Improved Hardness of Approximation for Geometric Bin Packing
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The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of d-dimensional rectangles, and the goal is to pack them into unit d-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for the problem, even when d=2. For general d, the best known approximation algorithm has an approximation guarantee exponential in d, while the best hardness of approximation is still a small constant inapproximability from the case when d=2. In this paper, we show that the problem cannot be approximated within d^1-ϵ factor unless NP=ZPP. Recently, d-dimensional Vector Bin Packing, a closely related problem to the GBP, was shown to be hard to approximate within Ω(log d) when d is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when d is fixed, we prove a couple of key properties of the Geometric Packing Dimension that highlight the difference between Geometric Packing Dimension and Packing Dimension.
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