Optimal covering tours with turn costs

Authors:
Esther M. Arkin;Michael A. Bender;Erik D. Demaine;Sándor P. Fekete;Joseph S. B. Mitchell;Saurabh Sethia
Affiliations:
Dept. of Applied Mathematics and Statistics, SUNY Stony Brook, NY;Dept. of Computer Science, SUNY Stony Brook, NY;Dept. of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada;Fachbereich Mathematik, TU Berlin, 10623 Berlin, Germany;Dept. of Applied Mathematics and Statistics, SUNY Stony Brook, NY;Dept. of Computer Science, SUNY Stony Brook, NY
Venue:
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Year:
2001

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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 not only on the length of the tour but also 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 lower bounds (NP-completeness of minimum-turn milling) and give efficient approximation algorithms for several natural versions of the problem, including a polynomial-time approximation scheme based on a novel adaptation of the m-guillotine method.