Improved hardness of approximation for Geometric Bin Packing

Abstract

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 d-dimensional unit 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 that is exponential in d. In contrast, 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 a $d^{1-ϵ}$ factor unless $\text{NP}=\text{P}$.

Recently, d-dimensional Vector Bin Packing, a problem closely related to the GBP, was shown to be hard to approximate within a $\Omega (\log d)$ factor 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 which highlight fundamental differences between Geometric Bin Packing and Vector Bin Packing.