Approximation Ineffectiveness of a Tour-Untangling Heuristic

We analyze a tour-uncrossing heuristic for the Travelling Salesperson Problem, showing that its worst-case approximation ratio is $\Omega(n)$ and its average-case approximation ratio is $\Omega(\sqrt{n})$ in expectation. We furthermore evaluate the approximation performance of this heuristic numerically on average-case instances, and find that it performs far better than the average-case lower bound suggests. This indicates a shortcoming in the approach we use for our analysis, which is a rather common approach in the analysis of local search heuristics.Comment: Accepted for presentation at WAOA 202

MetroSets: Visualizing Sets as Metro Maps

We propose MetroSets, a new, flexible online tool for visualizing set systems using the metro map metaphor. We model a given set system as a hypergraph $\mathcal{H} = (V, \mathcal{S})$, consisting of a set $V$ of vertices and a set $\mathcal{S}$, which contains subsets of $V$ called hyperedges. Our system then computes a metro map representation of $\mathcal{H}$, where each hyperedge $E$ in $\mathcal{S}$ corresponds to a metro line and each vertex corresponds to a metro station. Vertices that appear in two or more hyperedges are drawn as interchanges in the metro map, connecting the different sets. MetroSets is based on a modular 4-step pipeline which constructs and optimizes a path-based hypergraph support, which is then drawn and schematized using metro map layout algorithms. We propose and implement multiple algorithms for each step of the MetroSet pipeline and provide a functional prototype with \new{easy-to-use preset configurations.} % many real-world datasets. Furthermore, \new{using several real-world datasets}, we perform an extensive quantitative evaluation of the impact of different pipeline stages on desirable properties of the generated maps, such as octolinearity, monotonicity, and edge uniformity.Comment: 19 pages; accepted for IEEE INFOVIS 2020; for associated live system, see http://metrosets.ac.tuwien.ac.a

Short Flip Sequences to Untangle Segments in the Plane

A (multi)set of segments in the plane may form a TSP tour, a matching, a tree, or any multigraph. If two segments cross, then we can reduce the total length with the following flip operation. We remove a pair of crossing segments, and insert a pair of non-crossing segments, while keeping the same vertex degrees. The goal of this paper is to devise efficient strategies to flip the segments in order to obtain crossing-free segments after a small number of flips. Linear and near-linear bounds on the number of flips were only known for segments with endpoints in convex position. We generalize these results, proving linear and near-linear bounds for cases with endpoints that are not in convex position. Our results are proved in a general setting that applies to multiple problems, using multigraphs and the distinction between removal and insertion choices when performing a flip.Comment: 19 pages, 10 figure

Flip Distance to some Plane Configurations

We study an old geometric optimization problem in the plane. Given a perfect matching M on a set of n points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from M and adds two non-crossing edges. Let f(M) and F(M) denote the minimum and maximum lengths of a flip sequence on M, respectively. It has been proved by Bonnet and Miltzow (2016) that f(M)=O(n^2) and by van Leeuwen and Schoone (1980) that F(M)=O(n^3). We prove that f(M)=O(n Delta) where Delta is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound for point sets with sublinear spread. For a matching M on n points in convex position we prove that f(M)=n/2-1 and F(M)={{n/2} choose 2}; these bounds are tight. Any bound on F(*) carries over to the bichromatic setting, while this is not necessarily true for f(*). Let M\u27 be a bichromatic matching. The best known upper bound for f(M\u27) is the same as for F(M\u27), which is essentially O(n^3). We prove that f(M\u27)<=slant n-2 for points in convex position, and f(M\u27)= O(n^2) for semi-collinear points. The flip operation can also be defined on spanning trees. For a spanning tree T on a convex point set we show that f(T)=O(n log n)

Bounded-Angle Minimum Spanning Trees

Motivated by the connectivity problem in wireless networks with directional antennas, we study bounded-angle spanning trees. Let P be a set of points in the plane and let ? be an angle. An ?-ST of P is a spanning tree of the complete Euclidean graph on P with the property that all edges incident to each point p ? P lie in a wedge of angle ? centered at p. We study the following closely related problems for ? = 120 degrees (however, our approximation ratios hold for any ? ? 120 degrees). 1) The ?-minimum spanning tree problem asks for an ?-ST of minimum sum of edge lengths. Among many interesting results, Aschner and Katz (ICALP 2014) proved the NP-hardness of this problem and presented a 6-approximation algorithm. Their algorithm finds an ?-ST of length at most 6 times the length of the minimum spanning tree (MST). By adopting a somewhat similar approach and using different proof techniques we improve this ratio to 16/3. 2) To examine what is possible with non-uniform wedge angles, we define an ??-ST to be a spanning tree with the property that incident edges to all points lie in wedges of average angle ?. We present an algorithm to find an ??-ST whose largest edge-length and sum of edge lengths are at most 2 and 1.5 times (respectively) those of the MST. These ratios are better than any achievable when all wedges have angle ?. Our algorithm runs in linear time after computing the MST

Exploration autonome et efficiente de chantiers miniers souterrains inconnus avec un drone filaire

Abstract: Underground mining stopes are often mapped using a sensor located at the end of a pole that the operator introduces into the stope from a secure area. The sensor emits laser beams that provide the distance to a detected wall, thus creating a 3D map. This produces shadow zones and a low point density on the distant walls. To address these challenges, a research team from the Université de Sherbrooke is designing a tethered drone equipped with a rotating LiDAR for this mission, thus benefiting from several points of view. The wired transmission allows for unlimited flight time, shared computing, and real-time communication. For compatibility with the movement of the drone after tether entanglements, the excess length is integrated into an onboard spool, contributing to the drone payload. During manual piloting, the human factor causes problems in the perception and comprehension of a virtual 3D environment, as well as the execution of an optimal mission. This thesis focuses on autonomous navigation in two aspects: path planning and exploration. The system must compute a trajectory that maps the entire environment, minimizing the mission time and respecting the maximum onboard tether length. Path planning using a Rapidly-exploring Random Tree (RRT) quickly finds a feasible path, but the optimization is computationally expensive and the performance is variable and unpredictable. Exploration by the frontier method is representative of the space to be explored and the path can be optimized by solving a Traveling Salesman Problem (TSP) but existing techniques for a tethered drone only consider the 2D case and do not optimize the global path. To meet these challenges, this thesis presents two new algorithms. The first one, RRT-Rope, produces an equal or shorter path than existing algorithms in a significantly shorter computation time, up to 70% faster than the next best algorithm in a representative environment. A modified version of RRT-connect computes a feasible path, shortened with a deterministic technique that takes advantage of previously added intermediate nodes. The second algorithm, TAPE, is the first 3D cavity exploration method that focuses on minimizing mission time and unwound tether length. On average, the overall path is 4% longer than the method that solves the TSP, but the tether remains under the allowed length in 100% of the simulated cases, compared to 53% with the initial method. The approach uses a 2-level hierarchical architecture: global planning solves a TSP after frontier extraction, and local planning minimizes the path cost and tether length via a decision function. The integration of these two tools in the NetherDrone produces an intelligent system for autonomous exploration, with semi-autonomous features for operator interaction. This work opens the door to new navigation approaches in the field of inspection, mapping, and Search and Rescue missions.La cartographie des chantiers miniers souterrains est souvent réalisée à l’aide d’un capteur situé au bout d’une perche que l’opérateur introduit dans le chantier, depuis une zone sécurisée. Le capteur émet des faisceaux laser qui fournissent la distance à un mur détecté, créant ainsi une carte en 3D. Ceci produit des zones d’ombres et une faible densité de points sur les parois éloignées. Pour relever ces défis, une équipe de recherche de l’Université de Sherbrooke conçoit un drone filaire équipé d’un LiDAR rotatif pour cette mission, bénéficiant ainsi de plusieurs points de vue. La transmission filaire permet un temps de vol illimité, un partage de calcul et une communication en temps réel. Pour une compatibilité avec le mouvement du drone lors des coincements du fil, la longueur excédante est intégrée dans une bobine embarquée, qui contribue à la charge utile du drone. Lors d’un pilotage manuel, le facteur humain entraîne des problèmes de perception et compréhension d’un environnement 3D virtuel, et d’exécution d’une mission optimale. Cette thèse se concentre sur la navigation autonome sous deux aspects : la planification de trajectoire et l’exploration. Le système doit calculer une trajectoire qui cartographie l’environnement complet, en minimisant le temps de mission et en respectant la longueur maximale de fil embarquée. La planification de trajectoire à l’aide d’un Rapidly-exploring Random Tree (RRT) trouve rapidement un chemin réalisable, mais l’optimisation est coûteuse en calcul et la performance est variable et imprévisible. L’exploration par la méthode des frontières est représentative de l’espace à explorer et le chemin peut être optimisé en résolvant un Traveling Salesman Problem (TSP), mais les techniques existantes pour un drone filaire ne considèrent que le cas 2D et n’optimisent pas le chemin global. Pour relever ces défis, cette thèse présente deux nouveaux algorithmes. Le premier, RRT-Rope, produit un chemin égal ou plus court que les algorithmes existants en un temps de calcul jusqu’à 70% plus court que le deuxième meilleur algorithme dans un environnement représentatif. Une version modifiée de RRT-connect calcule un chemin réalisable, raccourci avec une technique déterministe qui tire profit des noeuds intermédiaires préalablement ajoutés. Le deuxième algorithme, TAPE, est la première méthode d’exploration de cavités en 3D qui minimise le temps de mission et la longueur du fil déroulé. En moyenne, le trajet global est 4% plus long que la méthode qui résout le TSP, mais le fil reste sous la longueur autorisée dans 100% des cas simulés, contre 53% avec la méthode initiale. L’approche utilise une architecture hiérarchique à 2 niveaux : la planification globale résout un TSP après extraction des frontières, et la planification locale minimise le coût du chemin et la longueur de fil via une fonction de décision. L’intégration de ces deux outils dans le NetherDrone produit un système intelligent pour l’exploration autonome, doté de fonctionnalités semi-autonomes pour une interaction avec l’opérateur. Les travaux réalisés ouvrent la porte à de nouvelles approches de navigation dans le domaine des missions d’inspection, de cartographie et de recherche et sauvetage

A Unified Framework for Integer Programming Formulation of Graph Matching Problems

Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program then solved using exact, heuristic and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding of complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The aim of the present study is to provide a unified framework for IP formulation of graph matching problems. Although there are many surveys on graph matching problems, however, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of difficult problems arising in practice, especially in pattern analysis

An Approach to Pathfinding for Real-World Situations

People plan their routes through new environments every day, but what factors influence these wayfinding decisions? In a world increasingly dependent on electronic navigation assistance devices, finding a way of automatically selecting routes suitable for pedestrian travel is an important challenge. With a greater freedom of movement than vehicular transport, and different requirements, an alternative approach should be taken to find an answer for pedestrian journeys than those taken in cars. Although previous research has produced a number of pedestrian route recommendation systems, the majority of these are restricted to a single route type or user group. The aim of this research was to develop an approach to route suggestion which could recommend routes according to the type of journey (everyday, leisure or tourist) a person is making. To achieve this aim, four areas of research were undertaken. Firstly, six experiments containing 450 participants were used to investigate the preference of seven different environment and route attributes (length, turns, decision points, vegetation, land use, dwellings and points of interest) for two attribute categories (simplicity and attractiveness) and three journey types (everyday, leisure and tourist). These empirically determined preferences were then used to find the rank-orders of the attributes, by comparing more of them simultaneously than earlier studies, and found either new rankings (for attractiveness, leisure journeys and tourist journey) or extended those already known (everyday journeys). Using these ranks and previously accepted relationships, an environment model was defined and built based on an annotated graph. This model can be built automatically from OpenStreetMap data, and is therefore simple enough to be applicable to many geographical areas, but it is detailed enough to allow route selection. Algorithms based on an extended version of Dijkstra’s shortest path algorithm were constructed. These used weighted minimum cost functions linked with attribute ranks, to select routes for different journey types. By avoiding the computational complexity of previous approaches, these algorithms could potentially be widely used in a variety of different platforms, and extended for different groups of users. Finally, the routes suggested by the algorithms were compared to participant recommendations for ‘simple’ routes with five start/end points, and for each of the three journey types (everyday, leisure and tourist). These comparisons determined that only length is required to select simple and everyday routes, but that the multi-attribute cost functions developed for leisure and tourist journeys select routes that are similar to those chosen by the participants. This indicates that the algorithms’ routes are appropriate for people to use in leisure and tourist journeys

Paths and compatible hamiltonian cycles

Aquest projecte està centrat en estudiar condicions que fan que un conjunt de punts tinguin un camí d'expansió que sigui compatible amb un cicle Hamiltonià. Hem demostrat que ser un camí monòton o self-approaching és condició suficient per asegurar que hi ha un cicle Hamiltonià compatible. A més, hem estudiat la condició de ser un camí que coincideix amb el MST del conjunt de punts i demostrat alguns resultats interessants per ajudar en futurs investigacions per demostrar que aquesta condició és suficient