Computing Partitions of Rectilinear Polygons with Minimum Stabbing Number

摘要

The stabbing number of a partition of a rectilinear polygon P into rectangles is the maximum number of rectangles stabbed by any axis-parallel line segment contained in P. We consider the problem of finding a rectangular partition with minimum stabbing number for a given rectilinear polygon P. First, we impose a conforming constraint on partitions: every vertex of every rectangle in the partition must lie on the polygon's boundary. We show that finding a conforming rectangular partition of minimum stabbing number is NP-hard for rectilinear polygons with holes. We present a rounding method based on a linear programming relaxation resulting in a polynomial-time 2-approximation algorithm. We give an O(n log n)-time algorithm to solve the problem exactly when P is a histogram (some edge in P can see every point in P) with n vertices. Next we relax the conforming constraint and show how to extend the first linear program to achieve a polynomial-time 2-approximation algorithm for the general problem, improving the approximation factor achieved by Abam, Aronov, de Berg, and Khosravi (ACM SoCC 2011).