Gálvez, Waldo and Grandoni, Fabrizio and Ingala, Salvatore and Heydrich, Sandy and Khan, Arindam and Wiese, Andreas (2021) Approximating Geometric Knapsack via L-packings. ACM Transactions on Algorithms, 17 (4). pp. 1-67. ISSN 1549-6325
Full text not available from this repository.Abstract
We study the two-dimensional geometric knapsack problem, in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is 2+ε [Jansen and Zhang, SODA 2004]. In this article we present a polynomial-time 17/9+ε < 1.89-approximation, which improves to 558/325+ε < 1.72 in the cardinality case. Prior results pack items into a constant number of rectangular containers that are filled via greedy strategies. We deviate from this setting and show that there exists a large profit solution where items are packed into a constant number of containers plus one L-shaped region at the boundary of the knapsack containing narrow-high items and thin-wide items. These items may interact in complex manners at the corner of the L. The best-known approximation ratio for the subproblem in the L-shaped region is 2+ε (via a trivial reduction to one-dimensional knapsack); hence, as a second major result we present a PTAS for this case that we believe might be of broader utility. We also consider the variant with rotations, where items can be rotated by 90 degrees. Again, the best-known polynomial-time approximation factor (even for the cardinality case) is 2+ε [Jansen and Zhang, SODA 2004]. We present a polynomial-time (3/2+ε)-approximation for this setting, which improves to 4/3+ε in the cardinality case.
| Item Type: | Scientific journal article, Newspaper article or Magazine article |
|---|---|
| Subjects: | Computer sciences > Computer science > Computational science foundations |
| Department/unit: | Dipartimento tecnologie innovative > Istituto Dalle Molle di studi sull’intelligenza artificiale USI-SUPSI |
| Depositing User: | Fabrizio Grandoni |
| Date Deposited: | 17 Jul 2023 10:03 |
| Last Modified: | 17 Jul 2023 10:09 |
| URI: | http://repository.supsi.ch/id/eprint/14315 |
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