Skip to main content
Log in

The symmetric travelling salesman problem I. New fast lower bounds for the problem of optimal 2-matching

  • Modeling of Behavior and Intelligence
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

To solve the symmetric travelling salesman problem we suggest a lower bound—the solution of an optimal 2-matching problem. The latter problem is solved (in a polynomial number of steps) not completely, but up to obtaining new stable lower bounds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. The Travelling Salesman Problem. A Guided Tour of Combinatorial Optimization, Lawler, E.L., Lenstra, J.K., Rinnoy Kan, A.H.G., and Shmoys, D.B., Eds., New York: Wiley, 1985.

    Google Scholar 

  2. Melamed, I.I., Sergeev, S.I., and Sigal, I.Kh., The Traveling Salesman Problem. I-III, Autom. Remote Control, 1989, no. 9, pp. 1147–1173; no. 10, pp. 1303–1324; no. 11, pp. 1459–1479.

  3. The Travelling Salesman Problem and Its Variations, Gutin, G. and Punnen, A.P., Eds., Dordrecht: Kluwer, 2002.

    Google Scholar 

  4. Held, M. and Karp, R., The Travelling Salesman Problem and Minimum Spanning Trees, Operat. Res., 1970, vol. 18, no. 6, pp. 1139–1162.

    Article  MathSciNet  Google Scholar 

  5. Balas, E. and Christofides, N., A Restricted Lagrangean Approach to the Travelling Salesman Problem, Math. Program., 1981, no. 1, pp. 19–46.

  6. Krotov, V.F. and Sergeev, S.I., Computing Algorithms for Solving Certain Problems in Linear and Linear Integer Programming. I–IV, Autom. Remote Control, 1980, no. 12, pp. 1693–1701; 1981, no. 1, pp. 67–75; no. 3, pp. 339–349; no. 4, pp. 494–500.

  7. Jonker, R., de Leve, G., van der Velde, J.R., and Volgenant, A., Rounding Symmetric Travelling Salesman Problem with an Asymmetric Assignment Problem, Operat. Res., 1980, vol. 28, no. 3, part I, pp. 623–627.

    Article  MATH  Google Scholar 

  8. Stewart, W.E., An Improved Assignment Lower Bound for the Euclidean Travelling Salesman Problem, Operat. Res. Lett., 1985, vol. 4, no. 6, pp. 55–60.

    Article  MATH  Google Scholar 

  9. Volgenant, A., Van Der Slus, H.J., and Jonker, R., Better Assignment Lower Bounds for the Euclidean Travelling Salesman Problem, Optimization, 1987, vol. 18, no. 3, pp. 393–404.

    Article  MATH  MathSciNet  Google Scholar 

  10. Christofides, N., The Travelling Salesman Problem, in Combinatorial Optimization, New York: Wiley, 1979, pp. 131–149.

    Google Scholar 

  11. Smith, T.H.C., Meyer, T.W.S., and Thompson, G.L., Lower Bounds for the Symmetric Travelling Salesman Problem from Lagrangean Relaxations, Discrete Appl. Math., 1980, vol. 26, pp. 209–217.

    Article  MathSciNet  Google Scholar 

  12. Bellmore, M. and Malone, J., Pathalogy of Travelling Salesman Subtour-Elimination Algorithms, Operat. Res., 1971, vol. 19, no. 2, pp. 278–301.

    Article  MATH  MathSciNet  Google Scholar 

  13. Christofides, N., Graph Theory. An Algorithmic Approach, New York: Academic, 1975. Translated under the title Teoriya grafov. Algoritmicheskii podkhod, Moscow: Mir, 1978.

    MATH  Google Scholar 

  14. Lovász, L. and Plummer, M.D., Matching theory, Providence: AMS, 2009.

    MATH  Google Scholar 

  15. Edmons, E., Paths, Trees and Flowers, Can. J. Math., 1965, vol. 17, pp. 449–467.

    Google Scholar 

  16. Ball, M.O. and Derigs, U., An Analysis of Alternate Strategies for Implementing Matching Algorithms Networks, 1983, vol. 13, pp. 517–549.

    Article  MATH  MathSciNet  Google Scholar 

  17. Grotschel, M. and Holland, O., A Cutting Plane Algorithm for Minimum Perfect 2-Matchings, Computing, 1987, vol. 39, pp. 327–344.

    Article  MathSciNet  Google Scholar 

  18. Noon, Ch., You, G.-M., and Chan, T.J., A Fast Lower Bound for the Minimum Cost Perfect 2-Matching Linear Program, Am. J. Math. Manag. Sci., 1993, vol. 13, no. 3–4, pp. 357–370.

    MATH  MathSciNet  Google Scholar 

  19. Galil, Z., Efficient Algorithms for Finding Maximum Matching in Graphs, Computing Surveys, 1986, vol. 18, no. 1, pp. 23–38.

    Article  MATH  MathSciNet  Google Scholar 

  20. Sergeev, S.I., Computational Algorithms for the Solution of the Salesman Problem. I, II, Autom. Remote Control, 1994, no. 5, pp. 669–680; no. 6, pp. 861–868.

  21. Sergeev, S.I., New Lower Bounds for the Triplanar Assignment Problem. I, II, in Tr. III Mezhdunar. konf. “Identifikatsiya sistem i zadachi upravleniya” (SICPRO’04) (Proc. of III International Conference “System Identification and Control Problems” (SICPRO’04)), Moscow: Inst. Probl. Upravlen., 2004, pp. 1708–1726; pp. 1727–1740.

    Google Scholar 

  22. Sergeev, S.I., The Three-Dimensional Assignment and Partition Problems. New Lower Bounds, Autom. Remote Control, 2006, no. 2, pp. 242–250.

  23. Krotov, V.F., Computational Algorithms for Solving and Optimizing Controlled Systems of Equations. I, II, Izv. Akad. Nauk SSSR, Tekhn. Kibern., 1975, no. 5, pp. 3–15; no. 6, pp. 3–13.

  24. Krotov, V.F. and Gurman, V.I., Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973.

    Google Scholar 

  25. Papadimitriou, C.H. and Steiglitz, K., Combinatorial Optimization: Algorithms and Complexity, Mineola: Dover, 1998.

    MATH  Google Scholar 

  26. Krotov, V.F. and Sergeev, S.I., Applying Sufficient Optimality Conditions to Solving Linear and Integer Linear Programming Problems, in Modelirovanie tekhniko-ekonomicheskikh protsessov (Modeling of Technical-economical Processes), Moscow: Mosk. Ekon.-Stat. Inst., 1978, pp. 3–42.

    Google Scholar 

  27. Rocafellar, R.T., Monotropic Programming: Descent Algorithms and Duality, in Nonlinear Programming-4, New Jersey: Academic, 1977, pp. 327–366.

    Google Scholar 

  28. Tseng, P. and Bertsekas, D., Relaxation Methods for Linear Programs, Math. Operat. Res., 1987, vol. 12, no. 4, pp. 569–596.

    Article  MATH  MathSciNet  Google Scholar 

  29. Bertsekas, D., A New Algorithm for the Assignment Problem, Math. Program., 1981, vol. 21, pp. 152–171.

    Article  MATH  MathSciNet  Google Scholar 

  30. Sergeev, S.I., On a Variant of the Residual Reducing Method, in Modelirovanie tekhniko-ekonomicheskikh protsessov (Modeling of Technical-economical Processes), Moscow: Mosk. Ekon.-Stat. Inst., 1976, pp. 44–52.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Original Russian Text © S.I. Sergeev, 2009, published in Avtomatika i Telemekhanika, 2009, No. 11, pp. 148–160.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sergeev, S.I. The symmetric travelling salesman problem I. New fast lower bounds for the problem of optimal 2-matching. Autom Remote Control 70, 1901–1912 (2009). https://doi.org/10.1134/S0005117909110149

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117909110149

PACS numbers

Navigation