The first computer implementation of the Dantzig-Fulkerson-Johnson cutting-plane 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

Abstract. We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the max-cut 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 21-dimensional 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

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.

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 branch-and-cut 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...

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, 52--76, 1995) first considered a maximum-likelihood 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 end-probes of clones, this model can be formulated as a weighted Betweenness Problem. This affords the significant advantage of allowing the well-developed tools of integer linear-programming and branch-and-cut 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...

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 well-developed tools of integer linearprogramming and branch-and-cut 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...

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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 object-oriented software framework for branch-and-cut-and-price algorithms for the solution of mixed integer and combinatorial optimization problems. This paper discusses some difficulties in the implementation of branch-and-cut-and-price algorithms for combinatorial optimization problems and shows how they are managed by ABACUS.

Analog performance space exploration identifies the range of feasible performance values of a given circuit topology. It is an extremely challenging task of great importance to topology selection and hierarchical sizing. In this paper, a novel technique for the efficient simulation-based exploration of high-dimensional performance spaces is presented. To this end, fundamental circuit design knowledge is described by constraint functions. Based on a linearization of the latter and of the circuit performance functions, a description of the feasible performance range in the form of a polytope is derived. Moreover, the approach is integrated into a hierarchical sizing method, where it propagates topological and technological constraints bottom-up. Practical application results demonstrate the efficiency and usefulness of the new method. 1.

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