Maximum Area Axis-Aligned Square Packings

TL;DR

This paper studies the problem of packing maximum area axis-aligned squares anchored at given points within a unit square, providing bounds, efficient computation, and NP-completeness results.

Contribution

It establishes a tight lower bound on the reach of anchored square packings, presents an $O(n ext{log}n)$ algorithm for computing the reach, and proves NP-completeness of finding maximum packings.

Findings

Area of reach $R(S)$ is at least 1/2 for any point set.

Reach $R(S)$ can be computed in $O(n\log n)$ time.

Finding maximum area anchored square packing is NP-complete.

Abstract

Given a point set $S=\{s_1,\ldots , s_n\}$ in the unit square $U=[0,1]^2$, an anchored square packing is a set of $n$ interior-disjoint empty squares in $U$ such that $s_i$ is a corner of the $i$th square. The reach $R(S)$ of $S$ is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in $S$. It is shown that area$(R(S))\geq \frac12$ for every finite set $S\subset U$, and this bound is the best possible. The region $R(S)$ can be computed in $O(n\log n)$ time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted.