Abstract
Two algorithms using cutting planes are developed for solving the Travelling Salesman Problem. In both algorithms the problem is started with a subset of the set of constraints that define the problem (apart from integrality requirements).
However, the two algorithms differ in the order in which the omitted constraints and the cutting planes that are required are generated.
The computational experience obtained suggests that cutting planes can provide a competitive approach to other efficient methods of solving the problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Bellmore and G.L. Nemhauser, “The travelling salesman problem: a survey”,Operations Research 16 (1968) 538–558.
G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, “Solution of a large scale travelling salesman problem”,Operations Research 2 (1954) 393–410.
G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, “On a linear programming combinatorial approach to the travelling salesman problem”,Operations Research 7 (1959) 58–66.
R. Gomory, “An algorithm for integer solutions to linear programs”, in: R.L. Graves and P. Wolfe, eds.,Recent advances in mathematical programming (McGraw-Hill, New York, 1963) pp. 269–302.
K. Helbig Hansen and J. Krarup, “Improvements of the Held—Karp algorithm for the symmetric travelling-salesman problem”,Mathematical Programming 7 (1974) 87–96.
M. Held and R.M. Karp, “A dynamic programming approach to sequencing problems”,Journal of the Society for Industrial and Applied Mathematics 10 (1962) 196–210.
M. Held and R.M. Karp, “The travelling salesman problem and minimum spanning trees, part II”,Mathematical Programming 1 (1971) 6–25.
L.L. Karg and G.L. Thompson, “A heuristic approach to solving travelling salesman problems”,Management Science 10 (1964) 225–248.
A. Land and S. Powell,Fortran codes for mathematical programming: linear, quadratic and discrete (Wiley, New York, 1973).
G.T. Martin, “An accelerated euclidean algorithm for integer linear programming”, in: R.L. Graves and P. Wolfe, eds.,Recent advances in mathematical programming (McGraw-Hill, New York, 1963) pp. 311–318.
G.T. Martin, “Solving the travelling salesman problem by integer linear programming”,CEIR, New York (1966).
P. Miliotis, “Integer programming approaches to the travelling salesman problem”,Mathematical Programming 10 (1976) 367–378.
P. Miliotis, “An all-integer arithmetic LP-cutting planes code applied to the travelling salesman problem”, London School of Economics, Department of Operational Research (1975).
J.D. Murchland, “A fixed matrix method for all shortest distances in a directed graph and for the inverse problem”, Ph.D. Dissertation, Karlsruhe (1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Miliotis, P. Using cutting planes to solve the symmetric Travelling Salesman problem. Mathematical Programming 15, 177–188 (1978). https://doi.org/10.1007/BF01609016
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01609016