Colorful bin packing

We study a variant of online bin packing, called colorful bin packing. In this problem, items that are presented one by one are to be packed into bins of size 1. Each item i has a size s_i ∈ [0,1] and a color c _i ∈ C, where C is a set of colors (that is not necessarily known in advance). The total size of items packed into a bin cannot exceed its size, thus an item i can always be packed into a new bin, but an item cannot be packed into a non-empty bin if the previous item packed into that bin has the same color, or if the occupied space in it is larger than 1 - s_i. This problem generalizes standard online bin packing and online black and white bin packing (where |C| = 2). We prove that colorful bin packing is harder than black and white bin packing in the sense that an online algorithm for zero size items that packs the input into the smallest possible number of bins cannot exist for |C| ≥ 3, while it is known that such an algorithm exists for |C| = 2. We show that natural generalizations of classic algorithms for bin packing fail to work for the case |C| ≥ 3, and moreover, algorithms that perform well for black and white bin packing do not perform well either, already for the case |C| = 3. Our main results are a new algorithm for colorful bin packing that we design and analyze, whose absolute competitive ratio is 4, and a new lower bound of 2 on the asymptotic competitive ratio of any algorithm, that is valid even for black and white bin packing.