 
Summary: OPTIMAL COVERING TOURS WITH TURN COSTS #
ESTHER M. ARKIN + , MICHAEL A. BENDER # , ERIK D. DEMAINE § ,
S ’
ANDOR P. FEKETE ¶ , JOSEPH S. B. MITCHELL + , AND SAURABH SETHIA #
Abstract. We give the first algorithmic study of a class of ``covering tour'' problems related
to the geometric Traveling Salesman Problem: Find a polygonal tour for a cutter so that it sweeps
out a specified region (``pocket''), in order to minimize a cost that depends mainly on the number of
turns. These problems arise naturally in manufacturing applications of computational geometry to
automatic tool path generation and automatic inspection systems, as well as arc routing (``postman'')
problems with turn penalties. We prove the NPcompleteness of minimumturn milling and give
e#cient approximation algorithms for several natural versions of the problem, including a polynomial
time approximation scheme based on a novel adaptation of the mguillotine method.
Key words. NC machining, manufacturing, traveling salesman problem, milling, lawn mowing,
covering, approximation algorithms, polynomialtime approximation scheme, mguillotine subdivi
sions, NPcompleteness, turn costs.
AMS subject classifications. 90C27, 68W25, 68Q25
1. Introduction. An important algorithmic problem in manufacturing is to
compute e#ective paths and tours for covering (``milling'') a given region (``pocket'')
with a cutting tool. The objective is to find a path or tour along which to move a
prescribed cutter in order that the sweep of the cutter covers the region, removing all
