# Some Examples of Difficult Traveling Salesman Problems

@article{Papadimitriou1978SomeEO, title={Some Examples of Difficult Traveling Salesman Problems}, author={Christos H. Papadimitriou and Kenneth Steiglitz}, journal={Oper. Res.}, year={1978}, volume={26}, pages={434-443} }

We construct instances of the symmetric traveling salesman problem with n = 8k cities that have the following property: There is exactly one optimal tour with cost n, and there are 2k-1k-1! tours that are next-best, have arbitrarily large cost, and cannot be improved by changing fewer than 3k edges. Thus, there are many local optima with arbitrarily high cost. It appears that local search algorithms are ineffective when applied to these problems. Even more catastrophic examples are available in…

## 76 Citations

The Traveling Salesman Problem

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The TSP is perhaps the best-studied NP-hard combinatorial optimization problem, and there are many techniques which have been applied, and so-called local search algorithms find better solutions for large instances although they do not have a finite performance ratio.

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This work considers the asymmetric traveling salesman problem for which the triangular inequality is satisfied, and constructs examples to show that the worst-case ratio of length of tour found to minimum length tour is (n) for n city problems.

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The results show that the algorithm performs remarkably well for some classes of problems, determining an optimal solution even for problems with large numbers of cities, yet for other classes, even small problems thwart determination of a provably optimal solution.

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This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler, and Shmoys and focuses on special cases of the traveling salesman problem that have been obtained during the decade 1985--1995.

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