# Tight approximation algorithms for geometric bin packing with skewed items

Khan, A and Sharma, E (2021) Tight approximation algorithms for geometric bin packing with skewed items. In: 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, 16-18 aug 2021, Seattle.

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## Abstract

In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan SODA'14. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko FOCS'05 obtained an APTAS for this problem. Let Î» be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by Î»opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 â�¤ Î» â�¤ 1.692. Bansal and Khan SODA'14 conjectured that Î» = 4/3. The conjecture, if true, will imply a (4/3 + Îµ)-approximation algorithm for 2BP. According to convention, for a given constant Î´ > 0, a rectangle is large if both its height and width are at least Î´, and otherwise it is called skewed. We make progress towards the conjecture by showing Î» = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on Î» was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS. Â© Arindam Khan and Eklavya Sharma; licensed under Creative Commons License CC-BY 4.0

Item Type: Conference Paper Leibniz International Proceedings in Informatics, LIPIcs Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing The copyright for this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing Geometry; Optimal systems, 'current; Approximation ratios; Best approximations; Bin packing; Geometric bin packing; Guillotine separability; Nonoverlapping; Optimal solutions; Two-dimensional; Unit squares, Approximation algorithms Division of Electrical Sciences > Computer Science & Automation 29 Nov 2021 09:44 29 Nov 2021 09:44 http://eprints.iisc.ac.in/id/eprint/70287