IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XXX, NO. XXX, JANUARY 200X 1 On the Curvature-Constrained Traveling Salesman Problem Jerome Le Ny, Member, IEEE, Eric Feron, Member, IEEE, and Emilio Frazzoli, Member, IEEE Abstract We study the traveling salesman problem for a Dubins vehicle. We prove that this problem is NP-hard, and provide lower bounds on the approximation ratio achievable by some recently proposed heuristics. We also describe new algorithms for this problem based on heading discretization, and evaluate their performance numerically. I. I NTRODUCTION In an instance of the traveling salesman problem (TSP) we are given the distances between any pair of n points. The problem is to find the shortest closed path (tour) visiting every point exactly once. We also call this problem the tour-TSP to distinguish it from the path-TSP, where the requirement that the vehicle must start and end at the same point is removed. This famously intractable problem is often encountered in robotics, and has traditionally been solved in two steps within the common layered controller architectures for mobile robots. At the higher decision- making level, the dynamics of the robot are usually not taken into account and the mission planner might typically chose to solve the TSP for the Euclidean metric (ETSP), i.e., using the Euclidean distances between waypoints. This determines the order in which the waypoints This work was supported by Air Force - DARPA - MURI award 009628-001-03-132 and Navy ONR award N00014-03-1-0171. Preliminary versions of this work appeared in [1], [2]. J. Le Ny is with the school of Electrical Engineering, University of Pennsylvania, PA 19104, USA jeromel@seas.upenn.edu. E. Frazzoli is with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139, USA frazzoli@mit.edu. E. Feron is with the School of Aerospace Engineering, Georgia Institute ot Technology, Atlanta, GA 30332, USA eric.feron@aerospace.gatech.edu. March 11, 2009 DRAFT