Dispersing and grouping points on planar segments

Motivated by (continuous) facility location, we study the problem of dispersing and grouping points on a set segments (of streets) in plane. In former problem, given n disjoint line plane, investigate computing point each such that minimum Euclidean distance between any two these is maximized. We prove this 2D dispersion NP-hard, fact, it NP-hard even if all are parallel unit length. This contrast to polynomial solvability corresponding 1D Li Wang (2016), where intervals disjoint. With result, also show Independent Set Colored Linear Unit Disk Graph (meaning convex hulls with same color form segments) remains parameterized version W[2]. latter plane maximum minimized. present factor-1.1547 approximation algorithm which runs O ( log ⁡ ) time. Our results can be generalized Manhattan distance. • A non-trivial reduction from planar rectilinear monotone 3-SAT problem. As byproduct (1), independent NP-hard. MIS Graph, number (or colors), factor −1.1547 for