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 form $wi(\lambda) = ai + \lambda \cdot bi$ for $i \in {1,\ldots,n}$ depending on a real-valued parameter $\lambda$. 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 $\varepsilon \in (0,1)$, computes $(1-\varepsilon)$-approximate solutions for all values of the parameter. Our FPTAS is based on two different approaches and achieves a running time of $\mathcal{O}(n^{3/\varepsilon2} \cdot \min{ \log² P, n² } \cdot \min{\log M, n \log (n/\varepsilon) / \log(n \log (n/\varepsilon) )})$ where $P$ is an upper bound on the optimal profit and $M := \max{W, n \cdot \max{ai,b_i: i \in {1,\ldots,n}}}$ for a knapsack with capacity $W$.