Approximation Algorithm for Multiple-Tool Milling

Milling is the mechanical process of removing material from a piece of stock through the use of a rapidly spinning circular milling tool in order to form some desired geometric shape. An important problem in computer-aided design and manufacturing is the automated generation of efficient milling plans for computerized numerically controlled (CNC) milling machines. Among the most common milling problems is simple 2-dimensional pocket milling: cut a given 2-dimensional region down to some constant depth using a given set of milling tools. Most of the research in this area has focused on generating such milling plans assuming that the machine has a tool of a single size. Since modern CNC milling machines typically have access to a number of milling tools of various sizes and the ability to change tools automatically, this raises the important optimization problem of generating efficient milling plans that take advantage of this capability to reduce the total milling time. We consider the following multiple-tool milling problem: Given a region in the plane and a set of tools of different sizes, determine how to mill the desired region with minimum cost. The problem is known to be NP-hard even when restricted to the case of a single tool. In this paper, we present a polynomial-time approximation algorithm for the multiple-tool milling problem. The running time and approximation ratio of our algorithm depend on the simple cover complexity (introduced by Mitchell, Mount, and Suri) of the milling region.