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：我们研究带旋转的二维（几何）背包问题（2DKR），其中给定一个方形背包和一组具有关联利润的矩形。目标是在不重叠且轴对齐的方式下，通过允许某些矩形旋转$90^{\circ}$，找到一组利润最大的可打包矩形子集。该问题已知最佳多项式时间算法对于任意常数$\epsilon>0$的近似比为$3/2+\epsilon$，在基数情形下改进至$4/3+\epsilon$，这归功于Gálvez等人（FOCS 2017, TALG 2021）。如Christensen等人（Computer Science Review, 2017）的调查所述，为该问题（即使仅在基数情形下）获取PTAS仍是多维打包问题设置中的一个重要公开问题。本文中，我们提出了2DKR基数情形下的一个PTAS。与无旋转设置相反，我们证明存在$(1+\epsilon)$-近似解，其中所有物品均以贪婪方式打包在常数个矩形“容器”内。我们的结果基于一个新的资源收缩引理，该引理可能具有独立意义。相比之下，对于一般加权情形，我们证明这种简单打包类型无法获得优于$1.5$的近似比。然而，我们打破这一结构障碍，为加权情形下的2DKR设计了一个$(1.497+\epsilon)$-近似算法。我们的论证还将（加权）无旋转情形下的最佳近似比改进至$13/7+\epsilon\approx 1.857+\epsilon$。最后，我们假设$k$-\textsc{Sum}猜想成立，为我们的问题（无论有无旋转）建立了任何$(1+\epsilon)$-近似算法运行时间的下界$n^{\Omega(1/\epsilon)}$——即使在基数情形下。