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. 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. The bin size vector can be chosen once in the beginning, under the constraint that the coordinate-wise bounds sum up to at most a given total bin size. This setting arises from a special resource constrained scheduling problem. We prove that the problem isNP-hard and we propose LP based bicriteria ( 1 ε , 1 1−ε )and (1, 2)-approximation algorithms. 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 very promising results on real world data.