Optimal Covering Tours with Turn Costs

TL;DR

This paper studies covering tour problems with turn costs, providing NP-completeness results and efficient approximation algorithms, including a polynomial-time approximation scheme, for applications in manufacturing and arc routing.

Contribution

It introduces the first algorithmic analysis of covering tours with turn costs, including NP-completeness proofs and a novel approximation scheme based on the m-guillotine method.

Findings

Proves NP-completeness of minimum-turn milling.

Develops efficient approximation algorithms for various problem versions.

Introduces a polynomial-time approximation scheme using m-guillotine method.

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 em 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 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.