Augmenting Online Algorithms for Knapsack Problem with Total Weight Information

Abstract

In this paper, we augment online algorithms for the knapsack problem using the total weight information. The conventional optimal online algorithm achieves the ln(U/L)+1 competitive ratio where L and U are the upper and lower bounds of the value-to-weight ratio. However, it does not consider that decision makers can know the total weight information or obtain it through machine-learned predictions. To fill this gap, we first propose the Known Weight Algorithm (KWA) which uses the exact total weight information to achieve a competitive ratio of W((U-L)/(eL))+1, where W denotes the Lambert-W function. We prove that it is optimal and tight. After that, we extend KWA to the Predicted Weight Algorithm (PWA), a learning-augmented online algorithm that uses predicted total weight. We show the consistency and robustness of PWA, and prove that its competitive ratio degrades gracefully as the prediction error grows. Finally, we introduce the Limited Volume Algorithm (LWA), which achieves a better competitive ratio than ln(U/L)+1 when the total weight is less than twice the capacity.