We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to write $P$ as a union of small pieces, and in partitioning, we furthermore require the pieces to be pairwise interior-disjoint. We show that these problems are in fact equivalent: Optimum covers and partitions have the same number of pieces. For covering, a natural local search algorithm repeatedly attempts to replace $k$ pieces from a candidate cover with $k-1$ pieces. In two dimensions and for sufficiently large $k$, we show that when no such swap is possible, the cover is a $1+O(1/\sqrt k)$-approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a $13$-approximation that only works for polygons without holes [Abrahamsen and Rasmussen, SODA 2025]. In contrast, in the three dimensional version of the problem, for a polyhedron $P$ of complexity $n$, we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in $n$, even if $P$ is simple, i.e., has genus $0$ and no holes.

翻译：我们研究使用最少数量的小片来覆盖或分割多边形 $P$（可能带孔）的问题，其中小片是包含于轴对齐单位正方形内的连通子多边形。对于覆盖问题，我们寻求将 $P$ 表示为小片的并集；而在分割问题中，我们还进一步要求这些小片两两内部不相交。我们证明这两个问题实际上是等价的：最优覆盖和最优分割所需的片数相同。对于覆盖问题，一种自然的局部搜索算法会反复尝试将候选覆盖中的 $k$ 片替换为 $k-1$ 片。在二维空间中，对于足够大的 $k$，我们证明当无法进行此类替换时，该覆盖是一个 $1+O(1/\sqrt k)$ 近似解，从而得到该问题的首个多项式时间近似方案 (PTAS)。在我们之前，已知的唯一算法是一个仅适用于无孔多边形的 $13$ 近似算法 [Abrahamsen and Rasmussen, SODA 2025]。相比之下，在问题的三维版本中，对于复杂度为 $n$ 的多面体 $P$，我们证明即使 $P$ 是简单的（即亏格为 $0$ 且无孔），要逼近最优覆盖或分割到 $n$ 的对数因子以内也是 NP-难的。