Sequential vector packing

We introduce a novel variant of the well known d-dimensional bin (or vector) packing problem. Given a sequence of non-negative d-dimensional vectors, the goal is to pack these into as few bins as possible, of the smallest possible size. In the classical problem, the bin size vector is given and the sequence can be partitioned arbitrarily. We study a variation where the vectors have to be packed in the order in which they arrive, and the bin size vector can be chosen once in the beginning. This setting gives rise to two combinatorial problems: one in which we want to minimize the number of bins used for a given total bin size, and one in which we want to minimize the total bin size for a given number of bins. We prove that both problems are NP-hard, and propose an LP based bicriteria -approximation algorithm. We give a 2-approximation algorithm for the version with a bounded number of bins. Furthermore, we investigate properties of natural greedy algorithms, and present an easy to implement heuristic, which is fast and performs well in practice. Experiments with the heuristic and an ILP formulation yield promising results on real world data.