The Minimum Guarding Tree Problem

Given a set L of non-parallel lines in the plane and a nonempty subset L′ ⊆ L, a guarding tree for L′ is a tree contained in the union of the lines in L such that if a mobile guard (agent) runs on the edges of the tree, all lines in L′ are visited by the guard. Similarly, given a connected arrangement S of line segments in the plane and a nonempty subset S ′ ⊆ S, we define a guarding tree for S . The minimum guarding tree problem for a given set of lines or line segments is to find a minimum-length guarding tree for the input set. We provide a simple alternative (to [29]) proof of NP-hardness of the problem of finding a guarding tree of minimum length for a set of orthogonal (axis-parallel) line segments in the plane. Then, we present two approximation algorithms with factors 2 and 3.98, respectively, for computing a minimum guarding tree for a subset of a set of n arbitrary non-parallel lines in the plane; their running times are O(n) and O(n logn), respectively. Finally, we show that this problem is NP-hard for lines in 3-space.