Online General Knapsack with Reservation Costs
Elisabet Burjons, Matthias Gehnen
TL;DR
This paper analyzes the online general knapsack problem under a reservation model in which decisions can be delayed at a cost proportional to either the item size or value. It develops two branches: (i) reservation costs proportional to size, where the authors design a densest-reserve strategy yielding a tight competitive ratio of $2$ with matching lower bounds, and (ii) reservation costs proportional to value, where they derive a lower bound for $0<\\alpha<0.5$ and propose a density-aware algorithm whose upper bound is a closed-form function of $\\alpha$ that approaches $2$ as $\\alpha$ tends to $0$ but becomes unbounded as $\\alpha$ approaches $1/2$. The results leverage the concept of the densest reserved subset and adversarial constructions to bound performance, providing practically meaningful guidance on the value of postponing decisions under cancellation or reservation fees. Overall, the work advances understanding of reservation-based online knapsack and offers concrete bounds and algorithms that are relevant for applications with postponement costs and cancellation penalties.
Abstract
In the online general knapsack problem, an algorithm is presented with an item $x=(s,v)$ of size $s$ and value $v$ and must irrevocably choose to pack such an item into the knapsack or reject it before the next item appears. The goal is to maximize the total value of the packed items without overflowing the knapsack's capacity. As this classical setting is way too harsh for many real-life applications, we will analyze the online general knapsack problem under the reservation model. Here, instead of accepting or rejecting an item immediately, an algorithm can delay the decision of whether to pack the item by paying a fraction $0\le α$ of the size or the value of the item. This models many practical applications, where, for example, decisions can be delayed for some costs e.g. cancellation fees. We present results for both variants: First, for costs depending on the size of the items and then for costs depending on the value of the items. If the reservation costs depend on the size of the items, we find a matching upper and lower bound of $2$ for every $α$. On the other hand, if the reservation costs depend on the value of the items, we find that no algorithm is competitive for reservation costs larger than $1/2$ of the item value, and we find upper and lower bounds for the rest of the reservation range $0\leα< 1/2$.