On-line bin Packing with Two Item Sizes

We study the on-line bin packing problem (BPP). In BPP, we are given a sequence B of items a1, a2, . . . , an and a sequence of their sizes (s1, s2, . . . , sn) (each size si ∈ (0, 1]) and are required to pack the items into a minimum number of unit-capacity bins. Let R∞ {α,β} be the minimal asymptotic competitive ratio of an on-line algorithm in the case when all items are only of two different sizes α and β. We prove that max{R∞ {α,β} : α, β ∈ (0, 1]} = 4/3. We also obtain an exact formula for R∞ {α,β} when max{α, β} > 1 2 . This result extends the result of Faigle, Kern and Turan (1989) that R∞ {α,β} = 4 3 for β = 1 2 − 2 and α = 12 + 2 for any fixed nonnegative 2 < 1 6 .