Computing Invariants of Simplicial Manifolds

This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book "Triangulated Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure

The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur

Combinatorial structure of rigid transformations in 2D digital images

International audienceRigid transformations are involved in a wide range of digital image processing applications. When applied on such discrete images, rigid transformations are however usually performed in their associated continuous space, then requiring a subsequent digitization of the result. In this article, we propose to study rigid transformations of digital images as a fully discrete process. In particular, we investigate a combinatorial structure modelling the whole space of digital rigid transformations on any subset of Z^2 of size N*N. We describe this combinatorial structure, which presents a space complexity O(N^9) and we propose an algorithm enabling to build it in linear time with respect to this space complexity. This algorithm, which handles real (i.e. non-rational) values related to the continuous transformations associated to the discrete ones, is however defined in a fully discrete form, leading to exact computation

Combinatorial properties of 2D discrete rigid transformations under pixel-invariance constraints

International audienceRigid transformations are useful in a wide range of digital image processing applications. In this context, they are generally considered as continuous processes, followed by discretization of the results. In recent works, rigid transformations on ℤ^2 have been formulated as a fully discrete process. Following this paradigm, we investigate --from a combinatorial point of view-- the effects of pixel-invariance constraints on such transformations. In particular we describe the impact of these constraints on both the combinatorial structure of the transformation space and the algorithm leading to its generation

Rigid transformations on 2D digital images : combinatorial and topological analysis

In this thesis, we study rigid transformations in the context of computer imagery. In particular, we develop a fully discrete framework for handling such transformations. Rigid transformations, initially defined in the continuous domain, are involved in a wide range of digital image processing applications. In this context, the induced digital rigid transformations present different geometrical and topological properties with respect to their continuous analogues. In order to overcome the issues raised by these differences, we propose to formulate rigid transformations on digital images in a fully discrete framework. In this framework, Euclidean rigid transformations producing the same digital rigid transformation are put in the same equivalence class. Moreover, the relationship between these classes can be modeled as a graph structure. We prove that this graph has a polynomial space complexity with respect to the size of the considered image, and presents useful structural properties. In particular, it allows us to generate incrementally all digital rigid transformations without numerical approximation. This structure constitutes a theoretical tool to investigate the relationships between geometry and topology in the context of digital images. It is also interesting from the methodological point of view, as we illustrate by its use for assessing the topological behavior of images under rigid transformationsDans cette thèse, nous étudions les transformations rigides dans le contexte de l'imagerie numérique. En particulier, nous développons un cadre purement discret pour traiter ces transformations. Les transformations rigides, initialement définies dans le domaine continu, sont impliquées dans de nombreuses applications de traitement d'images numériques. Dans ce contexte, les transformations rigides digitales induites présentent des propriétés géométriques et topologiques différentes par rapport à leurs analogues continues. Afin de s'affranchir des problèmes inhérents à ces différences, nous proposons de formuler ces transformations rigides dans un cadre purement discret. Dans ce cadre, les transformations rigides sont regroupées en classes correspondant chacune à une transformation digitale donnée. De plus, les relations entre ces classes de transformations peuvent être modélisées par une structure de graphe. Nous prouvons que ce graphe présente une complexité spatiale polynômiale par rapport à la taille de l'image. Il présente également des propriétés structurelles intéressantes. En particulier, il permet de générer de manière progressive toute transformation rigide digitale, et ce sans approximation numérique. Cette structure constitue un outil théorique pour l'étude des relations entre la géométrie et la topologie dans le contexte de l'imagerie numérique. Elle présente aussi un intérêt méthodologique, comme l'illustre son utilisation pour l'évaluation du comportement topologique des images sous des transformations rigide

Efficient algorithms for tensor scaling, quantum marginals and moment polytopes

We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our algorithm provides an efficient weak membership oracle for the associated moment polytopes, an important family of implicitly-defined convex polytopes with exponentially many facets and a wide range of applications. These include the entanglement polytopes from quantum information theory (in particular, we obtain an efficient solution to the notorious one-body quantum marginal problem) and the Kronecker polytopes from representation theory (which capture the asymptotic support of Kronecker coefficients). Our algorithm can be applied to succinct descriptions of the input tensor whenever the marginals can be efficiently computed, as in the important case of matrix product states or tensor-train decompositions, widely used in computational physics and numerical mathematics. We strengthen and generalize the alternating minimization approach of previous papers by introducing the theory of highest weight vectors from representation theory into the numerical optimization framework. We show that highest weight vectors are natural potential functions for scaling algorithms and prove new bounds on their evaluations to obtain polynomial-time convergence. Our techniques are general and we believe that they will be instrumental to obtain efficient algorithms for moment polytopes beyond the ones consider here, and more broadly, for other optimization problems possessing natural symmetries