Between Subgraph Isomorphism and Maximum Common Subgraph

When a small pattern graph does not occur inside a larger target graph, we can ask how to find "as much of the pattern as possible" inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths. We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph

Hybrid mapping for static and non-static indoor environments

Mención Internacional en el título de doctorIndoor environments populated by humans, such as houses, offices or universities, involve a great complexity due to the diversity of geometries and situations that they may present. Apart from the size of the environment, they can contain multiple rooms distributed into floors and corridors, repetitive structures and loops, and they can get as complicated as one can imagine. In addition, the structure and situations that the environment present may vary over time as objects could be moved, doors can be frequently opened or closed and places can be used for different purposes. Mobile robots need to solve these challenging situations in order to successfully operate in the environment. The main tools that a mobile robot has for dealing with these situations relate to navigation and perception and comprise mapping, localization, path planning and map adaptation. In this thesis, we try to address some of the open problems in robot navigation in non-static indoor environments. We focus on house-like environments as the work is framed into the HEROITEA research project that aims attention at helping elderly people with their everyday-life activities at their homes. This thesis contributes to HEROITEA with a complete robotic mapping system and map adaptation that grants safe navigation and understanding of the environment. Moreover, we provide localization and path planning strategies within the resulting map to further operate in the environment. The first problem tackled in this thesis is robot mapping in static indoor environments. We propose a hybrid mapping method that structures the information gathered from the environment into several maps. The hybrid map contains diverse knowledge of the environment such as its structure, the navigable and blocked paths, and semantic knowledge, such as the objects or scenes in the environment. All this information is separated into different components of the hybrid map that are interconnected so the system can, at any time, benefit from the information contained in every component. In addition to the conceptual conception of the hybrid map, we have also developed building procedures and an exploration algorithm to autonomous build the hybrid map. However, indoor environments populated by humans are far from being static as the environment may change over time. For this reason, the second problem tackled in this thesis is the adaptation of the map to non-static environments. We propose an object-based probabilistic map adaptation that calculates the likelihood of moving or remaining in its place for the different objects in the environment. Finally, a map is just a description of the environment whose importance is mostly related to how the map is used. In addition, map representations are more valuable as long as they offer a wider range of applications. Therefore, the third problem that we approach in this thesis is exploiting the intrinsic characteristics of the hybrid map in order to enhance the performance of localization and path planning methods. The particular objectives of these approaches are precision for robot localization and efficiency for path planning in terms of execution time and traveled distance. We evaluate our proposed methods in a diversity of simulated and real-world indoor environments. In this extensive evaluation, we show that hybrid maps can be efficiently built and maintained over time and they open up for new possibilities for localization and path planning. In this thesis, we show an increase in localization precision and robustness and an improvement in path planning performance. In sum, this thesis makes several contributions in the context of robot navigation in indoor environments, and especially in hybrid mapping. Hybrid maps offer higher efficiency during map building and other applications such as localization and path planning. In addition, we highlight the necessity of dealing with the dynamics of indoor environments and the benefits of combining topological, semantic and metric information to the autonomy of a mobile robot.Los entornos de interiores habitados por personas, como casas, oficinas o universidades, entrañan una gran complejidad por la diversidad de geometrías y situaciones que pueden ocurrir. Aparte de las diferencias en tamaño, estos entornos pueden contener muchas habitaciones organizadas en diferentes plantas o pasillos, pueden presentar estructuras repetitivas o bucles de tal forma que los entornos pueden llegar a ser tan complejos como uno se pueda imaginar. Además, la estructura y el estado del entorno pueden variar con el tiempo, ya que los objetos pueden moverse, las puertas pueden estar cerradas o abiertas y diferentes espacios pueden ser usados para diferentes propósitos. Los robots móviles necesitan resolver estas situaciones difíciles para poder funcionar de una forma satisfactoria. Las principales herramientas que tiene un robot móvil para manejar estas situaciones están relacionadas con la navegación y la percepción y comprenden el mapeado, la localización, la planificación de trayectorias y la adaptación del mapa. En esta tesis, abordamos algunos de los problemas sin resolver de la navegación de robots móviles en entornos de interiores no estáticos. Nos centramos en entornos tipo casa ya que este trabajo se enmarca en el proyecto de investigación HEROITEA que se enfoca en ayudar a personas ancianas en tareas cotidianas del hogar. Esta tesis contribuye al proyecto HEROITEA con un sistema completo de mapeado y adaptación del mapa que asegura una navegación segura y la comprensión del entorno. Además, aportamos métodos de localización y planificación de trayectorias usando el mapa construido para realizar nuevas tareas en el entorno. El primer problema que se aborda en esta tesis es el mapeado de entornos de interiores estáticos por parte de un robot. Proponemos un método de mapeado híbrido que estructura la información capturada en varios mapas. El mapa híbrido contiene información sobre la estructura del entorno, las trayectorias libres y bloqueadas y también incluye información semántica, como los objetos y escenas en el entorno. Toda esta información está separada en diferentes componentes del mapa híbrido que están interconectados de tal forma que el sistema puede beneficiarse en cualquier momento de la información contenida en cada componente. Además de la definición conceptual del mapa híbrido, hemos desarrollado unos procedimientos para construir el mapa y un algoritmo de exploración que permite que esta construcción se realice autónomamente. Sin embargo, los entornos de interiores habitados por personas están lejos de ser estáticos ya que pueden cambiar a lo largo del tiempo. Por esta razón, el segundo problema que intentamos solucionar en esta tesis es la adaptación del mapa para entornos no estáticos. Proponemos un método probabilístico de adaptación del mapa basado en objetos que calcula la probabilidad de que cada objeto en el entorno haya sido movido o permanezca en su posición anterior. Para terminar, un mapa es simplemente una descripción del entorno cuya importancia está principalmente relacionada con su uso. Por ello, los mapas más valiosos serán los que ofrezcan un rango mayor de aplicaciones. Para abordar este asunto, el tercer problema que intentamos solucionar es explotar las características intrínsecas del mapa híbrido para mejorar el desempeño de métodos de localización y de planificación de trayectorias usando el mapa híbrido. El objetivo principal de estos métodos es aumentar la precisión en la localización del robot y la eficiencia en la planificación de trayectorias en relación al tiempo de ejecución y la distancia recorrida. Hemos evaluado los métodos propuestos en una variedad de entornos de interiores simulados y reales. En esta extensa evaluación, mostramos que los mapas híbridos pueden construirse y mantenerse en el tiempo de forma eficiente y que dan lugar a nuevas posibilidades en cuanto a localización y planificación de trayectorias. En esta tesis, mostramos un aumento en la precisión y robustez en la localización y una mejora en el desempeño de la planificación de trayectorias. En resumen, esta tesis lleva a cabo diversas contribuciones en el ámbito de la navegación de robots móviles en entornos de interiores, y especialmente en mapeado híbrido. Los mapas híbridos ofrecen más eficiencia durante la construcción del mapa y en otras tareas como la localización y la planificación de trayectorias. Además, resaltamos la necesidad de tratar los cambios en entornos de interiores y los beneficios de combinar información topológica, semántica y métrica para la autonomía del robot.Programa de Doctorado en Ingeniería Eléctrica, Electrónica y Automática por la Universidad Carlos III de MadridPresidente: Carlos Balaguer Bernaldo de Quirós.- Secretario: Javier González Jiménez.- Vocal: Nancy Marie Amat

Experimental Evaluation of Subgraph Isomorphism Solvers

International audienceSubgraph Isomorphism (SI) is an NP-complete problem which is at the heart of many structural pattern recognition tasks as it involves finding a copy of a pattern graph into a target graph. In the pattern recognition community, the most well-known SI solvers are VF2, VF3, and RI. SI is also widely studied in the constraint programming community, and many constraint-based SI solvers have been proposed since Ullman, such as LAD and Glasgow, for example. All these SI solvers can solve very quickly some large SI instances, that involve graphs with thousands of nodes. However, McCreesh et al. have recently shown how to randomly generate SI instances the hardness of which can be controlled and predicted, and they have built small instances which are computationally challenging for all solvers. They have also shown that some small instances, which are predicted to be easy and are easily solved by constraint-based solvers, appear to be challenging for VF2 and VF3. In this paper, we widen this study by considering a large test suite coming from eight benchmarks. We show that, as expected for an NP-complete problem, the solving time of an instance does not depend on its size, and that some small instances coming from real applications are not solved by any of the considered solvers. We also show that, if RI and VF3 can solve very quickly a large number of easy instances, for which Glasgow or LAD need more time, they fail at solving some other instances that are quickly solved by Glasgow or LAD, and they are clearly outperformed by Glasgow on hard instances. Finally, we show that we can easily combine solvers to take benefit of their complementarity

Spatial arrangements in architecture and mechanical engineering: some aspects of their representation and construction

Spatial arrangements in architecture and mechanical engineering are represented by incidence structures and classified according to properties of these incidence structures. The relationships between classes are given by ornamentation operations and the construction of elements in fundamental classes by substructure replacement operations. Thus representations of the spatial arrangements for possible designs are generated. Planar maps represent spatial arrangements in architecutral plans. The edges correspond to walls and vertices to incidence between walls. Plans represented by 3-vertex connected maps are ornamented by rooting and extension operations. Further ornamentation specifies access between regions. Plans with all regions adjacent to the exterior correspond to outerplane maps. Trivalent maps represent an important class of plans. Fundamental plans with r internal regions and s regions adjacent to the exterior are represented by [r,s] triangulations. Ornamentations of simple [r,s] triangulations are specified which represent plans with rectangular regions. Plans with walls aligned along two directions are represented by rectangular shapes whose maximal lines correspond to contiguous aligned walls. Rules of construction for various classes are given and the incidence structures of maximal lines and regions are characterized. Spatial arrangements in machines are represented by systems whose blocks correspond to links and vertices to joints. The dual systems are also used. Coplanar kinematic chains with revolute pairs are classified according to mobility and connectedness. Two fundamental classes are considered. First, the chains with binary joints, represented by simple graphs and constructed by two new methods: (i) suspended chain and cycle addition and (ii) subgraph replacement. Second, the chains with binary links which are constructed by subgraph replacement

Constraints for symmetry breaking in graph representation

Many complex combinatorial problems arising from a range of scientific applications (such as computer networks, mathematical chemistry and bioinformatics) involve searching for an undirected graph satisfying a given property. Since for any possible solution there can be a large number of isomorphic representations, these problems can quickly become intractable. One way to mitigate this problem is to eliminate as many isomorphic copies as possible by breaking symmetry during search - i.e. by introducing constraints that ensure that at least one representative graph is generated for each equivalence class, but not the entire class. The goal is to generate as few members of each class as possible - ideally exactly one: the symmetry break is said to be complete in this case. In this paper we introduce novel, effective and compact, symmetry breaking constraints for undirected graph search. While incomplete, these prove highly beneficial in pruning the search for a graph. We illustrate the application of symmetry breaking in graph representation to resolve several open instances in extremal graph theory. We also illustrate the application of our approach to graph edge coloring problems which exhibit additional symmetries due to the fact that the colors of the edges in any solution can be permuted

A Parallel, Backjumping Subgraph Isomorphism Algorithm Using Supplemental Graphs

This registry entry contains a reference to the code, data and experimental scripts needed to reproduce the subgraph isomorphism paper: Ciaran McCreesh and Patrick Prosser, "A Parallel, Backjumping Subgraph Isomorphism Algorithm using Supplemental Graphs". To appear at the 21st International Conference on Principles and Practice of Constraint Programming (CP 2015)