Path planning and routing in computational geometry are essential for optimizing a wide array of real-world applications, significantly impacting industries such as logistics, robotics, telecommunications, and urban planning. These problems often involve devising efficient routes or strategies to achieve specific goals, such as minimizing travel time, reducing costs, or ensuring safety. In logistics, advanced routing algorithms enhance the efficiency of delivery services by optimizing routes, thereby cutting operational expenses and improving customer satisfaction. In robotics, precise path planning enables autonomous systems to navigate complex environments safely and efficiently. Telecommunications benefit from optimized routing by improving data transmission across networks, leading to faster and more reliable communication. Urban planners use these computational techniques to design better traffic flow systems, reducing congestion and improving public transportation. Overall, advancements in path planning and routing drive technological innovation and contribute to societal progress by solving complex real-world challenges.

Emerging applications of path planning and routing in computational geometry are revolutionizing various cutting-edge fields. In autonomous driving, sophisticated path planning algorithms enable self-driving cars to navigate dynamically changing environments with precision and safety. In drone technology, efficient routing ensures optimal flight paths for tasks ranging from delivery services to agricultural monitoring and disaster response. The healthcareindustry is leveraging these advancements for robotic surgery, where precise path planning allows minimally invasive procedures with increased accuracy. In environmental monitoring, autonomous underwater vehicles (AUVs) use advanced routing to conduct extensive marine surveys, aiding in oceanographic research and resource management. Additionally, smart city initiatives utilize computational geometry for optimizing public transportation routes, reducing traffic congestion, and enhancing urban mobility. These applications demonstrate the expanding influence of path planning and routing in addressing complex, real-world challenges across diverse domains.

Recently, significant theoretical advancements have been made in the field of computational geometry to address certain path planning and routing problems. For instance, Wang developed an algorithm to compute Euclidean shortest paths in triangulated polygonal domains with optimal running time (JACM 2023). The longstanding two-point shortest path query data structures in polygonal domains, established more than two decades ago by Chiang and Mitchell, have been improved by S. de Berg, Miltzow, and Staals (SoCG 2024). Moreover, the traveling salesman problem in d-dimensional Euclidean space can now be solved exactly in time optimal up to the exponential-time hypothesis, thanks to the work of M. de Berg, Bodlaender, Kisfaludi-Bak, and Kolay (SICOMP 2023). These breakthroughs highlight the ongoing progress and potential of computational geometry in addressing complex path routing and planning problems.

This seminar aims to unite emerging and established experts in the field to share recent developments and identify future research directions. On the first day, overview lectures will cover significant recent developments, algorithmic methodologies, and emerging applications. The second and third days will focus on group discussions to pinpoint future research directions. On the fourth day, participants will be regrouped to exchange insights from the previous days and concentrate on more specific research questions. The final day will be dedicated to a wrapup discussion, consolidating ideas to foster future research collaborations.