  In the knapsack problem, we are given a knapsack of some capacity and a set
of items, each with a size and a value. The goal is to pack a selection of
these items fitting the knapsack that maximizes the total value. The online
version of this problem reveals the items one by one. For each item, the
algorithm must decide immediately whether to pack it or not. We consider a
natural variant of this problem, coined removable online knapsack. It differs
from the classical variant by allowing the removal of packed items. Repacking
is impossible, however: Once an item is removed, it is gone for good.
  We analyze the advice complexity of this problem. It measures how many advice
bits an omniscient oracle needs to provide for an online algorithm to reach any
given competitive ratio, which is, understood in its strict sense, just the
approximation factor. We show that the competitive ratio jumps from unbounded
without advice to near-optimal with just constantly many advice bits, a
behavior unique among all problems examined so far. We also examine algorithms
with barely any advice, for example just a single bit, and analyze the special
case of the proportional knapsack problem, where an item's size always equals
its value.
  We show that advice algorithms have various concrete applications and that
lower bounds on the advice complexity of any problem are exceptionally strong.
Our results improve some of the best known lower bounds on the competitive
ratio for randomized algorithms and even for deterministic deterministic
algorithms in established models such as knapsack with a resource buffer and
various problems with multiple knapsacks. The seminal paper introducing
knapsack with removability proposed such a problem for which we can even
establish a one-to-one correspondence with the advice model; this paper
therefore also provides a comprehensive analysis for this neglected problem.
