Truly asymptotic lower bounds for online vector bin packing

TL;DR

This paper establishes new, significantly improved asymptotic lower bounds for online vector bin packing, demonstrating that no algorithm can achieve a competitive ratio better than approximately d / log^2 d for large dimensions.

Contribution

The work provides the first asymptotic lower bounds matching the known upper bounds, using novel constructions including an adaptive approach for online coloring.

Findings

Lower bound of Ω(√d) established.

Asymptotic lower bound of Ω(d / log^2 d) proven.

Results hold for all dimensions d ≥ 3.

Abstract

In this work, we consider online vector bin packing. It is known that no algorithm can have a competitive ratio of $o(d/\log^2 d)$ in the absolute sense, though upper bounds for this problem were always shown in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds on the asymptotic competitive ratio. The existing lower bounds prior to this work were much smaller than $3$ even for very large dimensions. We significantly improve the best known lower bounds on the asymptotic competitive ratio (and as a byproduct, on the absolute competitive ratio) for online vector packing of vectors with $d \geq 3$ dimensions, for every such dimension $d$. To obtain these results, we use several different constructions, one of which is an adaptive construction showing a lower bound of $\Omega(\sqrt{d})$. Our main result is that the lower bound of $\Omega(d/\log^2 d)$ on the competitive ratio holds also in the asymptotic sense. The last result requires a careful adaptation of constructions for online coloring rather than simple black-box reductions.