Approximability of the two-stage stochastic knapsack problem with discretely distributed weights

Abstract

In this paper the two-stage knapsack problem with random weights is studied under the aspect of approximability. We assume finite probability distributions for the weights and show that, unless P=NP, the so obtained problem cannot be approximated in polynomial time within a better ratio than K−1/2 (where K is the number of second-stage scenarios). We further study the special cases where in the second stage items can only be added or only be removed, but not both. Positive approximation results are given for three particular cases, namely linearly dependent first- and second-stage rewards, the polynomial scenario model and the case where the number of scenarios is assumed to be a constant. To the best of our knowledge, this is the first study of a two-stage knapsack problem under the aspect of approximability and the first time a non-approximability result has been proven for a stochastic knapsack problem of any kind.