Online Colored Bin Packing

In the Colored Bin Packing problem a sequence of items of sizes up to 1 arrives to be packed into bins of unit capacity. Each item has one of c ≥ 2 colors and an additional constraint is that we cannot pack two items of the same color next to each other in the same bin. The objective is to minimize the number of bins. In the important special case when all items have size zero, we characterize the optimal value to be equal to color discrepancy. As our main result, we give an (asymptotically) 1.5-competitive algorithm which is optimal. In fact, the algorithm always uses at most d1.5 · OPTe bins and we show a matching lower bound of d1.5·OPTe for any value of OPT ≥ 2. In particular, the absolute ratio of our algorithm is 5/3 and this is optimal. For items of unrestricted sizes we give an asymptotically 3.5competitive algorithm. When the items have sizes at most 1/d for a real d ≥ 2 the asymptotic competitive ratio is 1.5 + d/(d − 1). We also show that classical algorithms First Fit, Best Fit and Worst Fit are not constant competitive, which holds already for three colors and