In the d-dimensional online bin packing problem, d-dimensional cubes of positive sizes no larger than 1 are presented one by one to be assigned to positions in d-dimensional unit cube bins. In this work, we provide improved upper bounds on the asymptotic competitive ratio for square and cube bin packing problems, where our bounds do not exceed 2.0885 and 2.5735 for square and cube packing, respectively. To achieve these results, we adapt and improve a previously designed harmonic-type algorithm, and apply a different method for defining weight functions. We detect deficiencies in the state-of-the-art results by providing counter-examples to the current best algorithms and the analysis, where the claimed bounds were 2.1187 for square packing and 2.6161 for cube packing.

翻译：在二维在线垃圾包装问题中,正尺寸不大于1的二维立方体逐个呈现出来,被分配到二维单元立方体箱中的位置。在这项工作中,我们为平方和立方体包装问题提供了无症状竞争比率的改进的上限,在平方和立方体包装问题上,我们的界限分别不超过2.0885和2.5735。为了实现这些结果,我们调整和改进了先前设计的调和型算法,并采用了不同的方法来界定重量函数。我们通过向当前最佳算法和分析提供反示例,发现最新结果中的缺陷,其中所称的平方包装界限为2.1187,立方体包装界限为2.6161。