Watchman Routes for Lines and Segments

Abstract

Given a set $\mathcal L$ of non-parallel lines, a watchman route (tour) for $\mathcal L$ is a closed curve contained in the union of the lines in $\mathcal L$ such that every point on any line is visible (along a line) from at least one point of the route; similarly, we define a watchman route (tour) for a connected set $\mathcal S$ of line segments. The watchman route problem for a given set of lines or line segments is to find a shortest watchman route for the input set, and these problems are natural special cases of the watchman route problem in multiply connected polygonal domains. In this paper, we show that the problem of computing a shortest watchman route for a set of n non-parallel lines in the plane is polynomially tractable, while it becomes NP-hard in 3D. Then, we reprove NP-hardness of this problem for line segments in the plane and provide a polynomial-time approximation algorithm with ratio O(log3n). Additionally, we consider some special cases of the watchman route problem on line segments, for which we provide improved approximation or exact algorithms.