A New Result on Packing Unit Squares into a Large Square
Shuang Wang, Tian Dong, Jiamin Li

TL;DR
This paper improves the upper bound on the uncovered area when packing unit squares into a large square, and refines the minimum number of squares needed for coverage, advancing previous mathematical results.
Contribution
It presents a tighter bound of O(x^{5/8}) for the uncovered area and the covering number, improving upon earlier bounds by Chung and Graham and others.
Findings
Uncovered area bound improved to O(x^{5/8})
Minimum covering squares bound refined to x^2 + O(x^{5/8})
Advances previous packing and covering bounds in geometric combinatorics.
Abstract
In their 2009 note: \emph{Packing equal squares into a large square}, Chung and Graham proved that the uncovered area of a large square of side length is after maximum number of non-overlapping unit squares are packed into it, which improved the earlier results of Erd\H{o}s-Graham, Roth-Vaughan, and Karabash-Soifer. Here we further improve the result to that also helps to improve the bound for the dual problem: finding the minimum number of unit squares needed for covering the large square, from to .
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Taxonomy
TopicsOptimization and Packing Problems · Advanced Manufacturing and Logistics Optimization · Manufacturing Process and Optimization
