We study the two-dimensional (geometric) knapsack problem with rotations (2DKR), in which we are given a square knapsack and a set of rectangles with associated profits. The objective is to find a maximum profit subset of rectangles that can be packed without overlap in an axis-aligned manner, possibly by rotating some rectangles by $90^{\circ}$. The best-known polynomial time algorithm for the problem has an approximation ratio of $3/2+ε$ for any constant $ε>0$, with an improvement to $4/3+ε$ in the cardinality case, due to G{á}lvez et al. (FOCS 2017, TALG 2021). Obtaining a PTAS for the problem, even in the cardinality case, has remained a major open question in the setting of multidimensional packing problems, as mentioned in the survey by Christensen et al. (Computer Science Review, 2017). In this paper, we present a PTAS for the cardinality case of 2DKR. In contrast to the setting without rotations, we show that there are $(1+ε)$-approximate solutions in which all items are packed greedily inside a constant number of rectangular {\em containers}. Our result is based on a new resource contraction lemma, which might be of independent interest. In contrast, for the general weighted case, we prove that this simple type of packing is not sufficient to obtain a better approximation ratio than $1.5$. However, we break this structural barrier and design a $(1.497+ε)$-approximation algorithm for 2DKR in the weighted case. Our arguments also improve the best-known approximation ratio for the (weighted) case {\em without rotations} to $13/7+ε\approx 1.857+ε$. Finally, we establish a lower bound of $n^{Ω(1/ε)}$ on the running time of any $(1+ε)$-approximation algorithm for our problem with or without rotations -- even in the cardinality setting, assuming the $k$-\textsc{Sum} Conjecture.

翻译：我们研究了带旋转的二维（几何）背包问题（2DKR），该问题中给定一个方形背包和一组带有相关利润的矩形。目标是找到一个利润最大化的矩形子集，使其能够以轴对齐的方式无重叠地打包，并允许通过$90^{\circ}$旋转部分矩形。该问题目前已知的最佳多项式时间算法对于任意常数$ε>0$具有$3/2+ε$的近似比，在基数情形下可改进至$4/3+ε$，该结果由Gálvez等人（FOCS 2017, TALG 2021）提出。正如Christensen等人（Computer Science Review, 2017）的综述所述，为该问题（即使在基数情形下）获取一个PTAS一直是多维打包问题领域的主要开放问题。本文针对2DKR的基数情形提出了一个PTAS。与无旋转情形相比，我们证明了存在$(1+ε)$-近似解，其中所有物品被贪心地打包在恒定数量的矩形“容器”内。该结果基于一个可能具有独立意义的新资源收缩引理。相反，对于一般加权情形，我们证明这种简单打包方式无法获得优于$1.5$的近似比。然而，我们突破了这一结构障碍，为加权情形下的2DKR设计了一个$(1.497+ε)$-近似算法。我们的论证还将（加权）无旋转情形下已知最佳近似比改进至$13/7+ε\approx 1.857+ε$。最后，我们针对本问题（无论是否带旋转）的任意$(1+ε)$-近似算法建立了运行时间的下界$n^{Ω(1/ε)}$——即使在基数情形下，该下界基于$k$-\textsc{Sum}猜想成立。