Edges Elimination for Traveling Salesman Problem Based on Frequency \(K_5\)s

  • Yong WangEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


We eliminate the useless edges for traveling salesman problem (TSP) based on frequency \(K_5\)s. A frequency \(K_5\) is computed with ten optimal five-vertex paths with given endpoints in a corresponding \(K_5\) in \(K_n\). A binomial distribution model is built based on frequency \(K_5\)s. As the frequency of each edge is computed with N frequency \(K_5\)s, the binomial distribution demonstrates that the frequency of an optimal Hamiltonian cycle edge is bigger than 4N on average. Thus, one can eliminate the edges with frequency below 4N to reduce the number of concerned edges for resolving TSP. A heuristic algorithm is given to eliminate the useless edges. After many useless edges are cut, the computation time of algorithms for TSP will be considerably reduced.


Traveling salesman problem Frequency \(K_5\) Binomial distribution Heuristic algorithm 


  1. 1.
    Johnson, D.S., McGeoch, L.-A.: The Traveling Salesman Problem and its Variations, Combinatorial Optimization. 1st edn. Springer Press, London (2004)Google Scholar
  2. 2.
    Karp, R.: On the computational complexity of combinatorial problems. Networks 5(1), 45–68 (1975)CrossRefGoogle Scholar
  3. 3.
    Held, M., Karp, R.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math 10(1), 196–210 (1962)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bellman, R.: Dynamic programming treatment of the traveling salesman problem. J. ACM 9(1), 61–63 (1962)CrossRefGoogle Scholar
  5. 5.
    Klerk, E.-D., Dobre, C.: A comparison of lower bounds for the symmetric circulant traveling salesman problem. Discret. Appl. Math 159(16), 1815–1826 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Applegate, D., Bixby, R., Chvátal, V., Cook, W., Espinoza, D.-G., Goycoolea, M., Helsgaun, K.: Certification of an optimal TSP tour through 85900 cities. Oper. Res. Lett. 37(1), 11–15 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Thomas, H.-C., Charles, E.-L., Ronald, L.-R., Clifford, S.: Introduction to Algorithm, 2nd edn. China Machine Press, Beijing (2006)Google Scholar
  8. 8.
    Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: FOCS 2011, pp. 560–569. IEEE, NY (2011)Google Scholar
  9. 9.
    Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sharir, M., Welzl, E.: On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput. 36(3), 695–720 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: The traveling salesman problem in bounded degree graphs. ACM T. Algorithms 8(2), 1–18 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Correa, J.-R., Larré, O., Soto, J.-A.: TSP tours in cubic graphs: beyond 4/3. SIAM J. Discret. Math. 29(2), 915–939 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Borradaile, G., Demaine, E.-D., Tazari, S.: Polynomial-time approximation schemes for subset-connectivity problems in bounded-genus graphs. Algorithmica 68(2), 287–311 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gharan, S.-O., Saberi, A.: The asymmetric traveling salesman problem on graphs with bounded genus. In: SODA 2011, pp. 23–25. ACM (2011)Google Scholar
  15. 15.
    Jonker, R., Volgenant, T.: Nonoptimal edges for the symmetric traveling salesman problem. Oper. Res. 32(4), 837–846 (1984)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hougardy, S., Schroeder, R.-T.: Edges elimination in TSP instances. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 275–286. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  17. 17.
    Taillard, \(\acute{E}\).-D., Helsgaun, K.: POPMUSIC for the traveling salesman problem. Eur. J. Oper. Res. 272(2), 420–429 (2019)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang, Y., Remmel, J.-B.: A binomial distribution model for the traveling salesman problem based on frequency quadrilaterals. J. Graph Algorithms Appl. 20(2), 411–434 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wang, Y., Remmel, J.-B.: An iterative algorithm to eliminate edges for traveling salesman problem based on a new binomial distribution. Appl. Intell. 48(11), 4470–4484 (2018)CrossRefGoogle Scholar
  20. 20.
    Wang, Y.: An approximate method to compute a sparse graph for traveling salesman problem. Expert Syst. Appl. 42(12), 5150–5162 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.North China Electric Power UniversityBeijingChina

Personalised recommendations