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.

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

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...

We describe a simple process for generating numerically safe cutting planes using floating-point arithmetic and the mixed-integer rounding (MIR) procedure. Applying this method to the rows of the simplex tableau permits the generation of Gomory mixed-integer cuts that are guaranteed to be satisfied by all feasible solutions to a mixed-integer programming problem. We report on tests with the MIPLIB 3.0 and MI-PLIB 2003 test collections, and with MIP instances derived from the TSPLIB traveling salesman library. 1

Throughout the history of integer programming, the field has been guided by research into solution approaches to combinatorial problems. We discuss some of the highlights and defining moments of this area. 1 Combinatorial integer programming Integer-programming models arise naturally in optimization problems over combinatorial structures, most notably in problems on graphs and general set systems. The translation from combinatorics to the language of integer programming is often straightforward, but the new rendering typically suggests direct lines of attack via linear programming. As an example, consider the stable-set problem in graphs. Given a graph G = (V, E) with vertices V and edges E, a stable set of G is a subset S ⊆ V such that no two vertices in S are joined by an edge. The stable-set problem is to find a maximum-cardinality stable set. To formulate this as an integer-programming (IP) problem, consider a vector of variables x = (xv: v ∈ V) and identify a set U ⊆ V with its characteristic vector ¯x, defined as ¯xv = 1 if v ∈ U and ¯xv = 0 otherwise. For e ∈ E write e = (u, v), where u and v are the ends of the edge. The stable-set problem is equivalent to the IP model