Abstract
Given a graph and a length function defined on its edge-set, the Traveling Salesman Problem can be described as the problem of finding a family of edges (an edge may be chosen several times) which forms a spanning Eulerian subgraph of minimum length. In this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints. This characterization is given in terms of excluded minors. A constructive characterization is also given which uses a small number of basic graphs.
Similar content being viewed by others
References
J.A. Bondy and U.S.R. Murty,Graph Theory with Applications (MacMillan, London, 1976).
G. Cornuejols, J. Fonlupt and D. Naddef, “The Traveling Salesman Problem on a graph and some related integer polyhedra,”Mathematical Programming 33 (1985) 1–27.
G. Cornuejols, D. Naddef and W. Pulleyblank, “The Traveling Salesman Problem on Graphs with 3-edge cutsets,”Journal of the A.C.M. 32 (1985) 383–410.
H. Crowder and M. Padberg, “Solving large scale symmetric Traveling Salesman Problem to optimality,”Management Science 26 (1980) 485–509.
G.A. Dirac, “Some results concerning the structure of graphs,”Canadian Mathematics Bulletin 6 (1963) 183–210.
J. Edmonds and R. Giles, “Total dual integrality of linear inequality system,” in: W.R. Pulleyblank, ed.,Progress in Combinatorial Optimization (Academic Press, New York, 1984).
A. Elnacheff and J. Fonlupt, “Un algorithme polynomial pour le problème du voyageur de commerce sur une classe de graphes,” RT6, ARTEMIS-IMAG Université Scientifique, Technologique et Médicale de Grenoble (Grenoble, 1986).
J. Fonlupt and A. Nachef, “Dynamic programming and the graphical traveling salesman problem,” to appear in:Journal of the A.C.M.
M. Grötschel, L. Lovasz and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,”Combinatorica 1 (1981) 169–197.
W.R. Pulleyblank, “Polyhedral combinatorics,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming: The State of the Art (Springer, Berlin, 1983).
D. Ratliff and A. Rosenthal, “Order-picking in a rectangular warehouse: a solvable case of the Traveling Salesman Problem,”Operations Research 31 (1983) 507–521.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fonlupt, J., Naddef, D. The traveling salesman problem in graphs with some excluded minors. Mathematical Programming 53, 147–172 (1992). https://doi.org/10.1007/BF01585700
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585700