Tight Bounds for Restricted Grid Scheduling

TL;DR

This paper establishes tight bounds for an online bin packing problem with restricted item sizes and bin sizes, providing asymptotically matching upper and lower bounds based on job size ratios.

Contribution

It introduces new bounds for the problem, improving understanding of the competitive ratio for specific item and bin size restrictions.

Findings

Lower bound of 5/4 on competitive ratio confirmed

Asymptotically matching upper bounds derived

Bounds depend on ratio of small to large jobs

Abstract

The following online bin packing problem is considered: Items with integer sizes are given and variable sized bins arrive online. A bin must be used if there is still an item remaining which fits in it when the bin arrives. The goal is to minimize the total size of all the bins used. Previously, a lower bound of 5/4 on the competitive ratio of this problem was achieved using jobs of size S and 2S-1. For these item sizes and maximum bin size 4S-3, we obtain asymptotically matching upper and lower bounds, which vary depending on the ratio of the number of small jobs to the number of large jobs.