Klee's measure problem (computing the volume of the union of $n$ axis-parallel boxes in $\mathbb{R}^d$) is well known to have $n^{\frac{d}{2}\pm o(1)}$-time algorithms (Overmars, Yap, SICOMP'91; Chan FOCS'13). Only recently, a conditional lower bound (without any restriction to ``combinatorial'' algorithms) could be shown for $d=3$ (K\"unnemann, FOCS'22). Can this result be extended to a tight lower bound for dimensions $d\ge 4$? In this paper, we formalize the technique of the tight lower bound for $d=3$ using a combinatorial object we call prefix covering design. We show that these designs, which are related in spirit to combinatorial designs, directly translate to conditional lower bounds for Klee's measure problem and various related problems. By devising good prefix covering designs, we give the following lower bounds for Klee's measure problem in $\mathbb{R}^d$, the depth problem for axis-parallel boxes in $\mathbb{R}^d$, the largest-volume/max-perimeter empty (anchored) box problem in $\mathbb{R}^{2d}$, and related problems: - $\Omega(n^{1.90476})$ for $d=4$, - $\Omega(n^{2.22222})$ for $d=5$, - $\Omega(n^{d/3 + 2\sqrt{d}/9-o(\sqrt{d})})$ for general $d$, assuming the 3-uniform hyperclique hypothesis. For Klee's measure problem and the depth problem, these bounds improve previous lower bounds of $\Omega(n^{1.777...}), \Omega(n^{2.0833...})$ and $\Omega(n^{d/3 + 1/3 + \Theta(1/d)})$ respectively. Our improved prefix covering designs were obtained by (1) exploiting a computer-aided search using problem-specific insights as well as SAT solvers, and (2) showing how to transform combinatorial covering designs known in the literature to strong prefix covering designs. In contrast, we show that our lower bounds are close to best possible using this proof technique.

翻译：Klee度量问题（计算$\mathbb{R}^d$中$n$个轴平行盒并集的体积）已知存在时间复杂度为$n^{\frac{d}{2}\pm o(1)}$的算法（Overmars, Yap, SICOMP'91；Chan FOCS'13）。直到最近，才针对$d=3$的情况证明了条件性下界（不限制为“组合”算法）（Künnemann, FOCS'22）。这一结果能否推广至$d\ge 4$维的紧下界？本文通过引入名为前缀覆盖设计的组合对象，形式化了$d=3\)维紧下界的技术。我们证明，这些在精神上与组合设计相关的结构可直接转化为Klee度量问题及若干相关问题的条件性下界。通过构造优秀的前缀覆盖设计，我们为$\mathbb{R}^d$中的Klee度量问题、$\mathbb{R}^d$中轴平行盒的深度问题、$\mathbb{R}^{2d}$中的最大体积/最大周长空（锚定）盒问题及相关问题给出如下下界：- 对于$d=4$，$\Omega(n^{1.90476})$；- 对于$d=5$，$\Omega(n^{2.22222})$；- 对于一般$d$，$\Omega(n^{d/3 + 2\sqrt{d}/9-o(\sqrt{d})})$，这些下界均基于3-均匀超团假设。对于Klee度量问题和深度问题，这些下界分别改进了先前的$\Omega(n^{1.777...})$、$\Omega(n^{2.0833...})$及$\Omega(n^{d/3 + 1/3 + \Theta(1/d)})$下界。我们改进的前缀覆盖设计通过以下方式获得：(1) 结合问题特定洞察与SAT求解器进行计算机辅助搜索；(2) 证明如何将文献中已知的组合覆盖设计转化为强前缀覆盖设计。相比之下，我们证明该证明技术所取得的下界已接近最优。