Biedl et al. introduced the minimum ply cover problem in CG 2021 following the seminal work of Erlebach and van Leeuwen in SODA 2008. They showed that determining the minimum ply cover number for a given set of points by a given set of axis-parallel unit squares is NP-hard, and gave a polynomial time $2$-approximation algorithm for instances in which the minimum ply cover number is bounded by a constant. Durocher et al. recently presented a polynomial time $(8 + \epsilon)$-approximation algorithm for the general case when the minimum ply cover number is $\omega(1)$, for every fixed $\epsilon > 0$. They divide the problem into subproblems by using a standard grid decomposition technique. They have designed an involved dynamic programming scheme to solve the subproblem where each subproblem is defined by a unit side length square gridcell. Then they merge the solutions of the subproblems to obtain the final ply cover. We use a horizontal slab decomposition technique to divide the problem into subproblems. Our algorithm uses a simple greedy heuristic to obtain a $(27+\epsilon)$-approximation algorithm for the general problem, for a small constant $\epsilon>0$. Our algorithm runs considerably faster than the algorithm of Durocher et al. We also give a fast $2$-approximation algorithm for the special case where the input squares are intersected by a horizontal line. The hardness of this special case is still open. Our algorithm is potentially extendable to minimum ply covering with other geometric objects such as unit disks, identical rectangles etc.

翻译：Biedl等人继Erlebach与van Leeuwen在SODA 2008的开创性工作之后，于CG 2021提出了最小层覆盖问题。他们证明，对于给定点集，用给定轴对齐单位正方形确定最小层覆盖数是NP难的，并给出了最小层覆盖数被常数界定时实例的多项式时间2-近似算法。Durocher等人近期针对最小层覆盖数为ω(1)的一般情形，对每个固定ε>0提出了多项式时间(8+ε)-近似算法。他们通过标准网格分解技术将问题划分为子问题，设计了复杂的动态规划方案来求解每个由单位边长正方形网格单元定义的子问题，随后合并子问题的解以获得最终层覆盖。我们采用水平条带分解技术划分问题，使用简单贪心启发式方法为一般问题提供(27+ε)-近似算法（其中ε>0为小常数）。我们的算法运行速度显著快于Durocher等人的算法。针对输入正方形与水平线相交的特殊情形，我们还给出了快速2-近似算法，该特殊情形的困难性仍为开放问题。我们的算法具有潜在扩展性，可应用于单位圆盘、相同矩形等其他几何对象的最小层覆盖问题。