Online Knapsack Problems with Estimates
Jakub Balabán, Matthias Gehnen, Henri Lotze, Finn Seesemann, Moritz Stocker
TL;DR
This work studies online knapsack problems with size estimates, introducing a distortion parameter $\delta$ that bounds how actual item sizes deviate from announced estimates. It provides tight, delta-dependent competitive-ratio bounds for two variants: the online simple knapsack with irrevocable decisions and the online simple knapsack with removability, including both additive and multiplicative accuracy analyses. The main contributions are lower-bound constructions and matching upper-bound algorithms that yield precise competitive ratios expressed in terms of $\delta$, such as $\tfrac{1}{\min(p,q)}$ with $p$ and $q$ defined via $k=\tfrac{2}{1-2\delta}$ for the additive case, and analogous results for removability with a threshold around $\delta=\tfrac{3}{4}-\sqrt{5}/4$. The results advance understanding of how prediction-like estimates influence online packing, demonstrate when estimates help or become ineffective, and connect to broader themes in robust optimization and learning-augmented online algorithms.
Abstract
Imagine you are a computer scientist who enjoys attending conferences or workshops within the year. Sadly, your travel budget is limited, so you must select a subset of events you can travel to. When you are aware of all possible events and their costs at the beginning of the year, you can select the subset of the possible events that maximizes your happiness and is within your budget. On the other hand, if you are blind about the options, you will likely have a hard time when trying to decide if you want to register somewhere or not, and will likely regret decisions you made in the future. These scenarios can be modeled by knapsack variants, either by an offline or an online problem. However, both scenarios are somewhat unrealistic: Usually, you will not know the exact costs of each workshop at the beginning of the year. The online version, however, is too pessimistic, as you might already know which options there are and how much they cost roughly. At some point, you have to decide whether to register for some workshop, but then you are aware of the conference fee and the flight and hotel prices. We model this problem within the setting of online knapsack problems with estimates: in the beginning, you receive a list of potential items with their estimated size as well as the accuracy of the estimates. Then, the items are revealed one by one in an online fashion with their actual size, and you need to decide whether to take one or not. In this article, we show a best-possible algorithm for each estimate accuracy $δ$ (i.e., when each actual item size can deviate by $\pm δ$ from the announced size) for both the simple knapsack and the simple knapsack with removability.