On the SQAP-Polytope

The study of the QAP-Polytope was started by Rijal (1995), Padberg and Rijal (1996), and Jünger and Kaibel (1996), investigating the structure of the feasible points of a (Mixed) Integer Linear Programming formulation of the QAP that provides good lower bounds by its continious relaxation. Rijal (1995), Padberg and Rijal (1996) propose an alternative (Mixed) Integer Linear Programming formulation for the case that the QAP-instance is symmetric in a certain sense and define analogously the SQAP-Polytope. They give a conjecture about the dimension of that polytope, whose proof is one part of this paper. Moreover, we investigate the trivial faces of the SQAP-Polytope and present a first class of non-trivial facets of it. The polyhedral results are used to compute lower bounds for symmetric QAPs

Box-Inequalities for Quadratic Assignment Polytopes

Linear Programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years. They have turned out to be quite good in practice. Investigations of the polytopes underlying the corresponding integer linear programming formulations (the non-symmetric and the symmetric quadratic assignment polytope) have been started by Rijal (1995), Padberg and Rijal (1996), and Jünger and Kaibel (1996, 1997). They have lead to basic knowledge on these polytopes concerning questions like their dimensions, affine hulls, and trivial facets. However, no large class of (facet-defining) inequalities that could be used in cutting plane procedures had been found. We present in this paper the first such class of inequalities, the box inequalities, which have an interesting origin in some well-known hypermetric inequalities for the cut polytope. Computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds. The most effective ones among the new inequalities turn out to be indeed facet-defining for both the non-symmetric as well as for the symmetric quadratic assignment polytope

Calculating Sparse and Dense Correspondences for Near-Isometric Shapes

Comparing and analysing digital models are basic techniques of geometric shape processing. These techniques have a variety of applications, such as extracting the domain knowledge contained in the growing number of digital models to simplify shape modelling. Another example application is the analysis of real-world objects, which itself has a variety of applications, such as medical examinations, medical and agricultural research, and infrastructure maintenance. As methods to digitalize physical objects mature, any advances in the analysis of digital shapes lead to progress in the analysis of real-world objects. Global shape properties, like volume and surface area, are simple to compare but contain only very limited information. Much more information is contained in local shape differences, such as where and how a plant grew. Sadly the computation of local shape differences is hard as it requires knowledge of corresponding point pairs, i.e. points on both shapes that correspond to each other. The following article thesis (cumulative dissertation) discusses several recent publications for the computation of corresponding points: - Geodesic distances between points, i.e. distances along the surface, are fundamental for several shape processing tasks as well as several shape matching techniques. Chapter 3 introduces and analyses fast and accurate bounds on geodesic distances. - When building a shape space on a set of shapes, misaligned correspondences lead to points moving along the surfaces and finally to a larger shape space. Chapter 4 shows that this also works the other way around, that is good correspondences are obtain by optimizing them to generate a compact shape space. - Representing correspondences with a “functional map” has a variety of advantages. Chapter 5 shows that representing the correspondence map as an alignment of Green’s functions of the Laplace operator has similar advantages, but is much less dependent on the number of eigenvectors used for the computations. - Quadratic assignment problems were recently shown to reliably yield sparse correspondences. Chapter 6 compares state-of-the-art convex relaxations of graphics and vision with methods from discrete optimization on typical quadratic assignment problems emerging in shape matching

FieldPlacer - A flexible, fast and unconstrained force-directed placement method for heterogeneous reconfigurable logic architectures

The field of placement methods for components of integrated circuits, especially in the domain of reconfigurable chip architectures, is mainly dominated by a handful of concepts. While some of these are easy to apply but difficult to adapt to new situations, others are more flexible but rather complex to realize. This work presents the FieldPlacer framework, a flexible, fast and unconstrained force-directed placement method for heterogeneous reconfigurable logic architectures, in particular for the ever important heterogeneous FPGAs. In contrast to many other force-directed placers, this approach is called ‘unconstrained’ as it does not require a priori fixed logic elements in order to calculate a force equilibrium as the solution to a system of equations. Instead, it is based on a free spring embedder simulation of a graph representation which includes all logic block types of a design simultaneously. The FieldPlacer framework offers a huge amount of flexibility in applying different distance norms (e. g., the Manhattan distance) for the force-directed layout and aims at creating adapted layouts for various objective functions, e. g., highest performance or improved routability. Depending on the individual situation, a runtime-quality trade-off can be considered to either produce a decent placement in a very short time or to generate an exceptionally good placement, which takes longer. An extensive comparison with the latest simulated annealing placement method from the well-known Versatile Place and Route (VPR) framework shows that the FieldPlacer approach can create placements of comparable quality much faster than VPR or, alternatively, generate better placements in the same time. The flexibility in defining arbitrary objective functions and the intuitive adaptability of the method, which, among others, includes different concepts from the field of graph drawing, should facilitate further developments with this framework, e. g., for new upcoming optimization targets like the energy consumption of an implemented design

On the SQAP-Polytope

The SQAP-polytope was associated to quadratic assignment problems with a certain symmetric objective function structure by Rijal (1995) and Padberg and Rijal (1996). We derive a technique for investigating the SQAP-polytope that is based on projecting the (low-dimensional) polytope into a lower dimensional vector-space, where the vertices have a "more convenient" coordinate structure. We exploit this technique in order to prove conjectures by Padberg and Rijal on the dimension of the SQAP-polytope as well as on its trivial facet

On the SQAP-Polytope

The study of the QAP-Polytope was started by Rijal (1995), Padberg and Rijal (1996), and Junger and Kaibel (1996), investigating the structure of the feasible points of a (Mixed) Integer Linear Programming formulation of the QAP that provides good lower bounds by its continious relaxation. Rijal (1995) and Padberg and Rijal (1996) propose an alternative (Mixed) Integer Linear Programming formulation for the case that the QAP-instance is symmetric in a certain sense and define analogously the SQAP-Polytope. They give a conjecture about the dimension of that polytope, whose proof is one part of this paper. Moreover, we investigate the trivial faces of the SQAP-Polytope and present a first class of non-trivial facets of it. The polyhedral results are used to compute lower bounds for symmetric QAPs