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26
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 28 (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
Classification of eight dimensional perfect forms
 MATH
, 2007
"... In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible. ..."
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Cited by 19 (9 self)
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In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.
The ABACUS System for BranchandCutandPrice Algorithms in Integer Programming and Combinatorial Optimization
, 1998
"... The development of new mathematical theory and its application in software systems for the solution of hard optimization problems have a long tradition in mathematical programming. In this tradition we implemented ABACUS, an objectoriented software framework for branchandcutandprice algorithms ..."
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Cited by 18 (0 self)
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The development of new mathematical theory and its application in software systems for the solution of hard optimization problems have a long tradition in mathematical programming. In this tradition we implemented ABACUS, an objectoriented software framework for branchandcutandprice algorithms for the solution of mixed integer and combinatorial optimization problems. This paper discusses some difficulties in the implementation of branchandcutandprice algorithms for combinatorial optimization problems and shows how they are managed by ABACUS.
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 15 (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...
On skeletons, diameters and volumes of metric polyhedra
 Combinatorics and Computer Science, Lecture
"... Abstract. We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the maxcut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency a ..."
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Cited by 15 (9 self)
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Abstract. We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the maxcut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incidence relm:ons and connectivity of the metric polytope and its relatives. In partic~dar, using its large symmetry group, we completely describe all the 13 o:bits which form the 275 840 vertices of the 21dimensional metric polytope on 7 nodes and their incidence and adjacency relations. The edge connectivity, the/skeletons and a lifting procedure valid for a large class of vertices of the metric polytope are also given. Finally, we present an ordering of the facets of a polytope, based on their adjacency relations, for the enumeration of its vertices by the double description method. 1
A BranchandCut Approach to Physical Mapping of Chromosomes By Unique EndProbes
, 1997
"... A fundamental problem in computational biology is the construction of physical maps of chromosomes from hybridization experiments between unique probes and clones of chromosome fragments in the presence of error. Alizadeh, Karp, Weisser and Zweig (Algorithmica 13:1/2, 5276, 1995) first considered ..."
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Cited by 14 (5 self)
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A fundamental problem in computational biology is the construction of physical maps of chromosomes from hybridization experiments between unique probes and clones of chromosome fragments in the presence of error. Alizadeh, Karp, Weisser and Zweig (Algorithmica 13:1/2, 5276, 1995) first considered a maximumlikelihood model of the problem that is equivalent to finding an ordering of the probes that minimizes a weighted sum of errors, and developed several effective heuristics. We show that by exploiting information about the endprobes of clones, this model can be formulated as a weighted Betweenness Problem. This affords the significant advantage of allowing the welldeveloped tools of integer linearprogramming and branchandcut algorithms to be brought to bear on physical mapping, enabling us for the first time to solve small mapping instances to optimality even in the presence of high error. We also show that by combining the optimal solution of many small overlapping Betweenness...
On semidefinite programming relaxations of the traveling salesman problem
, 2009
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A BranchandCut Approach to Physical Mapping With EndProbes
, 1997
"... A fundamental problem in computational biology is the construction of physical maps of chromosomes from hybridization experiments between unique probes and clones of chromosome fragments in the presence of error. Alizadeh, Karp, Weisser and Zweig [AKWZ94] first considered a maximumlikelihood model o ..."
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Cited by 12 (0 self)
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A fundamental problem in computational biology is the construction of physical maps of chromosomes from hybridization experiments between unique probes and clones of chromosome fragments in the presence of error. Alizadeh, Karp, Weisser and Zweig [AKWZ94] first considered a maximumlikelihood model of the problem that is equivalent to finding an ordering of the probes that minimizes a weighted sum of errors, and developed several effective heuristics. We show that by exploiting information about the endprobes of clones, this model can be formulated as a weighted Betweenness Problem. This affords the significant advantage of allowing the welldeveloped tools of integer linearprogramming and branchandcut algorithms to be brought to bear on physical mapping, enabling us for the first time to solve small mapping instances to optimality even in the presence of high error. We also show that by combining the optimal solution of many small overlapping Betweenness Problems, one can effectively...
The symmetric traveling salesman polytope: New facets from the graphical relaxation
 MATHEMATICS OF OPERATIONS RESEARCH
, 2007
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