The curvature-constrained traveling salesman problem for high point densities

We consider algorithms for the curvature-constrained traveling salesman problem, when the nonholonomic constraint is described by Dubins´ model. We indicate a proof of the NP-hardness of this problem. In the case of low point densities, i.e., when the Euclidean distances between the points are larger than the turning radius of the vehicle, various heuristics based on the Euclidean Traveling salesman problem are expected to perform well. In this paper we do not put a constraint on the minimum Euclidean distance. We show that any algorithm that computes a tour for the Dubins´ vehicle following an ordering of points optimal for the Euclidean TSP cannot have an approximation ratio better than Omega(n), where n is the number of points. We then propose an algorithm that is not based on the Euclidean solution and seems to behave differently. For this algorithm, we obtain an approximation guarantee of O (min {(1+rho/epsiv)log n, (1+rho/epsiv)²), where rho is the minimum turning radius, and epsiv is the minimum Euclidean distance between any two points.