Optimal Binary Space Partitions for Segments in the Plane

An optimal bsp for a set S of disjoint line segments in the plane is a bsp for S that produces the minimum number of cuts. We study optimal bsps for three classes of bsps, which differ in the splitting lines that can be used when partitioning a set of fragments in the recursive partitioning process: free bsps can use any splitting line, restricted bsps can only use splitting lines through pairs of fragment endpoints, and auto-partitions can only use splitting lines containing a fragment. We obtain the following two results: • It is np-hard to decide whether a given set of segments admits an auto-partition that does not make any cuts. • An optimal restricted bsp makes at most 2 times as many cuts as an optimal free bsp for the same set of segments.