 
Summary: Approximation Algorithms for Lawn Mowing and Milling
Esther M. Arkin
S´andor P. Fekete
Joseph S. B. Mitchell§
Abstract
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 NPhard in general. We give efficient constantfactor approximation algorithms for
both problems. In particular, we give a (3+ )approximation algorithm for the lawn mowing problem and
a 2.5approximation algorithm for the milling problem. Furthermore, we give a simple 6
5
approximation
algorithm for the TSP problem in simple grid graphs, which leads to an 11
5
approximation algorithm for
