Watchman routes for lines and line segments

Given a set L of non-parallel lines in the plane, a watchman route (tour) for G is a closed curve contained in the union of the lines in L such that every line is visited (intersected) by the route; we similarly define a watchman route (tour) for a connected set 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 a polygon with holes (a polygonal domain). 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. We give an alternative NP-hardness proof of this problem for line segments in the plane and obtain a polynomial-time approximation algorithm with ratio O(log(3) n). Additionally, we consider some special cases of the watchman route problem on line segments, for which we provide exact algorithms or improved approximations. (C) 2013 Elsevier B.V. All rights reserved.