Independent Sets and Hitting Sets of Bicolored Rectangular Families

Abstract

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 \(O(n^{2.5}\sqrt{\log n})\)-time, where \(n=|A |+|B |\). This result significantly reduces the gap between the \(\Omega (n^7)\)-time algorithm by Benczúr (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 \(\mathbf {NP}\)-hard, and provide efficient algorithms to solve it on important subclasses.