Approximation Schemes for Bin Packing

Keywords and Synonyms

Cutting stock problem        

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 offline analysis, and worst-case online analysis. This note considers the most basic variant mentioned above under the offline 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 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 bin packing problem is a set...