An FPTAS for the Knapsack Problem with Parametric Weights

In this paper, we investigate the parametric weight knapsack problem, in which the item weights are affine functions of the formwi(λ) = ai + λ ·bi for i ∈ {1, . . . ,n} depending on a real-valued parameter λ. The aim is to provide a solution for all values of the parameter. It is well-known that any exact algorithm for the problem may need to output an exponential number of knapsack solutions. We present the first fully polynomial-time approximation scheme (FPTAS) for the problem that, for any desired precision ε ∈ (0, 1), computes (1− ε)approximate solutions for all values of the parameter. Our FPTAS is based on two different approaches and achieves a running time of O(n/ε ·min{log P,n} ·min{logM,n log(n/ε)/ log(n log(n/ε))})where P is an upper bound on the optimal profit and M := max{W,n ·max{ai,bi : i ∈ {1, . . . ,n}}} for a knapsack with capacity W.