Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abu-Khzam, F.N.: A kernelization algorithm for d-Hitting Set. J. Comput. System Sci. 76(7), 524–531 (2010)
Babcock, W.: Intermodulation interference in radio systems. Bell Syst. Tech. J., 63–73 (1953)
Bloom, G., Golomb, S.: Applications of numbered undirected graphs. Proc. IEEE 65(4), 562–570 (1977)
Blum, E., Biraud, F., Ribes, J.: On optimal synthetic linear arrays with applications to radioastronomy. IEEE T. Antenn. Propag. 22, 108–109 (1974)
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)
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)
Distributed.net. Home page, http://www.distributed.net/ (accessed April 2012)
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)
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)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)
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)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)
Meyer, C., Papakonstantinou, P.A.: On the complexity of constructing Golomb rulers. Discrete Appl. Math. 157, 738–748 (2008)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)
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)
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)
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)
Sorge, M.: Algorithmic aspects of Golomb ruler construction. Studienarbeit, Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, Available electronically, arXiv:1005.5395v2 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sorge, M., Moser, H., Niedermeier, R., Weller, M. (2012). Exploiting a Hypergraph Model for Finding Golomb Rulers. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds) Combinatorial Optimization. ISCO 2012. Lecture Notes in Computer Science, vol 7422. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32147-4_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-32147-4_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32146-7
Online ISBN: 978-3-642-32147-4
eBook Packages: Computer ScienceComputer Science (R0)