We study a natural geometric variant of the classic Knapsack problem called 2D-Knapsack: we are given a set of axis-parallel rectangles and a rectangular bounding box, and the goal is to pack as many of these rectangles inside the box without overlap. Naturally, this problem is NP-complete. Recently, Grandoni et al. [ESA'19] showed that it is also 𝖶[1]-hard when parameterized by the size k of the sought packing, and they presented a parameterized approximation scheme (PAS) for the variant where we are allowed to rotate the rectangles by 90° before packing them into the box. Obtaining a PAS for the original 2D-Knapsack problem, without rotation, appears to be a challenging open question. In this work, we make progress towards this goal by showing a PAS under the following assumptions: - both the box and all the input rectangles have integral, polynomially bounded sidelengths; - every input rectangle is wide - its width is greater than its height; and - the aspect ratio of the box is bounded by a constant. Our approximation scheme relies on a mix of various parameterized and approximation techniques, including color coding, rounding, and searching for a structured near-optimum packing using dynamic programming.
@InProceedings{mari_et_al:LIPIcs.IPEC.2023.33, author = {Mari, Mathieu and Picavet, Timoth\'{e} and Pilipczuk, Micha{\l}}, title = {{A Parameterized Approximation Scheme for the Geometric Knapsack Problem with Wide Items}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {33:1--33:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.33}, URN = {urn:nbn:de:0030-drops-194529}, doi = {10.4230/LIPIcs.IPEC.2023.33}, annote = {Keywords: Parameterized complexity, Approximation scheme, Geometric knapsack, Color coding, Dynamic programming, Computational geometry} }
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