Qualitative reasoning with directional relations

AbstractQualitative spatial reasoning (QSR) pursues a symbolic approach to reasoning about a spatial domain. Qualitative calculi are defined to capture domain properties in relation operations, granting a relation algebraic approach to reasoning. QSR has two primary goals: providing a symbolic model for human common-sense level of reasoning and providing efficient means for reasoning. In this paper, we dismantle the hope for efficient reasoning about directional information in infinite spatial domains by showing that it is inherently hard to decide consistency of a set of constraints that represents positions in the plane by specifying directions from reference objects. We assume that these reference objects are not fixed but only constrained through directional relations themselves. Known QSR reasoning methods fail to handle this information

Topological challenges in multispectral image segmentation

Land cover classification from remote sensing multispectral images has been traditionallyconducted by using mainly spectral information associated with discrete spatial units (i.e. pixels).Geometric and topological characteristics of the spatial context close to every pixel have been either not fully treated or completely ignored.This article provides a review of the strategies used by a number of researchers in order to include spatial and topological properties in image segmentation.­­­It is shown how most of researchers have proposed to perform -previous to classification- a grouping or segmentation of nearby pixels by modeling neighborhood relationships as 4-connected, 8-connected and (a, b) – connected graphs.In this object-oriented approach, however, topological concepts such as neighborhood, contiguity, connectivity and boundary suffer from ambiguity since image elements (pixels) are two-dimensional entities composing a spatially uniform grid cell (i.e. there are not uni-dimensional nor zero-dimensional elements to build boundaries). In order to solve such topological paradoxes, a few proposals have been proposed. This review discusses how the alternative of digital images representation based on Cartesian complexes suggested by Kovalevsky (2008) for image segmentation in computer vision, does not present topological flaws, typical of conventional solutions based on grid cells. However, such a proposal has not been yet applied to multispectral image segmentation in remote sensing. This review is part of the PhD in Engineering research conducted by the first author under guidance of the second one. This review concludes suggesting the need to research on the potential of using Cartesian complexes for multispectral image segmentation

Determinantal Sieving

We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of rank $k$, both over a field $\mathbb{F}$ of characteristic 2, in $2^k$ evaluations we can sieve for those terms in the monomial expansion of $P$ which are multilinear and whose support is a basis for $M$. Alternatively, using $2^k$ evaluations of $P$ we can sieve for those monomials whose odd support spans $M$. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving $q$-Matroid Intersection in time $O^*(2^{(q-2)k})$ and $q$-Matroid Parity in time $O^*(2^{qk})$, improving on $O^*(4^{qk})$ (Brand and Pratt, ICALP 2021) 2. $T$-Cycle, Colourful $(s,t)$-Path, Colourful $(S,T)$-Linkage in undirected graphs, and the more general Rank $k$ $(S,T)$-Linkage problem, all in $O^*(2^k)$ time, improving on $O^*(2^{k+|S|})$ respectively $O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))})$ (Fomin et al., SODA 2023) 3. Many instances of the Diverse X paradigm, finding a collection of $r$ solutions to a problem with a minimum mutual distance of $d$ in time $O^*(2^{r(r-1)d/2})$, improving solutions for $k$-Distinct Branchings from time $2^{O(k \log k)}$ to $O^*(2^k)$ (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from $O^*(2^{2^{O(rd)}})$ to $O^*(2^{r^2d/2})$ (Fomin et al., STACS 2021) All matroids are assumed to be represented over a field of characteristic 2. Over general fields, we achieve similar results at the cost of using exponential space by working over the exterior algebra. For a class of arithmetic circuits we call strongly monotone, this is even achieved without any loss of running time. However, the odd support sieving result appears to be specific to working over characteristic 2

Qualitative Reasoning about Relative Directions : Computational Complexity and Practical Algorithm

Qualitative spatial reasoning (QSR) enables cognitive agents to reason about space using abstract symbols. Among several aspects of space (e.g., topology, direction, distance) directional information is useful for agents navigating in space. Observers typically describe their environment by specifying the relative directions in which they see other objects or other people from their point of view. As such, qualitative reasoning about relative directions, i.e., determining whether a given statement involving relative directions is true, can be advantageously used for applications, for example, robot navigation, computer-aided design and geographical information systems. Unfortunately, despite the apparent importance of reasoning about relative directions, QSR-research so far could not provide efficient decision procedures for qualitative reasoning about relative directions. Accordingly, the question about how to devise an efficient decision procedure for qualitative reasoning about relative directions has meanwhile turned to the question about whether an efficient decision procedure exists at all. Answering the latter existential question, which requires a formal analysis of relative directions from a computational complexity point of view, has remained an open problem in the field of QSR. The present thesis solves the open problem by proving that there is no efficient decision procedure for qualitative reasoning about relative directions, even if only left or right relations are involved. This is surprising as it contradicts the early premise of QSR believed by many researchers in and outside the field, that is, abstracting from an infinite domain to a finite set of relations naturally leads to efficient reasoning. As a consequence of this rather negative result, efficient reasoning with any of the well-known relative direction calculi (OPRAm, DCC, DRA, LR) is impossible. Indeed, the present thesis shows that all the relative direction calculi belong to one and the same class of ∃R-complete problems, which are the problems that can be reduced to the NP-hard decision problem of the existential theory of the reals, and vice versa. Nevertheless, in practice, many interesting computationally hard AI problems can be tackled by means of approximative algorithms and heuristics. In the same vein, the present thesis shows that qualitative reasoning about relative directions can also be tackled with approximative algorithms. In the thesis we develop the qualitative calculus SVm which allows for a practical algorithm for qualitative reasoning about relative directions. SVm also provides an effective semi-decision procedure for the OPRAm calculus, the most versatile one among the relative direction calculi. In this thesis we substantiate the usefulness of SVm by applying it in the marine navigation domain

Modelo de representación del espacio geográfico mediante matroides orientados

En el contexto de los Sistemas de Información Geográfica (SIG) se emplea una noción clásica del espacio que abarca, entre otros, dos conceptos: la continuidad y la externalidad. Estas nociones se encuentran presentes en las dos formas de representación basadas en campo y en objeto. Ya que cualquiera de estas dos formas conlleva básicamente un ejercicio de muestreo, el modelo de la realidad espacial así obtenido carece de exactitud y precisión, además de posición real, en muchas ocasiones. Dadas las deficiencias que tiene la representación de un modelo continuo (basado en los números reales ) implementado en un entorno de computación el cual es finito y no continuo (basado en un subconjunto de los números enteros ), han aparecido propuestas alternativas que pretenden subsanar dichas deficiencias. Una de las más prometedoras son los matroides orientados, esta propuesta está basada en estructuras combinatoriales complementadas con el concepto de convexidad. A lo largo de la presente revisión temática se exponen los conceptos básicos que subyacen a los matroides orientados y se relacionan algunas áreas de aplicación potencial en la representación de conceptos espaciales

Reconstructibility of matroid polytopes

We specify what is meant for a polytope to be reconstructible from its graph or dual graph, and we introduce the problem of class reconstructibility; i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present an O(n3) algorithm that computes the vertices of a matroid polytope from its n-vertex graph. Moreover, our proof includes a characterization of all matroids with isomorphic basis exchange graphs. © 2022 Society for Industrial and Applied Mathematic

Retos topológicos en la segmentación de imágenes multiespectrales

Land cover classification from remote sensing multispectral images has been traditionallyconducted by using mainly spectral information associated with discrete spatial units (i.e. pixels).Geometric and topological characteristics of the spatial context close to every pixel have been either not fully treated or completely ignored.This article provides a review of the strategies used by a number of researchers in order to include spatial and topological properties in image segmentation.­­­It is shown how most of researchers have proposed to perform -previous to classification- a grouping or segmentation of nearby pixels by modeling neighborhood relationships as 4-connected, 8-connected and (a, b) – connected graphs.In this object-oriented approach, however, topological concepts such as neighborhood, contiguity, connectivity and boundary suffer from ambiguity since image elements (pixels) are two-dimensional entities composing a spatially uniform grid cell (i.e. there are not uni-dimensional nor zero-dimensional elements to build boundaries). In order to solve such topological paradoxes, a few proposals have been proposed. This review discusses how the alternative of digital images representation based on Cartesian complexes suggested by Kovalevsky (2008) for image segmentation in computer vision, does not present topological flaws, typical of conventional solutions based on grid cells. However, such a proposal has not been yet applied to multispectral image segmentation in remote sensing.  This review is part of the PhD in Engineering research conducted by the first author under guidance of the second one. This review concludes suggesting the need to research on the potential of using Cartesian complexes for multispectral image segmentation.La clasificación de la cobertura terrestre a partir de imágenes multiespectrales de teledetección se ha llevado a cabo tradicionalmente utilizando información principalmente espectral asociada a unidades espaciales discretas (es decir, píxeles). Las características geométricas y topológicas del contexto espacial cercanas a cada píxel no se han tratado del todo o se han ignorado por completo. proporciona una revisión de las estrategias utilizadas por un número de investigadores para incluir propiedades espaciales y topológicas en la segmentación de imágenes. Se muestra cómo la mayoría de los investigadores han propuesto realizar, antes de la clasificación, una agrupación o segmentación de píxeles cercanos modelando el vecindario relaciones como 4 conectadas, 8 conectadas y (a, b) conectadas. Sin embargo, en este enfoque orientado a objetos, los conceptos topológicos como vecindad, contigüidad, conectividad y límite sufren de ambigüedad ya que los elementos de imagen (píxeles) son dos entidades tridimensionales que componen una celda de cuadrícula espacialmente uniforme (es decir, no hay uni-di elementos mensionales o de cero dimensiones para construir límites). Para resolver tales paradojas topológicas, se han propuesto algunas propuestas. Esta revisión discute cómo la alternativa de representación de imágenes digitales basada en complejos cartesianos sugerida por Kovalevsky (2008) para la segmentación de imágenes en visión artificial, no presenta fallas topológicas, típicas de soluciones convencionales basadas en celdas de grillas. Sin embargo, tal propuesta aún no se ha aplicado a la segmentación de imágenes multiespectrales en teledetección. Esta revisión es parte del doctorado en investigación de ingeniería conducida por el primer autor bajo la dirección del segundo. Esta revisión concluye sugiriendo la necesidad de investigar sobre el potencial del uso de complejos cartesianos para la segmentación de imágenes multiespectrales

Networks, (K)nots, Nucleotides, and Nanostructures

Designing self-assembling DNA nanostructures often requires the identification of a route for a scaffolding strand of DNA through the target structure. When the target structure is modeled as a graph, these scaffolding routes correspond to Eulerian circuits subject to turning restrictions imposed by physical constraints on the strands of DNA. Existence of such Eulerian circuits is an NP-hard problem, which can be approached by adapting solutions to a version of the Traveling Salesperson Problem. However, the author and collaborators have demonstrated that even Eulerian circuits obeying these turning restrictions are not necessarily feasible as scaffolding routes by giving examples of nontrivially knotted circuits which cannot be traced by the unknotted scaffolding strand. Often, targets of DNA nanostructure self-assembly are modeled as graphs embedded on surfaces in space. In this case, Eulerian circuits obeying the turning restrictions correspond to A-trails, circuits which turn immediately left or right at each vertex. In any graph embedded on the sphere, all A-trails are unknotted regardless of the embedding of the sphere in space. We show that this does not hold in general for graphs on the torus. However, we show this property does hold for checkerboard-colorable graphs on the torus, that is, those graphs whose faces can be properly 2-colored, and provide a partial converse to this result. As a consequence, we characterize (with one exceptional family) regular triangulations of the torus containing unknotted A-trails. By developing a theory of sums of A-trails, we lift constructions from the torus to arbitrary n-tori, and by generalizing our work on A-trails to smooth circuit decompositions, we construct all torus links and certain sums of torus links from circuit decompositions of rectangular torus grids. Graphs embedded on surfaces are equivalent to ribbon graphs, which are particularly well-suited to modeling DNA nanostructures, as their boundary components correspond to strands of DNA and their twisted ribbons correspond to double-helices. Every ribbon graph has a corresponding delta-matroid, a combinatorial object encoding the structure of the ribbon-graph\u27s spanning quasi-trees (substructures having exactly one boundary component). We show that interlacement with respect to quasi-trees can be generalized to delta-matroids, and use the resulting structure on delta-matroids to provide feasible-set expansions for a family of delta-matroid polynomials, both recovering well-known expansions of this type (such as the spanning-tree expansion of the Tutte polynnomial) as well as providing several previously unknown expansions. Among these are expansions for the transition polynomial, a version of which has been used to study DNA nanostructure self-assembly, and the interlace polynomial, which solves a problem in DNA recombination