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

In the textsc2-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 one L-shaped region and packs items into those in a structured way. We generalize this approach by allowing up to a constant number of 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.