The Design of (Almost) Disjunct Matrices by Evolutionary Algorithms

  • Karlo Knezevic
  • Stjepan Picek
  • Luca Mariot
  • Domagoj JakobovicEmail author
  • Alberto Leporati
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11324)


Disjunct Matrices (DM) are a particular kind of binary matrices which have been especially applied to solve the Non-Adaptive Group Testing (NAGT) problem, where the task is to detect any configuration of t defectives out of a population of N items. Traditionally, the methods used to construct DM leverage on error-correcting codes and other related algebraic techniques. Here, we investigate the use of Evolutionary Algorithms to design DM and two of their generalizations, namely Resolvable Matrices (RM) and Almost Disjunct Matrices (ADM). After discussing the basic encoding used to represent the candidate solutions of our optimization problems, we define three fitness functions, each measuring the deviation of a generic binary matrix from being respectively a DM, an RM or an ADM. Next, we employ Estimation of Distribution Algorithms (EDA), Genetic Algorithms (GA), and Genetic Programming (GP) to optimize these fitness functions. The results show that GP achieves the best performances among the three heuristics, converging to an optimal solution on a wider range of problem instances. Although these results do not match those obtained by other state-of-the-art methods in the literature, we argue that our heuristic approach can generate solutions that are not expressible by currently known algebraic techniques, and sketch some possible ideas to further improve its performance.


Evolutionary computing Disjunct matrices Resolvable matrices Almost disjunct matrices Group testing Estimation of distribution algorithms Genetic algorithms Genetic programming 



Parts of our work have been inspired by COST Action CA15140 supported by COST (European Cooperation in Science and Technology).


  1. 1.
    Balint, G., et al.: An investigation of \(d\)-separable, \(\bar{d}\)-separable, and \(d\)-disjunct binary matrices. Technical report, San Diego State University (2013).
  2. 2.
    Barg, A., Mazumdar, A.: Group testing schemes from codes and designs. IEEE Trans. Inf. Theory 63(11), 7131–7141 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Belazzougui, D., Gagie, T., Mäkinen, V., Previtali, M.: Fully dynamic De Bruijn graphs. In: Inenaga, S., Sadakane, K., Sakai, T. (eds.) SPIRE 2016. LNCS, vol. 9954, pp. 145–152. Springer, Cham (2016). Scholar
  4. 4.
    Colbourn, C.J., Dinitz, J.H.: Combinatorial designs. In: Handbook of Discrete and Combinatorial Mathematics. CRC Press (1999)Google Scholar
  5. 5.
    Colbourn, C.J., Ling, A.C.H., Syrotiuk, V.R.: Cover-free families and topology-transparent scheduling for manets. Des. Codes Cryptogr. 32(1–3), 65–95 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Damaschke, P., Schliep, A.: An optimization problem related to bloom filters with bit patterns. In: Tjoa, A.M., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds.) SOFSEM 2018. LNCS, vol. 10706, pp. 525–538. Springer, Cham (2018). Scholar
  7. 7.
    Du, D., Hwang, F.K., Hwang, F.: Combinatorial Group Testing and Its Applications, vol. 12. World Scientific, Singapore (2000)Google Scholar
  8. 8.
    Eiben, A.E., Smith, J.E.: Introduction to Evolutionary Computing. Natural Computing Series. Springer, Heidelberg (2015). Scholar
  9. 9.
    Kautz, W.H., Singleton, R.C.: Nonrandom binary superimposed codes. IEEE Trans. Inf. Theory 10(4), 363–377 (1964)CrossRefGoogle Scholar
  10. 10.
    Larrañaga, P., Karshenas, H., Bielza, C., Santana, R.: A review on probabilistic graphical models in evolutionary computation. J. Heuristics 18(5), 795–819 (2012)CrossRefGoogle Scholar
  11. 11.
    Mariot, L., Leporati, A.: Heuristic search by particle swarm optimization of boolean functions for cryptographic applications. In: Genetic and Evolutionary Computation Conference, GECCO 2015, Madrid, Spain, 11–15 July 2015, Companion Material Proceedings, pp. 1425–1426 (2015)Google Scholar
  12. 12.
    Mariot, L., Picek, S., Jakobovic, D., Leporati, A.: Evolutionary algorithms for the design of orthogonal latin squares based on cellular automata. In: Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2017, Berlin, Germany, 15–19 July 2017, pp. 306–313 (2017)Google Scholar
  13. 13.
    Mariot, L., Picek, S., Jakobovic, D., Leporati, A.: Evolutionary search of binary orthogonal arrays. In: Auger, A., Fonseca, C.M., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds.) PPSN 2018. LNCS, vol. 11101, pp. 121–133. Springer, Cham (2018). Scholar
  14. 14.
    Mazumdar, A.: On almost disjunct matrices for group testing. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 649–658. Springer, Heidelberg (2012). Scholar
  15. 15.
    Mühlenbein, H.: The equation for response to selection and its use for prediction. Evol. Comput. 5(3), 303–346 (1997). Scholar
  16. 16.
    Mühlenbein, H., Paaß, G.: From recombination of genes to the estimation of distributions I. Binary parameters. In: Voigt, H.-M., Ebeling, W., Rechenberg, I., Schwefel, H.-P. (eds.) PPSN 1996. LNCS, vol. 1141, pp. 178–187. Springer, Heidelberg (1996). Scholar
  17. 17.
    Pearl, J.: Causality: Models, Reasoning and Inference. Cambridge University Press, New York (2000)zbMATHGoogle Scholar
  18. 18.
    Poli, R., Langdon, W.B., McPhee, N.F.: A field guide to genetic programming (2008)., (With contributions by J. R. Koza)
  19. 19.
    Porat, E., Rothschild, A.: Explicit nonadaptive combinatorial group testing schemes. IEEE Trans. Inf. Theory 57(12), 7982–7989 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Safadi, R., Wang, R.: The use of genetic algorithms in the construction of mixed multilevel orthogonal arrays. Technical report, OLIN CORP CHESHIRE CT OLIN RESEARCH CENTER (1992)Google Scholar
  21. 21.
    Stinson, D.R., Van Trung, T., Wei, R.: Secure frameproof codes, key distribution patterns, group testing algorithms and related structures. J. Stat. Plan. Inference 86(2), 595–617 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Stinson, D.R., Wei, R.: Generalized cover-free families. Discret. Math. 279(1–3), 463–477 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Karlo Knezevic
    • 1
  • Stjepan Picek
    • 2
  • Luca Mariot
    • 3
  • Domagoj Jakobovic
    • 1
    Email author
  • Alberto Leporati
    • 3
  1. 1.Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia
  2. 2.Cyber Security Research GroupDelft University of TechnologyDelftThe Netherlands
  3. 3.Department of Informatics, Systems, and CommunicationUniversity of Milano-BicoccaMilanItaly

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