On approximations to minimum link visibility paths in simple polygons

摘要

We investigate a practical variant of the well-known polygonal visibility path (watchman) problem. For a polygon P, a minimum link visibility path is a polygonal visibility path in P that has the minimum number of links. The problem of finding a minimum link visibility path is NP-hard for simple polygons. If the link-length (number of links) of a minimum link visibility path (tour) is Opt for a simple polygon P with n vertices, we provide an algorithm with O(kn~2) runtime that produces polygonal visibility paths (or tours) of link-length at most (γ + a_1/(k - 1 ))Opt (or (γ + a_1/k)Opt), where k is a parameter dependent on P, a_1 is an output sensitive parameter and γ is the approximation factor of an O(k~3) time approximation algorithm for the geometric travelling salesman problem (path or tour version).