A Restarting Rule Based on the Schnabel Census for Genetic Algorithms

  • Anton V. EremeevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11353)


A new restart rule is proposed for Genetic Algorithms (GAs) with multiple restarts. The rule is based on the Schnabel Census method, transfered from the biometrics, where it was originally developed for the statistical estimation of a size of animal population. In this paper, the Schnabel Census method is applied to estimate the number of different solutions that may be visited with positive probability, given the current distribution of offspring. The rule consists in restarting the GA as soon as the maximum likelihood estimate reaches the number of different solutions observed at the recent iterations.We demonstrate how the new restart rule can be incorporated into a GA on the example of the Set Cover Problem. Computational experiments on benchmarks from OR-Library show a significant advantage of the GA with the new restarting rule over the original GA. On the unicost instances, the new rule also tends to be superior to the well-known rule, which restarts an algorithm when the current iteration number is twice the iteration number when the best incumbent was found.


Maximum likelihood Abundance of population Set cover Transfer of methods 



This research is supported by the Russian Science Foundation grant 17-18-01536.


  1. 1.
    Alexandrov, D., Kochetov, Y.: Behavior of the Ant Colony Algorithm for the Set Covering Problem. In: Inderfurth, K., Schwödiauer, G., Domschke, W., Juhnke, F., Kleinschmidt, P., Wäscher, G. (eds.) Operations Research Proceedings 1999. ORP, vol. 1999, pp. 255–260. Springer, Heidelberg (2000).
  2. 2.
    Balas, E., Niehaus, W.: Optimized crossover-based genetic algorithms for the maximum cardinality and maximum weight clique problems. J. Heuristics 4(2), 107–122 (1998)CrossRefGoogle Scholar
  3. 3.
    Beasley, J.E.: OR-Library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)CrossRefGoogle Scholar
  4. 4.
    Beasley, J.E., Chu, P.C.: A genetic algorithm for the set covering problem. Eur. J. Oper. Res. 94(2), 394–404 (1996)CrossRefGoogle Scholar
  5. 5.
    Brown, B.W., Hollander, M.: Statistics: A Biomedical Introduction. Wiley, New York (1977)Google Scholar
  6. 6.
    Caprara, A., Fischetti, M., Toth, P.: Heuristic method for the set covering problem. Oper. Res. 47(5), 730–743 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Craig, C.C.: Use of marked specimens in estimating populations. Biometrika 40(1–2), 170–176 (1953)CrossRefGoogle Scholar
  8. 8.
    Erdös, P.: On a combinatorial problem I. Nordisk Mat. Tidskrift 11, 5–10 (1963)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Eremeev, A.V.: A genetic algorithm with a non-binary representation for the set covering problem. In: Proceedings of OR’98, pp. 175–181. Springer-Verlag (1999)Google Scholar
  10. 10.
    Eremeev, A.V., Kovalenko, Y.V.: Genetic algorithm with optimal recombination for the Asymmetric Travelling Salesman Problem. In: Lirkov, I., Margenov, S. (eds.) LSSC 2017. LNCS, vol. 10665, pp. 332–339. Springer, Cham (2018). Scholar
  11. 11.
    Eremeev, A.V., Reeves, C.R.: Non-parametric estimation of properties of combinatorial landscapes. In: Cagnoni, S., Gottlieb, J., Hart, E., Middendorf, M., Raidl, G.R. (eds.) EvoWorkshops 2002. LNCS, vol. 2279, pp. 31–40. Springer, Heidelberg (2002).
  12. 12.
    Fulkerson, D.R., Nemhauser, G.L., Trotter, L.E.: Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. Math. Program. Stud. 2, 72–81 (1974)CrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of \(NP\)-Completeness. W.H. Freeman and Company, San Francisco (1979)Google Scholar
  14. 14.
    Garnier, J., Kallel, L.: How to detect all maxima of a function? In: Proceedings of the Second EVONET Summer School on Theoretical Aspects of Evolutionary Computing (Anvers, 1999), pp. 343–370 Springer, Berlin (2001)Google Scholar
  15. 15.
    Grossman, T., Wool, A.: Computational experience with approximation algorithms for the set covering problem. Eur. J. Oper. Res. 101(1), 81–92 (1997)CrossRefGoogle Scholar
  16. 16.
    Hampson, S., Kibler, D.: Plateaus and Plateau Search in Boolean Satisfiability Problems: When to Give Up Searching and Start Again, pp. 437–456. American Mathematical Society (1996)Google Scholar
  17. 17.
    Hernando, L., Mendiburu, A., Lozano, J.A.: An evaluation of methods for estimating the number of local optima in combinatorial optimization problems. Evol. Comput. 21(4), 625–658 (2013)CrossRefGoogle Scholar
  18. 18.
    Holland, J.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)Google Scholar
  19. 19.
    Hulin, M.: An optimal stop criterion for genetic algorithms: a Bayesian approach. In: Proceedings of the Seventh International Conference on Genetic Algorithms (ICGA ’97), pp. 135–143. Morgan Kaufmann (1997)Google Scholar
  20. 20.
    Jansen, T.: On the analysis of dynamic restart strategies for evolutionary algorithms. In: Guervós, J.J.M., Adamidis, P., Beyer, H.-G., Schwefel, H.-P., Fernández-Villacañas, J.-L. (eds.) PPSN 2002. LNCS, vol. 2439, pp. 33–43. Springer, Heidelberg (2002). Scholar
  21. 21.
    Johnson, N.L., Kotz, S.: Discrete Distributions. Wiley, New York (1969)Google Scholar
  22. 22.
    Luke, S.: When short runs beat long runs. In: Proceedings of he Genetic and Evolutionary Computation Conference (GECCO 2001), pp. 74–80. Morgan Kaufmann (2001)Google Scholar
  23. 23.
    Marti, L., Garcia, J., Berlanga, A. and Molina, J. M.: An approach to stopping criteria for multi-objective optimization evolutionary algorithms: the MGBM criterion. In: Proceedings of 2009 IEEE Congress on Evolutionary Computation, Trondheim, pp. 1263–1270. IEEE (2009)Google Scholar
  24. 24.
    Van Nimwegen, E., Crutchfield, J.P., Mitchell, M.: Finite populations induce metastability in evolutionary search. Phys. Lett. A 229(2), 144–150 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ostrowski, J., Linderoth, J.T., Rossi, F., Smriglio, S.: Solving Steiner triple covering problems. Oper. Res. Lett. 39, 127–131 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Paixao, T., Badkobeh, G., Barton, N., et al.: Toward a unifying framework for evolutionary processes. J. Theor. Biol. 383, 28–43 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pledger, S.: The performance of mixture models in heterogeneous closed population capture-recapture. Biometrics 61(3), 868–876 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Reeves, C.R.: The “crossover landscape” and the Hamming landscape for binary search spaces. In: Foundations of Genetic Algorithms, vol. 7, pp. 81–98. Morgan Kaufmann, San Francisco (2003)Google Scholar
  29. 29.
    Reeves, C.R.: Fitness landscapes and evolutionary algorithms. In: Fonlupt, C., Hao, J.-K., Lutton, E., Schoenauer, M., Ronald, E. (eds.) AE 1999. LNCS, vol. 1829, pp. 3–20. Springer, Heidelberg (2000).
  30. 30.
    Reeves, C.R.: Landscapes, operators and heuristic search. Ann. Oper. Res. 86, 473–490 (1999)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Reeves, C.R.: Direct statistical estimation of GA landscape properties. In: Foundations of Genetic Algorithms, vol. 6, pp. 91–108. Morgan Kaufmann, San Francisco (2001)Google Scholar
  32. 32.
    Reeves, C.R.: Estimating the number of optima in a landscape, part II: experimental investigations. Coventry University Technical Report SOR#01-04 (2001)Google Scholar
  33. 33.
    Schnabel, Z.E.: The estimation of the total fish population of a lake. Am. Math. Mon. 45, 348–352 (1938)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Seber, G.A.F.: The Estimation of Animal Abundance. Charles Griffin, London (1982)Google Scholar
  35. 35.
    Vose, M.D.: The Simple Genetic Algorithm: Foundations and Theory. MIT Press, Cambridge (1999)Google Scholar
  36. 36.
    Vose, M.D., Wright, A.H.: Stability of vertex fixed points and applications. In: Foundations of Genetic Algorithms, vol. 3, pp. 103–114. Morgan Kaufmann, San Mateo, CA (1995)Google Scholar
  37. 37.
    Wright, A.H., Rowe, J.E.: Continuous dynamical systems models of steady-state genetic algorithms. In: Foundations of Genetic Algorithms, vol. 6, pp. 209–226. Morgan Kaufmann, San Francisco, CA (2001)Google Scholar
  38. 38.
    Yagiura, M., Kishida, M., Ibaraki, T.: A 3-flip neighborhood local search for the set covering problem. Eur. J. Oper. Res. 172, 472–499 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics SB RASOmskRussia
  2. 2.Institute of Scientific Information on Social Sciences RASMoscowRussia

Personalised recommendations