Fast EAX Algorithm Considering Population Diversity for Traveling Salesman Problems

  • Yuichi Nagata
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3906)


This paper proposes an evolutionary algorithm (EA) that is applied to the traveling salesman problem (TSP). Existing approximation methods to address the TSP known to be state-of-the-art heuristics almost exclusively utilize Lin-Kernighan local search (LKLS) and its variants. We propose an EA that does not use LKLS, and demonstrate that it is comparable with these heuristics even though it does not use them. The proposed EA uses edge assembly crossover (EAX) that is known to be an efficient and effective crossover for solving TSPs. We first propose a modified EAX algorithm that can be executed more efficiently than the original, which is 2–7 times faster. We then propose a selection model that can efficiently maintain population diversity at negligible computational cost. The edge entropy measure is used as an indicator of population diversity.

The proposed method called EAX-1AB(ENT) is applied to TSP benchmarks up to instances of 13509 cities. Experimental results reveal that EAX-1AB(ENT) with a population of 200 can almost always find optimal solutions effectively in most TSP benchmarks up to instances of 5915 cities. In the experiments, a previously proposed EAs using EAX can find an optimal solution of usa13509 with reasonable computational cost due to the fast EAX algorithm proposed in this paper. We also demonstrate that EAX-1AB(ENT) is comparable to well-known LKLS methods when relatively small populations such as 30 are used.


Population Diversity Fast Algorithm Travel Salesman Problem Travel Salesman Problem Iterate Local Search 


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  1. 1.
    Johnson, D.S.: Local Optimization and the Traveling Salesman Problem, Automata, Languages and Programming. In: Schmidt, D.A., Main, M.G., Melton, A.C., Mislove, M.W. (eds.) MFPS 1989. LNCS, vol. 442, pp. 446–461. Springer, Heidelberg (1990)Google Scholar
  2. 2.
    8th DIMACS Implementation Challenge: The Traveling Salesman Problem,
  3. 3.
    Lin, S., Kernighan, B.: Effective heuristic algorithms for the traveling salesman problem. Oper. Res. 21, 498–516 (1973)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Applegate, D., Bixby, R., Chvatal, V., Cook, W.: Finding tours in the TSP. Technical Report 99885, Forschungsinstitut fur Diskrete Mathematik, Universitat Bonn (1999)Google Scholar
  5. 5.
    Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Nagata, Y., Kobayashi, S.: Edge Assembly Crossover: A High-power Genetic Algorithm for the Traveling Salesman Problem. In: Proc. of the 7th Int. Conference on Genetic Algorithms, pp. 450–457 (1997)Google Scholar
  7. 7.
    Tsai, H.K., Yang, J.M., Kao, C.Y.: Solving Traveling Salesman Problems by Combining Global and Local Search Mechanisms. In: Proc. of the the 2002 Congress on Evolutionary Computation, pp. 1290–1295 (2002)Google Scholar
  8. 8.
    Ikeda, K., Kobayashi, S.: Deterministic multi-step crossover fusion: A handy crossover composition for gAs. In: Guervós, J.J.M., Adamidis, P.A., Beyer, H.-G., Fernández-Villacañas, J.-L., Schwefel, H.-P. (eds.) PPSN 2002. LNCS, vol. 2439, pp. 162–171. Springer, Heidelberg (2002)Google Scholar
  9. 9.
    Watson, J.-P., Ross, C., Eisele, V., Denton, J., Bins, J., Guerra, C., Whitley, L.D., Howe, A.E.: The traveling salesrep problem, edge assembly crossover, and 2-opt. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 823–833. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Nagata, Y.: Criteria for Designing Crossovers for TSP. In: Proc. of the 2004 Congress on Evolutionary Computation, pp. 1465–1472 (2004)Google Scholar
  11. 11.
    Merz, P., Freisleben, B.: Genetic Local Search for the TSP: New Results. In: Proc. of the 1997 IEEE Int. Conf. on Evolutionary Computation, pp. 159–163 (1997)Google Scholar
  12. 12.
    Tsai, H.K., Yang, J.M., Tsai, Y.F., Kao, C.Y.: An Evolutionary Algorithm for Large Traveling Salesman Problem. IEEE Transaction on SMC-part B 34(4), 1718–1729 (2004)Google Scholar
  13. 13.
    Maekawa, K., Mori, N., Kita, H., Nishikawa, H.: A Genetic Solution for the Traveling Salesman Problem by Means of a Thermodynamical Selection Rule. In: Proc. 1996 IEEE Int. Conference on Evolutionary Computation, pp. 529–534 (1996)Google Scholar
  14. 14.
    Nagata, Y.: The EAX algorithm considering diversity loss. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) PPSN 2004. LNCS, vol. 3242, pp. 332–341. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yuichi Nagata
    • 1
  1. 1.Graduate School of Information SciencesJapan Advanced Institute of Science and TechnologyJapan

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