On variable sized vector packing

One of the open problems in on-line packing is the gap between the lower bound Ω(1) and the upper bound O(d) for vector packing of d-dimensional items into d-dimensional bins. We address a more general packing problem with variable sized bins. In this problem, the set of allowed bins contains the traditional "all-1" vector, but also a finite number of other d-dimensional vectors. The study of this problem can be seen as a first step towards solving the classical problem. It is not hard to see that a simple greedy algorithm achieves competitive ratio O(d) for every set of bins. We show that for all small ε > 0 there exists a set of bins for which the competitive ratio is 1 + ε. On the other hand we show that there exists a set of bins for which every deterministic or randomized algorithm has competitive ratio Ω(d). We also study one special case for d = 2.