Given a set $\mathcal{B}$ of d-dimensional boxes (i.e., axis-aligned hyperrectangles), a minimum coverage kernel is a subset of $\mathcal{B}$ of minimum size covering the same region as $\mathcal{B}$. Computing it is $\mathsf{NP}$-hard, but as for many similar $\mathsf{NP}$-hard problems (e.g., Box Cover, and Orthogonal Polygon Covering), the problem becomes solvable in polynomial time under restrictions on $\mathcal{B}$. We show that computing minimum coverage kernels remains $\mathsf{NP}$-hard even when restricting the graph induced by the input to a highly constrained class of graphs. Alternatively, we present two polynomial-time approximation algorithms for this problem: one deterministic with an approximation ratio within $O(\log n)$, and one randomized with an improved approximation ratio within $O(\lg \mathtt{OPT})$ (with high probability).

给定一套 $\mathcal{乙}$在d维盒（即轴对齐的超矩形）中，最小覆盖率内核是$\mathcal{乙}$ 最小尺寸覆盖与 $\mathcal{乙}$。计算它是$\mathsf{NP}$-硬，但对于许多类似 $\mathsf{NP}$-困难问题（例如Box Cover和Orthogonal Polygon Covering），在以下条件的限制下，该问题可以在多项式时间内解决$\mathcal{乙}$。我们证明了计算最小覆盖率内核仍然存在$\mathsf{NP}$-即使将输入所引起的图限制为高度约束的图类，也很难。或者，针对此问题，我们提出了两种多项式时间近似算法：一种是确定性的，其近似比率在$Ø（\mathrm{日志}ñ）$，并且随机分配一个具有改进的近似比的 $Ø（\lg \mathtt{选择}）$ （很有可能）。