Computing Partitions of Rectilinear Polygons with Minimum Stabbing Number
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(nlogn)-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 SoCG 2011).
KeywordsLine Segment Vertical Edge Optimal Partition Linear Program Relaxation Initial Partition
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- 1.Abam, M.A., Aronov, B., de Berg, M., Khosravi, A.: Approximation algorithms for computing partitions with minimum stabbing number of rectilinear and simple polygons. In: Proc. ACM SoCG, pp. 407–416 (2011)Google Scholar