Tight Bounds For Restricted Grid Scheduling

The following 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 items of size S and 2S-1. For these item sizes and maximum bin size M = 4S- 3, we obtain asymptotically matching upper and lower bounds, which vary depending on the ratio of the number of small items to the number of large items.