Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items

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 into the minimum number unit square bins. 2BP admits no APTAS current best approximation ratio 1.406 by Bansal Khan (ACM-SIAM symposium on discrete algorithms (SODA), pp 13–25, 2014. https://doi.org/10.1137/1.9781611973402.2 ). A well-studied variant Guillotine (G2BP), where must be packed in such a way that every rectangle can obtained applying sequence end-to-end axis-parallel cuts, also called guillotine cuts. et al. (Symposium foundations computer science (FOCS). IEEE, 657–666, 2005. https://doi.org/10.1109/SFCS.2005.10 ) gave for G2BP. Let $$\lambda $$ smallest constant set I items, bins optimal solution G2BP upper bounded {{\,\textrm{opt}\,}}(I) + c$$ , $${{\,\textrm{opt}\,}}(I)$$ c constant. It known $$4/3 \le \lambda 1.692$$ . (2014) conjectured = 4/3$$ The conjecture, if true, will imply $$(4/3+\varepsilon )$$ -approximation algorithm 2BP. Given small $$\delta > 0$$ large both its height width at least else it skewed. We make progress towards conjecture showing when all give skewed though general does not admit APTAS.