Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

The Unreasonable Success of Local Search: Geometric Optimization

What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude $1/\epsilon^c$ is an approximation scheme for the following problems in the Euclidian plane: TSP with random inputs, Steiner tree with random inputs, facility location (with worst case inputs), and bicriteria $k$-median (also with worst case inputs). The randomness assumption is necessary for TSP

Approximation Algorithms for Geometric Clustering and Touring Problems

Clustering and touring are two fundamental topics in optimization that have been studied extensively and have ``launched a thousand ships''. In this thesis, we study variants of these problems for Euclidean instances, in which clusters often correspond to sensors that are required to cover, measure or localize targets and tours need to visit locations for the purpose of item delivery or data collection. In the first part of the thesis, we focus on the task of sensor placement for environments in which localization is a necessity and in which its quality depends on the relative angle between the target and the pair of sensors observing it. We formulate a new coverage constraint that bounds this angle and consider the problem of placing a small number of sensors that satisfy it in addition to classical ones such as proximity and line-of-sight visibility. We present a general framework that chooses a small number of sensors and approximates the coverage constraint to arbitrary precision. In the second part of the thesis, we consider the task of collecting data from a set of sensors by getting close to them. This corresponds to a well-known generalization of the Traveling Salesman Problem (TSP) called TSP with Neighborhoods, in which we want to compute a shortest tour that visits at least one point from each unit disk centered at a sensor. One approach is based on an observation that relates the optimal solution with the optimal TSP on the sensors. We show that the associated bound can be improved unless we are in certain exceptional circumstances for which we can get better algorithms. Finally, we discuss Maximum Scatter TSP, which asks for a tour that maximizes the length of the shortest edge. While the Euclidean version admits an efficient approximation scheme and the problem is known to be NP-hard in three dimensions or higher, the question of getting a polynomial time algorithm for two dimensions remains open. To this end, we develop a general technique for the case of points concentrated around the boundary of a circle that we believe can be extended to more general cases

Generating approximate region boundaries from heterogeneous spatial information: an evolutionary approach

Spatial information takes different forms in different applications, ranging from accurate coordinates in geographic information systems to the qualitative abstractions that are used in artificial intelligence and spatial cognition. As a result, existing spatial information processing techniques tend to be tailored towards one type of spatial information, and cannot readily be extended to cope with the heterogeneity of spatial information that often arises in practice. In applications such as geographic information retrieval, on the other hand, approximate boundaries of spatial regions need to be constructed, using whatever spatial information that can be obtained. Motivated by this observation, we propose a novel methodology for generating spatial scenarios that are compatible with available knowledge. By suitably discretizing space, this task is translated to a combinatorial optimization problem, which is solved using a hybridization of two well-known meta-heuristics: genetic algorithms and ant colony optimization. What results is a flexible method that can cope with both quantitative and qualitative information, and can easily be adapted to the specific needs of specific applications. Experiments with geographic data demonstrate the potential of the approach

Placement and motion planning algorithms for robotic sensing systems

University of Minnesota Ph.D. dissertation. October 2014. Major: Computer Science. Advisor: Prof. Ibrahim Volkan Isler. I computer file (PDF); xxiii, 226 pages.Recent technological advances are making it possible to build teams of sensors and robots that can sense data from hard-to-reach places at unprecedented spatio-temporal scales. Robotic sensing systems hold the potential to revolutionize a diverse collection of applications such as agriculture, environmental monitoring, climate studies, security and surveillance in the near future. In order to make full use of this technology, it is crucial to complement it with efficient algorithms that plan for the sensing in these systems. In this dissertation, we develop new sensor planning algorithms and present prototype robotic sensing systems.In the first part of this dissertation, we study two problems on placing stationary sensors to cover an environment. Our objective is to place the fewest number of sensors required to ensure that every point in the environment is covered. In the first problem, we say a point is covered if it is seen by sensors from all orientations. The environment is represented as a polygon and the sensors are modeled as omnidirectional cameras. Our formulation, which builds on the well-known art gallery problem, is motivated by practical applications such as visual inspection and video-conferencing where seeing objects from all sides is crucial. In the second problem, we study how to deploy bearing sensors in order to localize a target in the environment. The sensors measure noisy bearings towards the target which can be combined to localize the target. The uncertainty in localization is a function of the placement of the sensors relative to the target. For both problems we present (i) lower bounds on the number of sensors required for an optimal algorithm, and (ii) algorithms to place at most a constant times the optimal number of sensors. In the second part of this dissertation, we study motion planning problems for mobile sensors. We start by investigating how to plan the motion of a team of aerial robots tasked with tracking targets that are moving on the ground. We then study various coverage problems that arise in two environmental monitoring applications: using robotic boats to monitor radio-tagged invasive fish in lakes, and using ground and aerial robots for data collection in precision agriculture. We formulate the coverage problems based on constraints observed in practice. We also present the design of prototype robotic systems for these applications. In the final problem, we investigate how to optimize the low-level motion of the robots to minimize their energy consumption and extend the system lifetime.This dissertation makes progress towards building robotic sensing systems along two directions. We present algorithms with strong theoretical performance guarantees, often by proving that our algorithms are optimal or that their costs are at most a constant factor away from the optimal values. We also demonstrate the feasibility and applicability of our results through system implementation and with results from simulations and extensive field experiments

Assouad-Nagata dimension and gap for ordered metric spaces

We prove that all spaces of finite Assouad-Nagata dimension admit a good order for Travelling Salesman Problem, and provide sufficient conditions under which the converse is true. We formulate a conjectural characterisation of spaces of finite $AN$-dimension, which would yield a gap statement for the efficiency of orders on metric spaces. Under assumption of doubling, we prove a stronger gap phenomenon about all orders on a given metric space.Comment: 27 pages. This paper appeared originally as the second part of first versions of arXiv:2011.01732. Now the paper is split it two parts, the first one "Spaces that can be ordered effectively: virtually free groups and hyperbolicity" arXiv:2011.01732, and the second part her

3D packing of balls in different containers by VNS

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel UniversityIn real world applications such as the transporting of goods products, packing is a major issue. Goods products need to be packed such that the smallest space is wasted to achieve the maximum transportation efficiency. Packing becomes more challenging and complex when the product is circular/spherical. This thesis focuses on the best way to pack three-dimensional unit spheres into the smallest spherical and cubical space. Unit spheres are considered in lieu of non-identical spheres because the search mechanisms are more difficult in the latter set up and any improvements will be due to the search mechanism not to the ordering of the spheres. The two-unit sphere packing problems are solved by approximately using a variable neighborhood search (VNS) hybrid heuristic. A general search framework belonging to the Artificial Intelligence domain, the VNS offers a diversification of the search space by changing neighborhood structures and intensification by thoroughly investigating each neighborhood. It is exible, easy to implement, adaptable to both continuous and discrete optimization problems and has been use to solve a variety of problems including large-sized real-life problems. Its runtime is usually lower than other meta heuristic techniques. A tutorial on the VNS and its variants along with recent applications and areas of applicability of each variant. Subsequently, this thesis considers several variations of VNS heuristics for the two problems at hand, discusses their individual efficiencies and effectiveness, their convergence rates and studies their robustness. It highlights the importance of the hybridization which yields near global optima with high precision and accuracy, improving many best- known solutions indicate matching some, and improving the precision and accuracy of others. Keywords: variable neighborhood search, sphere packing, three-dimensional packing, meta heuristic, hybrid heuristics, multiple start heuristics

Optimal UAS Assignments and Trajectories for Persistent Surveillance and Data Collection from a Wireless Sensor Network

This research developed a method for multiple Unmanned Aircraft Systems (UAS) to efficiently collect data from a Wireless Sensor Networks (WSN). WSN are composed of any number of fixed, ground-based sensors that collect and upload local environmental data to over flying UAS. The three-step method first uniquely assigns aircraft to specific sensors on the ground. Second, an efficient flight path is calculated to minimize the aircraft flight time required to verify their assigned sensors. Finally, sensors reporting relatively higher rates of local environmental activity are re-assigned to dedicated aircraft tasked with concentrating on only those sensors. This work was sponsored by the Air Force Research Laboratory, Control Sciences branch, at Wright Patterson AFB. Based on simulated scenarios and preliminary flight tests, optimal flight paths resulted in a 14 to 32 reduction in flight time and distance when compared to traditional flight planning methods

Hyperbolicity, Assouad-Nagata dimension and orders on metric spaces

We study asymptotic invariants of metric spaces, defined in terms of the travelling salesman problem, and our goal is to classify groups and spaces depending on how well they can be ordered in this context. We characterize virtually free groups as those admitting an order which has some efficiency on $4$-point subsets. We show that all $\delta$-hyperbolic spaces can be ordered extremely efficiently, for the question when the number of points of a subset tends to $\infty$. We prove that all spaces of finite Assouad-Nagata dimension admit a good order for the above mentioned problem, and under an additional hypothesis we prove the converse. Despite travelling salesman terminology, our paper does not aim at applications in computer science. Our goal is to study new properties of groups and metric spaces, and describe their connection with more traditional invariants, such as hyperbolicity, dimension, number of ends and doubling.Comment: title and abstract revised. minor correction

Design of Heuristic Algorithms for Hard Optimization

This open access book demonstrates all the steps required to design heuristic algorithms for difficult optimization. The classic problem of the travelling salesman is used as a common thread to illustrate all the techniques discussed. This problem is ideal for introducing readers to the subject because it is very intuitive and its solutions can be graphically represented. The book features a wealth of illustrations that allow the concepts to be understood at a glance. The book approaches the main metaheuristics from a new angle, deconstructing them into a few key concepts presented in separate chapters: construction, improvement, decomposition, randomization and learning methods. Each metaheuristic can then be presented in simplified form as a combination of these concepts. This approach avoids giving the impression that metaheuristics is a non-formal discipline, a kind of cloud sculpture. Moreover, it provides concrete applications of the travelling salesman problem, which illustrate in just a few lines of code how to design a new heuristic and remove all ambiguities left by a general framework. Two chapters reviewing the basics of combinatorial optimization and complexity theory make the book self-contained. As such, even readers with a very limited background in the field will be able to follow all the content