# Online Bin Covering with Advice

## TL;DR

This paper investigates the advice complexity of the online bin covering problem, establishing tight bounds on advice size needed for optimality and improved competitive ratios, revealing fundamental differences from bin packing.

## Contribution

It provides tight bounds on advice size for optimal solutions and improved ratios, and highlights the distinct advice complexity landscape compared to bin packing.

## Key findings

- Advice of o(log log n) bits cannot improve the competitive ratio beyond 0.5.
- O(log log n) advice bits can achieve a ratio close to 0.5333.
- Linear advice is needed to surpass a 15/16 competitive ratio.

## Abstract

The bin covering problem asks for covering a maximum number of bins with an online sequence of $n$ items of different sizes in the range $(0,1]$; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide tight bounds for the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size $o(\log \log n)$ has a competitive ratio of at most 0.5. In other words, advice of size $o(\log \log n)$ is useless for improving the competitive ratio of 0.5, attainable by an online algorithm without advice. This result highlights a difference between the bin covering and the bin packing problems in the advice model: for the bin packing problem, there are several algorithms with advice of constant size that outperform online algorithms without advice. Furthermore, we show that advice of size $O(\log \log n)$ is sufficient to achieve a competitive ratio that is arbitrarily close to $0.53\bar{3}$ and hence strictly better than the best ratio $0.5$ attainable by purely online algorithms. The technicalities involved in introducing and analyzing this algorithm are quite different from the existing results for the bin packing problem and confirm the different nature of these two problems. Finally, we show that a linear number of bits of advice is necessary to achieve any competitive ratio better than 15/16 for the online bin covering problem.