MiTS in Depth: An Analysis of Distinct Tabu Search Configurations for Constructing Mixed Covering Arrays

  • Loreto Gonzalez-Hernandez
  • Jose Torres-Jimenez
  • Nelson Rangel-Valdez
Part of the Studies in Computational Intelligence book series (SCI, volume 427)


Alan turing work is related with the first use of heuristic algorithms. His work on broking the Nazi code of the Enigma cipher was oriented by a guided search whose expected result in most of the times would be the deciphering of the codes, even though sometimes it might not work. This idea reflects the modern meaning of an heuristic, and represents the main relationship with this chapter, as it involves the use of metaheuristics to try to guide the search to find a solution faster, or a better solution of a problem. The metaheuristic is Tabu Search (TS), and it is used to solve the Mixed Covering Array Problem (MCAP). This problem focuses on the construction of optimal test sets for software testing. The metaheuristic is designed through a fine tuning process that involves the parameters: initialization function, tabu list size, stop criterion, and neighborhood functions. The contributions are: a) a more robust fine tune process to design a new TS approach; b) the analys is of parameter values of the TS; and, c) new bounds over a benchmark reported in the literature.


Tabu Search Test Suite Initialization Function Software Testing Diophantine Equation 
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.


  1. 1.
    Bracho-Rios, J., Torres-Jimenez, J., Rodriguez-Tello, E.: A New Backtracking Algorithm for Constructing Binary Covering Arrays of Variable Strength. In: Aguirre, A.H., Borja, R.M., Garciá, C.A.R. (eds.) MICAI 2009. LNCS, vol. 5845, pp. 397–407. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Bryce, R.C., Colbourn, C.J.: The density algorithm for pairwise interaction testing: Research articles. Software Testing, Verification and Reliability 17, 159–182 (2007)CrossRefGoogle Scholar
  3. 3.
    Bryce, R.C., Colbourn, C.J.: One-test-at-a-time heuristic search for interaction test suites. In: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation Conference, GECCO 2007, July 7-11, 2007, pp. 1082–1089. ACM, New York (2007)CrossRefGoogle Scholar
  4. 4.
    Burnstein, I.: Practical software testing: a process-oriented approach. Springer Professional Computing (2003) ISBN: 0-387-95131-8Google Scholar
  5. 5.
    Bush, K.A.: Orthogonal arrays of index unity. Annals of Mathematical Statistics 23(3), 426–434 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Changhai, N., Hareton, L.: A survey of combinatorial testing. ACM Computing Surveys (CSUR) 43, 11:1–11:29 (2011)Google Scholar
  7. 7.
    Cohen, D.M., Fredman, M.L., Patton, G.C.: The aetg system: An approach to testing based on combinatorial design. IEEE Transactions on Software Engineering 23(7), 437–444 (1997)CrossRefGoogle Scholar
  8. 8.
    Cohen, M.B., Gibbons, P.B., Mugridge, W.B., Colbourn, C.J.: Constructing test suites for interaction testing. In: Proceedings of the 25th International Conference on Software Engineering, ICSE 2003, May 3-10, pp. 38–48. IEEE Computer Society, Washington, DC (2003)Google Scholar
  9. 9.
  10. 10.
    Colbourn, C.J.: Covering arrays from cyclotomy. Designs, Codes and Cryptography 55, 201–219 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Colbourn, C.J., Cohen, M.B., Turban, R.C.: A deterministic density algorithm for pairwise interaction coverage. In: Proceedings of the IASTED International Conference on Software Engineering, February 17-19, pp. 345–352 (2004)Google Scholar
  12. 12.
    Colbourn, C.J., Dinitz, J.H.: The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton (1996) ISBN: 0-8493-8948-8zbMATHCrossRefGoogle Scholar
  13. 13.
    Colbourn, C.J., Martirosyan, S., Trung, T., Walker II., R.A.: Roux-type constructions for covering arrays of strengths three and four. Designs, Codes and Cryptography 41, 33–57 (2006), doi:10.1007/s10623-006-0020-8zbMATHCrossRefGoogle Scholar
  14. 14.
    Colbourn, C.J., Martirosyan, S.S., Mullen, G.L., Shasha, D., Sherwood, G.B., Yucas, J.L.: Products of mixed covering arrays of strength two. Journal of Combinatorial Designs 14(2), 124–138 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Forbes, M., Lawrence, J., Lei, Y., Kacker, R.N., Kuhn, D.R.: Refining the in-parameter-order strategy for constructing covering arrays. Journal of Research of the National Institute of Standards and Technology 113(5), 287–297 (2008)CrossRefGoogle Scholar
  16. 16.
    Glover, F., Laguna, M.: Tabu Search. Kluwer Academic Publishers (1998) ISBN 0-7923-9965-XGoogle Scholar
  17. 17.
    Gonzalez-Hernandez, L., Rangel-Valdez, N., Torres-Jimenez, J.: Construction of Mixed Covering Arrays of Variable Strength Using a Tabu Search Approach. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part I. LNCS, vol. 6508, pp. 51–64. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Gonzalez-Hernandez, L., Torres-Jimenez, J.: MiTS: A New Approach of Tabu Search for Constructing Mixed Covering Arrays. In: Sidorov, G., Hernández Aguirre, A., Reyes García, C.A. (eds.) MICAI 2010, Part II. LNCS, vol. 6438, pp. 382–393. Springer, Heidelberg (2010), CrossRefGoogle Scholar
  19. 19.
    Gonzalez-Hernandez, L., Torres-Jiménez, J., Rangel-Valdez, N.: An Exact Approach to Maximize the Number of Wild Cards in a Covering Array. In: Batyrshin, I., Sidorov, G. (eds.) MICAI 2011, Part I. LNCS (LNAI), vol. 7094, pp. 210–221. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Hartman, A., Raskin, L.: Problems and algorithms for covering arrays. Discrete Mathematics 284, 149–156 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hnich, B., Prestwich, S.D., Selensky, E., Smith, B.M.: Constraint models for the covering test problem. Constraints 11, 199–219 (2006), doi:10.1007/s10601-006-7094-9MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Walker II, R.A., Colbourn, C.J.: Tabu search for covering arrays using permutation vectors. Journal of Statistical Planning and Inference 139(1), 69–80 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Ji, L., Yin, J.: Constructions of new orthogonal arrays and covering arrays of strength three. Journal of Combinatorial Theory Series A 117, 236–247 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Katona, G.O.H.: Two applications (for search theory and truth functions) of sperner type theorems. Periodica Mathematica Hungarica 3, 19–26 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kleitmain, D.J., Spencer, J.: Families of k-independent sets. Discrete Mathematics 6(3), 255–262 (1973)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kuhn, D.R., Kacker, R.N., Lei, Y.: Practical combinatorial testing. Technical report, National Institute of Standards and Technology (October 2010)Google Scholar
  27. 27.
    Kuhn, D.R., Kacker, R.N., Lei, Y.: Advanced combinatorial test methods for system reliability. Technical report, 2010 Annual Technical Report of the IEEE Reliability Society, 2010 Annual Technical Report (January 2011)Google Scholar
  28. 28.
    Kuhn, D.R., Wallance, D.R., Gallo Jr., A.M.: Software fault interactions and implications for software testing. IEEE Transactions on Software Engineering 30, 418–421 (2004)CrossRefGoogle Scholar
  29. 29.
    Kuliamin, V., Petukhov, A.: Covering Arrays Generation Methods Survey. In: Margaria, T., Steffen, B. (eds.) ISoLA 2010. LNCS, vol. 6416, pp. 382–396. Springer, Heidelberg (2010), doi:10.1007/978-3-642-16561-0_36CrossRefGoogle Scholar
  30. 30.
    Kuliamin, V., Petukhov, A.: A survey of methods for constructing covering arrays. Programming and Computer Software 37, 121–146 (2011)zbMATHCrossRefGoogle Scholar
  31. 31.
    Lawrence, J., Kacker, R.N., Lei, Y., Kuhn, D.R., Forbes, M.: A survey of binary covering arrays. Electronic Journals of Combinatorics 18, P84 (2011)MathSciNetGoogle Scholar
  32. 32.
    Lions, J.L.: Ariane 5, flight 501, report of the inquiry board. European Space Agency (July 1996)Google Scholar
  33. 33.
    Martinez-Pena, J., Torres-Jimenez, J., Rangel-Valdez, N., Avila-George, H.: A heuristic approach for constructing ternary covering arrays using trinomial coefficients. In: Kuri-Morales, A., Simari, G.R. (eds.) IBERAMIA 2010. LNCS, vol. 6433, pp. 572–581. Springer, Heidelberg (2010), doi:10.1007/978-3-642-16952-6_58CrossRefGoogle Scholar
  34. 34.
    Nayeri, P., Colbourn, C.J., Konjevod, G.: Randomized postoptimization of covering arrays. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 408–419. Springer, Heidelberg (2009), doi:10.1007/978-3-642-10217-2_40CrossRefGoogle Scholar
  35. 35.
    Nurmela, K.J.: Upper bounds for covering arrays by tabu search. Discrete Applied Mathematics 138(1-2), 143–152 (2004); Optimal Discrete Structures and AlgorithmsMathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Rényi, A.: Foundations of Probability. Wiley (1971) ISBN: 0486462617Google Scholar
  37. 37.
    Rodrigues, L.C.A., Weller, T.R.: Cell Formation with Alternative Routings and Capacity Considerations: A Hybrid Tabu Search Approach. In: Gelbukh, A., Morales, E.F. (eds.) MICAI 2008. LNCS (LNAI), vol. 5317, pp. 482–491. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  38. 38.
    Rodriguez-Tello, E., Torres-Jimenez, J.: Memetic Algorithms for Constructing Binary Covering Arrays of Strength Three. In: Collet, P., Monmarché, N., Legrand, P., Schoenauer, M., Lutton, E. (eds.) EA 2009. LNCS, vol. 5975, pp. 86–97. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  39. 39.
    Shiba, T., Tsuchiya, T., Kikuno, T.: Using artificial life techniques to generate test cases for combinatorial testing. In: Proceedings of the 28th Annual International Computer Software and Applications Conference, COMPSAC 2004, September 27-30, vol. 1, pp. 72–77. IEEE Computer Society, Washington, DC (2004)CrossRefGoogle Scholar
  40. 40.
    Stardom, J.: Metaheuristics and the search for covering and packing arrays. Master’s thesis, Simon Fraser University (2001)Google Scholar
  41. 41.
    Tai, K.C., Lei, Y.: A test generation strategy for pairwise testing. IEEE Transactions on Software Engineering 28, 109–111 (2002)CrossRefGoogle Scholar
  42. 42.
    Tassey, G.: The economic impacts of inadequate infrastructure for software testing. Technical report, National Institute of Standards and Technology (May 2002)Google Scholar
  43. 43.
    Torres-Jimenez, J., Rodriguez-Tello, E.: Simulated annealing for constructing binary covering arrays of variable strength. In: IEEE Congress on Evolutionary Computation, CEC 2010, July 18-23, pp. 1–8 (2010)Google Scholar
  44. 44.
    Torres-Jimenez, J., Rodriguez-Tello, E.: New bounds for binary covering arrays using simulated annealing. Information Sciences 185(1), 137–152 (2012)CrossRefGoogle Scholar
  45. 45.
    Turing, A.M.: On computable numbers, with an application to the entscheidungsproblem. Proceedings of the London Mathematical Society 42, 230–265 (1936)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Williams, A.W.: Determination of test configurations for pair-wise interaction coverage. In: TestCom 2000: Proceedings of the IFIP TC6/WG6.1 13th International Conference on Testing Communicating Systems, August 29-September 1, pp. 59–74. B.V. Kluwer, Deventer (2000)Google Scholar
  47. 47.
    Williams, A.W., Probert, R.L.: A practical strategy for testing pair-wise coverage of network interfaces. In: Proceedings of the The Seventh International Symposium on Software Reliability Engineering, ISSRE 1996, October 30-November 02, pp. 246–254. IEEE Computer Society, Washington, DC (1996)CrossRefGoogle Scholar
  48. 48.
    Yan, J., Zhang, J.: Backtracking algorithms and search heuristics to generate test suites for combinatorial testing. In: 30th Annual International on Computer Software and Applications Conference, COMPSAC 2006, September 17-21, vol. 1, pp. 385–394. IEEE Computer Society, Washington, DC (2006)Google Scholar
  49. 49.
    Yan, J., Zhang, J.: A backtraking search tool for constructing combinatorial test suites. The Journal of Systems and Software 81, 1681–1693 (2008)CrossRefGoogle Scholar
  50. 50.
    Zekaoui, L.: Mixed covering arrays on graphs and tabu search algorithms. Master’s thesis, Ottawa-Carleton Institute for Computer Science at the University of Ottawa (2006)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2013

Authors and Affiliations

  • Loreto Gonzalez-Hernandez
    • 1
  • Jose Torres-Jimenez
    • 1
  • Nelson Rangel-Valdez
    • 2
  1. 1.CINVESTAV-TamaulipasCd. VictoriaMexico
  2. 2.Universidad Politécnica de VictoriaCd. VictoriaMexico

Personalised recommendations