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

Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More