Let $S$ be a point set in the plane such that each of its elements is colored either red or blue. A matching of $S$ with rectangles is any set of pairwise-disjoint axis-aligned rectangles such that each rectangle contains exactly two points of $S$. Such a matching is monochromatic if every rectangle contains points of the same color, and is bichromatic if every rectangle contains points of different colors. In this paper we study the following two problems: 1. Find a maximum monochromatic matching of $S$ with rectangles. 2. Find a maximum bichromatic matching of $S$ with rectangles. For each problem we provide a polynomial-time approximation algorithm that constructs a matching with at least $1/4$ of the number of rectangles of an optimal matching. We show that the first problem is $\mathsf{NP}$-hard even if either the matching rectangles are restricted to axis-aligned segments or $S$ is in general position, that is, no two points of $S$ share the same $x$ or $y$ coordinate. We further show that the second problem is also $\mathsf{NP}$-hard, even if $S$ is in general position. These $\mathsf{NP}$-hardness results follow by showing that deciding the existence of a perfect matching is $\mathsf{NP}$-complete in each case. The approximation results are based on a relation of our problem with the problem of finding a maximum independent set in a family of axis-aligned rectangles. With this paper we extend previous ones on matching one-colored points with rectangles and squares, and matching two-colored points with segments. Furthermore, using our techniques, we prove that it is $\mathsf{NP}$-complete to decide a perfect matching with rectangles in the case where all points have the same color, solving an open problem of Bereg, Mutsanas, and Wolff [CGTA (2009)].
翻译:在平面上设置了 $S 的点值, 这样每个矩形都含有不同颜色的点 。 在本文中, 我们研究以下两个问题 : 1. 找到一个最大单色相匹配 $S 和矩形 。 2 找到一个最高双色相匹配 $S 和矩形 。 对于每一个矩形都含有两个精确的点 $S 。 如果每个矩形都含有相同颜色的点, 这样的匹配是单色的, 而且如果每个矩形都含有不同颜色的点, 则是双色相匹配 。 即使匹配的正色相与正方格 。 找到一个最大单色相匹配的 $C 。 找到一个最大双色相匹配 $, 美元 直方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方上。