Online Results for Black and White Bin Packing

In online bin packing problems, items of sizes in [0, 1] are to be partitioned into subsets of total size at most 1, called bins. We introduce a new variant where items are of two types, called black and white, and the item types must alternate in each bin, that is, two items of the same type cannot be assigned consecutively into a bin. This variant generalizes the standard online bin packing problem. We design an online algorithm with an absolute competitive ratio of 3. We further show that a number of well-known algorithms cannot have a better performance, even in the asymptotic sense. Interestingly, we show that this problem is harder than standard online bin packing by proving a general lower bound 1+12ln2≈1.7213$1+\frac {1}{2\ln 2}\approx 1.7213$ on the asymptotic competitive ratio of any deterministic or randomized online algorithm. This lower bound exceeds the upper bounds known for the absolute competitive ratio of standard online bin packing.