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… 

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References

SHOWING 1-10 OF 28 REFERENCES
A Method for Solving Traveling-Salesman Problems
TLDR
A method of solution for the traveling-salesman problem that is applicable to both symmetric and asymmetric problems with random elements, and does not use subjective decisions, so that it can be completely mechanized.
Computer solutions of the traveling salesman problem
TLDR
Two algorithms for solving the (symmetric distance) traveling salesman problem have been programmed for a high-speed digital computer and are based on a general heuristic approach believed to be of general applicability to various optimization problems.
On the Complexity of Local Search for the Traveling Salesman Problem
It is shown that, unless $P = NP$, local search algorithms for the traveling salesman problem having polynomial time complexity per iteration will generate solutions arbitrarily far from the optimal.
P-Complete Approximation Problems
For P-complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is
The Traveling Salesman Problem: A Survey
TLDR
A survey and synthesis of research on the traveling salesman problem is given and a general classification of the solution techniques and a detailed description of some of the proven methods are given.
The complexity of theorem-proving procedures
  • S. Cook
  • Mathematics, Computer Science
    STOC
  • 1971
It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a
Reducibility among combinatorial problems" in complexity of computer computations
TLDR
In his 1972 paper, Reducibility Among Combinatorial Problems, Richard Karp used Stephen Cooks 1971 theorem that the boolean satisfiability problem is.
KERNIGHAN, "An Effective Heuristic Algorithm for the Traveling Salesman Problem," Opns
  • Res
  • 1973
Worst Case Analysis of an Algorithm for the Traveling Salesman Problem," presented at the Carnegie-Mellon Conference on Algorithms and Complexity
  • 1976
KERNIGHAN, "An Effective Heuristic Algorithm for the Traveling Salesman Problem,
  • Opns. Res
  • 1973
...
...