Approximation Schemes and Structural Barriers for the Two-Dimensional Knapsack Problem with Rotations
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arXiv
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| Natura: | Preprint |
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2026
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| author | Kar, Debajyoti Khan, Arindam Wiese, Andreas |
| author_facet | Kar, Debajyoti Khan, Arindam Wiese, Andreas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Approximation Schemes and Structural Barriers for the Two-Dimensional Knapsack Problem with Rotations Kar, Debajyoti Khan, Arindam Wiese, Andreas Data Structures and Algorithms Computational Geometry 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. |
| title | Approximation Schemes and Structural Barriers for the Two-Dimensional Knapsack Problem with Rotations |
| topic | Data Structures and Algorithms Computational Geometry |
| url | https://arxiv.org/abs/2603.23970 |