Independent Sets and Hitting Sets of Bicolored Rectangular Families

A bicolored rectangular family BRF is the collection of all axis-parallel rectangles formed by selecting a bottom-left corner from a finite set of points A and an upper-right corner from a finite set of points B. We devise a combinatorial algorithm to compute the maximum independent set and the minimum hitting set of a BRF that runs in O(n(2.5) root log n)-time, where n = vertical bar A vertical bar + vertical bar B vertical bar. This result significantly reduces the gap between the Omega(n(7))-time algorithm by Benczur (Discrete Appl Math 129 (2-3):233-262, 2003) for the more general problem of finding directed covers of pairs of sets, and the O(n(2))-time algorithms of Franzblau and Kleitman (Inf Control 63(3):164-189, 1984) and Knuth (ACM J Exp Algorithm 1:1, 1996) for BRFs where the points of A lie on an anti-diagonal line. Furthermore, when the bicolored rectangular family is weighted, we show that the problem of finding the maximum weight of an independent set is NP-hard, and provide efficient algorithms to solve it on important subclasses.