 
Summary: Optimization Problems Related to
Zigzag Pocket Machining \Lambda
Esther M. Arkin (1) y Martin Held (2) z Christopher L. Smith (1) x
(1) Department of Applied Mathematics and Statistics
State University of New York, Stony Brook, NY 117943600, USA
(2) Institut f¨ur Computerwissenschaften
Universit¨at Salzburg, A5020 Salzburg, Austria
[submitted to: Algorithmica] [(revised) September 5, 1997]
Abstract
A fundamental problem of manufacturing is to produce mechanical parts from billets by
clearing areas within specified boundaries from the material. Based on a graphtheoretical
formulation, the algorithmic handling of one particular machining problem  `zigzag pocket
machining'  is investigated. We present a lineartime algorithm that ensures that every re
gion of the pocket is machined exactly once, while attempting to minimize the number of
tool retractions required. This problem is shown to be NPhard for pockets with holes. Our
algorithm is provably good in the sense that the machining path generated for a pocket with
h holes requires at most 5 \Delta OPT + 6 \Delta h retractions, where OPT is the (unknown) minimum
number of retractions required by any algorithm. The algorithm has been implemented, and
practical tests for pockets without holes suggest that one can expect an approximation factor
of about 1.5 for practical examples, rather than the factor 5 as proved by our analysis.
