The adjacency relation on the traveling salesman polytope is NP-Complete

We consider the problem of determining whether two traveling salesman tours correspond to non-adjacent vertices of the convex polytope associated with the traveling salesman problem. This problem is shown to be NP-Complete for both the symmetric and nonsymmetric traveling salesman problem. Several implications are discussed.

[1]  Teofilo F. Gonzalez,et al.  P-Complete Problems and Approximate Solutions , 1974, SWAT.

[2]  George L. Nemhauser,et al.  The Traveling Salesman Problem: A Survey , 1968, Oper. Res..

[3]  Jack Edmonds,et al.  Matroids and the greedy algorithm , 1971, Math. Program..

[4]  David S. Johnson,et al.  The Planar Hamiltonian Circuit Problem is NP-Complete , 1976, SIAM J. Comput..

[5]  M. R. Rao,et al.  The travelling salesman problem and a class of polyhedra of diameter two , 1974, Math. Program..

[6]  Katta G. Murty,et al.  On the Tours of a Traveling Salesman , 1969 .

[7]  David S. Johnson,et al.  Some simplified NP-complete problems , 1974, STOC '74.

[8]  M. R. Rao,et al.  Adjacency of the Traveling Salesman Tours and $0 - 1$ Vertices , 1976 .

[9]  Teofilo F. Gonzalez,et al.  P-Complete Approximation Problems , 1976, J. ACM.

[10]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[11]  A. Bagchi,et al.  Neighborhood Search Algorithms for Guaranteeing Optimal Traveling Salesman Tours Must Be Inefficient , 1976, J. Comput. Syst. Sci..

[12]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[13]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[14]  Shen Lin.  Computer solutions of the traveling salesman problem , 1965 .

[15]  Egon Balas,et al.  Facets of the knapsack polytope , 1975, Math. Program..

[16]  K. Steiglitz,et al.  The Design of Minimum-Cost Survivable Networks , 1969 .

[17]  Katta G. Murty.  A fundamental problem in linear inequalities with applications to the travelling salesman problem , 1972, Math. Program..

[18]  Kenneth Steiglitz,et al.  On the Complexity of Local Search for the Traveling Salesman Problem , 1977, SIAM J. Comput..

[19]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[20]  Christos H. Papadimitriou,et al.  The Euclidean Traveling Salesman Problem is NP-Complete , 1977, Theor. Comput. Sci..

[21]  Richard M. Karp,et al.  On the Computational Complexity of Combinatorial Problems , 1975, Networks.