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202
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 225 (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.
An effective implementation of the linkernighan traveling salesman heuristic
 European Journal of Operational Research
, 2000
"... This report describes an implementation of the LinKernighan heuristic, one of the most successful methods for generating optimal or nearoptimal solutions for the symmetric traveling salesman problem. Computational tests show that the implementation is highly effective. It has found optimal solution ..."
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Cited by 188 (1 self)
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This report describes an implementation of the LinKernighan heuristic, one of the most successful methods for generating optimal or nearoptimal solutions for the symmetric traveling salesman problem. Computational tests show that the implementation is highly effective. It has found optimal solutions for all solved problem instances we have been able to obtain, including a 7397city problem (the largest nontrivial problem instance solved to optimality today). Furthermore, the algorithm has improved the best known solutions for a series of largescale problems with unknown optima, among these an 85900city problem. 1.
A BranchandCut Algorithm for the Symmetric Generalized Travelling Salesman Problem
, 1995
"... We consider 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. This NPhard problem is known in the literature as the symmetric Generalized Travelling Salesman Problem (GT ..."
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Cited by 75 (4 self)
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We consider 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. This NPhard problem is known in the literature as the symmetric Generalized Travelling Salesman Problem (GTSP), and finds practical applications in routing, scheduling and locationrouting. In a companion paper [5] we modeled GTSP as an integer linear program, and studied the facial structure of two polytopes associated with the problem. Here we propose exact and heuristic separation procedures for some classes of facetdefining inequalities, which are used within a branchandcut algorithm for the exact solution of GTSP. Heuristic procedures are also described. Extensive computational results for instances taken from the literature and involving up to 442 nodes are reported.
The sample average approximation method applied to stochastic routing problems: a computational study
 Computational Optimization and Applications
"... Abstract. The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. ..."
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Cited by 66 (8 self)
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Abstract. The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps. We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and firststage integer variables. For each of the three problem classes, we use decomposition and branchandcut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0 % of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably nearoptimal solutions to these difficult stochastic programs using only a moderate amount of computation time. Keywords: salesman stochastic optimization, stochastic programming, stochastic routing, shortest path, traveling 1.
Numerical experience with lower bounds for MIQP branchandbound
, 1995
"... The solution of convex Mixed Integer Quadratic Programming (MIQP) problems with a general branchandbound framework is considered. It is shown how lower bounds can be computed efficiently during the branchandbound process. Improved lower bounds such as the ones derived in this paper can reduc ..."
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Cited by 65 (0 self)
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The solution of convex Mixed Integer Quadratic Programming (MIQP) problems with a general branchandbound framework is considered. It is shown how lower bounds can be computed efficiently during the branchandbound process. Improved lower bounds such as the ones derived in this paper can reduce the number of QP problems that have to be solved. The branchandbound approach is also shown to be superior to other approaches to solving MIQP problems. Numerical experience is presented which supports these conclusions. Key words : Integer Programming, Mixed Integer Quadratic Programming, BranchandBound AMS subject classification: 90C10, 90C11, 90C20 1 Introduction One of the most successful methods for solving mixedinteger nonlinear problems is branchandbound. Land and Doig [16] first introduced a branchandbound algorithm for the travelling salesman problem. Dakin [3] introduced the now common branching dichotomy and was the first to realize that it is possible to so...
Automatic Data Layout Using 01 Integer Programming
 In Proceedings of the International Conference on Parallel Architectures and Compilation Techniques (PACT94
, 1994
"... : The goal of languages like Fortran D or High Performance Fortran (HPF) is to provide a simple yet efficient machineindependent parallel programming model. By shifting much of the burden of machinedependent optimization to the compiler, the programmer is able to write dataparallel programs that ..."
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Cited by 64 (5 self)
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: The goal of languages like Fortran D or High Performance Fortran (HPF) is to provide a simple yet efficient machineindependent parallel programming model. By shifting much of the burden of machinedependent optimization to the compiler, the programmer is able to write dataparallel programs that can be compiled and executed with good performance on many different architectures. However, the choice of a good data layout is still left to the programmer. Even the most sophisticated compiler may not be able to compensate for a poorly chosen data layout since many compiler decisions are driven by the data layout specified in the program. The choice of a good data layout depends on many factors, including the target machine architecture, the compilation system, the problem size, and the number of processors available. The option of remapping arrays at specific points in the program makes the choice even harder. Current programming tools provide little or no support for this difficult sele...
Minimum cost capacity installation for multicommodity network flows
 MATHEMATICAL PROGRAMMING
, 1998
"... Consider a directed graph G = (V; A), and a set of traffic demands to be shipped between pairs of nodes in V. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installatio ..."
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Cited by 63 (12 self)
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Consider a directed graph G = (V; A), and a set of traffic demands to be shipped between pairs of nodes in V. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs. We study two different approaches for solving problems of this type. The first one is based on the idea of metric inequalities (see Onaga and Kakusho[1971]), and uses a formulation with only jAj variables. The second uses an aggregated multicommodity flow formulation and has jV j \Delta jAj variables. We first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on 3 nodes. Next we explain our solution methods for both of the approaches in detail and present computational results. Our computational experience shows that the two formulations are comparable and yield effective algorithms for solving reallife problems.
Memetic Algorithms for Combinatorial Optimization Problems: Fitness Landscapes and Effective Search Strategies
, 2001
"... ..."
Gomory Cuts Revisited
, 1996
"... In this paper, we investigate the use of Gomory's mixed integer cuts within a branchandcut framework. It has been argued in the literature that "a marriage of classical cutting planes and tree search is out of the question as far as the solution of largescale combinatorial optimization ..."
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Cited by 54 (5 self)
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In this paper, we investigate the use of Gomory's mixed integer cuts within a branchandcut framework. It has been argued in the literature that "a marriage of classical cutting planes and tree search is out of the question as far as the solution of largescale combinatorial optimization problems is concerned" [16] because the cuts generated at one node of the search tree need not be valid at other nodes. We show in this paper that it is possible, using a simple lifting procedure, to make Gomory cuts generated in a node of the enumeration tree globally valid in the case of mixed 01 programs. Other issues addressed in this paper are of computational nature, such as strategies for generating the cutting planes, deciding between branching and cutting, etc. The result is a robust mixed integer program solver. 1 Introduction In the late fifties and early sixties, Gomory [6], [7], [8] proposed to solve integer programs by using cutting planes, thus reducing integer programming to the solu...