# Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

# Lower Bounds for Online Bin Packing

• Rob van Stee
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_493

## Years and Authors of Summarized Original Work

• 1992; van Vliet

• 2012; Balogh, Békési, Galambos

## Problem Definition

In the online bin packing problem, a sequence of items with sizes in the interval (0, 1] arrive one by one and need to be packed into bins, so that each bin contains items of total size at most 1. Each item must be irrevocably assigned to a bin before the next item becomes available. The algorithm has no knowledge about future items. There is an unlimited supply of bins available, and the goal is to minimize the total number of used bins (bins that receive at least one item).

The most common performance measure for online bin packing algorithms is the asymptotic performance ratio, or asymptotic competitive ratio, which is defined as
$$\displaystyle{ R_{\mathrm{ASY}}(A) :=\mathop{\lim \sup }\limits_{ n \rightarrow \infty }\!\left \{\!\mathop{\max }\limits_{L}\!\left \{\left .\!\!\frac{A(L)} {n} \right \vert \!\mbox{ OPT}(L)\! =\! n\right \}\!\right \}\!. }$$

## Keywords

Bin packing Competitive analysis Lower bounds Online algorithms
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