Removable Online Knapsack and Advice

TL;DR

This work analyzes advice complexity for Removable Online Knapsack, introducing Prop-Rem-Knap where items may be removed at most once and capacity is fixed at $1$. It proves that optimality requires near-linear advice bits, while even a single bit can yield strong constant-factor guarantees, and a refined size-class scheme attains $\sqrt{2}$-competitiveness with one bit. The results highlight a sharp contrast between removability-enabled models and classical online knapsack, with broad implications for randomness and deterministic variants in multi-knapsack settings. Overall, the paper offers a comprehensive framework linking advice complexity to competitive performance in online knapsack problems and demonstrates the practical impact of limited advice on robust algorithm design.

Abstract

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.

Figures (2)

• Figure 1: A schematic plot of the advice complexity behavior of the classical online knapsack problem in light gray and the relaxed variant with removability in dark gray. For the proportional version without removability there are two large plateaus; removability collapses to a single vast expanse. For the general version, in which an item's value may differ from its size, there is only one but a more extreme jump directly from an unbounded competitive ratio to near optimality; with removability, this jump is occurring earlier and even steeper.
• Figure 2: The partition of the interval $(0,1]$ of possible sizes into the five subintervals used in the proof of Theorem \ref{['thm:prop_one_bit_upper']}---namely $(0,a]$, $(a,b]$, $(b,c]$, $(c,d]$, and $(d,1]$---plus the corresponding class names. The values are $a=1-1/\sqrt2\approx0.293$, and $b=\sqrt2-1\approx0.414$, and $c=1/2$, and $d=1/\sqrt2\approx0.707$.

Theorems & Definitions (6)

• Definition 1: Removable Knapsack Problem
• Definition 2: Competitive Ratio
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6