Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through the work of M. Grötschel, S. Hong, M. Junger, P. Miliotis, D. Naddef, M. Padberg, W.R. Pulleyblank, G. Reinelt, G. Rinaldi, and others. We enumerate some of its refinements that led to the solution of a 13,509-city instance.

Dantzig, Fulkerson, and Johnson (1954) introduced the cutting-plane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et al.'s method that is suitable for TSP instances having 1,000,000 or more cities. Our aim is to use the study of the TSP as a step towards understanding the applicability and limits of the general cutting-plane method in large-scale applications.

The first computer implementation of the Dantzig-Fulkerson-Johnson cutting-plane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Grötschel, Padberg, and Hong advocated a different policy, where the template paradigm is the only source of cuts; furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Grötschel and Holland, and in the work of Padberg and Rinaldi; their computer codes produced the most impressive computational TSP successes of the nineteen eighties. Eventually, the template paradigm became the standard frame of reference for cutting planes in the TSP. The purpose of this paper is to describe a technique

Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.

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We describe a computer code and data that together certify the optimality of a solution to the 85,900-city traveling salesman problem pla85900, the largest instance in the TSPLIB collection of challenge problems.

The early computer implementation of the cutting-plane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory's type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Grotschel, Padberg, and Hong advocated a di#erent policy, where the template paradigm is the only source of cuts; furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Grotschel and Holland, and in the work of Padberg and Rinaldi; their computer codes produced the most impressive computational successes of the nineteen eighties. Eventually, the template paradigm had become the standard frame of reference for cutting planes in the TSP. The purpose of this paper i...

Increasing competition in the industrial and service sectors has led to a demand for optimal or provably near-optimal solutions to large scale optimization problems, and the exploration of a much larger range of alternatives in the race to improve productivity and efficiency. Many such decision problems involve the choice between a finite set of alternatives. Combinatorial optimization involves careful modeling of such problems, the mathematical analysis of the resulting discrete structures, the development of algorithms and their analysis, and the implementation of software to produce practical solutions.