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49
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 166 (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
The linear programming approach to approximate dynamic programming
 Operations Research
, 2001
"... The curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of largescale stochastic control problems. We study an efficient method based on linear programming for approximating solutions to such problems. The approach “fits ” a linear ..."
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Cited by 146 (17 self)
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The curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of largescale stochastic control problems. We study an efficient method based on linear programming for approximating solutions to such problems. The approach “fits ” a linear combination of preselected basis functions to the dynamic programming costtogo function. We develop error bounds that offer performance guarantees and also guide the selection of both basis functions and “staterelevance weights ” that influence quality of the approximation. Experimental results in the domain of queueing network control provide empirical support for the methodology. (Dynamic programming/optimal control: approximations/largescale problems. Queues, algorithms: control of queueing networks.)
Evolution in time and space  the parallel genetic algorithm
 FOUNDATIONS OF GENETIC ALGORITHMS
, 1991
"... The parallel genetic algorithm (PGA) uses two major modifications compared to the genetic algorithm. Firstly, selection for mating is distributed. Individuals live in a 2D world. Selection of a mate is done by each individual independently in its neighborhood. Secondly, each individual may improve ..."
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Cited by 109 (13 self)
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The parallel genetic algorithm (PGA) uses two major modifications compared to the genetic algorithm. Firstly, selection for mating is distributed. Individuals live in a 2D world. Selection of a mate is done by each individual independently in its neighborhood. Secondly, each individual may improve its fitness during its lifetime by e.g. local hillclimbing. The PGA is totally asynchronous, running with maximal efficiency on MIMD parallel computers. The search strategy of the PGA is based on a small number of active and intelligent individuals, whereas a GA uses a large population of passive individuals. We will investigate the PGA with deceptive problems and the traveling salesman problem. We outline why and when the PGA is succesful. Abstractly, a PGA is a parallel search with information exchange between the individuals. If we represent the optimization problem as a fitness landscape in a certain configuration space, we see, that a PGA tries to jump from two local minima to a third, still better local minima, by using the crossover operator. This jump is (probabilistically) successful, if the fitness landscape has a certain correlation. We show the correlation for the traveling salesman problem by a configuration space analysis. The PGA explores implicitly the above correlation.
On constraint sampling in the linear programming approach to approximate dynamic programming
 Mathematics of Operations Research
, 2004
"... doi 10.1287/moor.1040.0094 ..."
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 62 (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...
Solving RealWorld Linear Programs: A Decade and More of Progress
 Operations Research
, 2002
"... This paper is an invited contribution to the 50th anniversary issue of the journal Operations Research, published by the Institute of Operations Research and Management Science (INFORMS). It describes one persons perspective on the development of computational tools for linear programming. The pape ..."
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Cited by 59 (1 self)
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This paper is an invited contribution to the 50th anniversary issue of the journal Operations Research, published by the Institute of Operations Research and Management Science (INFORMS). It describes one persons perspective on the development of computational tools for linear programming. The paper begins with a short, personal history, followed by historical remarks covering the some 40 years of linearprogramming developments that predate my own involvement in this subject. It concludes with a more detailed look at the evolution of computational linear programming since 1987. 2
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 59 (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.
Packing Steiner Trees: A Cutting Plane Algorithm and Computational Results
 Mathematical Programming
, 1992
"... In this paper we describe a cutting plane algorithm for the Steiner tree packing problem. We use our algorithm to solve some switchbox routing problems of VLSIdesign and report on our computational experience. This includes a brief discussion of separation algorithms, a new LPbased primal heuristi ..."
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Cited by 31 (12 self)
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In this paper we describe a cutting plane algorithm for the Steiner tree packing problem. We use our algorithm to solve some switchbox routing problems of VLSIdesign and report on our computational experience. This includes a brief discussion of separation algorithms, a new LPbased primal heuristic and implementation details. The paper is based on the polyhedral theory for the Steiner tree packing polyhedron developed in our companion paper [GMW92] and meant to turn this theory into an algoritmic tool for the solution of practical problems.
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 27 (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.
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