# A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems

@article{Padberg1991ABA, title={A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems}, author={Manfred W. Padberg and Giovanni Rinaldi}, journal={SIAM Rev.}, year={1991}, volume={33}, pages={60-100} }

An algorithm is described for solving large-scale instances of the Symmetric Traveling Salesman Problem (STSP) to optimality. The core of the algorithm is a “polyhedral” cutting-plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph. The cuts are generated by several identification procedures that have been described in a companion paper. Whenever the cutting-plane procedure does…

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## References

SHOWING 1-10 OF 46 REFERENCES

Facet identification for the symmetric traveling salesman polytope

- MathematicsMath. Program.
- 1990

Exact and heuristic shrinking conditions for the input graph are given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities.

A branch-and-cut approach to a traveling salesman problem with side constraints

- Computer Science
- 1989

A software system AIAA/SOLVER is described that is implemented to solve the problem to optimality under an apparently weak assumption about its stochastic cost structure using branch-and-cut.

The Traveling-Salesman Problem and Minimum Spanning Trees

- Computer ScienceOper. Res.
- 1970

It is shown that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers.

Improvements of the Held—Karp algorithm for the symmetric traveling-salesman problem

- Computer ScienceMath. Program.
- 1974

A highly efficient algorithm (HK) devised by Held and Karp for solving the symmetric traveling-salesman problem was presented at the 7th Mathematical Programming Symposium in 1970 and published in…

Solving Large-Scale Symmetric Travelling Salesman Problems to Optimality

- Computer Science
- 1980

The present study convincingly establishes the usefulness of mathematically proven good cutting-planes as an invaluable algorithmic tool for difficult combinatorial optimization problems.

Clique Tree Inequalities and the Symmetric Travelling Salesman Problem

- MathematicsMath. Oper. Res.
- 1986

A new class of inequalities clique tree inequalities valid for the travelling salesman polytope is defined which properly contains many of the known classes of inequalities like subtour elimination constraints, 2-matching constraints, comb inequalities, and it is shown that all these new inequalities induce facets of the Traveller's salesmanpolytope.

The traveling-salesman problem and minimum spanning trees: Part II

- BusinessMath. Program.
- 1971

An efficient iterative method for approximating this bound closely from below is presented, and a branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it.

Solution of large-scale symmetric travelling salesman problems

- Computer ScienceMath. Program.
- 1991

The implementation is based on a fast LP-solver (IBM's MPSX) and makes effective use of polyhedral results on the symmetric travelling salesman polytope and describes the important ingredients of the code.

An Effective Heuristic Algorithm for the Traveling-Salesman Problem

- BusinessOper. Res.
- 1973

This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric traveling-salesman problem. The procedure is based on a general…

On the symmetric travelling salesman problem: Solution of a 120-city problem

- Computer Science
- 1980

It is reported how the shortest roundtrip through 120 German cities was found using a commercial linear programming code and adding facetial cutting planes in an interactive way.