An FPTAS for the knapsack problem with parametric weights

Abstract

In this paper, we investigate the parametric weight knapsack problem, in which the item weights are affine functions of the form wi(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 schemes (FPTASs) for the problem that, for any desired precision ε∈(0,1), compute (1−ε)-approximate solutions for all values of the parameter. Among others, we present a strongly polynomial FPTAS running in On5ε2⋅log3nεlogn time and a weakly polynomial FPTAS with a running time of On3ε2⋅log2nε⋅logP⋅logMlogn, 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.