Online Bin Packing with Item Size Estimates
Matthias Gehnen, Andreas Usdenski
TL;DR
This work studies online bin packing with item-size estimates: each item comes with an estimate $c'(a)$ and a known accuracy $\delta$, while the actual size $c(a)$ lies in $[c'(a)(1-\delta), \min(c'(a)(1+\delta),1)]$. The authors prove a fundamental lower bound of $\frac{4}{3}$ on the competitive ratio for any $\delta>0$ and show a $\frac{3}{2}$ barrier when prediction errors are large, specifically for $\delta>\frac{41}{43}$. They then introduce the Planned-Harmonic (PH) algorithm, which is $1.5$-competitive for all $\delta\leq \frac{1}{35}$, by combining a planning phase with a harmonic classification of item sizes into $I_k$ classes and an update phase that adapts to revealed actual sizes. In addition, they present Delayed-Best-Fit (DBF), an approach tailored to the two-items-per-bin setting that achieves a $\frac{4}{3}$-competitive ratio, showing that predictions can yield substantial improvements in restricted variants. Overall, the results delineate how the quality of size estimates (via $\delta$) governs the potential gains over classical online bin packing and highlight concrete algorithms that exploit predictions to approach optimal packing in complementary regimes.
Abstract
Imagine yourself moving to another place, and therefore, you need to pack all of your belongings into moving boxes with some capacity. In the classical bin packing model, you would try to minimize the number of boxes, knowing the exact size of each item you want to pack. In the online bin packing problem, you need to start packing the first item into a box, without knowing what other stuff is upcoming. Both settings are somewhat unrealistic, as you are likely not willing to measure the exact size of all your belongings before packing the first item, but you are not completely clueless about what other stuff you have when you start packing. In this article, we introduce the online bin packing with estimates model, where you start packing with a rough idea about the upcoming item sizes in mind. In this model, an algorithm receives a size estimate for every item in the input list together with an accuracy factor $δ$ in advance. Just as for regular online bin packing the items are then presented iteratively. The actual sizes of the items are allowed to deviate from the size estimate by a factor of $δ$. Once the actual size of an item is revealed the algorithm has to make an irrevocable decision on the question where to place it. This is the first time online bin packing is studied under this model. This article has three main results: First, no algorithm can achieve a competitive ratio of less than $\frac{4}{3}$, even for an arbitrary small factor $δ>0$. Second, we present an algorithm that is $1.5$-competitive for all $δ\leq \frac{1}{35}$. Finally, we design a strategy that yields a competitive ratio of $\frac{4}{3}$ under the assumption that not more than two items can be placed in the same bin, which is best possible in this setting.