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24
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
WorstCase Comparison of Valid Inequalities for the TSP
 MATH. PROG
, 1995
"... We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worstcase improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor gr ..."
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Cited by 25 (1 self)
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We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worstcase improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10/9. The corresponding factor for the class of clique tree inequalities is 8/7, while it is 4/3 for the path configuration inequalities.
SEPARATING A SUPERCLASS OF COMB INEQUALITIES IN PLANAR GRAPHS
, 2000
"... Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedu ..."
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Cited by 23 (6 self)
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Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x∗ , nds one or more inequalities in the class which are violated by x , or proves that none exist. Such algorithms are at the core of the highly successful branchandcut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We de ne a generalization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges with x ∗ e ¿0 is planar. The separation algorithm runs in O(n³) time, where n is the number of vertices in the graph.
The Symmetric Generalized Travelling Salesman Polytope
, 1995
"... The symmetric Generalized Travelling Salesman Problem (GTSP) is a variant of the classical symmetric Travelling Salesman Problem, in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster. A different version of the problem, called EGTSP, aris ..."
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Cited by 21 (5 self)
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The symmetric Generalized Travelling Salesman Problem (GTSP) is a variant of the classical symmetric Travelling Salesman Problem, in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster. A different version of the problem, called EGTSP, arises when exactly one node for each cluster has to be visited. Both GTSP and EGTSP are NPhard problems, and find practical applications in routing and scheduling. In this paper we model GTSP and EGTSP as integer linear programs, and study the facial structure of the corresponding polytopes. In a companion paper (Fischetti, Salazar and Toth [5]), the results described in this work have been used to design a branchandcut algorithm for the exact solution of instances up to 442 nodes.
The Circuit Polytope: Facets
, 1994
"... Given an undirected graph G = (V; E) and a cost vector c 2 IR E , the weighted girth problem is to find a circuit in G having minimum total cost. This problem is in general NPhard since the traveling salesman problem can be reduced to it. A promising approach to hard combinatorial optimization p ..."
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Cited by 15 (1 self)
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Given an undirected graph G = (V; E) and a cost vector c 2 IR E , the weighted girth problem is to find a circuit in G having minimum total cost. This problem is in general NPhard since the traveling salesman problem can be reduced to it. A promising approach to hard combinatorial optimization problems is given by the socalled cutting plane methods. These involve linear programming techniques based on a partial description of the convex hull of the incidence vectors of possible solutions. We consider the weighted girth problem in the case where G is the complete graph K n and study the facial structure of the circuit polytope P n C and some related polyhedra. In the appendix we give complete characterizations of P n C for n up to 6.
Decomposition and Parallelization Techniques for Enumerating the Facets of 0/1Polytopes
 Int. J. Comput. Geom. Appl
, 1998
"... A convex polytope can either be described as convex hull of vertices or as solution set of a finite number of linear inequalities and equations. Whereas both representations are equivalent from a theoretical point of view, they are not when optimization problems over the polytope have to be solved. ..."
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Cited by 13 (2 self)
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A convex polytope can either be described as convex hull of vertices or as solution set of a finite number of linear inequalities and equations. Whereas both representations are equivalent from a theoretical point of view, they are not when optimization problems over the polytope have to be solved. Moreover, it is a challenging task in practical computation to convert one description into the other. In this paper we address the efficient computation of the facet structure of polytopes given by their vertices and present new computational results for polytopes which are of interest in combinatorial optimization. Keywords: polytope, convex hull, combinatorial optimization 1 Introduction Hard combinatorial optimization problems are often attacked with branchandcut methods. These methods strongly rely on knowledge about the structure of the polytope that is defined as convex hull of the 0/1 incidence vectors of feasible solutions. In particular, knowledge about linear equations and ineq...
The symmetric traveling salesman polytope: New facets from the graphical relaxation
 MATHEMATICS OF OPERATIONS RESEARCH
, 2007
"... ..."
Efficient Separation Routines for the Symmetric Traveling Salesman Problem II: Separating multi Handle Inequalities
, 2001
"... This paper is the second in a series of two papers dedicated to the separation problem in the symmetric traveling salesman polytope. The first one gave the basic ideas behind the separation procedures and applied them to the separation of Comb inequalities. We here address the problem of separating ..."
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Cited by 8 (2 self)
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This paper is the second in a series of two papers dedicated to the separation problem in the symmetric traveling salesman polytope. The first one gave the basic ideas behind the separation procedures and applied them to the separation of Comb inequalities. We here address the problem of separating inequalities which are all, in a way or another, a generalization of Comb inequalities. These are namely Clique Trees, Path, Ladder inequalities. Computational results are reported for the solution of instances of the TSPLib using the branch and cut framework ABACUS.