Dispersing and Grouping Points on Segments in the Plane

Motivated by (continuous) facility location, we consider the problem of dispersing and grouping points on a set of segments (of streets) in the plane. In the former problem, given a set of n disjoint line segments in the plane, we consider the problem of computing a point on each of the n segments such that the minimum Euclidean distance between any two of these points is maximized. We prove that this 2D dispersion problem is NP-hard, in fact, the problem is NP-hard even if all the segments are parallel and are of unit length. This is in contrast to the polynomial solvability of the corresponding 1D problem by Li and Wang (2016), where the intervals are in 1D and are all disjoint. With this result, we also show that the Independent Set problem on Colored Linear Unit Disk Graph (meaning the convex hulls of points with the same color form disjoint line segments) remains NP-hard, and the parameterized version of it is inW[2]. In the latter problem, given a set of n disjoint line segments in the plane, we consider the problem of computing a point on each of the n segments such that the maximum Euclidean distance between any two of these points is minimized. We present a factor-1.1547 approximation algorithm which runs in O(n log n) time. Our results can be generalized to the Manhattan distance.