There is no EPTAS for two-dimensional knapsack

In the d-dimensional 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 (European J. of Operational Research, 100–109, 1984 ) 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 f(1/ε)|I| √ , 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.