# Approximation Schemes for Bin Packing

**DOI:**https://doi.org/10.1007/978-1-4939-2864-4_31

## Years and Authors of Summarized Original Work

1982; Karmarker, Karp

## Problem Definition

In the bin-packing problem, the input consists of a collection of items specified by their sizes. There are also identical bins, which without loss of generality can be assumed to be of size 1, and the goal is to pack these items using the minimum possible number of bins.

Bin packing is a classic optimization problem, and hundreds of its variants have been defined and studied under various settings such as average case analysis, worst-case off-line analysis, and worst-case online analysis. This note considers the most basic variant mentioned above under the off line model where all the items are given in advance. The problem is easily seen to be NP-hard by a reduction from the partition problem. In fact, this reduction implies that unless P = NP, it is impossible to determine in polynomial time whether the items can be packed into two bins or whether they need three bins.

### Notations

The input to the...

## Keywords

Cutting-stock problem## Recommended Reading

- 1.Coffman EG, Garey MR, Johnson DS (1996) Approximation algorithms for bin packing: a survey. In: Hochbaum D (ed) Approximation algorithms for NP-hard problems. PWS, Boston, pp 46–93Google Scholar
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- 3.Fernandez de la Vega W, Lueker G (1981) Bin packing can be solved within 1 +
*ɛ*in linear time. Combinatorica 1:349–355MathSciNetzbMATHCrossRefGoogle Scholar - 4.Karmarkar N, Karp RM (1982) An efficient approximation scheme for the one-dimensional bin-packing problem. In: Proceedings of the 23rd IEEE symposium on foundations of computer science (FOCS), Chicago, pp 312–320Google Scholar
- 5.Rothvoss T (2013) Approximating bin packing withing O(log OPT log log OPT) bins. In: Proceedings of the 54th IEEE symposium on foundations of computer science (FOCS), Berkeley, pp 20–29Google Scholar