Given a set of squares and a strip of bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from $\Omega(n!)$ to $O(2^n)$, with $n$ the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard, and then we propose a fully polynomial-time approximation scheme (FPTAS) to solve it. We also propose three mathematical programming formulations based on different solution representations and confirm the performance of these algorithms through computational experiments. Finally, we discuss several extensions that are relevant to practical applications.

翻译：给定一组正方形以及一个宽度有限、高度无限的条带，我们考虑一种正方形条带打包问题，并将其称为独立正方形打包问题（SIPP），其目标是最小化条带高度，使得所有正方形被打包到由水平和垂直划分分隔的独立单元格中。针对SIPP，我们首先研究了高效的解表示方法，并提出了一种紧凑表示，将搜索空间从 $\Omega(n!)$ 降低到 $O(2^n)$（其中 $n$ 为给定正方形的数量），同时保证存在对应于最优解的解表示。基于该解表示，我们证明了该问题是NP难的，进而提出了一种完全多项式时间近似方案（FPTAS）来求解该问题。我们还基于不同解表示提出了三种数学规划模型，并通过计算实验验证了这些算法的性能。最后，我们讨论了与实际问题应用相关的若干扩展。