Results 1  10
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25
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
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
SCIP  a framework to integrate constraint and mixed integer programming
, 2004
"... 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 int ..."
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Cited by 24 (1 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.
On the Separation of Split Cuts and Related Inequalities
 Mathematical Programming
"... 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 20 (1 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. Key words: Cutting planes, separation, complexity, travelling salesman problem, comb inequalities. 1
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 19 (2 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......
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 12 (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...
The symmetric traveling salesman polytope: New facets from the graphical relaxation
 MATHEMATICS OF OPERATIONS RESEARCH
, 2007
"... ..."
Restricted 2Factor Polytopes
, 1998
"... The optimal krestricted 2factor problem consists of finding, in a complete undirected graph K n , a minimum cost 2factor (subgraph having degree 2 at every node) with all components having more than k nodes. The problem is a relaxation of the wellknown symmetric travelling salesman problem, and ..."
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Cited by 8 (0 self)
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The optimal krestricted 2factor problem consists of finding, in a complete undirected graph K n , a minimum cost 2factor (subgraph having degree 2 at every node) with all components having more than k nodes. The problem is a relaxation of the wellknown symmetric travelling salesman problem, and is equivalent to it when n 2 k n \Gamma 1. We study the krestricted 2factor polytope. We present a large class of valid inequalities, called bipartition inequalities, and describe some of their properties; some of these results are new even for the travelling salesman polytope. For the case k = 3, the trianglefree 2factor polytope, we derive a necessary and sufficient condition for such inequalities to be facet inducing. Research partially supported by a grant from N.S.E.R.C. of Canada 1 Introduction A 2factor of an undirected graph G = (V; E) is a spanning subgraph H of G that has degree 2 at each node. Equivalently, it is a set of nodedisjoint circuits that include all of t...
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.