Improved Online Square-into-Square Packing

In this paper, we show an improved bound and new algorithm for the online square-into-square packing problem. This two-dimensional packing problem involves packing an online sequence of squares into a unit square container without any two squares overlapping. The goal is to find the largest area $\alpha$ such that any set of squares with total area $\alpha$ can be packed. We show an algorithm that can pack any set of squares with total area $\alpha \leq 3/8$ into a unit square in an online setting, improving the previous bound of $11/32$.