Online variable-sized bin packing

AbstractThe classical bin packing problem is one of the best-known and most widely studied problems of combinatorial optimization. Efficient offline approximation algorithms have recently been designed and analyzed for the more general and realistic model in which bins of differing capacities are allowed (Friesen and Langston (1986)). In this paper, we consider fast online algorithms for this challenging model. Selecting either the smallest or the largest available bin size to begin a new bin as pieces arrive turns out to yield a tight worst-case ratio of 2. We devise a slightly more complicated scheme that uses the largest available bin size for small pieces, and selects bin sizes for large pieces based on a user-specified fill factor ƒ≥12, and prove that this strategy guarantees a worst-case bound not exceeding 1.5+ƒ2

On variable sized vector packing

One of the open problems in on-line packing is the gap between the lower bound Ω(l) 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