Online bin packing of squares and cubes

TL;DR

This paper improves the upper bounds on the asymptotic competitive ratio for online square and cube bin packing problems, using enhanced algorithms and analysis techniques, and identifies deficiencies in previous results.

Contribution

It presents tighter bounds for online square and cube packing ratios and introduces improved algorithms and weight function methods.

Findings

Upper bound of 2.0885 for square packing

Upper bound of 2.5735 for cube packing

Counter-examples to previous bounds of 2.1187 and 2.6161

Abstract

In the d-dimensional online bin packing problem, d-dimensional cubes of positive sizes no larger than 1 are presented one by one to be assigned to positions in d-dimensional unit cube bins. In this work, we provide improved upper bounds on the asymptotic competitive ratio for square and cube bin packing problems, where our bounds do not exceed 2.0885 and 2.5735 for square and cube packing, respectively. To achieve these results, we adapt and improve a previously designed harmonic-type algorithm, and apply a different method for defining weight functions. We detect deficiencies in the state-of-the-art results by providing counter-examples to the current best algorithms and the analysis, where the claimed bounds were 2.1187 for square packing and 2.6161 for cube packing.