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Square Packing with Asymptotically Smallest Waste Only Needs Good Squares

Hong Duc Bui

TL;DR

The paper addresses the asymptotic wasted area $W(x)$ when packing unit squares into a large square by showing $W(x)=\Theta(W^*(x))$, where $W^*(x)$ restricts to good squares with tilt $\theta \le 10^{-10}$. It combines a max-flow min-cut construction with a path-based geometric waste bound to remove bad squares with only $O(f)$ additional waste, where $f$ counts disjoint bad-cell paths, thus proving that good-square packings suffice for the asymptotic analysis. This reduces the complexity of lower-bound proofs like Roth and Vaughan's by enabling them to operate within the good-square regime and provides a framework that generalizes to nearly-right-angled quadrilateral packings. The work also discusses algorithmic implementations and potential refinements to tilt bounds, offering a concrete method to translate combinatorial bounds into packing guarantees with practical implications for related geometric packing problems.

Abstract

We consider the problem of packing a large square with nonoverlapping unit squares. Let $W(x)$ be the minimum wasted area when a large square of side length $x$ is packed with unit squares. In Roth and Vaughan's paper that proves the lower bound $W(x) \notin o(x^{1/2})$, a good square is defined to be a square with inclination at most $10^{-10}$ with respect to the large square. In this article, we prove that in calculating the asymptotic growth of the wasted space, it suffices to only consider packings with only good squares. This allows the lower bound proof in Roth and Vaughan's paper to be simplified by not having to handle bad squares.

Square Packing with Asymptotically Smallest Waste Only Needs Good Squares

TL;DR

The paper addresses the asymptotic wasted area when packing unit squares into a large square by showing , where restricts to good squares with tilt . It combines a max-flow min-cut construction with a path-based geometric waste bound to remove bad squares with only additional waste, where counts disjoint bad-cell paths, thus proving that good-square packings suffice for the asymptotic analysis. This reduces the complexity of lower-bound proofs like Roth and Vaughan's by enabling them to operate within the good-square regime and provides a framework that generalizes to nearly-right-angled quadrilateral packings. The work also discusses algorithmic implementations and potential refinements to tilt bounds, offering a concrete method to translate combinatorial bounds into packing guarantees with practical implications for related geometric packing problems.

Abstract

We consider the problem of packing a large square with nonoverlapping unit squares. Let be the minimum wasted area when a large square of side length is packed with unit squares. In Roth and Vaughan's paper that proves the lower bound , a good square is defined to be a square with inclination at most with respect to the large square. In this article, we prove that in calculating the asymptotic growth of the wasted space, it suffices to only consider packings with only good squares. This allows the lower bound proof in Roth and Vaughan's paper to be simplified by not having to handle bad squares.

Paper Structure

This paper contains 13 sections, 6 theorems, 1 equation, 6 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Notations Pertained to the Geometry of Square Packing
  4. Notations Pertained to the Combinatorial Tools
  5. Combinatorial Tools
  6. Geometric Tools
  7. Main Results
  8. Discussion
  9. Generalization of the Wasted Area Along Path Proposition
  10. Strengthening of the Result
  11. Other Algorithmic Considerations
  12. Conclusion
  13. Example Grid where Greedy Multi-source BFS Gives Suboptimal Result

Key Result

Theorem 1

$W(x) \in \Theta(W^*(x))$.

Figures (6)

  • Figure 1: Example of a $3 \times 4$ grid. Some edges are colored in red, and a boundary cell is colored in blue.
  • Figure 2: Example of a rectangle on a grid. The rectangle is marked in red.
  • Figure 4: Illustration of merging two rectangles that have at least a point in common. Here, they have a grid edge in common, and the total perimeter decreases by $2$ after merging.
  • Figure 5: Illustration for \ref{['angle_diff_over_path']} (drawing not to scale, $r=5$ in this article).
  • Figure 6: Illustration for the construction of circles.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Definition 2: Packing
  • Definition 3: $S_i$ as a set of points
  • Definition 4: Wasted area of a packing $W(\mathcal{A})$
  • Definition 5: The waste function $W(x)$
  • Definition 6: Wasted area in any shape
  • Definition 7: Angle between two squares
  • Definition 8: Distance between two objects
  • Definition 9: The open ball around an object
  • Definition 10: Path on a plane
  • ...and 13 more