Approximation Algorithms for the Two-Watchman Route in a Simple Polygon

The two-watchman route problem is that of computing a pair of closed tours in an environment so that the two tours together see the whole environment and some length measure on the two tours is minimized. Two standard measures are: the minmax measure, where we want the tours where the longest of them has smallest length, and the minsum measure, where we want the tours for which the sum of their lengths is the smallest. It is known that computing a minmax two-watchman route is NP-hard for simple rectilinear polygons and thus also for simple polygons. Also, any c-approximation algorithm for the minmax two-watchman route is automatically a 2c-approximation algorithm for the minsum two-watchman route. We exhibit two constant factor approximation algorithms for computing minmax two-watchman routes in simple polygons with approximation factors 5.969 and 11.939, having running times O(n^8) and O(n^4) respectively, where n is the number of vertices of the polygon. We also use the same techniques to obtain a 6.922-approximation for the fixed two-watchman route problem running in O(n^2) time, i.e., when two starting points of the two tours are given as input.Comment: 36 pages, 14 figure

Visibility-Based Pursuit-Evasion In The Plane

As technological advances further increase the amount of memory and computing power available to mobile robots, we are seeing an unprecedented explosion in the utilization of deployable robots for various tasks. The speed at which robots begin to enter various domains is largely dependent on the availability of robust and efficient algorithms that are capable of solving the complex planning problems inherent to the given domain. One such domain which is experiencing unprecedented growth in recent years requires a robot to detect and/or track a mobile agent or group of agents. In these scenarios, there are typically two players with diametrically opposed goals. For matters of security, we have a guard and an intruder. The guard’s goal is to ensure that if an intruder enters the premises they are caught in a timely manner. Analogously, the intruder wishes to evade detection for as long as possible. Search and rescue operations are often framed as a two-player game between rescuers and survivors. Though the survivors are unlikely to behave antagonistically, an agnostic model is useful for the rescuers to guarantee that the survivors are found, regardless of their movements. Both of these tasks, are at their core, pursuit-evasion problems. There are many variants of the pursuit-evasion problem, the common theme amongst them is that one group of agents, the “pursuers”, attempts to track members of another group, the “evaders”. Geometric formulations of the pursuit-evasion problem require a pursuer(s) to systematically search an environment to locate one or more evaders ensuring that all evaders will be captured by the pursuer(s) in a finite time. The visibility-based pursuit-evasion problem is a geometric variant of the pursuit-evasion problem that defines a visibility-region which corresponds to the region of the environment that the pursuer(s) can actively perceive. If an evader lies within this visibility region then it is captured (detected). This thesis contains four novel contributions that solve various visibility-based pursuit-evasion problems. The first contribution is an algorithm that computes the optimal (minimal path length) pursuer trajectory for a single pursuer. The second contribution is an algorithm that generates a joint motion strategy for multiple pursuers. Motivated by the result of the second contribution, the third result is a sampling-based algorithm for the multiple pursuer scenario. The fourth contribution is a complete algorithm that computes a trajectory for a pursuer that has a very limited sensor footprint

Local-Global Results on Discrete Structures

Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Émery curvature has been one such successful notion of curvature. In this thesis we use combinatorial implications of Bakry- Émery curvature on graphs to prove a sort of local discrepancy inequality. This then allows us to derive a number of results regarding the local structure of graphs, dependent only on a curvature lower bound. For instance, it turns out that a curvature lower bound implies a nontrivial lower bound on graph connectivity. We also use these results to consider the curvature of strongly regular graphs, a well studied and important class of graphs. In this regard, we give a partial solution to an open conjecture: all SRGs satisfy the curvature condition CD(∞, 2). Finally we transition to consider a facility location problem motivated by using Unmanned Aerial Vehicles (UAVs) to guard a border. Here, we find a greedy algorithm, acting on local geometric information, which finds a near optimal placement of base stations for the guarding of UAVs

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Multi Robot Intruder Search

The aim of this work is the development and analysis of methods and algorithms to allow a multi robot system to cooperatively search a closed, 2-dimensional environment for a human intruder. The underlying problem corresponds to the game-theoretic concept of a classical pursuit evasion game, whereas the focus is set to the generation of plans for the group of pursuers. While the main aspect of of this work lies in the field of probabilistic robotics, concepts and ideas are incorporated from differential game theory, algorithmic geometry and graph theory. The probabilistic basis allows the integration of sensor error as well as nondeterministic robot motion. The main contributions of this work can be divided into three major parts: The first part deals with the development and implementation of probabilistic human models. Depending on the specific behavior of an intruder, ranging from uncooperative to unaware, different classes of intruders are identified. Models are proposed for two of these classes. For the case of a clever and uncooperative intruder who actively evades detection, we propose a model based on the concept of contamination. The second class corresponds to a person who is unaware of the pursuit. We show that simple Markov models, which are often proposed in literature, are not suited for modeling realistic human motion and develop advanced Markov models, which conform to random waypoint motion models. The second part, which is also the most extensive part of this work, deals with the problem of finding an uncooperative and clever intruder. A solution is presented, which projects the problem on a graph structure, which is then searched by a highly optimized A* planner. The solution for the corresponding graph problem is afterwards projected back to the original search space and can be executed by the robotic pursuers. By means of the models proposed in the first part, the performance and correctness of the method is shown. We present experiments in simulation as on real robots to show the practicability and efficiency of the method. The third part deals with the problem of finding an intruder who is unaware of the search. Based on the advanced Markov model previously discussed, a greedy algorithm is proposed, which aims at maximizing the probability to find the intruder in the near future. Experimental results for this method are shown and comparisons to simpler methods are given.Mehrroboter-Eindringlings-Suche Ziel dieser Arbeit ist die Entwicklung und Analyse von Methoden und Algorithmen, die einem kooperativen Mehrrobotersystem erlauben nach einem Eindringling in einer zweidimensionalen, geschlossenen Umgebung zu suchen. Das dem zugrunde liegende Problem entspricht dem spieltheoretischen Konzept eines Suche und Ausweichen Spieles (pursuit evasion game), wobei der Fokus auf der Generierung von Plänen für die Verfolger liegt. Der Hauptaspekt dieser Arbeit liegt dabei im Feld der probabilistischen Robotik, wobei Konzepte und Ideen aus dem Gebiet der differentiellen Spieltheorie, der algorithmischen Geometrie und der Graph Theorie verwendet werden. Die probabilistische Modellierung erlaubt dabei die Integration von Sensorfehlern wie auch nichtdeterministische Roboter-Bewegung. Die Arbeit gliedert sich in drei Hauptteile: Der erste Teil beschäftigt sich mit dem Entwurf und der Implementierung von probabilistischen Modellen für menschliche Bewegung. Abhängig vom angenommenen Verhalten eines Eindringlings, von aktiv ausweichend bis zu ignorant, werden verschiedene Klassen von menschlichem Verhalten unterschieden. Für zwei dieser Klassen stellen wir Modelle auf: Für den Fall einer intelligenten und aktiv ausweichenden Person, generieren wir ein Modell basierend auf dem Konzept von Kontamination. Das zweite Modell entspricht einem Eindringling, der sich der Suche nach ihm nicht bewusst ist. Wir zeigen, dass einfache Markov-Modelle, wie sie in der Vergangenheit oft vorgeschlagen worden sind, ungeeignet sind, um realistische Bewegung zu abzubilden und entwickeln entsprechend erweiterte Markov-Modelle für eine realistischere Modellierung. Der zweite Teil der Arbeit beschäftigt sich mit der Frage, wie man einen intelligente und aktiv ausweichenden Eindringling aufspüren kann. Die vorgestellte Lösung basiert auf der Projektion des Problems auf einen Graphen, der anschließend von einem hoch optimierten A*-Planungsalgorithmus durchsucht werden kann. Diese Lösung kann anschließend auf den ursprünglichen Raum rückprojeziert werden und kann als direkter Plan von den verfolgenden Robotern ausgeführt werden. Mittels der Modelle aus dem ersten Teil wird die Korrektheit und Effizienz der Lösung gezeigt. Der letzte Teil befasst sich mit der Frage, wie ein Eindringling gefunden werden kann, der sich neutral zur Suche verhält. Basierend auf den erweiterten Markov-Modellen aus dem ersten Teil, wird eine Lösung durch gierige Suche präsentiert, die die Wahrscheinlichkeit eine Person im nächsten Zeitschritt aufzuspüren, maximiert. Wie im zweiten Teil werden Experimente diskutiert und diese mit der Proformanz simplerer Methoden verglichen

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Shortest watchman tours in weak visibility polygons

An $O(n^4loglogn)$ algorithm for shortest watchman tour (SWT) problem for simple polygons, given a starting point on the boundary of the polygon is proposed by W.P. Chin and S. Ntafos. The problem of finding the SWT in general polygons when the starting point is not specified is open. We develop an $O(n^{10})$ algorithm for the SWT problem in weak visibility polygons, with no assumption on the starting point

Policies, methods and tools for visitor management proceedings of the second International Conference on Monitoring and Management of Visitor Flows in Recreational and Protected Areas, June 16 20, 2004, Rovaniemi, Finland

The second International Conference on Monitoring and Management of Visitor Flows in recreational and protected areas (MMV 2) -conference provided a forum for research presentations and for exchange of information and experience of managerial policies, problems, practices and solutions regarding issues related to monitoring and management of visitor flows in recreational and protected areas. These proceedings cover ten research topics, which were chosen to reflect current on-going research work internationally in the field of visitor monitoring and management. Monitoring visitor flows and also other types of recreational inventories are discussed in 16 articles and four posters on visitor monitoring methods, experiences of national, regional and on-site visitor inventories and visitor flow modeling and data management. Nineteen papers and three posters are discussing visitor management research from several perspectives. Articles related to issues of visitor conflicts, implementation of visitor information in management processes, different aspects of sustainability and carrying capacity issues in recreational settings make the largest group of papers. The third major subject group of articles (16) deal with visitor management policy issues, and nature tourism policies in recreational and protected areas. The last topics include economic and social impacts of recreation and nature tourism in the surroundings communities, regions and countries