In the \textsc{2-Dimensional Knapsack} problem (2DK) we are given a square knapsack and a collection of $n$ rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed non-overlappingly into the knapsack. The currently best known polynomial-time approximation factor for 2DK is $17/9+\varepsilon<1.89$ and there is a $(3/2+\varepsilon)$-approximation algorithm if we are allowed to rotate items by 90 degrees~{[}G\'alvez et al., FOCS 2017{]}. In this paper, we give $(4/3+\varepsilon)$-approximation algorithms in polynomial time for both cases, assuming that all input data are {integers polynomially bounded in $n$}. G\'alvez et al.'s algorithm for 2DK partitions the knapsack into a constant number of rectangular regions plus \emph{one} L-shaped region and packs items into those {in a structured way}. We generalize this approach by allowing up to a \emph{constant} number of {\emph{more general}} regions that can have the shape of an L, a U, a Z, a spiral, and more, and therefore obtain an improved approximation ratio. {In particular, we present an algorithm that computes the essentially optimal structured packing into these regions. }

翻译：在 & textsc{ 2 { dimensional Knapsack} 问题 (2DK) 中, 我们得到了一个正方位的折叠和集合的重方形项目。 我们的目标是找到最有利可图的子集, 可以不重叠地包装到 knapsack 中。 目前已知的 2DK 的多元时间近似系数是 17/9 ⁇ varepsilon < 1. 89$, 如果允许我们以 90 ° { [} G\ alvez et al., FOCS 2017}, 我们得到一个正方位的折叠成值的折叠合方形算法 。 假设所有输入数据都是 { Integers minymlationalslated to $nqual_ groupations a constalem roupal commational asional 。 Grequestations the we\'als a pasions a max a matical a commaticle.