| Abstrakt: |
In Two-dimensional Bin Packing (2BP), we are given n rectangles as input and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. 2BP admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan (ACM-SIAM symposium on discrete algorithms (SODA), pp 13–25, 2014. https://doi.org/10.1137/1.9781611973402.2). A well-studied variant of 2BP is Guillotine Two-dimensional Bin Packing (G2BP), where rectangles must be packed in such a way that every rectangle in the packing can be obtained by applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal et al. (Symposium on foundations of computer science (FOCS). IEEE, pp 657–666, 2005. https://doi.org/10.1109/SFCS.2005.10) gave an APTAS for G2BP. Let λ 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 λ 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 ≤ λ ≤ 1.692 . Bansal and Khan (2014) conjectured that λ = 4 / 3 . The conjecture, if true, will imply a (4 / 3 + ε) -approximation algorithm for 2BP. Given a small constant δ > 0 , a rectangle is called large if both its height and width are at least δ , else it is called skewed. We make progress towards the conjecture by showing that λ = 4 / 3 when all input rectangles are skewed. We also give an APTAS for 2BP for skewed items, though general 2BP does not admit an APTAS. [ABSTRACT FROM AUTHOR] |
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