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

TL;DR

This paper introduces a Polynomial Time Approximation Scheme (PTAS) for the 2D knapsack problem with rotations in the cardinality case, and improves approximation ratios for the weighted case, addressing longstanding open questions.

Contribution

It presents the first PTAS for the cardinality case of 2DKR and advances approximation algorithms for the weighted case, overcoming structural barriers and establishing complexity bounds.

Findings

PTAS achieved for the cardinality case of 2DKR.

Improved approximation ratio of 1.497+ε for the weighted case.

Lower bound of n^{Ω(1/ε)} on approximation algorithms assuming the k-Sum conjecture.

Abstract

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.