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On the Solution of Traveling Salesman Problems
 DOC. MATH. J. DMV
, 1998
"... 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 TS ..."
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Cited by 164 (7 self)
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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. Jünger , P. Miliotis , D. Naddef , M. Padberg
Implementing the DantzigFulkersonJohnson Algorithm for Large Traveling Salesman Problems
, 2003
"... Dantzig, Fulkerson, and Johnson (1954) introduced the cuttingplane 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 ..."
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Cited by 36 (6 self)
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Dantzig, Fulkerson, and Johnson (1954) introduced the cuttingplane 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 cuttingplane method in largescale applications.
TSP cuts which do not conform to the template paradigm
 IN COMPUTATIONAL COMBINATORIAL OPTIMIZATION
, 2001
"... The first computer implementation of the DantzigFulkersonJohnson cuttingplane 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 c ..."
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Cited by 25 (1 self)
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The first computer implementation of the DantzigFulkersonJohnson cuttingplane 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
Certification of an optimal TSP tour through 85,900 cities
, 2007
"... We describe a computer code and data that together certify the optimality of a solution to the 85,900city traveling salesman problem pla85900, the largest instance in the TSPLIB collection of challenge problems. ..."
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Cited by 9 (1 self)
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We describe a computer code and data that together certify the optimality of a solution to the 85,900city traveling salesman problem pla85900, the largest instance in the TSPLIB collection of challenge problems.
FiftyPlus Years of Combinatorial Integer Programming
, 2009
"... 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 Integerprogramming models arise naturally in optimization ..."
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Cited by 2 (0 self)
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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 Integerprogramming 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 stableset 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 stableset problem is to find a maximumcardinality stable set. To formulate this as an integerprogramming (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 stableset problem is equivalent to the IP model
TSP cuts outside the template paradigm
, 2000
"... The early computer implementation of the cuttingplane 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 cut ..."
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Cited by 1 (0 self)
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The early computer implementation of the cuttingplane 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...
ISSN 16148835 (Print) The Asymmetric Quadratic Traveling Salesman Problem ∗
, 2011
"... Abstract. The quadratic traveling salesman problem asks for a tour of minimal costs where the costs are associated with each two arcs that are traversed in succession. This structure arises, e. g., if the succession of two arcs represents the costs of loading processes in transport networks or a swi ..."
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Abstract. The quadratic traveling salesman problem asks for a tour of minimal costs where the costs are associated with each two arcs that are traversed in succession. This structure arises, e. g., if the succession of two arcs represents the costs of loading processes in transport networks or a switch between different technologies in communication networks. Based on a quadratic integer program we present a linearized integer programming formulation and study the corresponding polyhedral structure of the asymmetric quadratic traveling salesman problem (AQTSP), where the costs may depend on the direction of traversal. The constructive approach that is used to establish the dimension of the underlying polyhedron allows to prove the facetness of several classes of valid inequalities. Some of them are related to the Boolean quadric polytope. Two new classes are presented that exclude conflicting configurations. Among these the first one is separable in polynomial time, the separation problem for the second class is NPcomplete under certain conditions. We provide a general strengthening approach that allows to lift valid inequalities for the asymmetric