🤖 AI Summary
This work proposes a novel variant of the Traveling Salesman Problem in which two agents must collaboratively traverse a given set of points, minimizing the Fréchet distance between their respective trajectories while simultaneously optimizing total path length and load balance. It introduces the Fréchet distance—commonly used to measure curve similarity—into multi-agent path planning for the first time, uncovering new applications in routing and network design. For the discrete Fréchet distance, the authors devise an efficient near-linear-time algorithm; in contrast, they rigorously establish the NP-hardness of the continuous case. By integrating techniques from computational geometry and combinatorial optimization, the study systematically analyzes multiple problem variants, carefully balancing theoretical complexity with practical tractability.
📝 Abstract
The Fréchet distance is a well-studied distance measure between two curves. In this work, we demonstrate that the merit of Fréchet distance extends beyond evaluating similarity, and introduce a new setting in which it proves useful. Consider a situation where two agents are required to visit a given set of sites, while staying close to each other throughout their traversal. In this paper, we study problems where the goal is to construct two curves whose vertices are from a given set of points, under the constraint that the Fréchet distance between the curves is kept as small as possible. This problem can be viewed as a variant of the Traveling Salesman Problem (TSP), and thus may be of interest in routing, network planning and more. We present a near-linear algorithm for this problem under the discrete Fréchet distance, and explore several variants of the problem, including minimizing the lengths of the curves and balancing the number of sites assigned to each agent. Lastly, we prove that the problem is NP-hard under the continuous Fréchet Distance.