An Approximation Scheme for the Two-Stage, Two-Dimensional Bin Packing Problem

Abstract

We present an asymptotic PTAS for Two-Dimensional Bin Packing, which requires packing (or cutting) a given set of rectangles from the minimum number of square bins, with the further restriction that cutting the rectangles from the bins can be done in two stages, as is frequently the case in real-world applications. To the best of our knowledge, this is the first approximation scheme for a nontrivial two-dimensional (and real-world) generalization of classical one-dimensional Bin Packing in which rectangles have to be packed in (finite) squares. A simplification of our method yields an asymptotic PTAS for the two-stage packing of rectangles in a bin of unit width and infinite height. Moreover, we point out that our method may lead to a better approximation guarantee for Two-Dimensional Bin Packing without stage restrictions, provided some structural property holds.