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 = N P , 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 )  In this paper we resolve this question by showing that there is no EPTAS for ddimensional knapsack, already for d = 2, unless W [1] = F P T . 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| o( √ 1/ε) , 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.
In fact, there is no efficient polynomial-time approximation scheme (EPTAS) for 2-KP unless W[1] = FPT (See REF ).
There is no EPTAS for two-dimensional knapsack
Inf. Process. Lett
Mathematics Computer Science