On Fr\'echet Traveling Salesmen Problems
The Fr\'echet distance is a well-studied distance measure between two curves. In this work, we demonstrate that the merit of Fr\'echet 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 Fr\'echet 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 Fr\'echet 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 Fr\'echet Distance.