Optimal on-line bin packing with two item sizes
Gregory Gutin, Tommy Jensen and Anders Yeo
Algorithmic Oper. Res. Volume 1, pp. 72-78, 2006.

## Abstract

We study the on-line bin packing problem (BPP). In BPP, we are given a sequence $B$ of items $a_1,a_2,\ldots, a_n$ and a sequence of their sizes $(s_1,s_2,\ldots,s_n)$ (each size $s_i\in (0,1]$) and are required to pack the items into a minimum number of unit-capacity bins. Let $\rr$ be the minimal asymptotic competitive ratio of an on-line algorithm in the case when all items are only of two different sizes $\al$ and $\bb$. We prove that $\max\{ \rr :\ \al,\bb \in (0,1]\} = 4/3.$ We also obtain an exact formula for $\rr$ when $\max\{\al,\bb\}>\frac{1}{2}$. This result extends the result of Faigle, Kern and Turan (1989) that $\rr = \frac{4}{3}$ for $\be=\frac{1}{2}-\epsilon$ and $\al=\frac{1}{2}+\epsilon$ for any fixed nonnegative $\epsilon < \frac{1}{6}.$