Fixed Set Search Applied to the Traveling Salesman Problem

  • Raka Jovanovic
  • Milan Tuba
  • Stefan Voß
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11299)


In this paper we present a new population based metaheuristic called the fixed set search (FSS). The proposed approach represents a method of adding a learning mechanism to the greedy randomized adaptive search procedure (GRASP). The basic concept of FSS is to avoid focusing on specific high quality solutions but on parts or elements that such solutions have. This is done through fixing a set of elements that exist in such solutions and dedicating computational effort to finding near optimal solutions for the underlying subproblem. The simplicity of implementing the proposed method is illustrated on the traveling salesman problem. Our computational experiments show that the FSS manages to find significantly better solutions than the GRASP it is based on, the dynamic convexized method and the ant colony optimization combined with a local search.


Metaheuristic Traveling salesman problem GRASP 


  1. 1.
    Banks, A., Vincent, J., Anyakoha, C.: A review of particle swarm optimization. Part I: background and development. Nat. Comput. 6(4), 467–484 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Banks, A., Vincent, J., Anyakoha, C.: A review of particle swarm optimization. Part II: hybridisation, combinatorial, multicriteria and constrained optimization, and indicative applications. Nat. Comput. 7(1), 109–124 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bentley, J.J.: Fast algorithms for geometric traveling salesman problems. ORSA J. Comput. 4(4), 387–411 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Blum, C., Pinacho, P., López-Ibáñez, M., Lozano, J.A.: Construct, merge, solve & adapt a new general algorithm for combinatorial optimization. Comput. Oper. Res. 68, 75–88 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Blum, C., Puchinger, J., Raidl, G.R., Roli, A.: Hybrid metaheuristics in combinatorial optimization: a survey. Appl. Soft Comput. 11(6), 4135–4151 (2011)zbMATHCrossRefGoogle Scholar
  6. 6.
    Caserta, M., Voß, S.: Metaheuristics: intelligent problem solving. In: Maniezzo, V., Stützle, T., Voß, S. (eds.) Matheuristics: Hybridizing Metaheuristics and Mathematical Programming, vol. 10, pp. 1–38. Springer, Boston (2010). Scholar
  7. 7.
    Concorde: Concorde TSP solver (2015).
  8. 8.
    Croes, G.A.: A method for solving traveling-salesman problems. Oper. Res. 6(6), 791–812 (1958)MathSciNetCrossRefGoogle Scholar
  9. 9.
    De Boer, P.T., Kroese, D.P., Mannor, S., Rubinstein, R.Y.: A tutorial on the cross-entropy method. Ann. Oper. Res. 134(1), 19–67 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dorigo, M., Blum, C.: Ant colony optimization theory: a survey. Theor. Comput. Sci. 344(2–3), 243–278 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP. Algorithmica 68(1), 190–264 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Feo, T.A., Resende, M.G.: Greedy randomized adaptive search procedures. J. Global Optim. 6(2), 109–133 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Festa, P., Resende, M.G.C.: Hybridizations of GRASP with path-relinking. In: Talbi, E.G. (ed.) Hybrid Metaheuristics. Studies in Computational Intelligence, vol. 434, pp. 135–155. Springer, Heidelberg (2013). Scholar
  14. 14.
    Fister, I., Yang, X.S., Fister, D., Fister, I.: Cuckoo search: a brief literature review. In: Yang, X.S. (ed.) Cuckoo Search and Firefly Algorithm: Theory and Applications, vol. 516, pp. 49–62. Springer, Cham (2014). Scholar
  15. 15.
    Glover, F.: Tabu search-part I. ORSA J. Comput. 1(3), 190–206 (1989)zbMATHCrossRefGoogle Scholar
  16. 16.
    Glover, F.: Tabu search-part II. ORSA J. Comput. 2(1), 4–32 (1990)zbMATHCrossRefGoogle Scholar
  17. 17.
    Hansen, P., Mladenović, N.: Variable neighborhood search: principles and applications. Eur. J. Oper. Res. 130(3), 449–467 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hart, J., Shogan, A.: Semi-greedy heuristics: an empirical study. Oper. Res. Lett. 6, 107–114 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Jovanovic, R., Bousselham, A., Voß, S.: Partitioning of supply/demand graphs with capacity limitations: an ant colony approach. J. Comb. Optim. 35(1), 224–249 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Jovanovic, R., Tuba, M.: An ant colony optimization algorithm with improved pheromone correction strategy for the minimum weight vertex cover problem. Appl. Soft Comput. 11(8), 5360–5366 (2011)CrossRefGoogle Scholar
  21. 21.
    Jovanovic, R., Tuba, M., Voß, S.: An ant colony optimization algorithm for partitioning graphs with supply and demand. Appl. Soft Comp. 41, 317–330 (2016)CrossRefGoogle Scholar
  22. 22.
    Karaboga, D., Gorkemli, B., Ozturk, C., Karaboga, N.: A comprehensive survey: artificial bee colony (ABC) algorithm and applications. Artif. Intell. Rev. 42(1), 21–57 (2014)CrossRefGoogle Scholar
  23. 23.
    van Laarhoven, P.J.M., Aarts, E.H.L.: Simulated annealing. In: van Laarhoven, P.J.M., Aarts, E.H.L., et al. (eds.) Simulated Annealing: Theory and Applications, vol. 37, pp. 7–15. Springer, Dordrecht (1987). Scholar
  24. 24.
    Lin, S.: Computer solutions of the traveling salesman problem. Bell Syst. Tech. J. 44(10), 2245–2269 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Marinakis, Y., Marinaki, M., Dounias, G.: Honey bees mating optimization algorithm for the Euclidean traveling salesman problem. Inf. Sci. 181(20), 4684–4698 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Marinakis, Y., Migdalas, A., Pardalos, P.M.: Expanding neighborhood GRASP for the traveling salesman problem. Comput. Optim. Appl. 32(3), 231–257 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1998)zbMATHGoogle Scholar
  28. 28.
    Neri, F., Tirronen, V.: Recent advances in differential evolution: a survey and experimental analysis. Artif. Intell. Rev. 33(1–2), 61–106 (2010)CrossRefGoogle Scholar
  29. 29.
    Reinelt, G.: TSPLIB—a traveling salesman problem library. ORSA J. Comput. 3(4), 376–384 (1991)zbMATHCrossRefGoogle Scholar
  30. 30.
    Sondergeld, L., Voß, S.: Cooperative intelligent search using adaptive memory techniques. In: Voß, S., Martello, S., Osman, I., Roucairol, C. (eds.) Meta-Heuristics: Advances and Trends in Local Search Paradigms for Optimization, pp. 297–312. Springer, Boston (1999). Scholar
  31. 31.
    Sörensen, K.: Metaheuristics - the metaphor exposed. Int. Trans. Oper. Res. 22(1), 3–18 (2015). Scholar
  32. 32.
    Stützle, T., Hoos, H.: Max-min ant system and local search for the traveling salesman problem, pp. 309–314. IEEE (1997)Google Scholar
  33. 33.
    Taillard, E., Voß, S.: POPMUSIC - a partial optimization metaheuristic under special intensification conditions. In: Ribeiro, C., Hansen, P. (eds.) Essays and Surveys in Metaheuristics. Operations Research/Computer Science Interfaces Series, vol. 15, pp. 613–629. Kluwer, Boston (2002). Scholar
  34. 34.
    Tsai, C.F., Tsai, C.W., Tseng, C.C.: A new hybrid heuristic approach for solving large traveling salesman problem. Inf. Sci. 166(1), 67–81 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Voß, S., Gutenschwager, K.: A chunking based genetic algorithm for the Steiner tree problem in graphs. In: Pardalos, P., Du, D.Z. (eds.) Network Design: Connectivity and Facilities Location. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 40, pp. 335–355. AMS, Princeton (1998)CrossRefGoogle Scholar
  36. 36.
    Woodruff, D.: Proposals for chunking and tabu search. Eur. J. Oper. Res. 106, 585–598 (1998)zbMATHCrossRefGoogle Scholar
  37. 37.
    Zhu, M., Chen, J.: Computational comparison of GRASP and DCTSP methods for the Traveling Salesman Problem, pp. 1044–1048 (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Qatar Environment and Energy Research Institute (QEERI)Hamad Bin Khalifa UniversityDohaQatar
  2. 2.Department of Technical SciencesState University of Novi PazarNovi PazarSerbia
  3. 3.Institute of Information SystemsUniversity of HamburgHamburgGermany
  4. 4.Escuela de Ingenieria IndustrialPontificia Universidad Católica de ValparaísoValparaísoChile

Personalised recommendations