The problem of tool path routing for the CNC thermal cutting machines is considered. Pierce points are located at the parts bounding contours, consisting of straight-line segments and circular arcs. Continuous cutting technique is used, each contour is cut out entirely, and no pre-sampling occurs, so cutting can start from any point on the contour. General problem of minimizing the route length is reduced to minimizing the air move length. It is shown to be equivalent to finding the shortest polyline with vertices on the contours. New algorithm for constructing such a broken line for fixed order of contour traversing is proposed. The resulting solution is shown to be a local minimum. Some sufficient conditions are described for the it to be also a global minimum, which can be easily verified numerically, and some even visually. A technique is described for automatically taking into account precedence constraints for the practically important case of nested contours. This also decreases the size of the problem, which has a positive effect on the optimization time. A heuristic routing algorithm based on the variable neighborhood search (VNS) is proposed. Alternative approaches to the use of other discrete optimization methods along with the proposed algorithm for constructing the shortest polyline for solving the complete problem of continuous cutting, and the resulting difficulties of both theoretical and practical nature are described. The generalization of the problem of continuous cutting to a wider class of problems of (generalized) segment cutting is described, which makes it possible to advance in solving the problem of intermittent cutting. The scheme of application of the proposed algorithm for solving problems of generalized segment cutting is described. The results of numerical experiments are considered in comparison with the exact solution of the GTSP problem.