Near-Optimal Coverage Path Planning with Turn Costs

Coverage path planning is a fundamental challenge in robotics, with diverse applications in aerial surveillance, manufacturing, cleaning, inspection, agriculture, and more. The main objective is to devise a trajectory for an agent that efficiently covers a given area, while minimizing time or energy consumption. Existing practical approaches often lack a solid theoretical foundation, relying on purely heuristic methods, or overly abstracting the problem to a simple Traveling Salesman Problem in Grid Graphs. Moreover, the considered cost functions only rarely consider turn cost, prize-collecting variants for uneven cover demand, or arbitrary geometric regions. In this paper, we describe an array of systematic methods for handling arbitrary meshes derived from intricate, polygonal environments. This adaptation paves the way to compute efficient coverage paths with a robust theoretical foundation for real-world robotic applications. Through comprehensive evaluations, we demonstrate that the algorithm also exhibits low optimality gaps, while efficiently handling complex environments. Furthermore, we showcase its versatility in handling partial coverage and accommodating heterogeneous passage costs, offering the flexibility to trade off coverage quality and time efficiency

A Parallel Distributed Strategy for Arraying a Scattered Robot Swarm

We consider the problem of organizing a scattered group of $n$ robots in two-dimensional space, with geometric maximum distance $D$ between robots. The communication graph of the swarm is connected, but there is no central authority for organizing it. We want to arrange them into a sorted and equally-spaced array between the robots with lowest and highest label, while maintaining a connected communication network. In this paper, we describe a distributed method to accomplish these goals, without using central control, while also keeping time, travel distance and communication cost at a minimum. We proceed in a number of stages (leader election, initial path construction, subtree contraction, geometric straightening, and distributed sorting), none of which requires a central authority, but still accomplishes best possible parallelization. The overall arraying is performed in $O(n)$ time, $O(n^2)$ individual messages, and $O(nD)$ travel distance. Implementation of the sorting and navigation use communication messages of fixed size, and are a practical solution for large populations of low-cost robots

Minimum Scan Cover and Variants - Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph G that is embedded in Euclidean space. The edges of G need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances

Probing a Set of Trajectories to Maximize Captured Information

We study a trajectory analysis problem we call the Trajectory Capture Problem (TCP), in which, for a given input set T of trajectories in the plane, and an integer k? 2, we seek to compute a set of k points ("portals") to maximize the total weight of all subtrajectories of T between pairs of portals. This problem naturally arises in trajectory analysis and summarization. We show that the TCP is NP-hard (even in very special cases) and give some first approximation results. Our main focus is on attacking the TCP with practical algorithm-engineering approaches, including integer linear programming (to solve instances to provable optimality) and local search methods. We study the integrality gap arising from such approaches. We analyze our methods on different classes of data, including benchmark instances that we generate. Our goal is to understand the best performing heuristics, based on both solution time and solution quality. We demonstrate that we are able to compute provably optimal solutions for real-world instances

Publishing and sharing multi-dimensional image data with OMERO

Imaging data are used in the life and biomedical sciences to measure the molecular and structural composition and dynamics of cells, tissues, and organisms. Datasets range in size from megabytes to terabytes and usually contain a combination of binary pixel data and metadata that describe the acquisition process and any derived results. The OMERO image data management platform allows users to securely share image datasets according to specific permissions levels: data can be held privately, shared with a set of colleagues, or made available via a public URL. Users control access by assigning data to specific Groups with defined membership and access rights. OMERO’s Permission system supports simple data sharing in a lab, collaborative data analysis, and even teaching environments. OMERO software is open source and released by the OME Consortium at www.openmicroscopy.org

Minimum Scan Cover with Angular Transition Costs

We provide a comprehensive study of a natural geometric optimization problem motivated by questions in the context of satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs (msc), we are given a graph G that is embedded in Euclidean space. The edges of G need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex takes some time proportional to the corresponding turn angle. Our goal is to minimize the time until all scans are completed, i.e., to compute a schedule of minimum makespan. We show that msc is closely related to both graph coloring and the minimum (directed and undirected) cut cover problem; in particular, we show that the minimum scan time for instances in 1D and 2D lies in ?(log ?(G)), while for 3D the minimum scan time is not upper bounded by ?(G). We use this relationship to prove that the existence of a constant-factor approximation implies P=NP, even for one-dimensional instances. In 2D, we show that it is NP-hard to approximate a minimum scan cover within less than a factor of 3/2, even for bipartite graphs; conversely, we present a 9/2-approximation algorithm for this scenario. Generally, we give an O(c)-approximation for k-colored graphs with k ? ?(G)^c. For general metric cost functions, we provide approximation algorithms whose performance guarantee depend on the arboricity of the graph

Algorithm Engineering für schwere Probleme in Computational Geometry

Many practically relevant problems in Computational Geometry are provably hard to solve. In this thesis, we consider a series of these problems and tackle them with techniques from algorithm engineering, e.g., mixed integer programming or deep reinforcement learning. Additionally, we provide a number of results on complexity and algorithms for these problems. We start with a set of problems occurring in the context of large satellite systems, where communication requires a rotation of directional antennas. Here, we discover a close relationship to the vertex coloring problem, use constraint programming to obtain optimal solutions, and provide a practical auction-based algorithm that achieves good results in a realistic simulation. Then we continue with (partial) coverage path planning problems involving turn costs, which require different approaches than when minimizing only the traveled distance. We engineer an approximation algorithm based on linear relaxation and minimum-weight perfect matching to solve large instances in grid graphs. This approach is then generalized to complex polygonal instances, which allow modeling of difficult real-world scenarios. Afterward, we focus on probing a set of trajectories to maximize the captured information, where we successfully apply mixed integer programming and a set of heuristics. We also consider robots so small they need to be actuated by external forces like a magnetic field; applications for such robots include cancer treatment. Construction plans for miniature objects are computed using SAT-solvers, and control sequences to gather a swarm of these robots are optimized using deep reinforcement learning. Finally, we close the thesis with an excursion in hosting the CG:SHOP competition for three years and counting.Viele der praktisch relevanten Probleme in der Computational Geometry sind beweisbar schwer zu lösen. In dieser Ausarbeitung betrachten wir eine Reihe solcher Probleme und lösen sie mit Techniken aus dem Algorithm Engineering, z.B., Mixed Integer Programming oder Deep Reinforcement Learning. Zusätzlich erlangen wir etliche Erkenntnisse zur Komplexität und Algorithmen für diese Probleme. Wir beginnen mit diversen Problemen, die im Kontext von größeren Satellitensystemen auftreten und bei denen die Kommunikation das Rotieren von gerichteten Antennen erfordert. Hier entdecken wir einen engen Zusammenhang zum Graphenfärbungsproblem, nutzen Constraint Programming zur Berechnung von optimalen Lösungen und entwickeln einen praktischen, auktionsbasierten Algorithmus, der gute Ergebnisse in realistischen Simulationen erzielt. Danach fahren wir mit dem (Partial) Coverage Path Planning Problem unter der Berücksichtigung von Abbiegekosten fort, welches andere Ansätze braucht, als wenn wir nur die gefahrene Distanz minimieren wollen. Hierfür implementieren wir einen Approximationsalgorithmus - basierend auf linearer Relaxierung und Minimum-Weight Perfect Matching - auf eine Art, die es ermöglicht, auch große Instanzen im Gittergraphen zu optimieren. Dieser Ansatz wird anschließend von uns für komplexe polygonale Instanzen erweitert, was die Modellierung von schwierigen, realistischen Szenarien erlaubt. Anschließend fokussieren wir uns auf die Maximierung von eingeschlossenen Informationen aus einer Menge von Trajektorien, wo wir erfolgreich Mixed Integer Programming sowie eine Menge von Heuristiken anwenden. Wir betrachten auch Roboter, die so klein sind, dass sie nur durch äußere Kräfte - wie etwa ein magnetisches Feld - bewegt werden können und einen potentiellen Einsatz in der Tumorbehandlung haben. Konstruktionspläne für Miniaturobjekte berechnen wir mittels SAT-Solvern, und Kontrollsequenzen zum Sammeln eines solchen Roboterschwarms optimieren wir mittels Deep Reinforcement Learning. Letztlich schließen wir diese Arbeit mit einem Ausflug in die Organisation der CG:SHOP Challenges