There is no EPTAS for two-dimensional knapsack

In the d-dimensional (vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P=NP, there is no fully polynomial-time approximation scheme for d-dimensional knapsack, already for d=2. The best known result is a polynomial-time approximation scheme (PTAS) due to Frieze and Clarke [A.M. Frieze, M. Clarke, Approximation algorithms for the m-dimensional 0–1 knapsack problem: worst-case and probabilistic analyses, European J. Operat. Res. 15 (1) (1984) 100–109] for the case where d⩾2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS (EPTAS).In this paper we resolve this question by showing that there is no EPTAS for d-dimensional knapsack, already for d=2, unless W[1]=FPT. Furthermore, we show that unless all problems in SNP are solvable in sub-exponential time, there is no approximation scheme for two-dimensional knapsack whose running time is , for any function f. Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist.