More on online bin packing with two item sizes

We follow the work of [G. Gutin, T. Jensen, A. Yeo, On-line bin packing with two item sizes, Algorithmic Operations Research 1 (2) (2006)] and study the online bin packing problem, where every item has one of two possible sizes which are known in advance. We focus on the parametric case, where both item sizes are bounded from above by frac(1, k) for some natural number k ≥ 1. We show that for every possible pair of item sizes, there is an algorithm with competitive ratio of at most frac((k + 1)², k² + k + 1). We prove that this bound is tight for every k and, moreover, that it cannot be achieved if the two item sizes are not known in advance.