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.

双色矩形族BRF是通过从有限的一组点A选择左下角和从有限的一组点B选择右上角而形成的所有轴平行矩形的集合。我们设计了一种组合算法来计算在\（O（n ^ {2.5} \ sqrt {\ log n}）\） -time中运行的BRF的最大独立集和最小命中集，其中\（n = | A | + | B | \）。该结果显着减小了Benczúr的\（\ Omega（n ^ 7）\）时间算法之间的差距（Discrete Appl Math 129（2–3）：233–262，2003），这是查找定向覆盖的更普遍问题。集对和\（O（n ^ 2）\）BRF的Franzblau和Kleitman（Inf Control 63（3）：164–189，1984）和Knuth（ACM J Exp Algorithm 1：1，1996）的实时算法，其中A点位于反对角线上。此外，当对双色矩形族进行加权时，我们表明找到独立集的最大权重的问题是\（\ mathbf {NP} \）- hard，并提供了在重要子类上求解该问题的有效算法。