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35
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 226 (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 ... some of its refinements that led to the solution of a 13,509city instance.
MINIMUMWEIGHT TWOCONNECTED SPANNING NETWORKS
, 1990
"... We consider the problem of constructing a minimumweight, twoconnected network spanning all the points in a set V. We assume a symmetric, nonnegative distance function d ( ' ) defined on V x V which satisfies the triangle inequality. We obtain a structural characterization of optimal solution ..."
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Cited by 47 (3 self)
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We consider the problem of constructing a minimumweight, twoconnected network spanning all the points in a set V. We assume a symmetric, nonnegative distance function d ( ' ) defined on V x V which satisfies the triangle inequality. We obtain a structural characterization of optimal solutions. Specifically, there exists an optimal twoconnected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since we also show thar any twoconnected graph satisfying these conditions is the unique optimal solution for a particular choice of 'canonical' distances satisfying the triangle inequality. we use these properties to show that the weight of an optimal traveling salesman cycle i, uiort f times the weight of an optimal twoconnected solution; examples are provided which approach this bound arbiirarily closely. In addition, we obtain similar results for the variation of thii problem where the network need only span a prespecified subset of the points.
SCIP  a framework to integrate Constraint and Mixed Integer Programming
, 2005
"... Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today’s stateoftheart algorithms of both fields have several strategies in common, in particular the branchandbound process to recursively divide the problem in ..."
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Cited by 34 (2 self)
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Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today’s stateoftheart algorithms of both fields have several strategies in common, in particular the branchandbound process to recursively divide the problem into smaller subproblems. On the other hand, the main techniques to process each subproblem are different, and it was observed that they have complementary strengths. We present the programming framework Scip that integrates techniques from both fields in order to exploit the strengths of both, Constraint Programming and Mixed Integer Programming. In contrast to other proposals of recent years to combine both fields, Scip does not focus on easy implementation and rapid prototyping, but is tailored towards expert users in need of full, indepth control and high performance.
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 27 (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
A New BranchandCut Algorithm for the Capacitated Vehicle Routing Problem
 Mathematical Programming
, 2003
"... We present a new branchandcut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities, and classical Gomo ..."
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Cited by 27 (4 self)
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We present a new branchandcut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities, and classical Gomory mixedinteger cuts. For each of these classes of inequalities we descrine our separation algorithms in detail......
On the Separation of Split Cuts and Related Inequalities
 MATHEMATICAL PROGRAMMING
, 2001
"... The split cuts of Cook, Kannan and Schrijver are generalpurpose valid inequalities for integer programming which include a variety of other wellknown cuts as special cases. To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was ..."
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Cited by 24 (3 self)
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The split cuts of Cook, Kannan and Schrijver are generalpurpose valid inequalities for integer programming which include a variety of other wellknown cuts as special cases. To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was recently cited as an open problem by Cornuejols & Li [10]. In this paper we settle this question by proving strong NPcompleteness of separation for split cuts. As a byproduct we also show NPcompleteness of separation for several other classes of inequalities, including the MIRinequalities of Nemhauser and Wolsey and some new inequalities which we call balanced split cuts and binary split cuts. We also strengthen NPcompleteness results of Caprara & Fischetti [5] (for {0, 1 2 }cuts) and Eisenbrand [12] (for ChvatalGomory cuts). To compensate for this bleak picture, we also give a positive result for the Symmetric Travelling Salesman Problem. We show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle.
Separating Maximally Violated Comb Inequalities in Planar Graphs
 Math. Oper. Res
, 1997
"... The Traveling Salesman Problem (TSP) is a benchmark problem in combinatorial optimization. It was one of the very first problems used for developing and testing approaches to solving large integer programs, including cutting plane algorithms and branchandcut algorithms. Much of the research in thi ..."
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Cited by 14 (2 self)
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The Traveling Salesman Problem (TSP) is a benchmark problem in combinatorial optimization. It was one of the very first problems used for developing and testing approaches to solving large integer programs, including cutting plane algorithms and branchandcut algorithms. Much of the research in this area has been focused on finding new classes of facets for the TSP polytope, and much less attention has been paid to algorithms for separating from these classes of facets. In this paper, we consider the problem of finding violated comb inequalities. If there are no violated subtour constraints in a fractional solution of the TSP, a comb inequality may not be violated by more than 1. Given a fractional solution in the subtour elimination polytope whose graph is planar, we either find a violated comb inequality or determine that there are no comb inequalities violated by 1. Our algorithm runs in O(n + MC(n)) time, where MC(n) is the time to compute a cactus representation of all minimum cu...
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 13 (5 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.