Approximation Schemes and Structural Barriers for the Two-Dimensional Knapsack Problem with Rotations

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 $3/2+\epsilon$ for any constant $\epsilon>0$, with an improvement to $4/3+\epsilon$ in the cardinality case, due to G{\'a}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 $(1+\epsilon)$-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 $(1.497+\epsilon)$-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 $13/7+\epsilon \approx 1.857+\epsilon$. Finally, we establish a lower bound of $n^{\Omega(1/\epsilon)}$ on the running time of any $(1+\epsilon)$-approximation algorithm for our problem with or without rotations -- even in the cardinality setting, assuming the $k$-\textsc{Sum} Conjecture.