Approximation algorithms for lawn mowing and milling

We study the problem of finding shortest tours/paths for "lawn mowing" and "milling" problems: Given a region in the plane, and given the shape of a "cutter" (typically, a circle or a square), find a shortest tour/path for the cutter such that every point within the region is covered by the cutter at some position along the tour/path. In the milling version of the problem, the cutter is constrained to stay within the region. The milling problem arises naturally in the area of automatic tool path generation for NC pocket machining. The lawn mowing problem arises in optical inspection, spray painting, and optimal search planning. Both problems are NP-hard in general. We give efficient constant-factor approximation algorithms for both problems. In particular, we give a (3+ε)-approximation algorithm for the lawn mowing problem and a 2.5-approximation algorithm for the milling problem. Furthermore, we give a simple 65-approximation algorithm for the TSP problem in simple grid graphs, which leads to an 115-approximation algorithm for milling simple rectilinear polygons.