关于锚定矩形填充几乎半个正方形的构造性方法
On Packing Almost Half of a Square with Anchored Rectangles: A Constructive Approach
摘要：
在本文中，我们考虑下面的几何难题，其起源可以追溯到Allan Freedman \cite{croft91,tutte69}在20世纪60年代由Dumitrescu和T {ó} th \cite{adriancasaba2011}.这个谜题最近由Peter Winkler \cite{Winkler2007}推广开来。令$P_{n}$是单位正方形$U = [0,1]^2$中的一组$n$点，包括原点。问题是在$U$中构造$n$轴平行且互不相交的矩形，使得每个矩形的左下角与$P_{n}$中的一个点重合，并且矩形覆盖的总面积最大化。我们将上述矩形称为\emph{anchored rectangles}。长期以来的猜想是，当锚定矩形正确放置时，至少可以覆盖$U$的一半。Dumitrescu和T {0} th \cite{Dumitrescu2012}已经示出了可以覆盖至少$0.09121$的构造方法，即，大约是该地区的$9\%$。
英文摘要：
In this paper, we consider the following geometric puzzle whose origin was traced to Allan Freedman \cite{croft91,tutte69} in the 1960s by Dumitrescu and T{ó}th \cite{adriancasaba2011}. The puzzle has been popularized of late by Peter Winkler \cite{Winkler2007}. Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. The problem is to construct $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each rectangle coincides with a point in $P_{n}$ and the total area covered by the rectangles is maximized. We would term the above rectangles as \emph{anchored rectangles}. The longstanding conjecture has been that at least half of $U$ can be covered when anchored rectangles are properly placed. Dumitrescu and T{ó}th \cite{Dumitrescu2012} have shown a construction method that can cover at least $0.09121$, i.e., roughly $9\%$ of the area.
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