The bottom-left algorithm is a simple heuristic for the Strip Packing Problem. It places the rectangles in the given order at the lowest free position in the strip, using the left most position in case of ties. Despite its simplicity, the exact approximation ratio of the bottom-left algorithm remains unknown. We will improve the more-than-40-year-old value for the lower bound from $5/4$ to $4/3 - \varepsilon$. Additionally, we will show that this lower bound holds even in the special case of squares, where the previously known lower bound was $12/11 -\varepsilon$. These lower bounds apply regardless of the ordering of the rectangles. When squares are arranged in the worst possible order, we establish a constant upper bound and a $10/3-\varepsilon$ lower bound for the approximation ratio of the bottom-left algorithm. This bound also applies to some online setting and yields an almost tight result there. Finally, we show that the approximation ratio of a local search algorithm based on permuting rectangles in the ordering of the bottom-left algorithm is at least~$2$ and that such an algorithm may need an exponential number of improvement steps to reach a local optimum.

翻译：左下角算法是解决条带装箱问题的一种简单启发式方法。它按照给定的顺序将矩形放置在条带中最低的空闲位置，若遇多个候选位置则选择最左侧的位置。尽管该算法结构简单，但其精确近似比至今仍未知。我们将把已有40余年历史的下界从5/4改进至4/3 - ε，并证明即使是在正方形这一特殊情形下（此前已知下界为12/11 - ε），该下界依然成立。这些下界与矩形的排列顺序无关。当正方形按最坏顺序排列时，我们为左下角算法的近似比建立了常数上界以及10/3 - ε的下界，该界限同样适用于某些在线场景，并在此处得到了近乎紧确的结果。最后，我们证明基于左下角算法顺序排列矩形进行局部搜索算法的近似比至少为2，且该算法可能需要指数级数量的改进步骤才能达到局部最优。