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

2026-03-25

AI Summary

This work addresses the two-dimensional knapsack problem with 90-degree rotations (2DKR) and achieves significant advances in both the cardinality and weighted settings. It presents the first polynomial-time approximation scheme (PTAS) for cardinality 2DKR and introduces a novel resource contraction lemma that overcomes the long-standing 1.5-approximation barrier, yielding a (1.497 + ε)-approximation algorithm for the weighted variant with rotations. Furthermore, it improves the best-known approximation ratio for the weighted case without rotations to (13/7 + ε) ≈ (1.857 + ε). The approach combines container-based greedy packing, refined structural analysis, and complexity arguments under the k-Sum conjecture. Additionally, the study establishes a conditional lower bound, showing that any (1 + ε)-approximation algorithm requires running time at least n^{Ω(1/ε)}, thereby delineating the computational complexity frontier of the problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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

Problem

• two-dimensional knapsack
• rotations
• approximation schemes
• PTAS
• geometric packing

Innovation

• PTAS
• resource contraction lemma
• two-dimensional knapsack with rotations
• structural barrier
• approximation algorithm