New bounds for variable-sized and resource augmented online bin packing

In the variable-sized online bin packing problem, one has to assign items to bins one by one. The bins are drawn from some fixed set of sizes, and the goal is to minimize the sum of the sizes of the bins used. We present new algorithms for this problem and show upper bounds for them which improve on the best previous upper bounds. We also show the first general lower bounds for this problem. The case where bins of two sizes, 1 and alpha is an element of (0, 1), are used is studied in detail. This investigation leads us to the discovery of several interesting fractal-like curves. Our techniques are also applicable to the closely related resource augmented online bin packing problem, where we have also obtained the first general lower bounds.