# Exploiting a hypergraph model for finding Golomb rulers

- 130 Downloads

## Abstract

Golomb rulers are special rulers where for any two marks it holds that the distance between them is unique. They find applications in radio frequency selection, radio astronomy, data encryption, 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 consider a natural hypergraph characterization of rulers and investigate the construction and structure of the corresponding hypergraphs. We exploit their properties to derive polynomial-time data reduction rules that reduce a given instance of GSR to an equivalent one with \({{\mathrm{O}}}(k^3)\) marks. Finally, we complement a recent computational complexity study of GSR by providing a simplified reduction that shows NP-hardness even when all integers are bounded by a polynomial in the input length.

## Notes

### Acknowledgments

We thank Falk Hüffner for assistance with Sect. 3.3 and we are grateful to anonymous referees for their valuable feedback leading to significant improvements of the presentation.

## References

- 1.Abu-Khzam, F.N.: A kernelization algorithm for \(d\)-hitting set. J. Comput. Syst. Sci.
**76**(7), 524–531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms
**7**(4), 567–583 (1986)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Babcock, W.: Intermodulation interference in radio systems. Bell Syst. Tech. J.
**32**, 63–73 (1953)CrossRefGoogle Scholar - 4.Bertram-Kretzberg, C., Lefmann, H.: The algorithmic aspects of uncrowded hypergraphs. SIAM J. Comput.
**29**(1), 201–230 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Bloom, G., Golomb, S.: Applications of numbered undirected graphs. Proc. IEEE
**65**(4), 562–570 (1977)CrossRefGoogle Scholar - 6.Blum, E., Biraud, F., Ribes, J.: On optimal synthetic linear arrays with applications to radioastronomy. IEEE Trans. Antennas Propag.
**22**, 108–109 (1974)CrossRefGoogle Scholar - 7.Bodlaender, H.L.: Kernelization: new upper and lower bound techniques. In: Proceedings of the 4th International Workshop on Parameterized and Exact Computation (IWPEC ’09). Lecture Notes in Computer Science, vol. 5917, pp. 17–37. Springer, Berlin (2009)Google Scholar
- 8.Cotta, C., Dotú, I., Fernández, A.J., Hentenryck, P.V.: Local search-based hybrid algorithms for finding Golomb rulers. Constraints
**12**, 263–291 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proceedings of the 42th Annual ACM Symposium on Theory of Computing (STOC ’10), pp. 251–260. ACM. Journal version to appear in Journal of the ACM (2010)Google Scholar
- 10.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 (2002)Google Scholar
- 11.Distributed.net. Home page. http://www.distributed.net/. Accessed May 2014
- 12.Dollas, A., Rankin, W.T., McCracken, D.: A new algorithm for Golomb ruler derivation and proof of the 19 mark ruler. IEEE Trans. Inf. Theory
**44**(1), 379–382 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 13.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)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
- 15.Drakakis, K.: A review of the available construction methods for Golomb rulers. Adv. Math. Commun.
**3**(3), 235–250 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Fellows, M.R., Jansen, B.M.P., Rosamond, F.A.: Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. Eur. J. Comb.
**34**(3), 541–566 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
- 18.Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News
**38**(1), 31–45 (2007)CrossRefGoogle Scholar - 19.Komusiewicz, C., Niedermeier, R., Uhlmann, J.: Deconstructing intractability—a multivariate complexity analysis of interval constrained coloring. J. Discrete Algorithms
**9**(1), 137–151 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Lokshtanov, D., Misra, N., Saurabh, S.: Kernelization—preprocessing with a guarantee. In: The Multivariate Algorithmic Revolution and Beyond—Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday. Lecture Notes in Computer Science, vol. 7370, pp. 129–161. Springer, Berlin (2012)Google Scholar
- 21.Malakonakis, P., Sotiriades, E., Dollas, A.: GE3: a single FPGA client-server architecture for Golomb ruler derivation. In: Proceedings of the International Conference on Field-Programmable Technology (FPT ’10), pp. 470–473. IEEE (2010)Google Scholar
- 22.Meyer, C., Papakonstantinou, P.A.: On the complexity of constructing Golomb rulers. Discrete Appl. Math.
**157**, 738–748 (2008)MathSciNetCrossRefGoogle Scholar - 23.Nicolas, F., Rivals, E.: Longest common subsequence problem for unoriented and cyclic strings. Theor. Comput. Sci.
**370**(1–3), 1–18 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
- 25.Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS ’10), volume 5 of Dagstuhl Seminar Proceedings, pp. 17–32. IBFI Dagstuhl, Germany (2010)Google Scholar
- 26.Pereira, F., Tavares, J., Costa, E.: Golomb rulers: the advantage of evolution. In: Proceedings of the 11th Portuguese Conference on Artificial Intelligence (EPIA ’03). Lecture Notes in Computer Science, vol. 2902, pp. 29–42. Springer, Berlin (2003)Google Scholar
- 27.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
- 28.Soliday, S.W., Homaifar, A., Lebby, G.L.: Genetic algorithm approach to the search for Golomb rulers. In: Proceedings of the 6th International Conference on Genetic Algorithms (ICGA ’95), pp. 528–535. Morgan Kaufmann, Burlington (1995)Google Scholar
- 29.Sorge, M.: Algorithmic Aspects of Golomb Ruler Construction. Studienarbeit, Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, 2010. Available electronically. arXiv:1005.5395v2
- 30.Sorge, M., Moser, H., Niedermeier, R., Weller, M.: Exploiting a hypergraph model for finding Golomb rulers. In: Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO ’12). Lecture Notes in Computer Science, vol. 7422, pp. 368–379. Springer, Berlin (2012)Google Scholar
- 31.Tavares, J., Pereira, F., Costa, E.: Golomb rulers: a fitness landscape analysis. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC ’08), pp. 3695–3701. IEEE (2008)Google Scholar
- 32.van Bevern, R.: Towards optimal and expressive kernelization for \(d\)-hitting set. Algorithmica (2013)Google Scholar
- 33.von zur Gathen, J., Sieveking, M.: A bound on solutions of linear integer equations and inequalities. Proc. Am. Math. Soc.
**72**, 155–158 (1978)CrossRefzbMATHGoogle Scholar