Given any set of points $S$ in the unit square that contains the origin, does a set of axis aligned rectangles, one for each point in $S$, exist, such that each of them has a point in $S$ as its lower-left corner, they are pairwise interior disjoint, and the total area that they cover is at least 1/2? This question is also known as Freedman's conjecture (conjecturing that such a set of rectangles does exist) and has been open since Allen Freedman posed it in 1969. In this paper, we improve the best known lower bound on the total area that can be covered from 0.09121 to 0.1039. Although this step is small, we introduce new insights that push the limits of this analysis. Our lower bound uses a greedy algorithm with a particular order of the points in $S$. Therefore, it also implies that this greedy algorithm achieves an approximation ratio of 0.1039. We complement the result with an upper bound of 3/4 on the approximation ratio for a natural class of greedy algorithms that includes the one that achieves the lower bound.

中文翻译：

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下左锚定矩形包装的更好下限

给定单位平方中包含原点的任何点$ S $，是否存在一组轴对齐的矩形（每个$ S $中的一个点），因此每个矩形都有一个以$ S $作为点的矩形。左下角，它们是成对的内部不相交的，并且它们覆盖的总面积至少为1/2？这个问题也称为Freedman猜想（猜想确实存在这样的一组矩形），并且自Allen Freedman于1969年提出该问题以来一直是开放的。在本文中，我们改进了可以覆盖的总面积上最著名的下界从0.09121到0.1039。尽管这一步很小，但是我们引入了新的见解，这些见解推动了这一分析的极限。我们的下限使用贪婪算法，其点的特定顺序为$ S $。因此，这也意味着该贪婪算法的近似比为0。