Online Bin Stretching with Three Bins

TL;DR

This paper introduces an online bin stretching algorithm for three bins with a stretching factor of 1.375, and establishes lower bounds close to this factor, advancing understanding of semi-online bin packing.

Contribution

It presents the first algorithm achieving a 1.375 stretching factor for three bins and provides new lower bounds for three, four, and five bins, including computationally discovered bounds.

Findings

Algorithm with 1.375 stretching factor for three bins.

Lower bound of 1.3636 for three bins.

Lower bounds of 19/14 for four and five bins.

Abstract

Online Bin Stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for Online Bin Stretching with a stretching factor of $11/8 = 1.375$ for three bins. Additionally, we present a lower bound of $45/33 = 1.\overline{36}$ for Online Bin Stretching on three bins and a lower bound of $19/14$ for four and five bins that were discovered using a computer search.