The d-dimensional vector packing problem is a well-known generalization of the classical bin packing problem: For a given list of vectors in [ 0, 1 ] d, the goal is to partition the list into the minimum number of subsets, such that in every subset the sum of all vectors is at most one in every coordinate. For the case d = 1, Femandez de la Vega and Lueker ( 1981) designed an asymptotic polynomial time approximation scheme. In this note we prove that already for the case d = 2, the existence of an asymptotic polynomial time approximation scheme would imply P = NP. The proof is very simple and uses no new ideas. @ 1997 Elsevier Science B.V.
For multi-dimensional vector packing Woeginger REF ruled out an approximation scheme even for the case d = 2.
There is no asymptotic PTAS for twodimensional vector packing
Information Processing Letters 64
Mathematics Computer Science