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.
Similar content being viewed by others
References
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.
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.
The Travelling Salesman Problem and Its Variations, Gutin, G. and Punnen, A.P., Eds., Dordrecht: Kluwer, 2002.
Held, M. and Karp, R., The Travelling Salesman Problem and Minimum Spanning Trees, Operat. Res., 1970, vol. 18, no. 6, pp. 1139–1162.
Balas, E. and Christofides, N., A Restricted Lagrangean Approach to the Travelling Salesman Problem, Math. Program., 1981, no. 1, pp. 19–46.
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.
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.
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.
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.
Christofides, N., The Travelling Salesman Problem, in Combinatorial Optimization, New York: Wiley, 1979, pp. 131–149.
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.
Bellmore, M. and Malone, J., Pathalogy of Travelling Salesman Subtour-Elimination Algorithms, Operat. Res., 1971, vol. 19, no. 2, pp. 278–301.
Christofides, N., Graph Theory. An Algorithmic Approach, New York: Academic, 1975. Translated under the title Teoriya grafov. Algoritmicheskii podkhod, Moscow: Mir, 1978.
Lovász, L. and Plummer, M.D., Matching theory, Providence: AMS, 2009.
Edmons, E., Paths, Trees and Flowers, Can. J. Math., 1965, vol. 17, pp. 449–467.
Ball, M.O. and Derigs, U., An Analysis of Alternate Strategies for Implementing Matching Algorithms Networks, 1983, vol. 13, pp. 517–549.
Grotschel, M. and Holland, O., A Cutting Plane Algorithm for Minimum Perfect 2-Matchings, Computing, 1987, vol. 39, pp. 327–344.
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.
Galil, Z., Efficient Algorithms for Finding Maximum Matching in Graphs, Computing Surveys, 1986, vol. 18, no. 1, pp. 23–38.
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.
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.
Sergeev, S.I., The Three-Dimensional Assignment and Partition Problems. New Lower Bounds, Autom. Remote Control, 2006, no. 2, pp. 242–250.
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.
Krotov, V.F. and Gurman, V.I., Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973.
Papadimitriou, C.H. and Steiglitz, K., Combinatorial Optimization: Algorithms and Complexity, Mineola: Dover, 1998.
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.
Rocafellar, R.T., Monotropic Programming: Descent Algorithms and Duality, in Nonlinear Programming-4, New Jersey: Academic, 1977, pp. 327–366.
Tseng, P. and Bertsekas, D., Relaxation Methods for Linear Programs, Math. Operat. Res., 1987, vol. 12, no. 4, pp. 569–596.
Bertsekas, D., A New Algorithm for the Assignment Problem, Math. Program., 1981, vol. 21, pp. 152–171.
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.
Author information
Authors and Affiliations
Additional information
Original Russian Text © S.I. Sergeev, 2009, published in Avtomatika i Telemekhanika, 2009, No. 11, pp. 148–160.
Rights 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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117909110149