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.


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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  1. 1.
    Bloom, G.S., Golomb, S.W.: Applications of numbered undirected graphs. Proceedings of the IEEE 65(4), 562–570 (1977)CrossRefGoogle Scholar
  2. 2.
    Moffet, A.T.: Minimum-redundancy linear arrays. IEEE Transactions on Anntennas and Propagation AP-16(2), 172–175 (1968)CrossRefGoogle Scholar
  3. 3.
    Gagliardi, R., Robbins, J., Taylor, H.: Acquisition sequences in PPM communications. IEEE Transactions on Information Theory IT-33(5), 738–744 (1987)CrossRefGoogle Scholar
  4. 4.
    Robinson, J.P., Bernstein, A.J.: A class of binary recurrent codes with limited error propagation. IEEE Transactions on Information Theory IT-13(1), 106–113 (1967)zbMATHCrossRefGoogle Scholar
  5. 5.
    Smith, B., Stergiou, K., Walsh, T.: Modelling the Golomb ruler problem. In: IJCAI Workshop on Non-binary Constraints (1999)Google Scholar
  6. 6.
    Galinier, P., Jaumard, B., Morales, R., Pesant, G.: A constraint-based approach to the Golomb ruler problem. In: Third International Workshop on the Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CPAIOR) (2001), A more recent version (June 11, 2007) can be downloaded from
  7. 7.
    Singer, J.: A theorem in finite projective geometry and some applications to number theory. Transactions of the American Mathematical Society 43(3), 377–385 (1938)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Drakakis, K., Gow, R., O’Carroll, L.: On some properties of costas arrays generated via finite fields. In: 2006 40th Annual Conference on Information Sciences and Systems, pp. 801–805. IEEE (2006)Google Scholar
  9. 9.
    Soliday, S.W., Homaifar, A., Lebby, G.L.: Genetic algorithm approach to the search for Golomb rulers. In: 6th International Conference on Genetic Algorithms (ICGA 1995), pp. 528–535. Morgan Kaufmann (1995)Google Scholar
  10. 10.
    Prestwich, S.: Trading completeness for scalability: Hybrid search for cliques and rulers. In: Third International Workshop on the Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, CPAIOR (2001)Google Scholar
  11. 11.
    Dotú, I., Van Hentenryck, P.: A simple hybrid evolutionary algorithm for finding Golomb rulers. In: The IEEE Congress on Evolutionary Computation, pp. 2018–2023. IEEE (2005)Google Scholar
  12. 12.
    Lorentzen, R., Nilsen, R.: Application of linear programming to the optimal difference triangle set problem. IEEE Trans. Inf. Theor. 37(5), 1486–1488 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hansen, P., Jaumard, B., Meyer, C.: On lower bounds for numbered complete graphs. Discrete Applied Mathematics 94(13), 205–225 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Shearer, J.B.: Improved LP lower bounds for difference triangle sets. Journal of Combinatorics 6 (1999)Google Scholar
  15. 15.
    Meyer, C., Jaumard, B.: Equivalence of some LP-based lower bounds for the Golomb ruler problem. Discrete Appl. Math. 154(1), 120–144 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Sellmann, M., Fahle, T.: Constraint programming based Lagrangian relaxation for the automatic recording problem. Annals of Operations Research 118(1-4), 17–33 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Cronholm, W., Ajili, F.: Strong cost-based filtering for Lagrange decomposition applied to network design. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 726–730. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Sellmann, M.: Theoretical foundations of CP-based Lagrangian relaxation. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 634–647. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Gellermann, T., Sellmann, M., Wright, R.: Shorter path constraints for the resource constrained shortest path problem. In: Barták, R., Milano, M. (eds.) CPAIOR 2005. LNCS, vol. 3524, pp. 201–216. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Khemmoudj, M.O.I., Bennaceur, H., Nagih, A.: Combining arc-consistency and dual Lagrangean relaxation for filtering CSPs. In: Barták, R., Milano, M. (eds.) CPAIOR 2005. LNCS, vol. 3524, pp. 258–272. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Menana, J., Demassey, S.: Sequencing and counting with the multicost-regular constraint. In: van Hoeve, W.-J., Hooker, J.N. (eds.) CPAIOR 2009. LNCS, vol. 5547, pp. 178–192. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Cambazard, H., O’Mahony, E., O’Sullivan, B.: Hybrid methods for the multileaf collimator sequencing problem. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR 2010. LNCS, vol. 6140, pp. 56–70. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  23. 23.
    Benchimol, P., van Hoeve, W.J., Régin, J.C., Rousseau, L.M., Rueher, M.: Improved filtering for weighted circuit constraints. Constraints 17(3), 205–233 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Régin, J.-C.: Solving problems with CP: Four common pitfalls to avoid. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 3–11. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  25. 25.
    Régin, J.C.: A filtering algorithm for constraints of difference in CSPs. In: Proceedings of AAAI, pp. 362–367. AAAI Press (1994)Google Scholar
  26. 26.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley (1988)Google Scholar
  27. 27.
    Held, M., Karp, R.M.: The travelling salesman problem and minimum spanning trees. Operations Research 18, 1138–1162 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Held, M., Wolfe, P., Crowder, H.: Validation of subgradient optimization. Mathematical Programming 6, 62–88 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Gervet, C.: Constraints over structured domains. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming. Elsevier Science Inc. (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marla R. Slusky
    • 1
  • Willem-Jan van Hoeve
    • 2
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityUSA
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityUSA

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