In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $\rho$ of the online bounded space variant is $\Omega(\log d)$ and $O(d/\log d)$, and conjectured that it is $\Theta(\log d)$. We show that $\rho$ is in fact $\Theta(d/\log d)$, using probabilistic arguments.

中文翻译：

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在线有界空间超立方体装箱问题的紧下界

在 $d$ 维超立方体装箱问题中，必须将给定的 $d$ 维超立方体列表装入最小数量的超立方体箱中。爱泼斯坦和范斯蒂 [SIAM J. Comput. 35 (2005)]表明在线有界空间变体的渐近性能比$\rho$为$\Omega(\log d)$和$O(d/\log d)$，并推测为$\ Theta(\log d)$。我们使用概率论证明 $\rho$ 实际上是 $\Theta(d/\log d)$。