Relaxation Analysis for the Dynamic Knapsack Problem with Stochastic Item Sizes

We consider a version of the knapsack problem in which an item size is random and revealedonly when the decision maker attempts to insert it. After every successful insertion the decisionmaker can dynamically choose the next item based on the remaining capacity and available items,while an unsuccessful insertion terminates the process. We build on a semi-infinite relaxationintroduced in previous work, known as the Multiple Choice Knapsack (MCK) bound. Our firstcontribution is an asymptotic analysis of MCK showing that it is asymptotically tight underappropriate assumptions. In our second contribution, we examine a new, improved relaxationbased on a quadratic value function approximation, which introduces the notion of diminishingreturns by encoding interactions between remaining items. We compare this bound to othersfrom the literature, including the best known pseudo-polynomial bound. The quadratic boundis theoretically more efficient than the pseudo-polynomial bound, yet empirically comparable toit in both value and running time.