There is no asymptotic PTAS for two-dimensional vector packing

Abstract

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, Fernandez 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.