Abstract
In this study we formulate the dual of the traveling salesman problem, which extends in a natural way the dual problem of Held and Karp to the nonsymmetric case. The dual problem is solved by a subgradient optimization technique. A branch and bound scheme with stepped fathoming is then used to find optimal and suboptimal tours. Computational experience for the algorithm is presented.
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References
M.S. Bazaraa and J.J. Goode, “A survey of various tactics for generating lagrangian multipliers in the context of lagrangian duality”, Georgia Institute of Technology, Atlanta, Georgia (August, 1974).
M.S. Bazaraa and A.N. Elshafei, “On the use of fictitious bounds in tree search algorithms”,Management Science 23 (1977) 904–908.
M. Bellmore and J.C. Malone, “Pathology of traveling-salesman subtour-elimination algorithms”,Operations Research 19 (1971) 278–307.
G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, “Solution of a large scale traveling salesman problem”,Operations Research 2 (1954) 393–410.
E.W. Dijkstra, “A note on two problems in connection with graphs”,Numerische Mathematik 1 (1956) 269–271.
M.L. Fisher, W.D. Northup and J.F. Shapiro, “Using duality to solve discrete optimization problems: theory and computational experience”,Mathematical Programming Study 3 (1975) 56–94.
A.M. Geoffrion, “Lagrangian relaxation and its use in integer programming”,Mathematical Programming Study 2 (1974) 82–114.
K.H. Hansen and J. Krarup, “Improvements of the Held—Karp algorithm for the symmetric traveling-salesman problem”,Mathematical Programming 7 (1974) 87–96.
M. Held and R.M. Karp, “The traveling salesman problem and minimum spanning trees”,Operations Research 18 (1970) 1138–1162.
M. Held and R.M. Karp, “The traveling salesman problem and minimum spanning trees: Part II”,Mathematical Programming 1 (1971) 6–25.
M. Held, P. Wolfe and H.D. Crowder, “Validation of subgradient optimization”,Mathematical Programming 6 (1974) 62–88.
L.L. Karg and G.L. Thompson, “A heuristic approach to solving traveling salesman problems”,Management Science 10 (1964) 225–248.
J.B. Kruskal, “On the shortest spanning subtree of the graph and the traveling salesman problem”,Proceedings of the American Mathematical Society 1 (1956) 48–50.
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This author's work was partially supported by NSF Grant #GK-38337.
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Bazaraa, M.S., Goode, J.J. The traveling salesman problem: A duality approach. Mathematical Programming 13, 221–237 (1977). https://doi.org/10.1007/BF01584338
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DOI: https://doi.org/10.1007/BF01584338