Dominik Michael Krupke

We provide a comprehensive study of a natural graph optimization problem that arises from transition costs between incident edges. 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. A real-world motivation arises in the context of satellite communication and astrophysics. 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 \Θ(\log χ(G)) while for 3D the minimum scan time is not upper bounded by \χ(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 \\frac32 even for bipartite graphs; conversely, we present a 9/2-approximation algorithm for this scenario. Generally, we give an Ø(c)\-approximation for \k\-colored graphs with \k≤χ(G)^c\. For general metric cost functions, we provide approximation algorithms whose performance guarantees depend on the arboricity of the graph.

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.

Modern software systems are typically configurable, a fundamental prerequisite for wide applicability and reusability. This flexibility poses an extraordinary challenge for quality assurance, as the enormous number of possible configurations makes it impractical to test each of them separately. This is where t-wise interaction sampling can be used to systematically cover the configuration space and detect unknown feature interactions. Over the last two decades, numerous algorithms for computing small interaction samples have been studied, providing improvements for a range of heuristic results; nevertheless, it has remained unclear how much these results can still be improved.We present a significant breakthrough: a fundamental framework, based on the mathematical principle of duality, for combining near-optimal solutions with provable lower bounds on the required sample size. This implies that we no longer need to work on heuristics with marginal or no improvement, but can certify the solution quality by establishing a limit on the remaining gap; in many cases, we can even prove optimality of achieved solutions. This theoretical contribution also provides extensive practical improvements: Our algorithm SampLNS was tested on 47 small and medium-sized configurable systems from the existing literature. SampLNS can reliably find samples of smaller size than previous methods in 85 % of the cases; moreover, we can achieve and prove optimality of solutions for 63 % of all instances. This makes it possible to avoid cumbersome efforts of minimizing samples by researchers as well as practitioners, and substantially save testing resources for most configurable systems.