在线包装以最小化面积或周长
Online Packing to Minimize Area or Perimeter
摘要：
我们考虑在线包装问题，我们得到一个轴平行的矩形流。矩形必须放置在平面上而不重叠，并且每个矩形必须在不知道后续矩形的情况下放置。目标是最小化矩形的轴平行限界框的周长或面积。我们要么允许旋转90度，要么只允许平移。对于周边版本，我们给出了算法的绝对竞争比略小于4时，只允许平移，旋转时也允许。然后，我们把我们的注意力最小化的区域，并表明，任何算法的渐近竞争比至少是$Ω(\sqrt{n})$，其中$n$是流中的矩形的数量，这与旋转和不旋转。然后，我们提出的算法，在这两种情况下，这一界限相匹配。我们还表明，竞争比不能作为一个功能的OPT.Then，我们考虑两种特殊情况下有界。第一种情况是所有给定的矩形的长宽比都受某个常数的限制。所有矩形都是正方形并且我们希望最小化边界正方形的面积的特定变体之前已经研究过，并且已经给出了竞争比为8的算法[Fekete和Hoffmann，Monumica，2017]。我们改进了算法的分析，并表明该比率最多为6，这是紧的。第二种特殊情况是所有边的长度至少为1。在这里，$Ω(\sqrt n)$下限仍然成立，我们把我们的注意力的下限取决于OPT.We表明，任何算法具有渐近竞争力的比例至少$Ω(\sqrt{OPT})$的平移的情况下，$Ω(\sqrt[4]{OPT})$时，旋转是允许的。对于这两个版本，我们给出的算法，匹配各自的下限。
英文摘要：
We consider online packing problems where we get a stream of axis-parallel rectangles. The rectangles have to be placed in the plane without overlapping, and each rectangle must be placed without knowing the subsequent rectangles. The goal is to minimize the perimeter or the area of the axis-parallel bounding box of the rectangles. We either allow rotations by 90 degrees or translations only. For the perimeter version we give algorithms with an absolute competitive ratio slightly less than 4 when only translations are allowed and when rotations are also allowed. We then turn our attention to minimizing the area and show that the asymptotic competitive ratio of any algorithm is at least $Ω(\sqrt{n})$, where $n$ is the number of rectangles in the stream, and this holds with and without rotations. We then present algorithms that match this bound in both cases. We also show that the competitive ratio cannot be bounded as a function of OPT. We then consider two special cases. The first is when all the given rectangles have aspect ratios bounded by some constant. The particular variant where all the rectangles are squares and we want to minimize the area of the bounding square has been studied before and an algorithm with a competitive ratio of 8 has been given [Fekete and Hoffmann, Algorithmica, 2017]. We improve the analysis of the algorithm and show that the ratio is at most 6, which is tight. The second special case is when all edges have length at least 1. Here, the $Ω(\sqrt n)$ lower bound still holds, and we turn our attention to lower bounds depending on OPT. We show that any algorithm has an asymptotic competitive ratio of at least $Ω(\sqrt{OPT})$ for the translational case and $Ω(\sqrt[4]{OPT})$ when rotations are allowed. For both versions, we give algorithms that match the respective lower bounds.
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