioning of radio channels, radio astronomy, communication networks, and bioinformatics. An important subproblem in constructing “compact” Golomb rulers is Golomb Subruler (GSR), which asks whether it is possible to make a given ruler Golomb by removing at most k marks. We initiate a study of GSR from a parameterized complexity perspective. In particular, we develop a hypergraph characterization of rulers and consider the construction and structure of the corresponding hypergraphs. We exploit their properties to derive polynomial-time data reduction rules that lead to a problem kernel for GSR with O(k 3) marks. Finally, we provide a simplified NP-hardness construction for GSR.


Hitting Set NP-Hardness Parameterized Complexity Data Reduction Problem Kernel Forbidden Subgraph Characterization 


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  1. 1.
    Abu-Khzam, F.N.: A kernelization algorithm for d-Hitting Set. J. Comput. System Sci. 76(7), 524–531 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Babcock, W.: Intermodulation interference in radio systems. Bell Syst. Tech. J., 63–73 (1953)Google Scholar
  3. 3.
    Bloom, G., Golomb, S.: Applications of numbered undirected graphs. Proc. IEEE 65(4), 562–570 (1977)CrossRefGoogle Scholar
  4. 4.
    Blum, E., Biraud, F., Ribes, J.: On optimal synthetic linear arrays with applications to radioastronomy. IEEE T. Antenn. Propag. 22, 108–109 (1974)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L.: Kernelization: New Upper and Lower Bound Techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Dimitromanolakis, A.: Analysis of the Golomb ruler and the Sidon set problems, and determination of large, near-optimal Golomb rulers. Master’s thesis, Department of Electronic and Computer Engineering, Technical University of Crete (June 2002)Google Scholar
  7. 7. Home page, (accessed April 2012)
  8. 8.
    Dollas, A., Rankin, W.T., McCracken, D.: A new algorithm for Golomb ruler derivation and proof of the 19 mark ruler. IEEE T. Inform. Theory 44(1), 379–382 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R., Truss, A.: Fixed-parameter tractability results for feedback set problems in tournaments. J. Discrete Algorithms 8(1), 76–86 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  11. 11.
    Fellows, M.: Towards Fully Multivariate Algorithmics: Some New Results and Directions in Parameter Ecology. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 2–10. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  13. 13.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  14. 14.
    Meyer, C., Papakonstantinou, P.A.: On the complexity of constructing Golomb rulers. Discrete Appl. Math. 157, 738–748 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  16. 16.
    Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proc. 27th STACS. Dagstuhl Seminar Proceedings, vol. 5, pp. 17–32. IBFI Dagstuhl, Germany (2010)Google Scholar
  17. 17.
    Pereira, F.B., Tavares, J., Costa, E.: Golomb Rulers: The Advantage of Evolution. In: Pires, F.M., Abreu, S.P. (eds.) EPIA 2003. LNCS (LNAI), vol. 2902, pp. 29–42. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. 18.
    Rankin, W.T.: Optimal Golomb rulers: An exhaustive parallel search implementation. Master’s thesis, Department of Electrical Engineering, Duke University, Durham, Addendum by Aviral Singh (1993)Google Scholar
  19. 19.
    Sorge, M.: Algorithmic aspects of Golomb ruler construction. Studienarbeit, Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, Available electronically, arXiv:1005.5395v2 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Manuel Sorge
    • 1
  • Hannes Moser
    • 1
  • Rolf Niedermeier
    • 1
  • Mathias Weller
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

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