Online Knapsack of Unknown Capacity:

Abstract

We propose a new variant of the more standard online knapsack problem where the only information missing to the provided instances is the capacity $B$ of the knapsack. We refer to this problem as the online Knapsack of Unknown Capacity problem. Any algorithm must provide a strategy for ordering the items that are inserted in the knapsack in an online fashion, until the actual capacity of the knapsack is revealed and the last inserted item might not fit in. Apart from the interest in a new version of the fundamental knapsack problem, the motivations that lead to define this new variant come from energy consumption constraints in smartphone communications. We do provide lower and upper bounds to the problem for various cases. In general, we design an optimal algorithm admitting a $\frac{1}{2}$-competitive ratio. When all items admit uniform ratio of profit over size, our algorithm provides a $\frac{49}{86}=.569\ldots$ competitive ratio that leaves some gap with the provided bound of $\frac{1}{\varphi }=.618\ldots$, the inverse of the golden number. We then conduct experimental analysis for the competitive ratio guaranteed algorithms compared to the optimum and to various heuristics.

Research partially supported by the Research Grant 2010N5K7EB ‘PRIN 2010’ ARS TechnoMedia (Algoritmica per le Reti Sociali Tecno-mediate) from the Italian Ministry of University and Research.