Improved Online Square-into-Square Packing

TL;DR

This paper presents an improved algorithm for online square-into-square packing, increasing the maximum packable total area from 11/32 to 3/8, enhancing efficiency in two-dimensional packing problems.

Contribution

The paper introduces a new algorithm that improves the bound for online square packing from 11/32 to 3/8, advancing the theoretical understanding of packing limits.

Findings

Achieved a packing bound of 3/8 for total area.

Improved the previous bound of 11/32.

Provided an efficient online packing algorithm.

Abstract

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 $11/32$.