Mathematical Programming

, Volume 33, Issue 1, pp 1–27

# The traveling salesman problem on a graph and some related integer polyhedra

• Gérard Cornuéjols
• Jean Fonlupt
Article

## Abstract

Given a graphG = (N, E) and a length functionl: E → ℝ, the Graphical Traveling Salesman Problem is that of finding a minimum length cycle goingat least once through each node ofG. This formulation has advantages over the traditional formulation where each node must be visited exactly once. We give some facet inducing inequalities of the convex hull of the solutions to that problem. In particular, the so-called comb inequalities of Grötschel and Padberg are generalized. Some related integer polyhedra are also investigated. Finally, an efficient algorithm is given whenG is a series-parallel graph.

## Key words

Traveling Salesman Problem Integer Polyhedron Facet Graph Series-Parallel Graph Steiner Tree Polynomial Algorithm

## References

1. [1]
J.A. Bondy and U.S.R. Murty,Graph theory with applications (North-Holland, Amsterdam, 1976).Google Scholar
2. [2]
G. Cornuéjols, D. Naddef and W.R. Pulleyblank, “Halin graphs and the travelling salesman problem”,Mathematical Programming 26 (1983) 287–294.Google Scholar
3. [3]
J. Edmonds, “Matroids and the greedy algorithm”,Mathematical Programming 1 (1971) 127–136.Google Scholar
4. [4]
B. Fleischmann, “Linear programming approaches to travelling salesman and vehicle scheduling problems”, Technical Report, Universität Hamburg (paper presented at the XI. International Symposium on Mathematical Programming, Bonn, 1982).Google Scholar
5. [5]
B. Fleischmann, “A new class of cutting planes for the symmetric travelling salesman problem”, Report No. QM-03-83, Institut für Unternehmensforschung, Universität Hamburg (1983).Google Scholar
6. [6]
G.N. Frederikson and J. Jager, “On the relationship between biconnectivity and travelling salesman problems”,Theoretical Computer Science 19 (1982) 189–201.Google Scholar
7. [7]
M. Grötschel and M.W. Padberg, “On the symmetric travelling salesman problem I, II”,Mathematical Programming 16 (1979) 265–302.Google Scholar
8. [8]
W. Pulleyblank, “Faces of matching polyhedra”, Ph.D. Thesis, University of Waterloo (Waterloo, Ontario, 1973).Google Scholar
9. [9]
D. M. Ratliff and A.S. Rosenthal, “Order-picking in a rectangular warehouse: A solvable case of the travelling salesman problem”,Operations Research 31 (1983) 507–521.Google Scholar
10. [10]
K. Takamizawa, T. Nishizeki and N. Saito, “Linear time computability of combinatorial problems on series-parallel graphs”,Journal of the ACM 29 (1982) 632–641.Google Scholar
11. [11]
J.A. Wald and C.J. Colbourn, “Steiner trees, partial 2-trees and minimum IFI methods”,Networks 13 (1983) 159–167.Google Scholar

© The Mathematical Programming Society, Inc. 1985

## Authors and Affiliations

• Gérard Cornuéjols
• 1
• Jean Fonlupt
• 2